research article broadband doa estimation based on nested...

Research ArticleBroadband DOA Estimation Based on Nested Arrays

Zhi-bo Shen Chun-xi Dong Yang-yang Dong Guo-qing Zhao and Long Huang

The Key Laboratory of Electronic Information Countermeasure and Simulation Technology Ministry of EducationXidian University Xirsquoan Shaanxi 710071 China

Correspondence should be addressed to Chun-xi Dong chxdongmailxidianeducn

Received 21 October 2014 Accepted 24 March 2015

Academic Editor Diego Masotti

Copyright copy 2015 Zhi-bo Shen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Direction of arrival (DOA) estimation is a crucial problem in electronic reconnaissance A novel broadband DOA estimationmethod utilizing nested arrays is devised in this paper which is capable of estimating the frequencies and DOAs of multiplenarrowband signals in broadbands even though they may have different carrier frequencies The proposed method convertsthe DOA estimation of multiple signals with different frequencies into the spatial frequency estimation Then the DOAs andfrequencies are pair matched by sparse recovery It is possible to significantly increase the degrees of freedom (DOF) with thenested arrays and the number of sources can bemore than that of sensor array In addition the method can achieve high estimationprecision without the two-dimensional search process in frequency and angle domain The validity of the proposed method isverified by theoretic analysis and simulation results

1 Introduction

Direction of arrival (DOA) has been an active researcharea playing an important role in electronic reconnaissance[1] as it is a crucial parameter for sorting and recognitionof sources directing jamming and passive location Unlikeradar receiver there is no priori information of signals forthe reconnaissance receiver Therefore the reconnaissancereceiver tends to have large instantaneous bandwidth in orderto cover a wide spectrum [2ndash4] This leads to two problemsin DOA estimation one is that the array aperture must beincreased to satisfy multiple signals of different frequencieswithin the reconnaissance band range and the other is that itmust be possible to achieve the frequency and DOA estima-tion ofmultiple signals simultaneously in broadband In [5] anested array structurewas proposed to increase the degrees offreedom by vectorizing the covariance matrix of the receivedsignals among different sensors Then Pal and Vaidyanathanincrease degrees of freedom of the coarray by extending thenesting strategy to multiple levels in [6] namely the 2119902th-order nested array whose 2119902th-order difference coarray isproved to contain a uniform linear arraywith119874(1198732119902) sensorsIt can be viewed as a virtual arraywith awider aperture whichis capable of being utilized to improve the DOA estimation

performance of the multiple signals with same frequencyHowever when the frequencies of the signals are differentand unknown this technique cannot be applied directlyBy exploiting the four-level nested array a novel estimationapproach is proposed in [7] which can be extended towideband scenarios However for each subband signal thevectorization is performed twice to construct a fourth orderdifference coarray and the spectrum peak search processis conducted in the whole spatial domain which suffersfrom the complex computation Signal subspace techniquessuch as MUSIC [8] need two-dimensional search process infrequency and angle domain which leads to large amountof computation In [9] an angle-frequency joint dictionaryis established to achieve the frequency and DOA estimationby sparse recovery The results suggest that sparse recoverybecomes more difficult and complicated while the length ofdictionary increases

A broadband DOA estimation method is investigated inthis paper A nested array is used to extend the array apertureand the spatial frequency which contains the informationof both frequency and angle is defined In this way theDOAestimation ofmultiple signals with different frequenciesis converted into the spatial frequency estimation Thenthe received signals of sensor array are transformed into

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 974634 7 pageshttpdxdoiorg1011552015974634

2 International Journal of Antennas and Propagation

d1 d2middot middot middot middot middot middot

N1 N2

Figure 1 A 2-level nested array with 1198731sensors in the inner

ULA and 1198732sensors in the outer ULA with spacings 119889

1and 119889

2

respectively

frequency domain By sparse recovery approach each of thesignals with different frequency can find its correspondingspatial frequency in the dictionary composed of spatialfrequencies that have been estimatedOnce the frequency andspatial frequency are pair matched DOA estimation can beimplemented simply by the definition of spatial frequency

This paper is organized as follows In Section 2 weintroduce the nested array model In Section 3 we proposethe broadband DOA estimation method based on nestedarray and analyze the performance Section 4 presents thesimulation results and conclusions are made in Section 5

2 The Nested Array Model

A two-level nested array [10] with 119873 sensors is basically aconcatenation of two uniform linear arrays (ULAs) innerand outer where the inner ULA has119873

1sensors with spacing

1198891and the outer ULA has 119873

2sensors with spacing 119889

2=

(1198731+ 1)119889

1 as shown in Figure 1 The resulting 119873 sensors

are positioned at 119878 = 11989911198891 1198991= 0 1 119873

1minus 1 cup 119899

21198892minus

1198891 1198992= 1 2 119873

2 in which the first sensor is set to be the

referenceAssume that there are119870 sources imping on the array from

directions 1205791 1205792 120579

119870with carrier frequency 119891

1 1198912 119891

119870

respectivelyThe steering vector a(120579119894 119891119894) corresponding to the

119894th source signal is expressed as

a (120579119894 119891119894) = [1 119890

minus11989521205871198891119891119894sin 120579119894119888 119890

minus1198952120587(1198731minus1)1198891119891119894sin 120579119894119888

119890minus1198952120587(119889

2minus1198891)119891119894sin 120579119894119888 119890

minus1198952120587(11987321198892minus1198891)119891119894sin 120579119894119888]T

(1)

where 119888 is the velocity of wave propagation The data vectorreceived at the nested array is expressed as

x (119905) =119870

sum

119894=1

a (120579119894 119891119894) 119904119894(119905) + n (119905) = As (119905) + n (119905) (2)

where

A = [a (1205791 1198911) a (1205792 1198912) sdot sdot sdot a (120579

119870 119891119870)] (3)

s (119905) = [1199041 (119905) 1199042(119905) sdot sdot sdot 119904

119870(119905)]

T (4)

where A denotes the array manifold matrix and s(119905) denotesthe source signal vector n(119905) is the white noise which isuncorrelated from the sources


31 Spatial Frequency Estimation From (1)ndash(4) it is obviousthat there is two-dimensional information in broadbandDOA estimation namely frequency and angleTherefore weattempt to estimate spatial frequency instead of frequencyand angle and the spatial frequencyΩ is defined as

Ω =119891 sin 120579119888

(5)

Combining (1) and (3) we obtain

A = [a (Ω1) a (Ω2) sdot sdot sdot a (Ω

119870)] (6)

where

a (Ω119894) = [1 119890

minus11989521205871198891Ω119894 119890

minus1198952120587(1198731minus1)1198891Ω119894 119890minus1198952120587(119889

2minus1198891)Ω119894

119890minus1198952120587(119873

21198892minus1198891)Ω119894]T

(7)

Here we assume that the sources are mutually uncorrelatedand each source signal 119904

119894(119905) is wide-sense quasistationary [11]

with frame length 119871 that is

119864 1003816100381610038161003816119904119894 (119905)

1003816100381610038161003816

2

= 1205902

119898119894

forall119905 isin [(119898 minus 1) 119871119898119871 minus 1] 119898 = 1 2 119872

(8)

Then the autocorrelation matrix of the 119898th frame receivedsignal is

R119898= 119864 [x (119905) x119867 (119905)] = AD

119898A119867 + 1205902

119899I

forall119905 isin [(119898 minus 1) 119871119898119871 minus 1]

(9)

whereD119898= diag(1205902

1198981 1205902

1198982 120590

2

119898119870) is the source covariance

matrix at frame 119898 1205902119898119894

is the power of the 119894th source and 1205902119899

is the power of noise Vectorizing R119898 we get the following

1198732times 1 vector

y119898= vec (R

119898) = (Alowast ⊙ A) p

119898+ 1205902

1198991119898 (10)

where p119898= [1205902

1198981 1205902

1198982 120590

2

119898119870]T and 1

119898= [eT1 eT2 eT

119873]T

with eT1being a column vector of all zeros except a 1 at the

119894th position Comparing it with (2) we can say that y119898in

(10) behaves like the data vector received at an array whosemanifold is given byAlowast ⊙A where ⊙ denotes the Khatri-Rao(KR) product

By stacking [y1 y2 y

119872] ≜ Y we have that

Y = (Alowast ⊙ A)P + 1205902119899E (11)

where P = [p1 p2 p

119872] and E = [1

1 12 1

119872]

The noise subspaces U119899can be obtained by applying

singular value decomposition (SVD) on Y which satisfiesthat

U119867119899(alowast (Ω

119894) otimes a (Ω

119894)) = 0 119894 = 1 2 119870 (12)

International Journal of Antennas and Propagation 3

Then the conventional subspace-based DOA estimationapproach as MUSIC can be exploited as follows

119875music (Ω)

=1

(alowast (Ω) otimes a (Ω))119867U119899U119867119899(alowast (Ω) otimes a (Ω))

(13)

It is interesting to note that spectrum peak search processis applied in the spatial frequency domain instead of the angledomain The source signals with different frequencies andangles correspond to respective spatial frequency thereforethey can be separated in the spatial frequency domain Sincethe spatial frequency includes both frequency and angleinformation we will show how to determine their matchingrelation in the following part

32 Pair Matching According to the definition of spatialfrequency in (5) once we know the frequency of each signalandmatch it to the corresponding spatial frequencyDOAcanbe obtained by simple computationHence the pairmatchingof frequency and angle is equivalent to that of frequency andspatial frequency

Consider the frequency domain model of (2) which isgiven by

X (119891119895) = A (119891

119895) S (119891

119895) + N (119891

119895) 119895 = 1 2 119869 (14)

where X(119891119895) is the received data at the array with frequency

119891119895 which is achieved by the Fourier transform of x(119905) Our

pair matching process is targeted at finding one or morespatial frequencies corresponding to X(119891

119895) among those that

have been already estimated We can treat this problem as asparse recovery problem [12] that is

minimize k0

subject to X (119891119895) = ADictk

(15)

where

Adict = [a (Ω1) a (Ω

2) a (Ω

119870)] (16)

Adict is an overcomplete dictionary composed of the steeringvector generated by the estimated values of spatial frequencyv is the sparse vector whose position of nonzero elementsrepresents the spatial frequency matched to X(119891

119895) However

the 1198970-minimization is NP-hard in general Thus it is often

converted into 1198971-minimization [13] as follows

minimize k1


(17)

The support of k can be obtained by convention sparserecovery algorithms such as Basis Pursuit (BP) [14] andOrthogonal Matching Pursuit (OMP) [15] Considering thescenario that some of the source signals are with the samefrequency the sparse recovery problem in the case of highsparsity ratio may exist Kwon et al proposed a multipath

matching pursuit (MMP) algorithm [16] which has beenproved effective in high sparsity ratio case Therefore theMMP sparse recovery algorithm is selected

As it is described above it is possible to find the matchedspatial frequency in the dictionary Adict for each X(119891

119895) with

the MMP algorithm

33 Frequency and DOA Estimation The set of 119891119895119869

119895=1

achieved by Fourier transform on the rows of x(119905) is regardedas the estimation of frequency Since the pair matching forthe set of 119891

119895119869

119895=1and Ω

119894119870

119894=1has been already implemented

the DOA estimation is given by

120579119894= arcsin(

119888Ω119894

119891(119894)

119895

) (18)

where119891(119894)119895

denotes the frequency corresponding to Ω119894 As 120579119894is

obtained by thematched119891119895and Ω

119894 120579119894and119891119895are pairmatched

automatically

34 Performance Analysis The implementation steps of theproposed method are summarized as follows

Given a received signal sequence x(119905)119879minus1119905=0

119879 = 119872119871 a sourcenumber 119870 and a frame length 119871

Step 1 Compute R119898 y119898

= vec(R119898) for 119898 = 1 2 119872

Then form a data matrix Y = [y1 y2 y

119872]

Step 2 Perform SVD on Y in order to obtain the noisesubspace matrix U

119899 Then compute the spatial frequency

spectrum by (13) and obtain Ω119894 for 119894 = 1 2 119870

Step 3 Compute the Fourier transform of x(119905) and pick upthe data X(119891

119895) for each frequency 119891

119895 for 119895 = 1 2 119869

Step 4 Compute the support of X(119891119895) in ADict by sparse

recovery and determine the corresponding Ω119894to 119891119895

Step 5 Compute 120579119894by (18) with the matched Ω

119894and 119891

119895

Step 6 Repeat Steps 4 and 5 for eachX(119891119895) and obtain the set

120579119894119870

119894=1as the DOA estimation

It is obvious that the performance of DOA estimation isrelated to the spatial frequency and frequency The precisionof frequency estimation is due to discrete Fourier transform(DFT) points 119873

119871 whose maximum value is the number of

snapshot From (18) we have

119889120579

119889119891=

Ω119888

1198912radic1 minus Ω

211988821198912

(19)

This means that the influence of frequency estimationerror on the DOA estimation decreases with the growth offrequencyTherefore theDOA estimation precision ismainly


due to the spatial frequency estimation precision when thecarrier frequency is high As the nested array which is capableof extending the array aperture effectively is utilized inspatial frequency estimation we can achieve high estimationprecision compared to the ULA If the spatial frequency isestimated accurately theDOAestimationwith high precisionfollows

In fact the broadband DOA estimation problem includestwo-dimensional parameter frequency and angle It is nor-mally implemented with a two-dimensional search processin frequency and angle domain Compared with traditionalmethod the proposedmethod only needs to conduct spectralsearching process once in the spatial frequency domain anda simple pair matching process However the traditionalmethod which uses band-pass filters for each element andobtains narrowband DOA estimation by MUSIC needsmultiple search processes for the signals with differentfrequencies Therefore the proposed method reduces thecomputational complexity to some extent

4 Simulation Results

In this section we provide several sets of simulation resultsto demonstrate the performance of the proposed methodIn all the simulation examples below we suppose that 119891 isin

(01 15)GHz 120579 isin (minus90∘ 90∘) for all the source signals The

root mean square error (RMSE) of angle from 119871Monte Carlotrials is used as our performance measure and it is defined as

RMSE = radic1

119870119871

119870

sum

119894=1

10038161003816100381610038161003816120579119894minus 120579119894

10038161003816100381610038161003816

2

(20)

where 120579119894and 120579

119894denote the true and estimation DOAs

respectively

41 Simulation and Setting

Example 1 This simulation example considers an under-determined case where 13 uncorrelated source signals (119870 =

13) impinge on the nested array with 6 sensors (1198731= 1198732=

3) The number of snapshots 119879 (119879 = 119872119871) and SNR are setto 119872 times 119871 (119872 = 32 119871 = 1024) and 15 dB respectively Thetrue DOAs are given by 120579

1 1205792 120579

119870 = minus60

∘ minus51∘ minus40∘

minus30∘ minus20∘ minus7∘ 5∘ 18∘ 25∘ 31∘ 45∘ 53∘ 60∘ with the carrier

frequency 1198911 1198912 119891

119870 = 115 103 103 057 135 08

103 135 057 135 08 103 115GHz Figures 2 and 3show the spatial frequency and frequency estimated resultsrespectively and Figure 4 shows the pair matching results offrequency and angle by MMP algorithm

Example 2 In this simulation example we consider thespatial frequency estimation performance with respect toSNR Five source signal angles and frequencies are setto 10 09 11 12 13GHz and minus30

∘ minus15∘ 5∘ 15∘ 30∘

respectively The SNR varies from minus10 dB to 15 dB in 25 dBintervals Figure 5 shows the RMSE of the spatial frequencyestimation as a function of SNR averaged over 1000 MonteCarlo trials

minus5 minus3 minus1 1 3 50

02

04

06

08

10

Spatial frequency

Spec

trum

(nor

mal

ized

)

Figure 2 Spatial frequency estimated results of 13 source signals119873 = 6119872 = 32 119871 = 1024 and SNR = 15 dB

05 07 09 11 13 150

02

04

06

08

10

Frequency (GHz)

Spec

trum

(nor

mal

ized

)

Figure 3 Frequency estimated results119873119871= 4096 and SNR = 15 dB

05 07 09 11 13 15

minus60

minus30

0

30

60

Frequency (GHz)

Ang

le (d

eg)

Figure 4 Pair mathcing results of 13 source signals SNR = 15 dB


minus10 minus5 0 5 10 15SNR (dB)

Proposed methodRootCRB(6)RootCRB(23)

100

10minus2

10minus4

RMSE

Figure 5 RMSE of spatial frequency versus SNR119873 = 6 and119870 = 5

01 03 05 07 09 11 13 15Frequency (GHz)

RMSE

(deg

)

100

10minus2

10minus1

120590f = 05MHz120590f = 1MHz120590f = 2MHz

Figure 6 RMSE of angle versus frequencies 119873 = 6 119870 = 3 andSNR = 15 dB

Example 3 In this simulation example we consider theinfluence of frequency estimated error on RMSE of angleThree DOAs are set to minus15∘ 30∘ and 45∘with SNR = 15 dBThe frequency varies from 01 GHz to 15 GHz with interval02 GHz Every fixed frequency conducts 1000 Monte Carlotrials Figure 6 shows the RMSE of angles as a function offrequency with different 120590

119891(standard deviation of frequency

estimation)

Example 4 In this example we examine the DOA estimationperformance with respect to SNR and snapshots The sim-ulation conditions are similar to those of Example 1 except

minus5 0 5 10 15 200

05

10

15

SNR (dB)

RMSE

(deg

)

L = 1024M = 32

L = 1024M = 16

L = 512M = 32

L = 512M = 16

Figure 7 RMSE of angle versus SNR and snapshots 119873 = 6 and119870 = 13

minus5 0 5 10 150

02

04

06

08

10

SNR (dB)

MUSICProposed method

RMSE

(deg

)

Figure 8 RMSE comparison119873 = 4 119870 = 5 and SNR = 15 dB

the SNR and snapshots The SNR varies from minus5 dB to 20 dBin 25 dB intervals The RMSE of angle as a function of SNRwith different snapshots is shown in Figure 7

Example 5 In this example we compare with traditionalmethod which uses band-pass filters for each element andobtains narrowband DOA estimation by MUSIC The SNRvaries from minus5 dB to 15 dB in 25 dB intervals We choosefive source signals with different frequencies and DOAsEvery fixed SNR conducts 1000 Monte Carlo trials Figure 8compares the RMSE of the proposed method and MUSICmethod based on band-pass filtering The CPU runtime ofMUSIC and the proposed method as a function of sourcenumber are shown in Figure 9



2 3 4 5 6 7 8 90

01

02

03

04

05

Source number

CPU

tim

e (s)

Figure 9 CPU time comparison119873 = 6 119871 = 1024 and119872 = 32

42 Discussion and Analysis From Figures 2 and 3 we seethat the proposed method is effective for spatial frequencyand frequency estimation In Figure 4 e and I denote thetrue values and estimated values respectively which provethat the frequency and angle can be pair matched correctlyby MMP algorithm

The RootCRB(119873) in Figure 5 denotes the square root ofCramer-Rao bound with 119873 sensors It is interesting to notethat the RootCRB is for spatial frequency instead of angleSince the nested array increases the DOF from 119873 to (1198732 minus2)2+119873 (119873 is even) [4] we compare it with the RootCRB(6)and RootCRB(23) From Figure 5 we observe that the nestedarray with 6 sensors yields better RMSE of spatial frequencythan the ULA with 6 sensors but still not as good as the ULAwith 23 sensors

The results of Figure 6 show that the smaller 120590119891is the

better RMSE we can obtain However with the same 120590119891

the RMSE of angle will decrease when the frequency variesfrom 01 GHz to 15 GHz Therefore we can conclude thatthe frequency estimation error has less influence at higherfrequencies and vice versa for lower frequencies which isequal to the analysis in (19)

From Figure 7 we see that better RMSE of angle can beobtained by increasing 119872 with fixed 119871 or increasing 119871 withfixed119872 as it increases the number of snapshots 119879 = 119872119871 inboth scenarios However when the number of snapshots isfixed at 119879 = 16384 the case of 119871 = 512 and119872 = 32 providesbetter RMSE than that of 119871 = 1024 and119872 = 16 Hence thisset of empirical results suggests that for a fixed 119879 trying toobtain more frames 119872 by decreasing the frame length 119871 (119871should not be overly small) tends to be a better option forperformance improvement than the opposite

From Figure 8 we see that the RMSE of the proposedmethod is better than MUSIC This is because it extendsthe array aperture by nested array which improves the DOAestimation precision The results of Figure 9 show that CPUtime of the proposed method is less than MUSIC and varies

little with the growth of source numberThe reason is that theproposed method can estimate the spatial frequencies of allthe signals in a single spectral search process which is lessaffected by source number However the times of spectralsearch in MUSIC are related to the source number whichequals the number of signals with different frequenciesTherefore the CPU runtime becomes longer significantlywhen the source number is large

5 Conclusion

This paper has addressed the broadband DOA estimationproblem with nested array The proposed method can obtainthe frequency and angle estimation of multiple source signalsby spatial frequency estimation and pair matching withsparse recovery The pair matching process is conducted inthe set of spatial frequency that have been estimated forreducing the computation Moreover the proposed methodcan work in broadband underdetermined and low SNRcases which is very suitable for electronic reconnaissancesystem

Conflict of Interests

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

Acknowledgments

This work is partially supported by the National BasicResearch ProgramunderGrant no 61XX81 NationalDefensePre-Research Program during the ldquo12th Five-Year Planrdquounder Grant no X110XX2030 and Fundamental ResearchFunds for the Central Universities of China under Grant noK5051202026

References

[1] N Yu X H Qi and X L Qiao ldquoMulti-channels widebanddigital reconnaissance receiver based on compressed sensingrdquoIET Signal Processing vol 7 no 8 pp 731ndash742 2013

[2] Z Liu X Wang G Zhao G Shi and J Lin ldquoWideband DOAestimation based on sparse representationrdquo in Proceedings of theIEEE International Conference on Signal Processing Communi-cations and Computing pp 1ndash4 Yunnan China August 2013

[3] D S Talagala W Zhang and T D Abhayapala ldquoBroadbandDOA estimation using sensor arrays on complex-shaped rigidbodiesrdquo IEEE Transactions on Audio Speech and LanguageProcessing vol 21 no 8 pp 1573ndash1585 2013

[4] M Agrawal and S Prasad ldquoBroadband DOA estimation usinglsquospatial-onlyrsquo modeling of array datardquo IEEE Transactions onSignal Processing vol 48 no 3 pp 663ndash670 2000

[5] P Pal and P P Vaidyanathan ldquoNested arrays a novel approachto array processing with enhanced degrees of freedomrdquo IEEETransactions on Signal Processing vol 58 no 8 pp 4167ndash41812010

[6] P Pal and P P Vaidyanathan ldquoMultiple level nested arrayan efficient geometry for 2qth order cumulant based array


processingrdquo IEEE Transactions on Signal Processing vol 60 no3 pp 1253ndash1269 2012

[7] S LiWHe X YangM Bao andYWang ldquoDirection-of-arrivalestimation of quasi-stationary signals using two-level Khatri-Rao subspace and four-level nested arrayrdquo Journal of CentralSouth University vol 21 no 7 pp 2743ndash2750 2014

[8] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[9] J-F Gu W-P Zhu and M N S Swamy ldquoCompressed sensingfor DOA estimation with fewer receivers than sensorsrdquo inProceedings of the IEEE International Symposium of Circuits andSystems (ISCAS rsquo11) pp 1752ndash1755 May 2011

[10] K Han and A Nehorai ldquoNested vector-sensor array processingvia tensor modelingrdquo IEEE Transactions on Signal Processingvol 62 no 10 pp 2542ndash2553 2014

[11] W-K Ma T-H Hsieh and C-Y Chi ldquoDOA estimation ofquasi-stationary signals with less sensors than sources andunknown spatial noise covariance a Khatri-Rao subspaceapproachrdquo IEEE Transactions on Signal Processing vol 58 no4 pp 2168ndash2180 2010

[12] E J Candes andT Tao ldquoNear-optimal signal recovery from ran-dom projections universal encoding strategiesrdquo IEEE Transac-tions on InformationTheory vol 52 no 12 pp 5406ndash5425 2006

[13] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationThe-ory vol 52 no 2 pp 489ndash509 2006

[14] S S Chen D L Donoho and M A Saunders ldquoAtomicdecomposition by basis pursuitrdquo SIAM Journal on ScientificComputing vol 20 no 1 pp 33ndash61 1998

[15] J A Tropp and A C Gilbert ldquoSignal recovery from randommeasurements via orthogonal matching pursuitrdquo IEEE Trans-actions on Information Theory vol 53 no 12 pp 4655ndash46662007

[16] S Kwon J Wang and B Shim ldquoMultipath matching pursuitrdquoIEEE Transactions on Information Theory vol 60 no 5 pp2986ndash3001 2014

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DistributedSensor Networks



d1 d2middot middot middot middot middot middot

N1 N2

Figure 1 A 2-level nested array with 1198731sensors in the inner

ULA and 1198732sensors in the outer ULA with spacings 119889

1and 119889

2

respectively

frequency domain By sparse recovery approach each of thesignals with different frequency can find its correspondingspatial frequency in the dictionary composed of spatialfrequencies that have been estimatedOnce the frequency andspatial frequency are pair matched DOA estimation can beimplemented simply by the definition of spatial frequency

This paper is organized as follows In Section 2 weintroduce the nested array model In Section 3 we proposethe broadband DOA estimation method based on nestedarray and analyze the performance Section 4 presents thesimulation results and conclusions are made in Section 5


A two-level nested array [10] with 119873 sensors is basically aconcatenation of two uniform linear arrays (ULAs) innerand outer where the inner ULA has119873

1sensors with spacing

1198891and the outer ULA has 119873

2sensors with spacing 119889

2=

(1198731+ 1)119889

1 as shown in Figure 1 The resulting 119873 sensors

are positioned at 119878 = 11989911198891 1198991= 0 1 119873

1minus 1 cup 119899

21198892minus

1198891 1198992= 1 2 119873

2 in which the first sensor is set to be the

referenceAssume that there are119870 sources imping on the array from

directions 1205791 1205792 120579

119870with carrier frequency 119891

1 1198912 119891

119870

respectivelyThe steering vector a(120579119894 119891119894) corresponding to the

119894th source signal is expressed as

a (120579119894 119891119894) = [1 119890

minus11989521205871198891119891119894sin 120579119894119888 119890

minus1198952120587(1198731minus1)1198891119891119894sin 120579119894119888

119890minus1198952120587(119889

2minus1198891)119891119894sin 120579119894119888 119890

minus1198952120587(11987321198892minus1198891)119891119894sin 120579119894119888]T

(1)

where 119888 is the velocity of wave propagation The data vectorreceived at the nested array is expressed as

x (119905) =119870

sum

119894=1

a (120579119894 119891119894) 119904119894(119905) + n (119905) = As (119905) + n (119905) (2)

where

A = [a (1205791 1198911) a (1205792 1198912) sdot sdot sdot a (120579

119870 119891119870)] (3)

s (119905) = [1199041 (119905) 1199042(119905) sdot sdot sdot 119904

119870(119905)]

T (4)

where A denotes the array manifold matrix and s(119905) denotesthe source signal vector n(119905) is the white noise which isuncorrelated from the sources


31 Spatial Frequency Estimation From (1)ndash(4) it is obviousthat there is two-dimensional information in broadbandDOA estimation namely frequency and angleTherefore weattempt to estimate spatial frequency instead of frequencyand angle and the spatial frequencyΩ is defined as

Ω =119891 sin 120579119888

(5)

Combining (1) and (3) we obtain

A = [a (Ω1) a (Ω2) sdot sdot sdot a (Ω

119870)] (6)

where

a (Ω119894) = [1 119890

minus11989521205871198891Ω119894 119890

minus1198952120587(1198731minus1)1198891Ω119894 119890minus1198952120587(119889

2minus1198891)Ω119894

119890minus1198952120587(119873

21198892minus1198891)Ω119894]T

(7)

Here we assume that the sources are mutually uncorrelatedand each source signal 119904

119894(119905) is wide-sense quasistationary [11]

with frame length 119871 that is

119864 1003816100381610038161003816119904119894 (119905)

1003816100381610038161003816

2

= 1205902

119898119894

forall119905 isin [(119898 minus 1) 119871119898119871 minus 1] 119898 = 1 2 119872

(8)

Then the autocorrelation matrix of the 119898th frame receivedsignal is

R119898= 119864 [x (119905) x119867 (119905)] = AD

119898A119867 + 1205902

119899I

forall119905 isin [(119898 minus 1) 119871119898119871 minus 1]

(9)

whereD119898= diag(1205902

1198981 1205902

1198982 120590

2

119898119870) is the source covariance

matrix at frame 119898 1205902119898119894

is the power of the 119894th source and 1205902119899

is the power of noise Vectorizing R119898 we get the following

1198732times 1 vector

y119898= vec (R

119898) = (Alowast ⊙ A) p

119898+ 1205902

1198991119898 (10)

where p119898= [1205902

1198981 1205902

1198982 120590

2

119898119870]T and 1

119898= [eT1 eT2 eT

119873]T

with eT1being a column vector of all zeros except a 1 at the

119894th position Comparing it with (2) we can say that y119898in

(10) behaves like the data vector received at an array whosemanifold is given byAlowast ⊙A where ⊙ denotes the Khatri-Rao(KR) product

By stacking [y1 y2 y

119872] ≜ Y we have that

Y = (Alowast ⊙ A)P + 1205902119899E (11)

where P = [p1 p2 p

119872] and E = [1

1 12 1

119872]

The noise subspaces U119899can be obtained by applying

singular value decomposition (SVD) on Y which satisfiesthat

U119867119899(alowast (Ω

119894) otimes a (Ω

119894)) = 0 119894 = 1 2 119870 (12)



119875music (Ω)

=1


(13)




X (119891119895) = A (119891

119895) S (119891

119895) + N (119891

119895) 119895 = 1 2 119869 (14)






minimize k0


(15)

where


2) a (Ω

119870)] (16)


119895) However



minimize k1


(17)




119895) with

the MMP algorithm


119895=1


119895119869

119895=1and Ω

119894119870



120579119894= arcsin(

119888Ω119894

119891(119894)

119895

) (18)

where119891(119894)119895




automatically





= vec(R119898) for 119898 = 1 2 119872


119872]






119895 for 119895 = 1 2 119869




119894and 119891

119895


120579119894119870





119889120579

119889119891=

Ω119888


211988821198912

(19)









RMSE = radic1

119870119871

119870

sum

119894=1

10038161003816100381610038161003816120579119894minus 120579119894

10038161003816100381610038161003816

2

(20)

where 120579119894and 120579


respectively





1 1205792 120579

119870 = minus60



frequency 1198911 1198912 119891

119870 = 115 103 103 057 135 08



∘ minus15∘ 5∘ 15∘ 30∘



02

04

06

08

10

Spatial frequency

Spec

trum

(nor

mal

ized

)


05 07 09 11 13 150

02

04

06

08

10

Frequency (GHz)

Spec

trum

(nor

mal

ized

)


05 07 09 11 13 15

minus60

minus30

0

30

60

Frequency (GHz)

Ang

le (d

eg)





100

10minus2

10minus4

RMSE


01 03 05 07 09 11 13 15Frequency (GHz)

RMSE

(deg

)

100

10minus2

10minus1

120590f = 05MHz120590f = 1MHz120590f = 2MHz




estimation)


minus5 0 5 10 15 200

05

10

15

SNR (dB)

RMSE

(deg

)

L = 1024M = 32

L = 1024M = 16

L = 512M = 32

L = 512M = 16


minus5 0 5 10 150

02

04

06

08

10

SNR (dB)


RMSE

(deg

)






2 3 4 5 6 7 8 90

01

02

03

04

05

Source number

CPU

tim

e (s)










5 Conclusion




Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119875music (Ω)

=1


(13)




X (119891119895) = A (119891

119895) S (119891

119895) + N (119891

119895) 119895 = 1 2 119869 (14)






minimize k0


(15)

where


2) a (Ω

119870)] (16)


119895) However



minimize k1


(17)




119895) with

the MMP algorithm


119895=1


119895119869

119895=1and Ω

119894119870



120579119894= arcsin(

119888Ω119894

119891(119894)

119895

) (18)

where119891(119894)119895




automatically





= vec(R119898) for 119898 = 1 2 119872


119872]






119895 for 119895 = 1 2 119869




119894and 119891

119895


120579119894119870





119889120579

119889119891=

Ω119888


211988821198912

(19)









RMSE = radic1

119870119871

119870

sum

119894=1

10038161003816100381610038161003816120579119894minus 120579119894

10038161003816100381610038161003816

2

(20)

where 120579119894and 120579


respectively





1 1205792 120579

119870 = minus60



frequency 1198911 1198912 119891

119870 = 115 103 103 057 135 08



∘ minus15∘ 5∘ 15∘ 30∘



02

04

06

08

10

Spatial frequency

Spec

trum

(nor

mal

ized

)


05 07 09 11 13 150

02

04

06

08

10

Frequency (GHz)

Spec

trum

(nor

mal

ized

)


05 07 09 11 13 15

minus60

minus30

0

30

60

Frequency (GHz)

Ang

le (d

eg)





100

10minus2

10minus4

RMSE


01 03 05 07 09 11 13 15Frequency (GHz)

RMSE

(deg

)

100

10minus2

10minus1

120590f = 05MHz120590f = 1MHz120590f = 2MHz




estimation)


minus5 0 5 10 15 200

05

10

15

SNR (dB)

RMSE

(deg

)

L = 1024M = 32

L = 1024M = 16

L = 512M = 32

L = 512M = 16


minus5 0 5 10 150

02

04

06

08

10

SNR (dB)


RMSE

(deg

)






2 3 4 5 6 7 8 90

01

02

03

04

05

Source number

CPU

tim

e (s)










5 Conclusion




Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















RMSE = radic1

119870119871

119870

sum

119894=1

10038161003816100381610038161003816120579119894minus 120579119894

10038161003816100381610038161003816

2

(20)

where 120579119894and 120579


respectively





1 1205792 120579

119870 = minus60



frequency 1198911 1198912 119891

119870 = 115 103 103 057 135 08



∘ minus15∘ 5∘ 15∘ 30∘



02

04

06

08

10

Spatial frequency

Spec

trum

(nor

mal

ized

)


05 07 09 11 13 150

02

04

06

08

10

Frequency (GHz)

Spec

trum

(nor

mal

ized

)


05 07 09 11 13 15

minus60

minus30

0

30

60

Frequency (GHz)

Ang

le (d

eg)





100

10minus2

10minus4

RMSE


01 03 05 07 09 11 13 15Frequency (GHz)

RMSE

(deg

)

100

10minus2

10minus1

120590f = 05MHz120590f = 1MHz120590f = 2MHz




estimation)


minus5 0 5 10 15 200

05

10

15

SNR (dB)

RMSE

(deg

)

L = 1024M = 32

L = 1024M = 16

L = 512M = 32

L = 512M = 16


minus5 0 5 10 150

02

04

06

08

10

SNR (dB)


RMSE

(deg

)






2 3 4 5 6 7 8 90

01

02

03

04

05

Source number

CPU

tim

e (s)










5 Conclusion




Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












100

10minus2

10minus4

RMSE


01 03 05 07 09 11 13 15Frequency (GHz)

RMSE

(deg

)

100

10minus2

10minus1

120590f = 05MHz120590f = 1MHz120590f = 2MHz




estimation)


minus5 0 5 10 15 200

05

10

15

SNR (dB)

RMSE

(deg

)

L = 1024M = 32

L = 1024M = 16

L = 512M = 32

L = 512M = 16


minus5 0 5 10 150

02

04

06

08

10

SNR (dB)


RMSE

(deg

)






2 3 4 5 6 7 8 90

01

02

03

04

05

Source number

CPU

tim

e (s)










5 Conclusion




Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2 3 4 5 6 7 8 90

01

02

03

04

05

Source number

CPU

tim

e (s)










5 Conclusion




Acknowledgments


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









research article broadband doa estimation based on nested...

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