research article frequency diverse array radar cramér...

Research ArticleFrequency Diverse Array Radar Crameacuter-Rao Lower Bounds forEstimating Direction Range and Velocity

Yongbing Wang1 Wen-Qin Wang12 and Huaizong Shao1

1 School of Communication and Information Engineering University of Electronic Science and Technology of ChinaChengdu 611731 China

2Department of Electrical and Electronic Engineering Imperial College London London SW7 2AZUK UK

Correspondence should be addressed to Yongbing Wang wangyongbinghi163com

Received 27 January 2014 Accepted 19 February 2014 Published 8 April 2014

Academic Editor Frankie KitWing Chan

Copyright copy 2014 Yongbing Wang et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Different from phased-array radar frequency diverse array (FDA) radar offers range-dependent beampattern and thus providesnew application potentials But there is a fundamental question what estimation performance can achieve for an FDA radar Inthis paper we derive FDA radar Cramer-Rao lower bounds (CRLBs) for estimating direction range (time delay) and velocity(Doppler shift) Two different data models including pre- and postmatched filtering are investigated separately As the FDA radarhas range-angle coupling we use a simple transmit subaperturing strategy which divides the whole array into two subarrays eachuses a distinct frequency increment Assuming temporally white Gaussian noise and linear frequency modulated transmit signalextensive simulation examples are performed When compared to conventional phased-array radar FDA can yield better CRLBsfor estimating the direction range and velocity Moreover the impacts of the element number and frequency increment are alsoanalyzed Simulation results show that the CRLBs decrease with the increase of the elements number and frequency increment

1 Introduction

The main task of a radar is to detect the existence oftargets and estimate their unknown parameters for examplerange velocity and direction of arrival (DOA) [1ndash4] All theelements in a phased-array antenna transmit the same signalat a fixed carrier frequency By controlling the linear phaseprogression across elements the beam can be steered to thedesired direction precisely But the beam steering is fixed inan angle for all the ranges and cannot mitigate undesirablerange-dependent interferences For this reason the range andangle information of targets cannot be directly obtained for aconventional phased-array radar which offers only the angleinformation However in some applications it is desired thatthe beam pointing can be steerable to the range of interest

Recently frequency diverse array (FDA) is proposed in[5ndash7] as amethod to achieve range-dependent beamformingwhich can be used to suppress range-dependent interfer-ences and clutter [8] or improve detection performance

[9] The most important difference of FDA antenna from aconventional phased-array antenna is that a small amountof frequency increment compared to the carrier frequencyis used across the array elements The periodicity of FDAbeampattern in range angle and time is studied in [10] In[11] the FDA is investigated from a simulation perspectiveand a low-cost FDA is designedThe FDA using chirp (linearfrequency modulation) waveforms with different startingfrequencies is analyzed in [12] to characterize the associatedrange-dependent beampattern The application of FDA insynthetic aperture radar (SAR) is investigated in [13 14]Additional studies to exploit the range-dependent beampat-tern characteristics have been reported in [15]

FDA offers a range-angle-dependent beampattern whichis of great importance as this provides a potential for range-angle localization of targets Therefore this paper considersa fundamental question what estimation performance canbe achieved for an FDA radar Referring to conventionalphased-array range radar the Cramer-Rao lower bound

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 830869 15 pageshttpdxdoiorg1011552014830869

2 International Journal of Antennas and Propagation

(CRLB) expressions for estimating velocity direction andDoppler shift are derived in [16 17] By assuming a narrow-band signal model prior to matched filtering the CRLBs forestimating the time delay direction and Doppler shift with aphased-array radar are detailed in [18 19] Furthermore theCRLB for jointly estimating the attributes of a moving targetusing MIMO radar is derived in [20ndash22]

Different from the above literatures this paper derivesthe CRLBs for jointly estimating the range direction andDoppler shift with an FDA radar Two typical data mod-els namely pre- and postmatched filtering are analyzedseparately for static and moving targets respectively Sincethe range and angle of targets cannot be estimated directlydue to the range-angle coupling we use a simple transmitsubaperturing strategy for the FDA radar This methoddivides the whole array elements into two subarrays eachuses a different frequency increment In doing so the rangeand angle of targets can be solely estimated The corre-sponding CRLBs for estimating direction range and veloc-ity are derived Furthermore the impacts of the elementsnumber and frequency increment on the CRLBs are alsoinvestigated

The rest of the paper is organized as follows The FDAis briefly introduced in Section 2 In Section 3 we considerthe data model after matched filtering and derive the CRLBexpressions for estimating the range and direction In orderto decouple the range-angle coupling response we divide the

whole array into two subarrays The corresponding CRLBsare derived In Section 4 the data model prior to matchedfiltering is investigated and the CRLB expressions for estimat-ing the direction range and Doppler shift are derived wheretemporally white Gaussian noise and chirp pulse signal areassumed Finally extensive simulation results are provided inSection 5 This paper is concluded in Section 6

2 Frequency Diverse Array (FDA)

In conventional phased-array radars it is assumed that anidentical waveform is radiated from each element excludingthe amplitudes and phases Different from phased-arrayradars FDA elements can be either excited by the samewaveform or different waveforms For simplicity and withoutloss of generality we consider a uniform linear array (ULA)FDA and assume that the waveforms radiated from eachantenna element are identical but with a frequency incrementof Δ119891HzThat is the radiation frequency of the119898th elementis

119891119898= 119891

0+ (119898 minus 1) sdot Δ119891 119898 = 0 1 119872 minus 1 (1)

where 1198910is the radar carrier frequency and119872 is the number

of the elementsSupposing a narrowband transmit signal and taking the

first element as the reference for the array the steering vectoris given by the following equation [11]

a (120579 119903) = [1 119890minus119895((21205871198910119889 sin 120579119888)+((2120587Δ119891sdot119889 sin 120579)119888)minus(2120587119903Δ119891119888))

sdot sdot sdot 119890minus119895(((21205871198910(119872minus1)119889 sin 120579)119888)+((2120587sdot(119872minus1)

2Δ119891sdot119889 sin 120579)119888)minus((2120587119903(119872minus1)Δ119891)119888))]

119879

(2)

where 120579 is the direction 119903 is the range 119889 is the elementspacing 119888 is the speed of light and 119879 is the transpose Since1198910

≫ Δ119891 and 119903 ≫ (119872 minus 1)119889 sin(120579) in an amplitude

sense an approximation can be taken by ignoring the phaseterm 2120587 sdot (119898 minus 1)

2

Δ119891 sdot 119889 sin(120579)119888 because it has ignorableimpacts In this case the steering vector can be simplifiedto

a (120579 119903) asymp [1 119890minus119895((21205871198910119889 sin(120579)119888)minus((2120587sdotΔ119891sdot119903)119888))

sdot sdot sdot 119890minus119895(((21205871198910(119898minus1)119889 sin(120579))119888)minus((2120587sdot(119898minus1)Δ119891sdot119903)119888))

]119879

(3)

The FDA characteristics can be summarized as follows[7ndash12] (1) If the frequency offset Δ119891 is fixed the beamdirection will vary as a function of the range 119903 that is it isa range- dependent beam (2) If the range is 119903 fixed the beamdirection will vary as a function of Δ119891 This means that theFDA is frequency-increment-dependent (3) If the frequencyincrement across the array is not applied (ie Δ119891 = 0)the corresponding FDA is just a conventional ULA phasedarray

3 CRLB with Signal Model afterMatched Filtering

31 Data Models Suppose that a target is located at (120579 119903)After matched filtering the baseband equivalent of thecomplex valued signals at the receiver can be expressed as

y = 1205730a (120579 119903) + n = (120595) + n (4)

International Journal of Antennas and Propagation 3

where1205730is a constant for a given target (120595) = 120573

0a(120579 119903)with

120595 = [120579 119903]119879 and n is a zero-mean complex Gaussian white

noise with spatial covariance

119864 [nnH] = Rn = 120590

2

119899I119872 (5)

where 119864[sdot] is the expectation operator H is the conjugatetranspose Rn is the spatial noise covariance matrix 1205902

119899is the

noise power and I119872is an119872times119872 identity matrix

32 CRLBs of Angle and Range Estimations In the followingwe discuss the CRLBs for several different cases separately

321 Range Is Known and Angle Is Unknown In this case120595 = 120579 The corresponding Fisher information matrix (FIM)is I

120595120595[23]

I120595120595

= 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) (6)

where Resdot is the real part of the signal 120595119894is the 119894th element

of 120595 and the119863120595119894(120595) is [24]

119863120595119894(120595) =

120597 (120595)

120597120595119894

(7)

Under the signal model (4) the FDA FIM with respect to 120579 isexpressed as

119868120579120579FDA

= 81205872

1198892cos2 (120579)

sdot SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(8)

where SNR is the signal-to-noise ratio (SNR) Accordinglythe CRLB is

CRLB120579120579FDA

= 119868minus1

120579120579FDA (9)

where minus1 denotes the inverse matrix In particular whenΔ119891 = 0 it is simplified to

119868120579120579phased-array

= 8SNR1205872

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(10)

Proof See Appendix A

It can be easily proved that

CRLB120579120579FDA

lt CRLB120579120579phased-array

(11)

That is to say the FDA radar has better CRLB for angleestimation than phased-array radar

322 Angle Is Known and Range Is Unknown As phased-array radar has range-independent beam here we onlycalculate the range estimation for FDA radar The FIM withrespect to 119903 can be expressed as

119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(12)

The corresponding CRLB is

CRLB119903119903FDA

= 119868minus1

119903119903FDA (13)

Proof See Appendix B

323 Both Angle and Range Are Unknown In this case the 120579and 119903 CRLBs can be similarly determined as

CRLB120579120579FDA

= (1198882

sdot (

119872

sum

119898=1

(119898 minus 1)2

))

times (2SNR sdot 41205872

1198892

sdot (Δ119891)2cos2 (120579)

times (

119872

sum

119898=1

(119898 minus 1)4

119872

sum

119898=1

(119898 minus 1)2

minus(

119872

sum

119898=1

(119898 minus 1)3

)

2

))

minus1

CRLB119903119903FDA

= (1198884

sdot (sum

119872

119898=1(119898 minus 1)

2

1205822

+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888))


sdot (Δ119891)4

times (

119872

sum

119898=1

(119898 minus 1)4

119872

sum

119898=1

(119898 minus 1)2

minus(

119872

sum

119898=1

(119898minus1)3

)

2

))

minus1

(14)

Proof See Appendix C

When the targetrsquos range and angle are estimated jointlythe range and angle CRLBs will be significantly degradeddue to the range-angle coupling (which will be furtherinvestigated in Section 5) Consequently the range and angle


d

1 2 N

middot middot middot

1 2 N


Subarray 1 Subarray 2

Figure 1 Illustration of transmit subaperturing FDA radar

of targets cannot be estimated directly by a standard FDAradar To overcome this problem we present a transmitsubaperturing method for the FDA radar

324 CRLBs of Transmit Subaperturing FDA Radar Todecouple the range and angle peaks and estimate both therange and angle of target we divide the whole array into twoequal subarrays [25 26] Suppose that the number of elementsis 119872 each subarray has 119873 elements namely 119872 = 2119873The first subarray uses the frequency increment of Δ119891

1 and

the second subarray uses the frequency increment of Δ1198912

as shown in Figure 1 The resulting system is referred to astransmit subaperturing FDA (TS-FDA) radar

Taking the first element of the first subarray as thereference the new steering vector can be represented as thefollowing equation

b (120579 119903) = [1 119890minus1198951205931 sdot sdot sdot 119890

minus119895(119873minus1)1205931 119890minus119895120593119897 119890

minus119895(1205932+120593119873) sdot sdot sdot 119890minus119895(119873minus1)1205932+120593119873]

119879

(15)

where

1205931= (

2120587119889 sin (120579)120582

) minus (2120587119903 sdot Δ119891

1

119888) (16a)

120593119873= (

2120587119873119889 sin (120579)120582

) (16b)

1205932= (

2120587119889 sin (120579)120582

) minus (2120587119903 sdot Δ119891

2

119888) (16c)

The corresponding FIM can be derived as

ITS-FDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (17)

where

119868120579120579

= 2

10038161003816100381610038161205730

1003816100381610038161003816

2

1205902

119899

41205872

1198892cos2 (120579)1205822

2119873

sum

119898=1

(119898 minus 1)2

(18a)

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2

1205902

119899

41205872

(Δ1198912

1+ Δ119891

2

2)

1198882

119873

sum

119898=1

(119898 minus 1)2

(18b)

119868120579119903= 119868

119903120579= minus

210038161003816100381610038161205730

1003816100381610038161003816

2

1205902

119899

41205872

119889 cos (120579)120582119888

times [(Δ1198911+ Δ119891

2)

119873

sum

119898=1

(119898 minus 1)2

+ Δ1198912119873

119873

sum

119898=1

(119898 minus 1)]

(19)

The angle and range CRLBs are the first two diagonalelements of the inverse of the FIM

CRLB120579120579TS-FDA

= [Iminus1

TS-FDA]11

(20a)


= [Iminus1

TS-FDA]22

(20b)

where [sdot]119894119895is the element at the 119894th row and 119895th column of the

matrix

4 CRLB with Signal Model prior toMatched Filtering

41 Data Models Unlike the first data model the data modelprior to matched filtering allows to estimate the Doppler shift(velocity) Suppose an 119872-element antenna array receives atime delayed and Doppler-shifted echo of the transmittedsignal 119904(119905) exp(119895Ω

119888119905) where Ω

119888is the center frequency

Knowing the time delay and Doppler shift Ω119863(assuming

a target with constant radial velocity) the range and radialcomponent of velocity can be determined by 119903 = 1198881205912

and V = Ω119863119888(2Ω

119888) We denote the continuous-time signal

119904(119905) as 119904[119897] = 119904(119897 sdot Δ119905) where Δ119905 is the sampling intervalCorrespondingly the time delay and Doppler shift in thesampled signal domain are 119897

120591= 120591Δ119905 and 119908

119863= Ω

119863sdot Δ119905

respectively [27]After converting to baseband and sampling the received

signal at time 119897 sdot Δ119905 becomes

y [119899] = 120573 sdot a (120579 119903) sdot 119904 [119897 minus 119897120591]exp (119895119908

119863119897) + n [119897] 119897 = 1 119871

(21)

where 120573 is complex amplitude of the signal and n[119897] is theadditive noise [28]

We assume that the snapshots taken at 119897 = 1 119871 coverthe whole of a coherent processing interval (CPI) Thereforethe time duration of the CPI is 119879CPI = 119871 sdot Δ119905 Under thedata model of (21) the complex amplitude 120573 is assumed tobe an unknown deterministic constant during the CPI Tomodel the Doppler effect with a frequency shift the radialcomponent of the target velocity needs to be much smallerthan the propagation speed (ie V119888 ≪ 1) Then the time-bandwidth product of the complex envelope should be largerthan 1 In addition it is assumed that the propagation time ofthe signal across the array is much smaller than the reciprocal


of the signal bandwidth which is the narrowband arrayassumption in array processing

Define the vector of unknown target parameters as120581 = [Re120573 Imag120573 120579 120578119879

] Stacking all samples into a singlevector (21) can be rewritten as

z = 120573 sdot 120601 (120578) otimes a (120579 119903) + n = (120581) + n (22)

where otimes denotes the Kronecker product (120581) =

120573 sdot 120601(120578) otimes a(120579 119903)

120601 (120578) = [119904 [1 minus2119903

119888 sdot Δ119905] exp (119895119908

1198631)

119904 [2 minus2119903

119888 sdot Δ119905] exp (119895119908

1198632)

119904 [119871 minus2119903

119888 sdot Δ119905] exp (119895119908

119863119871)]

119879

(23)

The noise n is assumed to be zero-mean Gaussian spatiallyand temporally correlated with spatiotemporal covariance[29]

119864 [nnH] = Cn otimes Rn (24)

where Cn is the temporal noise covariance matrixUnder the above data model the signal and noise param-

eters are disjoint and satisfy the space-time separability [30]

42 CRLBs of Angle Range and Doppler Shift Suppose thereceived signal is completely covered by the observations119897 = 1 119871 and the sampling is dense (ie Δ119905 rarr 0) TheRn is assumed to be constant in the frequency band 119891 isin

(minus1(2Δ119905) 1(2Δ119905)) where 119891 denotes the frequency in thecontinuous-time domain The vector of Doppler shift andtime delay in the continuous-time domain is defined as

120578 = [119903 Ω119863]119879

(25)

For Cn = I119871and Rn = 120590

2

119899I119872

with I119871being an 119871 times 119871

identity matrix We define the signal power s = 120601H 120601 The

CRLBs expression for 120579 and 120578 follows fromAppendix D as thefollowing

CRLB120579120578120579120578FDA

=

1205902

119899

210038161003816100381610038161205731003816100381610038161003816

2s

times

[[[[[[[[[[[

[

12058721198892cos2 (120579)119872(119872

2minus 1)

[

[

1

31205822

+

(Δ119891)2[(2119872minus 1) (8119872

2minus 3119872minus 11)]

451198882(119872+ 1)

+

2Δ119891 (119872minus 1)

3119888120582

]

]

minus1205872119889Δ119891 cos (120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 0

minus1205872119889Δ119891 cos (120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 1205872(Δ119891)2119872(119872

2minus 1)

31198882

+

12058511

sImag 12058512

s

0

Imag 12058512

s12058522

s

]]]]]]]]]]]

]

minus1

(26)

where

12058511

= int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905

minus1

s[int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905]

2

(27a)

12058522

= int

infin

minusinfin

1199052

|119904 (119905 minus 120591)|2

119889119905

minus1

s[int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

2

(27b)

12058512

= int

infin

minusinfin

119905119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905

minus [1

sint

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905

sdot int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905]

(28)

The terms 12058511

and 12058522

are proportional to the rootmean squared (RMS) bandwidth and RMS duration of 119904(119905)respectively Note that the decoupling is a consequence ofthe assumed space-time separability of signal and noisemodels and the assumption of the complex amplitude 120573 as anunknown deterministic constant In addition since the signaland additive noise parameters are disjoint the above CRLBexpressions hold regardless of whether the spatiotemporalnoise covariance Cn otimes Rn is known or unknown

43 CRLBs When Chirp Pulse Signal Is Employed In thissection we derive the CRLB expressions for the rangeDoppler shift and direction estimation when 119904(119905) is a chirppulse signal with a large time-bandwidth product [31 32] Itcan be expressed as

119904 (119905) =

119875minus1

sum

119901=0

1199040(119905 minus 119901119879

119901) (29)


where 119879119901is the pulse repetition interval 119875 is the number of

chirp pulses and 1199040(119905) is expressed as [12]

1199040(119905) = exp [119895120587 119861

1198790

(119905 minus1

21198790)

2

] sdot [ℎ (119905) minus ℎ (119905 minus 1198790)]

(30)

where1198790is the chirp pulse duration119861 is the chirp bandwidth

and ℎ(119905) is the Heaviside step function

Assume the time-bandwidth product of the pulse is1198790sdot 119861 ≫ 1 Using the signal given in (29) and (30) in

continuous-time domain we obtain the signal power s =

1198751198790 120585

11= 1198754120587

2

1198612

11987903119888

2 Imag12058512 = minus(119875120587119861119879

2

03119888) and

12058522

= (119875MT3

012)[1 + (119879

0119879

119877)2

(1198752

minus 1)] Thus the CRLBexpressions of 120579 and 120578 for FDA are derived as the followingequation

CRLB120579120578120579120578FDA

=

1

2SNR

sdot

[[[[[[[[[[[[

[

12058721198892cos2(120579)119872(119872

2minus 1)

[

[

1

31205822+

(Δ119891)2[(2119872minus 1) (8119872

2minus3119872minus11)]

451198882119872(119872+ 1)

+

2Δ119891 (119872minus 1)

3119888120582

]

]

minus1205872119889Δ119891 cos(120579)119872(119872

2minus1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 0

minus1205872119889Δ119891 cos(120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 1205872(Δ119891)2119872(119872

2minus1)

31198882

+

412058721198612119872

31198882

minus

1198721205871198611198790

3119888

0 minus

1198721205871198611198790

3119888

1198721198792

0

12

[1+(

119879119877

1198790

)

2

(1198752minus1)]

]]]]]]]]]]]]

]

minus1

(31)

Specially for the phased array (ie Δ119891 = 0) the CRLB is

CRLB120579120578120579120578phased-array

=1

2SNR

times

[[[[[[[[

[

1205872

1198892cos2 (120579)119872(119872

2

minus 1)

312058220 0

04120587

2

1198612

119872

31198882minus119872120587119861119879

0

3119888

0 minus119872120587119861119879

0

3119888

1198721198792

0

12[1 + (

119879119877

1198790

)

2

(1198752

minus 1)]

]]]]]]]]

]

minus1

(32)

When only one pulse (119875 = 1) is used the CRLBin (32) will be infinite because the model is not identifi-able and the range and Doppler shift cannot be uniquelyestimated [18] since phase-array radar has no rangeidentity capability In contrast (31) does not have sucha problem

According to the previous discussion the angle and rangeof targets cannot be estimated jointly due to the range-anglecoupling Using the similar transmit subaperturing approachpresented in Section 3 and steering vector b(120579 119903) (15) theCRLBs of angle range and Doppler shift are derived as inthe following equation

CRLB120579120578120579120578TS-FDA =

1

2SNR

times

[[[[[[[[[[[[[[[[[[[

[

412058721198892cos2 (120579)1205822

2119873

sum

119898=1

(119898 minus 1)2

minus

41205872119889 cos (120579)120582119888

[[[[

[

(Δ1198911 + Δ1198912)

119873

sum

119898=1

(119898 minus 1)2

+119873Δ1198912

119873

sum

119898=1

(119898 minus 1)

]]]]

]

0

minus

41205872119889 cos (120579)120582119888

[[[[

[

(Δ1198911 + Δ1198912)

119873

sum

119898=1

(119898 minus 1)2

+119873Δ1198912

119873

sum

119898=1

(119898 minus 1)

]]]]

]

(Δ1198912

1+ Δ1198912

2)

1198882

119873

sum

119898=1

(119898 minus 1)2+

412058721198612119872

31198882

minus

1198721205871198611198790

3119888

0 minus

1198721205871198611198790

3119888

1198721198792

0

12

[1 + (

119879119877

1198790

)

2

(1198752minus 1)]

]]]]]]]]]]]]]]]]]]]

]

minus1

(33)


20151050

10minus2

10minus3

10minus47001 7002 7003

10minus248203

10minus248201

10minus248199

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

Figure 2 CRLB for estimating angle versus SNR when the range isknown

5 Simulation and Verification

In this section we consider several numerical examples thatcompare the CRLBs in different signal and noise modelsConsider an X-band FDA radar with the carrier frequency1198910

= 10GHz We assume a ULA of 119872 elements usedfor transmitting The array elements are spaced half of thewavelength apart from each other namely 119889 = 1205822 Onetarget of interest is supposed to reflect a plane wave thatimpinges on the array from direction of angle 120579 = 30

∘Under the signal model after matched filtering Figures

2 and 3 compare the CRLBs according to (9) and (13)respectively It can be noticed that the CRLBs are improvedwhen a larger number of elements are employed Howeverit has no significant difference when different frequencyincrements are used This is because sum

119872

119898=1(119898 minus 1)

2

1205822

≫

(Δ119891)2

sum119872

119898=1(119898 minus 1)

4

1198882

+ 2Δ119891sum119872

119898=1(119898 minus 1)

3

120582119888 thus thefrequency increment has a small impact on the CRLBsIn [33] a frequency offset selection strategy is derived toprecisely steer the beam toward a fixed range with a desiredangle

Figure 4 shows the CRLBs of angle and range whenboth the angle and range are unknown The CRLBs aresignificantly degraded due to the range-angle coupling Con-sequently the range and angle of targets cannot be estimateddirectly by the FDA radar However the CRLBs decreaseas the increase of the number of elements and frequencyincrement still holds Moreover generally more elementsmean that better CRLBs performance can be achieved for theFDA radar

To overcome the problem that the range and angle oftargets cannot be estimated directly by the FDA radar weuse the transmit subaperturing strategy on the transmitfrequency increments Figure 5 shows the corresponding

CRLB

of r

ange

estim

atio

n (m

)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

102

101

10020151050

SNR (dB)

Figure 3 CRLB for estimating range versus SNR when the angle isknown

CRLBs where 119872 = 32 is assumed It can be noticed that toobtain a lower CRLB the Δ119891

1and Δ119891

2should have inverse

signs that is one is passive and the other is negative Onereason is that in this case the FDA radar has a wider systembandwidth Figure 6 shows the CRLBs of angle and rangewhen 119872 = 20 is employed It can be noticed that theCRLBs performance improves with the increase of the sensornumber

Under the data model prior to matched filtering wesuppose the following signal parameters bandwidth 119861 =

10MHz repetition period 119879119901

= 1ms and pulse duration1198790

= 250 120583s In this case the approximate expressionsgiven in (31) are valid because the transmitted has a largetime-bandwidth product (119879

0sdot 119861 = 2500 ≫ 1) Figure 7

shows the CRLBs for direction range and Doppler shift asa function of SNR Note that when SNR = minus10 dB and119872 = 32 are employed we can get CRLB

119903119903FDA= 653m

and CRLBΩ119863Ω119863FDA

= 136937 rads that corresponds to theCRLB for velocity is 327 cms (since V = Ω

119863119888(2Ω

119888) and

Ω119888= 2120587119891

119888) Since (Δ119891)2 sum119872

119898=1(119898 minus 1)

4

1198882

≪ 1 the frequencyincrement has a small impact on the CRLBs In additionobserve that the CRLB for 120579 and 120578 is block-diagonal (see(32)) and therefore decoupled that is CRLB

120578120578FDAremains the

same whether or not 120579 is known and similarly CRLB120579120579FDA

is the same whether or not 120578 is known The decouplingis a consequence of the assumed space-time separability ofsignal and noise models and the assumption of the complexamplitude 120573 as an unknown deterministic constant

Figure 8 shows that the CRLBs versus SNR for differentcombinations of Δ119891

1and Δ119891

2 Comparing Figures 8 and

7 the CRLBs have been significantly improved Likewisecomparing Figures 8 and 9 theCRLBs performance improvesas the number of elements increases


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

104

103

102

101

CRLB

of a

ngle

estim

atio

n (d

eg)

(a) CRLB for estimating angle versus SNR

108

107

106

105

10420151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

CRLB

of r

ange

estim

atio

n (m

)

(b) CRLB for estimating range versus SNR

Figure 4 Both angle and range are unknown

6 Conclusion

In this paper we derive the CRLB to jointly estimate theattributes of a moving target using FDA radar and computethe corresponding CRLB expressions First we briefly intro-duce the FDA concept and make a summary on the FDAcharacteristics Then we consider two different data modelsnamely pre- and postmatched filtering Under differentsignal and noise models we compute the CRLB expressionsfor estimating the range direction and Doppler shift Wedemonstrate that the FDA radar beamforming is coupledin range and angle and that the targetrsquos range and anglecannot be estimated directly by the FDA radar To overcome

20151050

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)

Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz

Δf1 = 0kHz Δf2 = 30

Δf1 = 15kHz Δf2 = 40kHzkHz


20151050

SNR (dB)

CRLB

of r

ange

estim

atio

n (m

)102

101

100


Δf1 = 0kHz Δf2 = 30



Figure 5 CRLBs of TS-FDA radar with119872 = 32

this problem this paper proposes a transmit subaperturingstrategy for the FDA radar In doing so the range and angle oftargets are estimated from the transmit-receive beamformingoutput Moreover we also specialize the CRLB results tothe case of temporally white noise and a chirp pulse signalExtensive simulation results verify the correctness of thederived CRLBs It is shown that the CRLBs decrease with theincrease of the number of elements and frequency incrementThe CRLBs can be further improved through three aspectsincreasing the number of elements enhancing the systembandwidth by employing a larger frequency increment andusing transmit subaperturing strategy with more subarrays


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

20151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30



CRLB

of r

ange

estim

atio

n (m

)

103

102

101

10020151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30




Appendices

A Derive the CRLB for Angle WhenRange Is Known

To derive the CRLB we start with a well-known expressionfor the FIM under the data model in Section 3 We define thespatial noise covariance matrix as Rn = 120590

2

119899I119872

and signal-to-noise ratio (SNR) as SNR = |120573

0|2

1205902

119899 Suppose the target range

is known the FIM of 120579 is

119868120579120579

= 2Re 119863H120579(120595) (Rminus1

n )119863120579(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

(A1)

For a phased-array radar there is (120595) = 1205730a(120579) We then

have

120597a (120579)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579) (A2)

whereD = diag[0 1 119872 minus 1] and

120597aH (120579)

120597120579

120597a (120579)120597120579

=4120587

2

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A3)

The FIM of the phased-array radar is

119868120579120579phased-array

= 2SNR41205872

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A4)

Similarly for the FDA radar there is (120595) = 1205730a(120579 119903)The

derivation of a(120579 119903) with respect to 120579 is

120597a (120579 119903)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579 119903)

minus 1198952120587119889Δ119891 cos (120579)

119888

times diag [0 1 (119872 minus 1)2

] a (120579 119903)

120597aH (120579 119903)

120597120579

120597a (120579 119903)120597120579

= 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822

+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A5)

Accordingly the FIM of 120579 for the FDA can be expressedas

119868120579120579FDA

= 2SNR sdot 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A6)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

CRLB

of r

ange

estim

atio

n (m

)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)

(c) CRLB for Doppler shift versus SNR

Figure 7 General CRLB results of FDA radar

B Derive the CRLB for Range WhenAngle Is Known

Under the data model in Section 3 when the direction 120579 isknown the parameter to be estimated is 119903 The FIM of 119903 is

119868119903119903FDA

= 2Re 119863H119903(120595) (Rminus1

n )119863119903(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597aH (120579 119903)

120597119903Rminus1

n120597a (120579 119903)

120597119903

(B1)

The derivation of a(120579 119903) with respect to 119903 for FDA is

120597a (120579 119903)120597119903

= 1198952120587Δ119891

119888Da (120579 119903)

120597aH (120579 119903)

120597119903

120597a (120579 119903)120597119903

=4120587

2

Δ1198912

1198882

119872

sum

119898=1

(119898 minus 1)2

(B2)

The FIM of 119903 is thus given by

119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(B3)


C Derive the CRLB for Range and Angle

Under the data model in Section 3 both the angle and rangeare unknown The range and angle of targets are estimatedjointly The FIM for parameters 120579 and 119903 can be expressed as

IFDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (C1)

We then have

IFDA = 210038161003816100381610038161205730

1003816100381610038161003816

2

times

[[[[

[

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

]]]]

]

(C2)

Since (120597H(120595)120597119903)Rminus1

n (120597(120595)120597120579) = (120597H(120595)

120597120579)Rminus1

n (120597(120595)120597119903) then 119868120579119903= 119868

119903120579 We can get

119868120579120579

= 210038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

= 81205872

1198892cos2 (120579)

times SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

119868120579119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

= minus81205872

119889Δ119891 cos (120579)

times SNR[sum

119872

119898=1(119898 minus 1)

2

120582119888+Δ119891sum

119872

119898=1(119898 minus 1)

3

1198882]

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

= SNR8120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(C3)

Since CRLB120579120579FDA

= [Iminus1

FDA]11 CRLB119903119903FDA

= [Iminus1

FDA]22 Theexpressions for the CRLB of angle and range estimationsgiven in (14) can be obtained by substituting (C3) into (C2)

D General CRLB Results

Under the data model in Section 4 we derive the continuousCRLB expressions for temporally and spatially white noiseand denote SNR = |120573|

2

s1205902

119899with s being the power We

also start with FIM

I120581119894120581119895

= 2Re 119863H120581119894

(120581) (Cminus1

n otimes Rminus1

n )119863120581119895(120581) (D1)

where 120581119894is the 119894th element of 120581 and 119863

120581119894(120595) = 120597(120581)120597120581

119894

Consider

IFDA =

[[[[[

[

119868120573120573

119868119879

120579120573119868119879

120578120573

119868120579120573

119868120579120579

119868119879

120578120579

119868120578120573

119868120578120579

119868120578120578

]]]]]

]

(D2)

For clarity we rewrite Fisherrsquos information matrix I as

IFDA = [A UV B] (D3)

where

V = U119879

(D4a)

V = [119868120579120573

119868120578120573

] (D4b)

B = [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] (D4c)

According to the matrix inversion lemma the inversematrix of IFDA is

Iminus1

FDA = [

[

(A minus UBminus1V)minus1

minusAminus1U(B minus VAminus1U)minus1

minusBminus1V(A minus UBminus1V)minus1

(B minus VAminus1U)minus1

]

]

(D5)

where the submatrix (A minus UBminus1V)minus1 corresponds to theCRLBof120573which is nuisance and (B minus VAminus1U)minus1 correspondsto the CRLB of 120579 and 120578 = [119903 Ω

119863]119879 which are of interest

CRLB120579120578120579120578FDA

= (B minus VAminus1U)minus1

= [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] minus [119868120579120573

119868120578120573

] 119868minus1

120573120573[119868

119879

120579120573119868119879

120578120573]

minus1

(D6)

where

119868120573120573

= 2 sdot[[[

[

s sdot 1198721205902

119899

0

0s sdot 1198721205902

119899

]]]

]

(D7a)

119868120579120573

= 2 sdot Re[1 119895] otimess1205902

119899

sdot 120573lowast

1198601 (D7b)

119868120578120573

= 2 sdot Re[1 119895] otimes120573

lowast

1205902

119908

sdot [119872 sdot 1198603+ s sdot 119860

4] (D7c)

119868120578120578

= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

[119872 sdot 1198606+ 119860

H3119860

4+ 119860

3119860

H4+ s sdot 119860

7]

(D7d)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

20151050minus5minus10

SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)

SNR (dB)


Δf1 = 0kHz Δf2 = 30

Δf1 = 15kHz Δf2 = 40kHz

70001 70001 70001 70002100286514

100286515

100286515

kHz


Figure 8 General CRLB results of TS-FDA radar with119872 = 32

119868120579120579minus (119868

120579120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198602minus119860

1119860

H1

119872) (D8a)

119868120578120579minus (119868

120578120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198605minus119860

4119860

H1

119872) (D8b)

119868120578120578

minus (119868120578120573119868119879

120578120573)

1205902

119899

s119872

= 2

10038161003816100381610038161205731003816100381610038161003816

2

s1205902

119899

(119872119860

6

s+ 119860

7minus119872119860

3119860

H3

s2minus119860

4119860

H4

119872)

(D8c)

1198601= minus119895(

2120587119889 cos (120579)sum119872

119898=1(119898 minus 1)

120582

+2120587119889Δ119891 cos (120579)sum119872

119898=1(119898 minus 1)

2

119888)

(D9a)

1198602= 4120587

2

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(D9b)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant


41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities

References

[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012

[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010

[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011

[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012

[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006

[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006

[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006

[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006

[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008

[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007

[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008

[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009

[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007

[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008

[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013

[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998

[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995

[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001

[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999

[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009

[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012

[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010

[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011

[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002

[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012

[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014

[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011

[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011

[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013

[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000

[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012


[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013

[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



(CRLB) expressions for estimating velocity direction andDoppler shift are derived in [16 17] By assuming a narrow-band signal model prior to matched filtering the CRLBs forestimating the time delay direction and Doppler shift with aphased-array radar are detailed in [18 19] Furthermore theCRLB for jointly estimating the attributes of a moving targetusing MIMO radar is derived in [20ndash22]

Different from the above literatures this paper derivesthe CRLBs for jointly estimating the range direction andDoppler shift with an FDA radar Two typical data mod-els namely pre- and postmatched filtering are analyzedseparately for static and moving targets respectively Sincethe range and angle of targets cannot be estimated directlydue to the range-angle coupling we use a simple transmitsubaperturing strategy for the FDA radar This methoddivides the whole array elements into two subarrays eachuses a different frequency increment In doing so the rangeand angle of targets can be solely estimated The corre-sponding CRLBs for estimating direction range and veloc-ity are derived Furthermore the impacts of the elementsnumber and frequency increment on the CRLBs are alsoinvestigated

The rest of the paper is organized as follows The FDAis briefly introduced in Section 2 In Section 3 we considerthe data model after matched filtering and derive the CRLBexpressions for estimating the range and direction In orderto decouple the range-angle coupling response we divide the

whole array into two subarrays The corresponding CRLBsare derived In Section 4 the data model prior to matchedfiltering is investigated and the CRLB expressions for estimat-ing the direction range and Doppler shift are derived wheretemporally white Gaussian noise and chirp pulse signal areassumed Finally extensive simulation results are provided inSection 5 This paper is concluded in Section 6

2 Frequency Diverse Array (FDA)

In conventional phased-array radars it is assumed that anidentical waveform is radiated from each element excludingthe amplitudes and phases Different from phased-arrayradars FDA elements can be either excited by the samewaveform or different waveforms For simplicity and withoutloss of generality we consider a uniform linear array (ULA)FDA and assume that the waveforms radiated from eachantenna element are identical but with a frequency incrementof Δ119891HzThat is the radiation frequency of the119898th elementis

119891119898= 119891

0+ (119898 minus 1) sdot Δ119891 119898 = 0 1 119872 minus 1 (1)

where 1198910is the radar carrier frequency and119872 is the number

of the elementsSupposing a narrowband transmit signal and taking the

first element as the reference for the array the steering vectoris given by the following equation [11]

a (120579 119903) = [1 119890minus119895((21205871198910119889 sin 120579119888)+((2120587Δ119891sdot119889 sin 120579)119888)minus(2120587119903Δ119891119888))

sdot sdot sdot 119890minus119895(((21205871198910(119872minus1)119889 sin 120579)119888)+((2120587sdot(119872minus1)

2Δ119891sdot119889 sin 120579)119888)minus((2120587119903(119872minus1)Δ119891)119888))]

119879

(2)

where 120579 is the direction 119903 is the range 119889 is the elementspacing 119888 is the speed of light and 119879 is the transpose Since1198910

≫ Δ119891 and 119903 ≫ (119872 minus 1)119889 sin(120579) in an amplitude

sense an approximation can be taken by ignoring the phaseterm 2120587 sdot (119898 minus 1)

2

Δ119891 sdot 119889 sin(120579)119888 because it has ignorableimpacts In this case the steering vector can be simplifiedto

a (120579 119903) asymp [1 119890minus119895((21205871198910119889 sin(120579)119888)minus((2120587sdotΔ119891sdot119903)119888))

sdot sdot sdot 119890minus119895(((21205871198910(119898minus1)119889 sin(120579))119888)minus((2120587sdot(119898minus1)Δ119891sdot119903)119888))

]119879

(3)

The FDA characteristics can be summarized as follows[7ndash12] (1) If the frequency offset Δ119891 is fixed the beamdirection will vary as a function of the range 119903 that is it isa range- dependent beam (2) If the range is 119903 fixed the beamdirection will vary as a function of Δ119891 This means that theFDA is frequency-increment-dependent (3) If the frequencyincrement across the array is not applied (ie Δ119891 = 0)the corresponding FDA is just a conventional ULA phasedarray

3 CRLB with Signal Model afterMatched Filtering

31 Data Models Suppose that a target is located at (120579 119903)After matched filtering the baseband equivalent of thecomplex valued signals at the receiver can be expressed as

y = 1205730a (120579 119903) + n = (120595) + n (4)



0a(120579 119903)with



119864 [nnH] = Rn = 120590

2

119899I119872 (5)


119899is the




120595120595[23]

I120595120595

= 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) (6)


of 120595 and the119863120595119894(120595) is [24]

119863120595119894(120595) =

120597 (120595)

120597120595119894

(7)


119868120579120579FDA

= 81205872

1198892cos2 (120579)

sdot SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(8)


CRLB120579120579FDA

= 119868minus1

120579120579FDA (9)


119868120579120579phased-array

= 8SNR1205872

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(10)



CRLB120579120579FDA


(11)



119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(12)


CRLB119903119903FDA

= 119868minus1

119903119903FDA (13)



CRLB120579120579FDA

= (1198882

sdot (

119872

sum

119898=1

(119898 minus 1)2

))


1198892

sdot (Δ119891)2cos2 (120579)

times (

119872

sum

119898=1

(119898 minus 1)4

119872

sum

119898=1

(119898 minus 1)2

minus(

119872

sum

119898=1

(119898 minus 1)3

)

2

))

minus1

CRLB119903119903FDA

= (1198884

sdot (sum

119872

119898=1(119898 minus 1)

2

1205822

+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888))


sdot (Δ119891)4

times (

119872

sum

119898=1

(119898 minus 1)4

119872

sum

119898=1

(119898 minus 1)2

minus(

119872

sum

119898=1

(119898minus1)3

)

2

))

minus1

(14)




d

1 2 N


1 2 N






1 and







119879

(15)

where

1205931= (

2120587119889 sin (120579)120582

) minus (2120587119903 sdot Δ119891

1

119888) (16a)

120593119873= (

2120587119873119889 sin (120579)120582

) (16b)

1205932= (

2120587119889 sin (120579)120582

) minus (2120587119903 sdot Δ119891

2

119888) (16c)


ITS-FDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (17)

where

119868120579120579

= 2

10038161003816100381610038161205730

1003816100381610038161003816

2

1205902

119899

41205872

1198892cos2 (120579)1205822

2119873

sum

119898=1

(119898 minus 1)2

(18a)

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2

1205902

119899

41205872

(Δ1198912

1+ Δ119891

2

2)

1198882

119873

sum

119898=1

(119898 minus 1)2

(18b)

119868120579119903= 119868

119903120579= minus

210038161003816100381610038161205730

1003816100381610038161003816

2

1205902

119899

41205872

119889 cos (120579)120582119888

times [(Δ1198911+ Δ119891

2)

119873

sum

119898=1

(119898 minus 1)2

+ Δ1198912119873

119873

sum

119898=1

(119898 minus 1)]

(19)



= [Iminus1

TS-FDA]11

(20a)


= [Iminus1

TS-FDA]22

(20b)


matrix



119888119905) where Ω




and V = Ω119863119888(2Ω



120591= 120591Δ119905 and 119908

119863= Ω

119863sdot Δ119905




119863119897) + n [119897] 119897 = 1 119871

(21)







z = 120573 sdot 120601 (120578) otimes a (120579 119903) + n = (120581) + n (22)


120573 sdot 120601(120578) otimes a(120579 119903)

120601 (120578) = [119904 [1 minus2119903

119888 sdot Δ119905] exp (119895119908

1198631)

119904 [2 minus2119903

119888 sdot Δ119905] exp (119895119908

1198632)

119904 [119871 minus2119903

119888 sdot Δ119905] exp (119895119908

119863119871)]

119879

(23)







120578 = [119903 Ω119863]119879

(25)

For Cn = I119871and Rn = 120590

2

119899I119872




CRLB120579120578120579120578FDA

=

1205902

119899

210038161003816100381610038161205731003816100381610038161003816

2s

times

[[[[[[[[[[[

[

12058721198892cos2 (120579)119872(119872

2minus 1)

[

[

1

31205822

+

(Δ119891)2[(2119872minus 1) (8119872


451198882(119872+ 1)

+

2Δ119891 (119872minus 1)

3119888120582

]

]

minus1205872119889Δ119891 cos (120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 0

minus1205872119889Δ119891 cos (120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 1205872(Δ119891)2119872(119872

2minus 1)

31198882

+

12058511

sImag 12058512

s

0

Imag 12058512

s12058522

s

]]]]]]]]]]]

]

minus1

(26)

where

12058511

= int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905

minus1

s[int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905]

2

(27a)

12058522

= int

infin

minusinfin

1199052

|119904 (119905 minus 120591)|2

119889119905

minus1

s[int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

2

(27b)

12058512

= int

infin

minusinfin

119905119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905

minus [1

sint

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905

sdot int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905]

(28)

The terms 12058511

and 12058522



119904 (119905) =

119875minus1

sum

119901=0

1199040(119905 minus 119901119879

119901) (29)




1199040(119905) = exp [119895120587 119861

1198790

(119905 minus1

21198790)

2


(30)





1198751198790 120585

11= 1198754120587

2

1198612

11987903119888

2 Imag12058512 = minus(119875120587119861119879

2

03119888) and

12058522

= (119875MT3

012)[1 + (119879

0119879

119877)2

(1198752


CRLB120579120578120579120578FDA

=

1

2SNR

sdot

[[[[[[[[[[[[

[

12058721198892cos2(120579)119872(119872

2minus 1)

[

[

1

31205822+

(Δ119891)2[(2119872minus 1) (8119872


451198882119872(119872+ 1)

+

2Δ119891 (119872minus 1)

3119888120582

]

]

minus1205872119889Δ119891 cos(120579)119872(119872

2minus1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 0

minus1205872119889Δ119891 cos(120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 1205872(Δ119891)2119872(119872

2minus1)

31198882

+

412058721198612119872

31198882

minus

1198721205871198611198790

3119888

0 minus

1198721205871198611198790

3119888

1198721198792

0

12

[1+(

119879119877

1198790

)

2

(1198752minus1)]

]]]]]]]]]]]]

]

minus1

(31)



=1

2SNR

times

[[[[[[[[

[

1205872

1198892cos2 (120579)119872(119872

2

minus 1)

312058220 0

04120587

2

1198612

119872

31198882minus119872120587119861119879

0

3119888

0 minus119872120587119861119879

0

3119888

1198721198792

0

12[1 + (

119879119877

1198790

)

2

(1198752

minus 1)]

]]]]]]]]

]

minus1

(32)



CRLB120579120578120579120578TS-FDA =

1

2SNR

times

[[[[[[[[[[[[[[[[[[[

[

412058721198892cos2 (120579)1205822

2119873

sum

119898=1

(119898 minus 1)2

minus

41205872119889 cos (120579)120582119888

[[[[

[

(Δ1198911 + Δ1198912)

119873

sum

119898=1

(119898 minus 1)2

+119873Δ1198912

119873

sum

119898=1

(119898 minus 1)

]]]]

]

0

minus

41205872119889 cos (120579)120582119888

[[[[

[

(Δ1198911 + Δ1198912)

119873

sum

119898=1

(119898 minus 1)2

+119873Δ1198912

119873

sum

119898=1

(119898 minus 1)

]]]]

]

(Δ1198912

1+ Δ1198912

2)

1198882

119873

sum

119898=1

(119898 minus 1)2+

412058721198612119872

31198882

minus

1198721205871198611198790

3119888

0 minus

1198721205871198611198790

3119888

1198721198792

0

12

[1 + (

119879119877

1198790

)

2

(1198752minus 1)]

]]]]]]]]]]]]]]]]]]]

]

minus1

(33)


20151050

10minus2

10minus3

10minus47001 7002 7003

10minus248203

10minus248201

10minus248199

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz







119872

119898=1(119898 minus 1)

2

1205822

≫

(Δ119891)2

sum119872

119898=1(119898 minus 1)

4

1198882

+ 2Δ119891sum119872

119898=1(119898 minus 1)

3




CRLB

of r

ange

estim

atio

n (m

)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

102

101

10020151050

SNR (dB)



1and Δ119891







0sdot 119861 = 2500 ≫ 1) Figure 7


119903119903FDA= 653m



119863119888(2Ω

119888) and

Ω119888= 2120587119891

119888) Since (Δ119891)2 sum119872

119898=1(119898 minus 1)

4

1198882






1and Δ119891




20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

104

103

102

101

CRLB

of a

ngle

estim

atio

n (d

eg)


108

107

106

105

10420151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

CRLB

of r

ange

estim

atio

n (m

)



6 Conclusion


20151050

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)


Δf1 = 0kHz Δf2 = 30



20151050

SNR (dB)

CRLB

of r

ange

estim

atio

n (m

)102

101

100


Δf1 = 0kHz Δf2 = 30






10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

20151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30



CRLB

of r

ange

estim

atio

n (m

)

103

102

101

10020151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30




Appendices



2

119899I119872


0|2

1205902



119868120579120579

= 2Re 119863H120579(120595) (Rminus1

n )119863120579(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

(A1)


have

120597a (120579)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579) (A2)


120597aH (120579)

120597120579

120597a (120579)120597120579

=4120587

2

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A3)


119868120579120579phased-array

= 2SNR41205872

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A4)



120597a (120579 119903)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579 119903)

minus 1198952120587119889Δ119891 cos (120579)

119888


] a (120579 119903)

120597aH (120579 119903)

120597120579

120597a (120579 119903)120597120579

= 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822

+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A5)


119868120579120579FDA

= 2SNR sdot 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A6)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

CRLB

of r

ange

estim

atio

n (m

)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)





119868119903119903FDA

= 2Re 119863H119903(120595) (Rminus1

n )119863119903(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597aH (120579 119903)

120597119903Rminus1

n120597a (120579 119903)

120597119903

(B1)


120597a (120579 119903)120597119903

= 1198952120587Δ119891

119888Da (120579 119903)

120597aH (120579 119903)

120597119903

120597a (120579 119903)120597119903

=4120587

2

Δ1198912

1198882

119872

sum

119898=1

(119898 minus 1)2

(B2)


119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(B3)




IFDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (C1)

We then have

IFDA = 210038161003816100381610038161205730

1003816100381610038161003816

2

times

[[[[

[

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

]]]]

]

(C2)

Since (120597H(120595)120597119903)Rminus1

n (120597(120595)120597120579) = (120597H(120595)

120597120579)Rminus1

n (120597(120595)120597119903) then 119868120579119903= 119868

119903120579 We can get

119868120579120579

= 210038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

= 81205872

1198892cos2 (120579)

times SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

119868120579119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

= minus81205872

119889Δ119891 cos (120579)

times SNR[sum

119872

119898=1(119898 minus 1)

2

120582119888+Δ119891sum

119872

119898=1(119898 minus 1)

3

1198882]

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

= SNR8120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(C3)


= [Iminus1

FDA]11 CRLB119903119903FDA

= [Iminus1




2

s1205902


also start with FIM

I120581119894120581119895

= 2Re 119863H120581119894

(120581) (Cminus1

n otimes Rminus1

n )119863120581119895(120581) (D1)


120581119894(120595) = 120597(120581)120597120581

119894

Consider

IFDA =

[[[[[

[

119868120573120573

119868119879

120579120573119868119879

120578120573

119868120579120573

119868120579120579

119868119879

120578120579

119868120578120573

119868120578120579

119868120578120578

]]]]]

]

(D2)



where

V = U119879

(D4a)

V = [119868120579120573

119868120578120573

] (D4b)

B = [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] (D4c)


Iminus1

FDA = [

[





]

]

(D5)



CRLB120579120578120579120578FDA


= [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] minus [119868120579120573

119868120578120573

] 119868minus1

120573120573[119868

119879

120579120573119868119879

120578120573]

minus1

(D6)

where

119868120573120573

= 2 sdot[[[

[

s sdot 1198721205902

119899

0

0s sdot 1198721205902

119899

]]]

]

(D7a)

119868120579120573

= 2 sdot Re[1 119895] otimess1205902

119899

sdot 120573lowast

1198601 (D7b)

119868120578120573

= 2 sdot Re[1 119895] otimes120573

lowast

1205902

119908

sdot [119872 sdot 1198603+ s sdot 119860

4] (D7c)

119868120578120578

= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

[119872 sdot 1198606+ 119860

H3119860

4+ 119860

3119860

H4+ s sdot 119860

7]

(D7d)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)

SNR (dB)


Δf1 = 0kHz Δf2 = 30


70001 70001 70001 70002100286514

100286515

100286515

kHz



119868120579120579minus (119868

120579120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198602minus119860

1119860

H1

119872) (D8a)

119868120578120579minus (119868

120578120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198605minus119860

4119860

H1

119872) (D8b)

119868120578120578

minus (119868120578120573119868119879

120578120573)

1205902

119899

s119872

= 2

10038161003816100381610038161205731003816100381610038161003816

2

s1205902

119899

(119872119860

6

s+ 119860

7minus119872119860

3119860

H3

s2minus119860

4119860

H4

119872)

(D8c)

1198601= minus119895(

2120587119889 cos (120579)sum119872

119898=1(119898 minus 1)

120582

+2120587119889Δ119891 cos (120579)sum119872

119898=1(119898 minus 1)

2

119888)

(D9a)

1198602= 4120587

2

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(D9b)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)



Acknowledgments




References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0a(120579 119903)with



119864 [nnH] = Rn = 120590

2

119899I119872 (5)


119899is the




120595120595[23]

I120595120595

= 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) (6)


of 120595 and the119863120595119894(120595) is [24]

119863120595119894(120595) =

120597 (120595)

120597120595119894

(7)


119868120579120579FDA

= 81205872

1198892cos2 (120579)

sdot SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(8)


CRLB120579120579FDA

= 119868minus1

120579120579FDA (9)


119868120579120579phased-array

= 8SNR1205872

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(10)



CRLB120579120579FDA


(11)



119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(12)


CRLB119903119903FDA

= 119868minus1

119903119903FDA (13)



CRLB120579120579FDA

= (1198882

sdot (

119872

sum

119898=1

(119898 minus 1)2

))


1198892

sdot (Δ119891)2cos2 (120579)

times (

119872

sum

119898=1

(119898 minus 1)4

119872

sum

119898=1

(119898 minus 1)2

minus(

119872

sum

119898=1

(119898 minus 1)3

)

2

))

minus1

CRLB119903119903FDA

= (1198884

sdot (sum

119872

119898=1(119898 minus 1)

2

1205822

+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888))


sdot (Δ119891)4

times (

119872

sum

119898=1

(119898 minus 1)4

119872

sum

119898=1

(119898 minus 1)2

minus(

119872

sum

119898=1

(119898minus1)3

)

2

))

minus1

(14)




d

1 2 N


1 2 N






1 and







119879

(15)

where

1205931= (

2120587119889 sin (120579)120582

) minus (2120587119903 sdot Δ119891

1

119888) (16a)

120593119873= (

2120587119873119889 sin (120579)120582

) (16b)

1205932= (

2120587119889 sin (120579)120582

) minus (2120587119903 sdot Δ119891

2

119888) (16c)


ITS-FDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (17)

where

119868120579120579

= 2

10038161003816100381610038161205730

1003816100381610038161003816

2

1205902

119899

41205872

1198892cos2 (120579)1205822

2119873

sum

119898=1

(119898 minus 1)2

(18a)

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2

1205902

119899

41205872

(Δ1198912

1+ Δ119891

2

2)

1198882

119873

sum

119898=1

(119898 minus 1)2

(18b)

119868120579119903= 119868

119903120579= minus

210038161003816100381610038161205730

1003816100381610038161003816

2

1205902

119899

41205872

119889 cos (120579)120582119888

times [(Δ1198911+ Δ119891

2)

119873

sum

119898=1

(119898 minus 1)2

+ Δ1198912119873

119873

sum

119898=1

(119898 minus 1)]

(19)



= [Iminus1

TS-FDA]11

(20a)


= [Iminus1

TS-FDA]22

(20b)


matrix



119888119905) where Ω




and V = Ω119863119888(2Ω



120591= 120591Δ119905 and 119908

119863= Ω

119863sdot Δ119905




119863119897) + n [119897] 119897 = 1 119871

(21)







z = 120573 sdot 120601 (120578) otimes a (120579 119903) + n = (120581) + n (22)


120573 sdot 120601(120578) otimes a(120579 119903)

120601 (120578) = [119904 [1 minus2119903

119888 sdot Δ119905] exp (119895119908

1198631)

119904 [2 minus2119903

119888 sdot Δ119905] exp (119895119908

1198632)

119904 [119871 minus2119903

119888 sdot Δ119905] exp (119895119908

119863119871)]

119879

(23)







120578 = [119903 Ω119863]119879

(25)

For Cn = I119871and Rn = 120590

2

119899I119872




CRLB120579120578120579120578FDA

=

1205902

119899

210038161003816100381610038161205731003816100381610038161003816

2s

times

[[[[[[[[[[[

[

12058721198892cos2 (120579)119872(119872

2minus 1)

[

[

1

31205822

+

(Δ119891)2[(2119872minus 1) (8119872


451198882(119872+ 1)

+

2Δ119891 (119872minus 1)

3119888120582

]

]

minus1205872119889Δ119891 cos (120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 0

minus1205872119889Δ119891 cos (120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 1205872(Δ119891)2119872(119872

2minus 1)

31198882

+

12058511

sImag 12058512

s

0

Imag 12058512

s12058522

s

]]]]]]]]]]]

]

minus1

(26)

where

12058511

= int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905

minus1

s[int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905]

2

(27a)

12058522

= int

infin

minusinfin

1199052

|119904 (119905 minus 120591)|2

119889119905

minus1

s[int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

2

(27b)

12058512

= int

infin

minusinfin

119905119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905

minus [1

sint

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905

sdot int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905]

(28)

The terms 12058511

and 12058522



119904 (119905) =

119875minus1

sum

119901=0

1199040(119905 minus 119901119879

119901) (29)




1199040(119905) = exp [119895120587 119861

1198790

(119905 minus1

21198790)

2


(30)





1198751198790 120585

11= 1198754120587

2

1198612

11987903119888

2 Imag12058512 = minus(119875120587119861119879

2

03119888) and

12058522

= (119875MT3

012)[1 + (119879

0119879

119877)2

(1198752


CRLB120579120578120579120578FDA

=

1

2SNR

sdot

[[[[[[[[[[[[

[

12058721198892cos2(120579)119872(119872

2minus 1)

[

[

1

31205822+

(Δ119891)2[(2119872minus 1) (8119872


451198882119872(119872+ 1)

+

2Δ119891 (119872minus 1)

3119888120582

]

]

minus1205872119889Δ119891 cos(120579)119872(119872

2minus1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 0

minus1205872119889Δ119891 cos(120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 1205872(Δ119891)2119872(119872

2minus1)

31198882

+

412058721198612119872

31198882

minus

1198721205871198611198790

3119888

0 minus

1198721205871198611198790

3119888

1198721198792

0

12

[1+(

119879119877

1198790

)

2

(1198752minus1)]

]]]]]]]]]]]]

]

minus1

(31)



=1

2SNR

times

[[[[[[[[

[

1205872

1198892cos2 (120579)119872(119872

2

minus 1)

312058220 0

04120587

2

1198612

119872

31198882minus119872120587119861119879

0

3119888

0 minus119872120587119861119879

0

3119888

1198721198792

0

12[1 + (

119879119877

1198790

)

2

(1198752

minus 1)]

]]]]]]]]

]

minus1

(32)



CRLB120579120578120579120578TS-FDA =

1

2SNR

times

[[[[[[[[[[[[[[[[[[[

[

412058721198892cos2 (120579)1205822

2119873

sum

119898=1

(119898 minus 1)2

minus

41205872119889 cos (120579)120582119888

[[[[

[

(Δ1198911 + Δ1198912)

119873

sum

119898=1

(119898 minus 1)2

+119873Δ1198912

119873

sum

119898=1

(119898 minus 1)

]]]]

]

0

minus

41205872119889 cos (120579)120582119888

[[[[

[

(Δ1198911 + Δ1198912)

119873

sum

119898=1

(119898 minus 1)2

+119873Δ1198912

119873

sum

119898=1

(119898 minus 1)

]]]]

]

(Δ1198912

1+ Δ1198912

2)

1198882

119873

sum

119898=1

(119898 minus 1)2+

412058721198612119872

31198882

minus

1198721205871198611198790

3119888

0 minus

1198721205871198611198790

3119888

1198721198792

0

12

[1 + (

119879119877

1198790

)

2

(1198752minus 1)]

]]]]]]]]]]]]]]]]]]]

]

minus1

(33)


20151050

10minus2

10minus3

10minus47001 7002 7003

10minus248203

10minus248201

10minus248199

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz







119872

119898=1(119898 minus 1)

2

1205822

≫

(Δ119891)2

sum119872

119898=1(119898 minus 1)

4

1198882

+ 2Δ119891sum119872

119898=1(119898 minus 1)

3




CRLB

of r

ange

estim

atio

n (m

)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

102

101

10020151050

SNR (dB)



1and Δ119891







0sdot 119861 = 2500 ≫ 1) Figure 7


119903119903FDA= 653m



119863119888(2Ω

119888) and

Ω119888= 2120587119891

119888) Since (Δ119891)2 sum119872

119898=1(119898 minus 1)

4

1198882






1and Δ119891




20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

104

103

102

101

CRLB

of a

ngle

estim

atio

n (d

eg)


108

107

106

105

10420151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

CRLB

of r

ange

estim

atio

n (m

)



6 Conclusion


20151050

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)


Δf1 = 0kHz Δf2 = 30



20151050

SNR (dB)

CRLB

of r

ange

estim

atio

n (m

)102

101

100


Δf1 = 0kHz Δf2 = 30






10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

20151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30



CRLB

of r

ange

estim

atio

n (m

)

103

102

101

10020151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30




Appendices



2

119899I119872


0|2

1205902



119868120579120579

= 2Re 119863H120579(120595) (Rminus1

n )119863120579(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

(A1)


have

120597a (120579)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579) (A2)


120597aH (120579)

120597120579

120597a (120579)120597120579

=4120587

2

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A3)


119868120579120579phased-array

= 2SNR41205872

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A4)



120597a (120579 119903)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579 119903)

minus 1198952120587119889Δ119891 cos (120579)

119888


] a (120579 119903)

120597aH (120579 119903)

120597120579

120597a (120579 119903)120597120579

= 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822

+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A5)


119868120579120579FDA

= 2SNR sdot 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A6)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

CRLB

of r

ange

estim

atio

n (m

)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)





119868119903119903FDA

= 2Re 119863H119903(120595) (Rminus1

n )119863119903(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597aH (120579 119903)

120597119903Rminus1

n120597a (120579 119903)

120597119903

(B1)


120597a (120579 119903)120597119903

= 1198952120587Δ119891

119888Da (120579 119903)

120597aH (120579 119903)

120597119903

120597a (120579 119903)120597119903

=4120587

2

Δ1198912

1198882

119872

sum

119898=1

(119898 minus 1)2

(B2)


119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(B3)




IFDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (C1)

We then have

IFDA = 210038161003816100381610038161205730

1003816100381610038161003816

2

times

[[[[

[

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

]]]]

]

(C2)

Since (120597H(120595)120597119903)Rminus1

n (120597(120595)120597120579) = (120597H(120595)

120597120579)Rminus1

n (120597(120595)120597119903) then 119868120579119903= 119868

119903120579 We can get

119868120579120579

= 210038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

= 81205872

1198892cos2 (120579)

times SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

119868120579119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

= minus81205872

119889Δ119891 cos (120579)

times SNR[sum

119872

119898=1(119898 minus 1)

2

120582119888+Δ119891sum

119872

119898=1(119898 minus 1)

3

1198882]

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

= SNR8120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(C3)


= [Iminus1

FDA]11 CRLB119903119903FDA

= [Iminus1




2

s1205902


also start with FIM

I120581119894120581119895

= 2Re 119863H120581119894

(120581) (Cminus1

n otimes Rminus1

n )119863120581119895(120581) (D1)


120581119894(120595) = 120597(120581)120597120581

119894

Consider

IFDA =

[[[[[

[

119868120573120573

119868119879

120579120573119868119879

120578120573

119868120579120573

119868120579120579

119868119879

120578120579

119868120578120573

119868120578120579

119868120578120578

]]]]]

]

(D2)



where

V = U119879

(D4a)

V = [119868120579120573

119868120578120573

] (D4b)

B = [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] (D4c)


Iminus1

FDA = [

[





]

]

(D5)



CRLB120579120578120579120578FDA


= [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] minus [119868120579120573

119868120578120573

] 119868minus1

120573120573[119868

119879

120579120573119868119879

120578120573]

minus1

(D6)

where

119868120573120573

= 2 sdot[[[

[

s sdot 1198721205902

119899

0

0s sdot 1198721205902

119899

]]]

]

(D7a)

119868120579120573

= 2 sdot Re[1 119895] otimess1205902

119899

sdot 120573lowast

1198601 (D7b)

119868120578120573

= 2 sdot Re[1 119895] otimes120573

lowast

1205902

119908

sdot [119872 sdot 1198603+ s sdot 119860

4] (D7c)

119868120578120578

= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

[119872 sdot 1198606+ 119860

H3119860

4+ 119860

3119860

H4+ s sdot 119860

7]

(D7d)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)

SNR (dB)


Δf1 = 0kHz Δf2 = 30


70001 70001 70001 70002100286514

100286515

100286515

kHz



119868120579120579minus (119868

120579120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198602minus119860

1119860

H1

119872) (D8a)

119868120578120579minus (119868

120578120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198605minus119860

4119860

H1

119872) (D8b)

119868120578120578

minus (119868120578120573119868119879

120578120573)

1205902

119899

s119872

= 2

10038161003816100381610038161205731003816100381610038161003816

2

s1205902

119899

(119872119860

6

s+ 119860

7minus119872119860

3119860

H3

s2minus119860

4119860

H4

119872)

(D8c)

1198601= minus119895(

2120587119889 cos (120579)sum119872

119898=1(119898 minus 1)

120582

+2120587119889Δ119891 cos (120579)sum119872

119898=1(119898 minus 1)

2

119888)

(D9a)

1198602= 4120587

2

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(D9b)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)



Acknowledgments




References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










d

1 2 N


1 2 N






1 and







119879

(15)

where

1205931= (

2120587119889 sin (120579)120582

) minus (2120587119903 sdot Δ119891

1

119888) (16a)

120593119873= (

2120587119873119889 sin (120579)120582

) (16b)

1205932= (

2120587119889 sin (120579)120582

) minus (2120587119903 sdot Δ119891

2

119888) (16c)


ITS-FDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (17)

where

119868120579120579

= 2

10038161003816100381610038161205730

1003816100381610038161003816

2

1205902

119899

41205872

1198892cos2 (120579)1205822

2119873

sum

119898=1

(119898 minus 1)2

(18a)

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2

1205902

119899

41205872

(Δ1198912

1+ Δ119891

2

2)

1198882

119873

sum

119898=1

(119898 minus 1)2

(18b)

119868120579119903= 119868

119903120579= minus

210038161003816100381610038161205730

1003816100381610038161003816

2

1205902

119899

41205872

119889 cos (120579)120582119888

times [(Δ1198911+ Δ119891

2)

119873

sum

119898=1

(119898 minus 1)2

+ Δ1198912119873

119873

sum

119898=1

(119898 minus 1)]

(19)



= [Iminus1

TS-FDA]11

(20a)


= [Iminus1

TS-FDA]22

(20b)


matrix



119888119905) where Ω




and V = Ω119863119888(2Ω



120591= 120591Δ119905 and 119908

119863= Ω

119863sdot Δ119905




119863119897) + n [119897] 119897 = 1 119871

(21)







z = 120573 sdot 120601 (120578) otimes a (120579 119903) + n = (120581) + n (22)


120573 sdot 120601(120578) otimes a(120579 119903)

120601 (120578) = [119904 [1 minus2119903

119888 sdot Δ119905] exp (119895119908

1198631)

119904 [2 minus2119903

119888 sdot Δ119905] exp (119895119908

1198632)

119904 [119871 minus2119903

119888 sdot Δ119905] exp (119895119908

119863119871)]

119879

(23)







120578 = [119903 Ω119863]119879

(25)

For Cn = I119871and Rn = 120590

2

119899I119872




CRLB120579120578120579120578FDA

=

1205902

119899

210038161003816100381610038161205731003816100381610038161003816

2s

times

[[[[[[[[[[[

[

12058721198892cos2 (120579)119872(119872

2minus 1)

[

[

1

31205822

+

(Δ119891)2[(2119872minus 1) (8119872


451198882(119872+ 1)

+

2Δ119891 (119872minus 1)

3119888120582

]

]

minus1205872119889Δ119891 cos (120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 0

minus1205872119889Δ119891 cos (120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 1205872(Δ119891)2119872(119872

2minus 1)

31198882

+

12058511

sImag 12058512

s

0

Imag 12058512

s12058522

s

]]]]]]]]]]]

]

minus1

(26)

where

12058511

= int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905

minus1

s[int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905]

2

(27a)

12058522

= int

infin

minusinfin

1199052

|119904 (119905 minus 120591)|2

119889119905

minus1

s[int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

2

(27b)

12058512

= int

infin

minusinfin

119905119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905

minus [1

sint

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905

sdot int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905]

(28)

The terms 12058511

and 12058522



119904 (119905) =

119875minus1

sum

119901=0

1199040(119905 minus 119901119879

119901) (29)




1199040(119905) = exp [119895120587 119861

1198790

(119905 minus1

21198790)

2


(30)





1198751198790 120585

11= 1198754120587

2

1198612

11987903119888

2 Imag12058512 = minus(119875120587119861119879

2

03119888) and

12058522

= (119875MT3

012)[1 + (119879

0119879

119877)2

(1198752


CRLB120579120578120579120578FDA

=

1

2SNR

sdot

[[[[[[[[[[[[

[

12058721198892cos2(120579)119872(119872

2minus 1)

[

[

1

31205822+

(Δ119891)2[(2119872minus 1) (8119872


451198882119872(119872+ 1)

+

2Δ119891 (119872minus 1)

3119888120582

]

]

minus1205872119889Δ119891 cos(120579)119872(119872

2minus1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 0

minus1205872119889Δ119891 cos(120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 1205872(Δ119891)2119872(119872

2minus1)

31198882

+

412058721198612119872

31198882

minus

1198721205871198611198790

3119888

0 minus

1198721205871198611198790

3119888

1198721198792

0

12

[1+(

119879119877

1198790

)

2

(1198752minus1)]

]]]]]]]]]]]]

]

minus1

(31)



=1

2SNR

times

[[[[[[[[

[

1205872

1198892cos2 (120579)119872(119872

2

minus 1)

312058220 0

04120587

2

1198612

119872

31198882minus119872120587119861119879

0

3119888

0 minus119872120587119861119879

0

3119888

1198721198792

0

12[1 + (

119879119877

1198790

)

2

(1198752

minus 1)]

]]]]]]]]

]

minus1

(32)



CRLB120579120578120579120578TS-FDA =

1

2SNR

times

[[[[[[[[[[[[[[[[[[[

[

412058721198892cos2 (120579)1205822

2119873

sum

119898=1

(119898 minus 1)2

minus

41205872119889 cos (120579)120582119888

[[[[

[

(Δ1198911 + Δ1198912)

119873

sum

119898=1

(119898 minus 1)2

+119873Δ1198912

119873

sum

119898=1

(119898 minus 1)

]]]]

]

0

minus

41205872119889 cos (120579)120582119888

[[[[

[

(Δ1198911 + Δ1198912)

119873

sum

119898=1

(119898 minus 1)2

+119873Δ1198912

119873

sum

119898=1

(119898 minus 1)

]]]]

]

(Δ1198912

1+ Δ1198912

2)

1198882

119873

sum

119898=1

(119898 minus 1)2+

412058721198612119872

31198882

minus

1198721205871198611198790

3119888

0 minus

1198721205871198611198790

3119888

1198721198792

0

12

[1 + (

119879119877

1198790

)

2

(1198752minus 1)]

]]]]]]]]]]]]]]]]]]]

]

minus1

(33)


20151050

10minus2

10minus3

10minus47001 7002 7003

10minus248203

10minus248201

10minus248199

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz







119872

119898=1(119898 minus 1)

2

1205822

≫

(Δ119891)2

sum119872

119898=1(119898 minus 1)

4

1198882

+ 2Δ119891sum119872

119898=1(119898 minus 1)

3




CRLB

of r

ange

estim

atio

n (m

)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

102

101

10020151050

SNR (dB)



1and Δ119891







0sdot 119861 = 2500 ≫ 1) Figure 7


119903119903FDA= 653m



119863119888(2Ω

119888) and

Ω119888= 2120587119891

119888) Since (Δ119891)2 sum119872

119898=1(119898 minus 1)

4

1198882






1and Δ119891




20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

104

103

102

101

CRLB

of a

ngle

estim

atio

n (d

eg)


108

107

106

105

10420151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

CRLB

of r

ange

estim

atio

n (m

)



6 Conclusion


20151050

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)


Δf1 = 0kHz Δf2 = 30



20151050

SNR (dB)

CRLB

of r

ange

estim

atio

n (m

)102

101

100


Δf1 = 0kHz Δf2 = 30






10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

20151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30



CRLB

of r

ange

estim

atio

n (m

)

103

102

101

10020151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30




Appendices



2

119899I119872


0|2

1205902



119868120579120579

= 2Re 119863H120579(120595) (Rminus1

n )119863120579(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

(A1)


have

120597a (120579)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579) (A2)


120597aH (120579)

120597120579

120597a (120579)120597120579

=4120587

2

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A3)


119868120579120579phased-array

= 2SNR41205872

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A4)



120597a (120579 119903)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579 119903)

minus 1198952120587119889Δ119891 cos (120579)

119888


] a (120579 119903)

120597aH (120579 119903)

120597120579

120597a (120579 119903)120597120579

= 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822

+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A5)


119868120579120579FDA

= 2SNR sdot 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A6)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

CRLB

of r

ange

estim

atio

n (m

)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)





119868119903119903FDA

= 2Re 119863H119903(120595) (Rminus1

n )119863119903(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597aH (120579 119903)

120597119903Rminus1

n120597a (120579 119903)

120597119903

(B1)


120597a (120579 119903)120597119903

= 1198952120587Δ119891

119888Da (120579 119903)

120597aH (120579 119903)

120597119903

120597a (120579 119903)120597119903

=4120587

2

Δ1198912

1198882

119872

sum

119898=1

(119898 minus 1)2

(B2)


119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(B3)




IFDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (C1)

We then have

IFDA = 210038161003816100381610038161205730

1003816100381610038161003816

2

times

[[[[

[

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

]]]]

]

(C2)

Since (120597H(120595)120597119903)Rminus1

n (120597(120595)120597120579) = (120597H(120595)

120597120579)Rminus1

n (120597(120595)120597119903) then 119868120579119903= 119868

119903120579 We can get

119868120579120579

= 210038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

= 81205872

1198892cos2 (120579)

times SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

119868120579119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

= minus81205872

119889Δ119891 cos (120579)

times SNR[sum

119872

119898=1(119898 minus 1)

2

120582119888+Δ119891sum

119872

119898=1(119898 minus 1)

3

1198882]

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

= SNR8120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(C3)


= [Iminus1

FDA]11 CRLB119903119903FDA

= [Iminus1




2

s1205902


also start with FIM

I120581119894120581119895

= 2Re 119863H120581119894

(120581) (Cminus1

n otimes Rminus1

n )119863120581119895(120581) (D1)


120581119894(120595) = 120597(120581)120597120581

119894

Consider

IFDA =

[[[[[

[

119868120573120573

119868119879

120579120573119868119879

120578120573

119868120579120573

119868120579120579

119868119879

120578120579

119868120578120573

119868120578120579

119868120578120578

]]]]]

]

(D2)



where

V = U119879

(D4a)

V = [119868120579120573

119868120578120573

] (D4b)

B = [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] (D4c)


Iminus1

FDA = [

[





]

]

(D5)



CRLB120579120578120579120578FDA


= [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] minus [119868120579120573

119868120578120573

] 119868minus1

120573120573[119868

119879

120579120573119868119879

120578120573]

minus1

(D6)

where

119868120573120573

= 2 sdot[[[

[

s sdot 1198721205902

119899

0

0s sdot 1198721205902

119899

]]]

]

(D7a)

119868120579120573

= 2 sdot Re[1 119895] otimess1205902

119899

sdot 120573lowast

1198601 (D7b)

119868120578120573

= 2 sdot Re[1 119895] otimes120573

lowast

1205902

119908

sdot [119872 sdot 1198603+ s sdot 119860

4] (D7c)

119868120578120578

= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

[119872 sdot 1198606+ 119860

H3119860

4+ 119860

3119860

H4+ s sdot 119860

7]

(D7d)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)

SNR (dB)


Δf1 = 0kHz Δf2 = 30


70001 70001 70001 70002100286514

100286515

100286515

kHz



119868120579120579minus (119868

120579120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198602minus119860

1119860

H1

119872) (D8a)

119868120578120579minus (119868

120578120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198605minus119860

4119860

H1

119872) (D8b)

119868120578120578

minus (119868120578120573119868119879

120578120573)

1205902

119899

s119872

= 2

10038161003816100381610038161205731003816100381610038161003816

2

s1205902

119899

(119872119860

6

s+ 119860

7minus119872119860

3119860

H3

s2minus119860

4119860

H4

119872)

(D8c)

1198601= minus119895(

2120587119889 cos (120579)sum119872

119898=1(119898 minus 1)

120582

+2120587119889Δ119891 cos (120579)sum119872

119898=1(119898 minus 1)

2

119888)

(D9a)

1198602= 4120587

2

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(D9b)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)



Acknowledgments




References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













z = 120573 sdot 120601 (120578) otimes a (120579 119903) + n = (120581) + n (22)


120573 sdot 120601(120578) otimes a(120579 119903)

120601 (120578) = [119904 [1 minus2119903

119888 sdot Δ119905] exp (119895119908

1198631)

119904 [2 minus2119903

119888 sdot Δ119905] exp (119895119908

1198632)

119904 [119871 minus2119903

119888 sdot Δ119905] exp (119895119908

119863119871)]

119879

(23)







120578 = [119903 Ω119863]119879

(25)

For Cn = I119871and Rn = 120590

2

119899I119872




CRLB120579120578120579120578FDA

=

1205902

119899

210038161003816100381610038161205731003816100381610038161003816

2s

times

[[[[[[[[[[[

[

12058721198892cos2 (120579)119872(119872

2minus 1)

[

[

1

31205822

+

(Δ119891)2[(2119872minus 1) (8119872


451198882(119872+ 1)

+

2Δ119891 (119872minus 1)

3119888120582

]

]

minus1205872119889Δ119891 cos (120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 0

minus1205872119889Δ119891 cos (120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 1205872(Δ119891)2119872(119872

2minus 1)

31198882

+

12058511

sImag 12058512

s

0

Imag 12058512

s12058522

s

]]]]]]]]]]]

]

minus1

(26)

where

12058511

= int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905

minus1

s[int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905]

2

(27a)

12058522

= int

infin

minusinfin

1199052

|119904 (119905 minus 120591)|2

119889119905

minus1

s[int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

2

(27b)

12058512

= int

infin

minusinfin

119905119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905

minus [1

sint

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905

sdot int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905]

(28)

The terms 12058511

and 12058522



119904 (119905) =

119875minus1

sum

119901=0

1199040(119905 minus 119901119879

119901) (29)




1199040(119905) = exp [119895120587 119861

1198790

(119905 minus1

21198790)

2


(30)





1198751198790 120585

11= 1198754120587

2

1198612

11987903119888

2 Imag12058512 = minus(119875120587119861119879

2

03119888) and

12058522

= (119875MT3

012)[1 + (119879

0119879

119877)2

(1198752


CRLB120579120578120579120578FDA

=

1

2SNR

sdot

[[[[[[[[[[[[

[

12058721198892cos2(120579)119872(119872

2minus 1)

[

[

1

31205822+

(Δ119891)2[(2119872minus 1) (8119872


451198882119872(119872+ 1)

+

2Δ119891 (119872minus 1)

3119888120582

]

]

minus1205872119889Δ119891 cos(120579)119872(119872

2minus1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 0

minus1205872119889Δ119891 cos(120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 1205872(Δ119891)2119872(119872

2minus1)

31198882

+

412058721198612119872

31198882

minus

1198721205871198611198790

3119888

0 minus

1198721205871198611198790

3119888

1198721198792

0

12

[1+(

119879119877

1198790

)

2

(1198752minus1)]

]]]]]]]]]]]]

]

minus1

(31)



=1

2SNR

times

[[[[[[[[

[

1205872

1198892cos2 (120579)119872(119872

2

minus 1)

312058220 0

04120587

2

1198612

119872

31198882minus119872120587119861119879

0

3119888

0 minus119872120587119861119879

0

3119888

1198721198792

0

12[1 + (

119879119877

1198790

)

2

(1198752

minus 1)]

]]]]]]]]

]

minus1

(32)



CRLB120579120578120579120578TS-FDA =

1

2SNR

times

[[[[[[[[[[[[[[[[[[[

[

412058721198892cos2 (120579)1205822

2119873

sum

119898=1

(119898 minus 1)2

minus

41205872119889 cos (120579)120582119888

[[[[

[

(Δ1198911 + Δ1198912)

119873

sum

119898=1

(119898 minus 1)2

+119873Δ1198912

119873

sum

119898=1

(119898 minus 1)

]]]]

]

0

minus

41205872119889 cos (120579)120582119888

[[[[

[

(Δ1198911 + Δ1198912)

119873

sum

119898=1

(119898 minus 1)2

+119873Δ1198912

119873

sum

119898=1

(119898 minus 1)

]]]]

]

(Δ1198912

1+ Δ1198912

2)

1198882

119873

sum

119898=1

(119898 minus 1)2+

412058721198612119872

31198882

minus

1198721205871198611198790

3119888

0 minus

1198721205871198611198790

3119888

1198721198792

0

12

[1 + (

119879119877

1198790

)

2

(1198752minus 1)]

]]]]]]]]]]]]]]]]]]]

]

minus1

(33)


20151050

10minus2

10minus3

10minus47001 7002 7003

10minus248203

10minus248201

10minus248199

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz







119872

119898=1(119898 minus 1)

2

1205822

≫

(Δ119891)2

sum119872

119898=1(119898 minus 1)

4

1198882

+ 2Δ119891sum119872

119898=1(119898 minus 1)

3




CRLB

of r

ange

estim

atio

n (m

)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

102

101

10020151050

SNR (dB)



1and Δ119891







0sdot 119861 = 2500 ≫ 1) Figure 7


119903119903FDA= 653m



119863119888(2Ω

119888) and

Ω119888= 2120587119891

119888) Since (Δ119891)2 sum119872

119898=1(119898 minus 1)

4

1198882






1and Δ119891




20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

104

103

102

101

CRLB

of a

ngle

estim

atio

n (d

eg)


108

107

106

105

10420151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

CRLB

of r

ange

estim

atio

n (m

)



6 Conclusion


20151050

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)


Δf1 = 0kHz Δf2 = 30



20151050

SNR (dB)

CRLB

of r

ange

estim

atio

n (m

)102

101

100


Δf1 = 0kHz Δf2 = 30






10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

20151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30



CRLB

of r

ange

estim

atio

n (m

)

103

102

101

10020151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30




Appendices



2

119899I119872


0|2

1205902



119868120579120579

= 2Re 119863H120579(120595) (Rminus1

n )119863120579(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

(A1)


have

120597a (120579)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579) (A2)


120597aH (120579)

120597120579

120597a (120579)120597120579

=4120587

2

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A3)


119868120579120579phased-array

= 2SNR41205872

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A4)



120597a (120579 119903)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579 119903)

minus 1198952120587119889Δ119891 cos (120579)

119888


] a (120579 119903)

120597aH (120579 119903)

120597120579

120597a (120579 119903)120597120579

= 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822

+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A5)


119868120579120579FDA

= 2SNR sdot 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A6)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

CRLB

of r

ange

estim

atio

n (m

)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)





119868119903119903FDA

= 2Re 119863H119903(120595) (Rminus1

n )119863119903(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597aH (120579 119903)

120597119903Rminus1

n120597a (120579 119903)

120597119903

(B1)


120597a (120579 119903)120597119903

= 1198952120587Δ119891

119888Da (120579 119903)

120597aH (120579 119903)

120597119903

120597a (120579 119903)120597119903

=4120587

2

Δ1198912

1198882

119872

sum

119898=1

(119898 minus 1)2

(B2)


119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(B3)




IFDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (C1)

We then have

IFDA = 210038161003816100381610038161205730

1003816100381610038161003816

2

times

[[[[

[

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

]]]]

]

(C2)

Since (120597H(120595)120597119903)Rminus1

n (120597(120595)120597120579) = (120597H(120595)

120597120579)Rminus1

n (120597(120595)120597119903) then 119868120579119903= 119868

119903120579 We can get

119868120579120579

= 210038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

= 81205872

1198892cos2 (120579)

times SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

119868120579119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

= minus81205872

119889Δ119891 cos (120579)

times SNR[sum

119872

119898=1(119898 minus 1)

2

120582119888+Δ119891sum

119872

119898=1(119898 minus 1)

3

1198882]

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

= SNR8120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(C3)


= [Iminus1

FDA]11 CRLB119903119903FDA

= [Iminus1




2

s1205902


also start with FIM

I120581119894120581119895

= 2Re 119863H120581119894

(120581) (Cminus1

n otimes Rminus1

n )119863120581119895(120581) (D1)


120581119894(120595) = 120597(120581)120597120581

119894

Consider

IFDA =

[[[[[

[

119868120573120573

119868119879

120579120573119868119879

120578120573

119868120579120573

119868120579120579

119868119879

120578120579

119868120578120573

119868120578120579

119868120578120578

]]]]]

]

(D2)



where

V = U119879

(D4a)

V = [119868120579120573

119868120578120573

] (D4b)

B = [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] (D4c)


Iminus1

FDA = [

[





]

]

(D5)



CRLB120579120578120579120578FDA


= [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] minus [119868120579120573

119868120578120573

] 119868minus1

120573120573[119868

119879

120579120573119868119879

120578120573]

minus1

(D6)

where

119868120573120573

= 2 sdot[[[

[

s sdot 1198721205902

119899

0

0s sdot 1198721205902

119899

]]]

]

(D7a)

119868120579120573

= 2 sdot Re[1 119895] otimess1205902

119899

sdot 120573lowast

1198601 (D7b)

119868120578120573

= 2 sdot Re[1 119895] otimes120573

lowast

1205902

119908

sdot [119872 sdot 1198603+ s sdot 119860

4] (D7c)

119868120578120578

= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

[119872 sdot 1198606+ 119860

H3119860

4+ 119860

3119860

H4+ s sdot 119860

7]

(D7d)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)

SNR (dB)


Δf1 = 0kHz Δf2 = 30


70001 70001 70001 70002100286514

100286515

100286515

kHz



119868120579120579minus (119868

120579120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198602minus119860

1119860

H1

119872) (D8a)

119868120578120579minus (119868

120578120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198605minus119860

4119860

H1

119872) (D8b)

119868120578120578

minus (119868120578120573119868119879

120578120573)

1205902

119899

s119872

= 2

10038161003816100381610038161205731003816100381610038161003816

2

s1205902

119899

(119872119860

6

s+ 119860

7minus119872119860

3119860

H3

s2minus119860

4119860

H4

119872)

(D8c)

1198601= minus119895(

2120587119889 cos (120579)sum119872

119898=1(119898 minus 1)

120582

+2120587119889Δ119891 cos (120579)sum119872

119898=1(119898 minus 1)

2

119888)

(D9a)

1198602= 4120587

2

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(D9b)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)



Acknowledgments




References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1199040(119905) = exp [119895120587 119861

1198790

(119905 minus1

21198790)

2


(30)





1198751198790 120585

11= 1198754120587

2

1198612

11987903119888

2 Imag12058512 = minus(119875120587119861119879

2

03119888) and

12058522

= (119875MT3

012)[1 + (119879

0119879

119877)2

(1198752


CRLB120579120578120579120578FDA

=

1

2SNR

sdot

[[[[[[[[[[[[

[

12058721198892cos2(120579)119872(119872

2minus 1)

[

[

1

31205822+

(Δ119891)2[(2119872minus 1) (8119872


451198882119872(119872+ 1)

+

2Δ119891 (119872minus 1)

3119888120582

]

]

minus1205872119889Δ119891 cos(120579)119872(119872

2minus1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 0

minus1205872119889Δ119891 cos(120579)119872(119872

2minus 1) [

1

3120582119888

+

Δ119891 (119872minus 1)

31198882

] 1205872(Δ119891)2119872(119872

2minus1)

31198882

+

412058721198612119872

31198882

minus

1198721205871198611198790

3119888

0 minus

1198721205871198611198790

3119888

1198721198792

0

12

[1+(

119879119877

1198790

)

2

(1198752minus1)]

]]]]]]]]]]]]

]

minus1

(31)



=1

2SNR

times

[[[[[[[[

[

1205872

1198892cos2 (120579)119872(119872

2

minus 1)

312058220 0

04120587

2

1198612

119872

31198882minus119872120587119861119879

0

3119888

0 minus119872120587119861119879

0

3119888

1198721198792

0

12[1 + (

119879119877

1198790

)

2

(1198752

minus 1)]

]]]]]]]]

]

minus1

(32)



CRLB120579120578120579120578TS-FDA =

1

2SNR

times

[[[[[[[[[[[[[[[[[[[

[

412058721198892cos2 (120579)1205822

2119873

sum

119898=1

(119898 minus 1)2

minus

41205872119889 cos (120579)120582119888

[[[[

[

(Δ1198911 + Δ1198912)

119873

sum

119898=1

(119898 minus 1)2

+119873Δ1198912

119873

sum

119898=1

(119898 minus 1)

]]]]

]

0

minus

41205872119889 cos (120579)120582119888

[[[[

[

(Δ1198911 + Δ1198912)

119873

sum

119898=1

(119898 minus 1)2

+119873Δ1198912

119873

sum

119898=1

(119898 minus 1)

]]]]

]

(Δ1198912

1+ Δ1198912

2)

1198882

119873

sum

119898=1

(119898 minus 1)2+

412058721198612119872

31198882

minus

1198721205871198611198790

3119888

0 minus

1198721205871198611198790

3119888

1198721198792

0

12

[1 + (

119879119877

1198790

)

2

(1198752minus 1)]

]]]]]]]]]]]]]]]]]]]

]

minus1

(33)


20151050

10minus2

10minus3

10minus47001 7002 7003

10minus248203

10minus248201

10minus248199

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz







119872

119898=1(119898 minus 1)

2

1205822

≫

(Δ119891)2

sum119872

119898=1(119898 minus 1)

4

1198882

+ 2Δ119891sum119872

119898=1(119898 minus 1)

3




CRLB

of r

ange

estim

atio

n (m

)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

102

101

10020151050

SNR (dB)



1and Δ119891







0sdot 119861 = 2500 ≫ 1) Figure 7


119903119903FDA= 653m



119863119888(2Ω

119888) and

Ω119888= 2120587119891

119888) Since (Δ119891)2 sum119872

119898=1(119898 minus 1)

4

1198882






1and Δ119891




20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

104

103

102

101

CRLB

of a

ngle

estim

atio

n (d

eg)


108

107

106

105

10420151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

CRLB

of r

ange

estim

atio

n (m

)



6 Conclusion


20151050

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)


Δf1 = 0kHz Δf2 = 30



20151050

SNR (dB)

CRLB

of r

ange

estim

atio

n (m

)102

101

100


Δf1 = 0kHz Δf2 = 30






10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

20151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30



CRLB

of r

ange

estim

atio

n (m

)

103

102

101

10020151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30




Appendices



2

119899I119872


0|2

1205902



119868120579120579

= 2Re 119863H120579(120595) (Rminus1

n )119863120579(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

(A1)


have

120597a (120579)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579) (A2)


120597aH (120579)

120597120579

120597a (120579)120597120579

=4120587

2

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A3)


119868120579120579phased-array

= 2SNR41205872

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A4)



120597a (120579 119903)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579 119903)

minus 1198952120587119889Δ119891 cos (120579)

119888


] a (120579 119903)

120597aH (120579 119903)

120597120579

120597a (120579 119903)120597120579

= 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822

+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A5)


119868120579120579FDA

= 2SNR sdot 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A6)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

CRLB

of r

ange

estim

atio

n (m

)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)





119868119903119903FDA

= 2Re 119863H119903(120595) (Rminus1

n )119863119903(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597aH (120579 119903)

120597119903Rminus1

n120597a (120579 119903)

120597119903

(B1)


120597a (120579 119903)120597119903

= 1198952120587Δ119891

119888Da (120579 119903)

120597aH (120579 119903)

120597119903

120597a (120579 119903)120597119903

=4120587

2

Δ1198912

1198882

119872

sum

119898=1

(119898 minus 1)2

(B2)


119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(B3)




IFDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (C1)

We then have

IFDA = 210038161003816100381610038161205730

1003816100381610038161003816

2

times

[[[[

[

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

]]]]

]

(C2)

Since (120597H(120595)120597119903)Rminus1

n (120597(120595)120597120579) = (120597H(120595)

120597120579)Rminus1

n (120597(120595)120597119903) then 119868120579119903= 119868

119903120579 We can get

119868120579120579

= 210038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

= 81205872

1198892cos2 (120579)

times SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

119868120579119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

= minus81205872

119889Δ119891 cos (120579)

times SNR[sum

119872

119898=1(119898 minus 1)

2

120582119888+Δ119891sum

119872

119898=1(119898 minus 1)

3

1198882]

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

= SNR8120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(C3)


= [Iminus1

FDA]11 CRLB119903119903FDA

= [Iminus1




2

s1205902


also start with FIM

I120581119894120581119895

= 2Re 119863H120581119894

(120581) (Cminus1

n otimes Rminus1

n )119863120581119895(120581) (D1)


120581119894(120595) = 120597(120581)120597120581

119894

Consider

IFDA =

[[[[[

[

119868120573120573

119868119879

120579120573119868119879

120578120573

119868120579120573

119868120579120579

119868119879

120578120579

119868120578120573

119868120578120579

119868120578120578

]]]]]

]

(D2)



where

V = U119879

(D4a)

V = [119868120579120573

119868120578120573

] (D4b)

B = [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] (D4c)


Iminus1

FDA = [

[





]

]

(D5)



CRLB120579120578120579120578FDA


= [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] minus [119868120579120573

119868120578120573

] 119868minus1

120573120573[119868

119879

120579120573119868119879

120578120573]

minus1

(D6)

where

119868120573120573

= 2 sdot[[[

[

s sdot 1198721205902

119899

0

0s sdot 1198721205902

119899

]]]

]

(D7a)

119868120579120573

= 2 sdot Re[1 119895] otimess1205902

119899

sdot 120573lowast

1198601 (D7b)

119868120578120573

= 2 sdot Re[1 119895] otimes120573

lowast

1205902

119908

sdot [119872 sdot 1198603+ s sdot 119860

4] (D7c)

119868120578120578

= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

[119872 sdot 1198606+ 119860

H3119860

4+ 119860

3119860

H4+ s sdot 119860

7]

(D7d)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)

SNR (dB)


Δf1 = 0kHz Δf2 = 30


70001 70001 70001 70002100286514

100286515

100286515

kHz



119868120579120579minus (119868

120579120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198602minus119860

1119860

H1

119872) (D8a)

119868120578120579minus (119868

120578120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198605minus119860

4119860

H1

119872) (D8b)

119868120578120578

minus (119868120578120573119868119879

120578120573)

1205902

119899

s119872

= 2

10038161003816100381610038161205731003816100381610038161003816

2

s1205902

119899

(119872119860

6

s+ 119860

7minus119872119860

3119860

H3

s2minus119860

4119860

H4

119872)

(D8c)

1198601= minus119895(

2120587119889 cos (120579)sum119872

119898=1(119898 minus 1)

120582

+2120587119889Δ119891 cos (120579)sum119872

119898=1(119898 minus 1)

2

119888)

(D9a)

1198602= 4120587

2

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(D9b)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)



Acknowledgments




References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










20151050

10minus2

10minus3

10minus47001 7002 7003

10minus248203

10minus248201

10minus248199

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz







119872

119898=1(119898 minus 1)

2

1205822

≫

(Δ119891)2

sum119872

119898=1(119898 minus 1)

4

1198882

+ 2Δ119891sum119872

119898=1(119898 minus 1)

3




CRLB

of r

ange

estim

atio

n (m

)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

102

101

10020151050

SNR (dB)



1and Δ119891







0sdot 119861 = 2500 ≫ 1) Figure 7


119903119903FDA= 653m



119863119888(2Ω

119888) and

Ω119888= 2120587119891

119888) Since (Δ119891)2 sum119872

119898=1(119898 minus 1)

4

1198882






1and Δ119891




20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

104

103

102

101

CRLB

of a

ngle

estim

atio

n (d

eg)


108

107

106

105

10420151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

CRLB

of r

ange

estim

atio

n (m

)



6 Conclusion


20151050

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)


Δf1 = 0kHz Δf2 = 30



20151050

SNR (dB)

CRLB

of r

ange

estim

atio

n (m

)102

101

100


Δf1 = 0kHz Δf2 = 30






10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

20151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30



CRLB

of r

ange

estim

atio

n (m

)

103

102

101

10020151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30




Appendices



2

119899I119872


0|2

1205902



119868120579120579

= 2Re 119863H120579(120595) (Rminus1

n )119863120579(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

(A1)


have

120597a (120579)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579) (A2)


120597aH (120579)

120597120579

120597a (120579)120597120579

=4120587

2

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A3)


119868120579120579phased-array

= 2SNR41205872

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A4)



120597a (120579 119903)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579 119903)

minus 1198952120587119889Δ119891 cos (120579)

119888


] a (120579 119903)

120597aH (120579 119903)

120597120579

120597a (120579 119903)120597120579

= 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822

+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A5)


119868120579120579FDA

= 2SNR sdot 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A6)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

CRLB

of r

ange

estim

atio

n (m

)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)





119868119903119903FDA

= 2Re 119863H119903(120595) (Rminus1

n )119863119903(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597aH (120579 119903)

120597119903Rminus1

n120597a (120579 119903)

120597119903

(B1)


120597a (120579 119903)120597119903

= 1198952120587Δ119891

119888Da (120579 119903)

120597aH (120579 119903)

120597119903

120597a (120579 119903)120597119903

=4120587

2

Δ1198912

1198882

119872

sum

119898=1

(119898 minus 1)2

(B2)


119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(B3)




IFDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (C1)

We then have

IFDA = 210038161003816100381610038161205730

1003816100381610038161003816

2

times

[[[[

[

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

]]]]

]

(C2)

Since (120597H(120595)120597119903)Rminus1

n (120597(120595)120597120579) = (120597H(120595)

120597120579)Rminus1

n (120597(120595)120597119903) then 119868120579119903= 119868

119903120579 We can get

119868120579120579

= 210038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

= 81205872

1198892cos2 (120579)

times SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

119868120579119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

= minus81205872

119889Δ119891 cos (120579)

times SNR[sum

119872

119898=1(119898 minus 1)

2

120582119888+Δ119891sum

119872

119898=1(119898 minus 1)

3

1198882]

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

= SNR8120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(C3)


= [Iminus1

FDA]11 CRLB119903119903FDA

= [Iminus1




2

s1205902


also start with FIM

I120581119894120581119895

= 2Re 119863H120581119894

(120581) (Cminus1

n otimes Rminus1

n )119863120581119895(120581) (D1)


120581119894(120595) = 120597(120581)120597120581

119894

Consider

IFDA =

[[[[[

[

119868120573120573

119868119879

120579120573119868119879

120578120573

119868120579120573

119868120579120579

119868119879

120578120579

119868120578120573

119868120578120579

119868120578120578

]]]]]

]

(D2)



where

V = U119879

(D4a)

V = [119868120579120573

119868120578120573

] (D4b)

B = [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] (D4c)


Iminus1

FDA = [

[





]

]

(D5)



CRLB120579120578120579120578FDA


= [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] minus [119868120579120573

119868120578120573

] 119868minus1

120573120573[119868

119879

120579120573119868119879

120578120573]

minus1

(D6)

where

119868120573120573

= 2 sdot[[[

[

s sdot 1198721205902

119899

0

0s sdot 1198721205902

119899

]]]

]

(D7a)

119868120579120573

= 2 sdot Re[1 119895] otimess1205902

119899

sdot 120573lowast

1198601 (D7b)

119868120578120573

= 2 sdot Re[1 119895] otimes120573

lowast

1205902

119908

sdot [119872 sdot 1198603+ s sdot 119860

4] (D7c)

119868120578120578

= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

[119872 sdot 1198606+ 119860

H3119860

4+ 119860

3119860

H4+ s sdot 119860

7]

(D7d)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)

SNR (dB)


Δf1 = 0kHz Δf2 = 30


70001 70001 70001 70002100286514

100286515

100286515

kHz



119868120579120579minus (119868

120579120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198602minus119860

1119860

H1

119872) (D8a)

119868120578120579minus (119868

120578120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198605minus119860

4119860

H1

119872) (D8b)

119868120578120578

minus (119868120578120573119868119879

120578120573)

1205902

119899

s119872

= 2

10038161003816100381610038161205731003816100381610038161003816

2

s1205902

119899

(119872119860

6

s+ 119860

7minus119872119860

3119860

H3

s2minus119860

4119860

H4

119872)

(D8c)

1198601= minus119895(

2120587119889 cos (120579)sum119872

119898=1(119898 minus 1)

120582

+2120587119889Δ119891 cos (120579)sum119872

119898=1(119898 minus 1)

2

119888)

(D9a)

1198602= 4120587

2

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(D9b)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)



Acknowledgments




References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

104

103

102

101

CRLB

of a

ngle

estim

atio

n (d

eg)


108

107

106

105

10420151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

CRLB

of r

ange

estim

atio

n (m

)



6 Conclusion


20151050

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

SNR (dB)


Δf1 = 0kHz Δf2 = 30



20151050

SNR (dB)

CRLB

of r

ange

estim

atio

n (m

)102

101

100


Δf1 = 0kHz Δf2 = 30






10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

20151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30



CRLB

of r

ange

estim

atio

n (m

)

103

102

101

10020151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30




Appendices



2

119899I119872


0|2

1205902



119868120579120579

= 2Re 119863H120579(120595) (Rminus1

n )119863120579(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

(A1)


have

120597a (120579)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579) (A2)


120597aH (120579)

120597120579

120597a (120579)120597120579

=4120587

2

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A3)


119868120579120579phased-array

= 2SNR41205872

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A4)



120597a (120579 119903)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579 119903)

minus 1198952120587119889Δ119891 cos (120579)

119888


] a (120579 119903)

120597aH (120579 119903)

120597120579

120597a (120579 119903)120597120579

= 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822

+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A5)


119868120579120579FDA

= 2SNR sdot 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A6)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

CRLB

of r

ange

estim

atio

n (m

)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)





119868119903119903FDA

= 2Re 119863H119903(120595) (Rminus1

n )119863119903(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597aH (120579 119903)

120597119903Rminus1

n120597a (120579 119903)

120597119903

(B1)


120597a (120579 119903)120597119903

= 1198952120587Δ119891

119888Da (120579 119903)

120597aH (120579 119903)

120597119903

120597a (120579 119903)120597119903

=4120587

2

Δ1198912

1198882

119872

sum

119898=1

(119898 minus 1)2

(B2)


119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(B3)




IFDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (C1)

We then have

IFDA = 210038161003816100381610038161205730

1003816100381610038161003816

2

times

[[[[

[

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

]]]]

]

(C2)

Since (120597H(120595)120597119903)Rminus1

n (120597(120595)120597120579) = (120597H(120595)

120597120579)Rminus1

n (120597(120595)120597119903) then 119868120579119903= 119868

119903120579 We can get

119868120579120579

= 210038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

= 81205872

1198892cos2 (120579)

times SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

119868120579119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

= minus81205872

119889Δ119891 cos (120579)

times SNR[sum

119872

119898=1(119898 minus 1)

2

120582119888+Δ119891sum

119872

119898=1(119898 minus 1)

3

1198882]

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

= SNR8120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(C3)


= [Iminus1

FDA]11 CRLB119903119903FDA

= [Iminus1




2

s1205902


also start with FIM

I120581119894120581119895

= 2Re 119863H120581119894

(120581) (Cminus1

n otimes Rminus1

n )119863120581119895(120581) (D1)


120581119894(120595) = 120597(120581)120597120581

119894

Consider

IFDA =

[[[[[

[

119868120573120573

119868119879

120579120573119868119879

120578120573

119868120579120573

119868120579120579

119868119879

120578120579

119868120578120573

119868120578120579

119868120578120578

]]]]]

]

(D2)



where

V = U119879

(D4a)

V = [119868120579120573

119868120578120573

] (D4b)

B = [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] (D4c)


Iminus1

FDA = [

[





]

]

(D5)



CRLB120579120578120579120578FDA


= [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] minus [119868120579120573

119868120578120573

] 119868minus1

120573120573[119868

119879

120579120573119868119879

120578120573]

minus1

(D6)

where

119868120573120573

= 2 sdot[[[

[

s sdot 1198721205902

119899

0

0s sdot 1198721205902

119899

]]]

]

(D7a)

119868120579120573

= 2 sdot Re[1 119895] otimess1205902

119899

sdot 120573lowast

1198601 (D7b)

119868120578120573

= 2 sdot Re[1 119895] otimes120573

lowast

1205902

119908

sdot [119872 sdot 1198603+ s sdot 119860

4] (D7c)

119868120578120578

= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

[119872 sdot 1198606+ 119860

H3119860

4+ 119860

3119860

H4+ s sdot 119860

7]

(D7d)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)

SNR (dB)


Δf1 = 0kHz Δf2 = 30


70001 70001 70001 70002100286514

100286515

100286515

kHz



119868120579120579minus (119868

120579120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198602minus119860

1119860

H1

119872) (D8a)

119868120578120579minus (119868

120578120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198605minus119860

4119860

H1

119872) (D8b)

119868120578120578

minus (119868120578120573119868119879

120578120573)

1205902

119899

s119872

= 2

10038161003816100381610038161205731003816100381610038161003816

2

s1205902

119899

(119872119860

6

s+ 119860

7minus119872119860

3119860

H3

s2minus119860

4119860

H4

119872)

(D8c)

1198601= minus119895(

2120587119889 cos (120579)sum119872

119898=1(119898 minus 1)

120582

+2120587119889Δ119891 cos (120579)sum119872

119898=1(119898 minus 1)

2

119888)

(D9a)

1198602= 4120587

2

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(D9b)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)



Acknowledgments




References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)

20151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30



CRLB

of r

ange

estim

atio

n (m

)

103

102

101

10020151050

SNR (dB)


Δf1 = 0kHz Δf2 = 30




Appendices



2

119899I119872


0|2

1205902



119868120579120579

= 2Re 119863H120579(120595) (Rminus1

n )119863120579(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

(A1)


have

120597a (120579)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579) (A2)


120597aH (120579)

120597120579

120597a (120579)120597120579

=4120587

2

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A3)


119868120579120579phased-array

= 2SNR41205872

1198892cos2 (120579)1205822

119872

sum

119898=1

(119898 minus 1)2

(A4)



120597a (120579 119903)120597120579

= minus1198952120587119889 cos (120579)

120582Da (120579 119903)

minus 1198952120587119889Δ119891 cos (120579)

119888


] a (120579 119903)

120597aH (120579 119903)

120597120579

120597a (120579 119903)120597120579

= 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822

+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A5)


119868120579120579FDA

= 2SNR sdot 41205872

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(A6)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

CRLB

of r

ange

estim

atio

n (m

)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)





119868119903119903FDA

= 2Re 119863H119903(120595) (Rminus1

n )119863119903(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597aH (120579 119903)

120597119903Rminus1

n120597a (120579 119903)

120597119903

(B1)


120597a (120579 119903)120597119903

= 1198952120587Δ119891

119888Da (120579 119903)

120597aH (120579 119903)

120597119903

120597a (120579 119903)120597119903

=4120587

2

Δ1198912

1198882

119872

sum

119898=1

(119898 minus 1)2

(B2)


119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(B3)




IFDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (C1)

We then have

IFDA = 210038161003816100381610038161205730

1003816100381610038161003816

2

times

[[[[

[

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

]]]]

]

(C2)

Since (120597H(120595)120597119903)Rminus1

n (120597(120595)120597120579) = (120597H(120595)

120597120579)Rminus1

n (120597(120595)120597119903) then 119868120579119903= 119868

119903120579 We can get

119868120579120579

= 210038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

= 81205872

1198892cos2 (120579)

times SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

119868120579119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

= minus81205872

119889Δ119891 cos (120579)

times SNR[sum

119872

119898=1(119898 minus 1)

2

120582119888+Δ119891sum

119872

119898=1(119898 minus 1)

3

1198882]

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

= SNR8120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(C3)


= [Iminus1

FDA]11 CRLB119903119903FDA

= [Iminus1




2

s1205902


also start with FIM

I120581119894120581119895

= 2Re 119863H120581119894

(120581) (Cminus1

n otimes Rminus1

n )119863120581119895(120581) (D1)


120581119894(120595) = 120597(120581)120597120581

119894

Consider

IFDA =

[[[[[

[

119868120573120573

119868119879

120579120573119868119879

120578120573

119868120579120573

119868120579120579

119868119879

120578120579

119868120578120573

119868120578120579

119868120578120578

]]]]]

]

(D2)



where

V = U119879

(D4a)

V = [119868120579120573

119868120578120573

] (D4b)

B = [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] (D4c)


Iminus1

FDA = [

[





]

]

(D5)



CRLB120579120578120579120578FDA


= [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] minus [119868120579120573

119868120578120573

] 119868minus1

120573120573[119868

119879

120579120573119868119879

120578120573]

minus1

(D6)

where

119868120573120573

= 2 sdot[[[

[

s sdot 1198721205902

119899

0

0s sdot 1198721205902

119899

]]]

]

(D7a)

119868120579120573

= 2 sdot Re[1 119895] otimess1205902

119899

sdot 120573lowast

1198601 (D7b)

119868120578120573

= 2 sdot Re[1 119895] otimes120573

lowast

1205902

119908

sdot [119872 sdot 1198603+ s sdot 119860

4] (D7c)

119868120578120578

= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

[119872 sdot 1198606+ 119860

H3119860

4+ 119860

3119860

H4+ s sdot 119860

7]

(D7d)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)

SNR (dB)


Δf1 = 0kHz Δf2 = 30


70001 70001 70001 70002100286514

100286515

100286515

kHz



119868120579120579minus (119868

120579120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198602minus119860

1119860

H1

119872) (D8a)

119868120578120579minus (119868

120578120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198605minus119860

4119860

H1

119872) (D8b)

119868120578120578

minus (119868120578120573119868119879

120578120573)

1205902

119899

s119872

= 2

10038161003816100381610038161205731003816100381610038161003816

2

s1205902

119899

(119872119860

6

s+ 119860

7minus119872119860

3119860

H3

s2minus119860

4119860

H4

119872)

(D8c)

1198601= minus119895(

2120587119889 cos (120579)sum119872

119898=1(119898 minus 1)

120582

+2120587119889Δ119891 cos (120579)sum119872

119898=1(119898 minus 1)

2

119888)

(D9a)

1198602= 4120587

2

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(D9b)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)



Acknowledgments




References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

CRLB

of r

ange

estim

atio

n (m

)


20151050

SNR (dB)

M = 16 Δf = 15kHzM = 16 Δf = 30kHz

M = 32 Δf = 15kHzM = 32 Δf = 30kHz

minus5minus10

102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)





119868119903119903FDA

= 2Re 119863H119903(120595) (Rminus1

n )119863119903(120595)

= 210038161003816100381610038161205730

1003816100381610038161003816

2 Re120597aH (120579 119903)

120597119903Rminus1

n120597a (120579 119903)

120597119903

(B1)


120597a (120579 119903)120597119903

= 1198952120587Δ119891

119888Da (120579 119903)

120597aH (120579 119903)

120597119903

120597a (120579 119903)120597119903

=4120587

2

Δ1198912

1198882

119872

sum

119898=1

(119898 minus 1)2

(B2)


119868119903119903FDA

= 2SNR4120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(B3)




IFDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (C1)

We then have

IFDA = 210038161003816100381610038161205730

1003816100381610038161003816

2

times

[[[[

[

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

]]]]

]

(C2)

Since (120597H(120595)120597119903)Rminus1

n (120597(120595)120597120579) = (120597H(120595)

120597120579)Rminus1

n (120597(120595)120597119903) then 119868120579119903= 119868

119903120579 We can get

119868120579120579

= 210038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

= 81205872

1198892cos2 (120579)

times SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

119868120579119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

= minus81205872

119889Δ119891 cos (120579)

times SNR[sum

119872

119898=1(119898 minus 1)

2

120582119888+Δ119891sum

119872

119898=1(119898 minus 1)

3

1198882]

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

= SNR8120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(C3)


= [Iminus1

FDA]11 CRLB119903119903FDA

= [Iminus1




2

s1205902


also start with FIM

I120581119894120581119895

= 2Re 119863H120581119894

(120581) (Cminus1

n otimes Rminus1

n )119863120581119895(120581) (D1)


120581119894(120595) = 120597(120581)120597120581

119894

Consider

IFDA =

[[[[[

[

119868120573120573

119868119879

120579120573119868119879

120578120573

119868120579120573

119868120579120579

119868119879

120578120579

119868120578120573

119868120578120579

119868120578120578

]]]]]

]

(D2)



where

V = U119879

(D4a)

V = [119868120579120573

119868120578120573

] (D4b)

B = [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] (D4c)


Iminus1

FDA = [

[





]

]

(D5)



CRLB120579120578120579120578FDA


= [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] minus [119868120579120573

119868120578120573

] 119868minus1

120573120573[119868

119879

120579120573119868119879

120578120573]

minus1

(D6)

where

119868120573120573

= 2 sdot[[[

[

s sdot 1198721205902

119899

0

0s sdot 1198721205902

119899

]]]

]

(D7a)

119868120579120573

= 2 sdot Re[1 119895] otimess1205902

119899

sdot 120573lowast

1198601 (D7b)

119868120578120573

= 2 sdot Re[1 119895] otimes120573

lowast

1205902

119908

sdot [119872 sdot 1198603+ s sdot 119860

4] (D7c)

119868120578120578

= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

[119872 sdot 1198606+ 119860

H3119860

4+ 119860

3119860

H4+ s sdot 119860

7]

(D7d)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)

SNR (dB)


Δf1 = 0kHz Δf2 = 30


70001 70001 70001 70002100286514

100286515

100286515

kHz



119868120579120579minus (119868

120579120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198602minus119860

1119860

H1

119872) (D8a)

119868120578120579minus (119868

120578120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198605minus119860

4119860

H1

119872) (D8b)

119868120578120578

minus (119868120578120573119868119879

120578120573)

1205902

119899

s119872

= 2

10038161003816100381610038161205731003816100381610038161003816

2

s1205902

119899

(119872119860

6

s+ 119860

7minus119872119860

3119860

H3

s2minus119860

4119860

H4

119872)

(D8c)

1198601= minus119895(

2120587119889 cos (120579)sum119872

119898=1(119898 minus 1)

120582

+2120587119889Δ119891 cos (120579)sum119872

119898=1(119898 minus 1)

2

119888)

(D9a)

1198602= 4120587

2

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(D9b)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)



Acknowledgments




References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












IFDA = 2Re 119863H120595119894

(120595) (Rminus1

n )119863120595119895(120595) = [

119868120579120579

119868120579119903

119868119903120579

119868119903119903

] (C1)

We then have

IFDA = 210038161003816100381610038161205730

1003816100381610038161003816

2

times

[[[[

[

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597120579

120597H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

120597H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

]]]]

]

(C2)

Since (120597H(120595)120597119903)Rminus1

n (120597(120595)120597120579) = (120597H(120595)

120597120579)Rminus1

n (120597(120595)120597119903) then 119868120579119903= 119868

119903120579 We can get

119868120579120579

= 210038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597120579

= 81205872

1198892cos2 (120579)

times SNRsum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

119868120579119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597120579Rminus1

n120597 (120595)

120597119903

= minus81205872

119889Δ119891 cos (120579)

times SNR[sum

119872

119898=1(119898 minus 1)

2

120582119888+Δ119891sum

119872

119898=1(119898 minus 1)

3

1198882]

119868119903119903= 2

10038161003816100381610038161205730

1003816100381610038161003816

2120597

H(120595)

120597119903Rminus1

n120597 (120595)

120597119903

= SNR8120587

2

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

(C3)


= [Iminus1

FDA]11 CRLB119903119903FDA

= [Iminus1




2

s1205902


also start with FIM

I120581119894120581119895

= 2Re 119863H120581119894

(120581) (Cminus1

n otimes Rminus1

n )119863120581119895(120581) (D1)


120581119894(120595) = 120597(120581)120597120581

119894

Consider

IFDA =

[[[[[

[

119868120573120573

119868119879

120579120573119868119879

120578120573

119868120579120573

119868120579120579

119868119879

120578120579

119868120578120573

119868120578120579

119868120578120578

]]]]]

]

(D2)



where

V = U119879

(D4a)

V = [119868120579120573

119868120578120573

] (D4b)

B = [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] (D4c)


Iminus1

FDA = [

[





]

]

(D5)



CRLB120579120578120579120578FDA


= [119868120579120579

119868119879

120578120579

119868120578120579

119868120578120578

] minus [119868120579120573

119868120578120573

] 119868minus1

120573120573[119868

119879

120579120573119868119879

120578120573]

minus1

(D6)

where

119868120573120573

= 2 sdot[[[

[

s sdot 1198721205902

119899

0

0s sdot 1198721205902

119899

]]]

]

(D7a)

119868120579120573

= 2 sdot Re[1 119895] otimess1205902

119899

sdot 120573lowast

1198601 (D7b)

119868120578120573

= 2 sdot Re[1 119895] otimes120573

lowast

1205902

119908

sdot [119872 sdot 1198603+ s sdot 119860

4] (D7c)

119868120578120578

= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

[119872 sdot 1198606+ 119860

H3119860

4+ 119860

3119860

H4+ s sdot 119860

7]

(D7d)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)

SNR (dB)


Δf1 = 0kHz Δf2 = 30


70001 70001 70001 70002100286514

100286515

100286515

kHz



119868120579120579minus (119868

120579120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198602minus119860

1119860

H1

119872) (D8a)

119868120578120579minus (119868

120578120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198605minus119860

4119860

H1

119872) (D8b)

119868120578120578

minus (119868120578120573119868119879

120578120573)

1205902

119899

s119872

= 2

10038161003816100381610038161205731003816100381610038161003816

2

s1205902

119899

(119872119860

6

s+ 119860

7minus119872119860

3119860

H3

s2minus119860

4119860

H4

119872)

(D8c)

1198601= minus119895(

2120587119889 cos (120579)sum119872

119898=1(119898 minus 1)

120582

+2120587119889Δ119891 cos (120579)sum119872

119898=1(119898 minus 1)

2

119888)

(D9a)

1198602= 4120587

2

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(D9b)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)



Acknowledgments




References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)

SNR (dB)


Δf1 = 0kHz Δf2 = 30


70001 70001 70001 70002100286514

100286515

100286515

kHz



119868120579120579minus (119868

120579120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198602minus119860

1119860

H1

119872) (D8a)

119868120578120579minus (119868

120578120573119868119879

120579120573)

1205902

119899

s119872= 2

s 10038161003816100381610038161205731003816100381610038161003816

2

1205902

119899

(1198605minus119860

4119860

H1

119872) (D8b)

119868120578120578

minus (119868120578120573119868119879

120578120573)

1205902

119899

s119872

= 2

10038161003816100381610038161205731003816100381610038161003816

2

s1205902

119899

(119872119860

6

s+ 119860

7minus119872119860

3119860

H3

s2minus119860

4119860

H4

119872)

(D8c)

1198601= minus119895(

2120587119889 cos (120579)sum119872

119898=1(119898 minus 1)

120582

+2120587119889Δ119891 cos (120579)sum119872

119898=1(119898 minus 1)

2

119888)

(D9a)

1198602= 4120587

2

1198892cos2 (120579)

sdot sum

119872

119898=1(119898 minus 1)

2

1205822+(Δ119891)

2

sum119872

119898=1(119898 minus 1)

4

1198882

+2Δ119891sum

119872

119898=1(119898 minus 1)

3

120582119888

(D9b)


10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)



Acknowledgments




References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10minus1

10minus2

10minus3

10minus4

CRLB

of a

ngle

estim

atio

n (d

eg)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

CRLB

of r

ange

estim

atio

n (m

)


SNR (dB)


Δf1 = 0kHz Δf2 = 30



102

101

100

10minus1

CRLB

of D

oppl

er sh

ift es

timat

ion

(rad

s)


SNR (dB)


Δf1 = 0kHz Δf2 = 30




1198603= [int

infin

minusinfin

119904 (119905 minus 120591)119889119904(119905 minus 120591)

H

119889119905119889119905 int

infin

minusinfin

119905|119904 (119905 minus 120591)|2

119889119905]

119879

(D9c)

1198604= [

2120587Δ119891sum119872

119898=1(119898 minus 1)

1198880]

119879

(D9d)

1198605= [minus4120587

2119889Δ119891 cos(120579) [

sum119872

119898=1(119898 minus 1)

2

12120582119888

+

Δ119891sum119872

119898=1(119898 minus 1)

3

121198882

] 0]

119879

(D9e)

1198606=[[

[

int

infin

minusinfin

10038161003816100381610038161003816100381610038161003816

119889119904 (119905 minus 120591)

119889119905

10038161003816100381610038161003816100381610038161003816

2

119889119905 int

infin

minusinfin

119905119904H(119905 minus 120591)

119889119904 (119905 minus 120591)

119889119905

119889119905

int

infin

minusinfin

119905119904 (119905 minus 120591)

119889119904(119905 minus 120591)H

119889119905

119889119905 int

infin

minusinfin

1199052|119904 (119905 minus 120591)|

2119889119905

]]

]

(D9f)

1198607=[[

[

41205872

(Δ119891)2

1198882

119872

sum

119898=1

(119898 minus 1)2

0

0 0

]]

]

(D9g)



Acknowledgments




References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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