Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 952743 9 pageshttpdxdoiorg1011552013952743

Research ArticleTime-Domain Joint Parameter Estimation of Chirp SignalBased on SVR

Xueqian Liu and Hongyi Yu

Zhengzhou Information Science and Technology Institute Zhengzhou 450002 China

Correspondence should be addressed to Xueqian Liu liuxqpapertomcom

Received 12 July 2013 Revised 7 August 2013 Accepted 29 August 2013

Academic Editor Zhiqiang Ma

Copyright copy 2013 X Liu and H Yu 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

Parameter estimation of chirp signal such as instantaneous frequency (IF) instantaneous frequency rate (IFR) and initial phase(IP) arises in many applications of signal processing During the phase-based parameter estimation a phase unwrapping processis needed to recover the phase information correctly and impact the estimation performance remarkably Therefore we introducesupport vector regression (SVR) to predict the variation trend of instantaneous phase and unwrap phases efficiently Even thoughwith that being the case errors still exist in phase unwrapping process because of its ambiguous phase characteristic Furthermorewe propose an SVR-based joint estimation algorithm and make it immune to these error phases by means of setting the SVRrsquosparameters properly Our results show that compared with the other three algorithms of chirp signal not only does the proposedone maintain quality capabilities at low frequencies but also improves accuracy at high frequencies and decreases the impact withthe initial phase

1 Introduction

Chirp signals that is second-order polynomial phase signalsare common in various areas of science and engineeringFor example in a synthetic aperture radar (SAR) systemwhen the target is moving the regulated signals will changeinto chirp ones after being reflected [1] The SAR imagingquality may be degraded by shifting andor defocusingdue to inaccurate estimation of instantaneous frequency(IF) and instantaneous frequency rate (IFR) respectively[2] In optical communications coding or instability of thelaser diode results in chirp phenomenon [3] The researchon estimating these chirp parameters is divided into twoparts one is based on cubic phase function (CPF) evenhigh-order phase function (HPF) [4ndash6] and the other isbased on maximum likelihood (ML) [7ndash9] The former hasthe advantage in fast calculation of IFR but costs moreto find IF The latter tries its best to maximize an MLfunction mainly in frequency domain which involves a two-dimensional nonlinear optimization Unfortunately thereis no exact closed-form solution for solving this generalnonlinear programming problemThe solution either resortsto a burdensome numerical search or is approximated by

linearized techniques Djuric and Kay proposed an efficienttime-domainML estimator [10] which can achieve optimumestimation at a moderate complexity By extending the phasenoise model of [11] to chirp signals Li et al derived animproved ML estimator and analyzed its performance in thetime domain [12 13] However both of them are suitableonly at high signal-to-noise ratio (SNR) for the reason ofapproximations of noise phase model and imperfections inphase unwrapping process

By introducing structural risk minimization (SRM) prin-ciple support vector regression (SVR) exhibits excellentcapabilities for generalizing and learning From the viewpointof the quadratic relationship between absolute signal phaseand time series therefore this paper employs SVR to unwrapphases and estimate IF and IFR recursively We avoid takingthe rationality of phase noise model into account by notmaking any approximations in it At one time we reduce theestimation performancersquos dependence on phase unwrappingprocess with a proper choice of SVRrsquos parameters It has beenshown that except for the property of low sensitivity to initialphase and closely approaching the Cramer-Rao lower bound(CRLB) at low frequencies the proposed algorithm improvesits estimation performance at high frequencies

2 Mathematical Problems in Engineering

2 The Proposed Algorithm

21 Signal Model The signal model used here is similar tothat in [10 12] The complex baseband chirp signal pollutedby noise is modelled as

119903119899= 119886119899119860 exp 119895 [120601 + 2120587119891

119889119899119879119904+ 2120587119891

119903(119899119879119904)2

] + 119908119899

119899 = 0 119873 minus 1

(1)

Here 119886119899is an independent symbol 119860 gt 0 119891

119889 119891119903isin

[minus05 05) and 120601 isin [minus120587 120587) are the amplitude deterministicbut unknown IF IFR and initial phase respectively 119879

119904is the

sampling period 119873 is the sample size 119908119899is an independent

complex additive white Gaussian noise (AWGN) with zeromean and variance 1205902 For the sake of simplicity we set119879

119904= 1

and rearrange (1)

119903119899= exp [119895 (120601 + 2120587119891

119889119899 + 2120587119891

1199031198992

)]

times 119860 + 119886lowast

119899exp [minus119895 (120601 + 2120587119891

119889119899 + 2120587119891

1199031198992

)]119908119899

= exp [119895 (120601 + 2120587119891119889119899 + 2120587119891

1199031198992

)] (119860 + 1199081015840

119899)

119899 = 0 119873 minus 1

(2)

where 1199081015840119899= 119886lowast

119899exp[minus119895(120601 + 2120587119891

119889119899 + 2120587119891

1199031198992

)]119908119899is still an

independent complex AWGN with zero mean and variance1205902 Absolute phase of 119903

119899is presented as

ang119903119899= 120601 + 2120587119891

119889119899 + 2120587119891

1199031198992

+ang (119860 + 1199081015840

119899) 119899 = 0 119873 minus 1

(3)

When SNR = 1198602

1205902 is large enough the approximate

model in [10] is expressed as

ang (119860 + 1199081015840

119899) asymp arctan

Im (1199081015840

119899)

119860 + Re (1199081015840119899)

asymp arctanIm (119908

1015840

119899)

119860

asymp

Im (1199081015840

119899)

119860

119899 = 0 119873 minus 1

(4)

A more reasonable model proposed in [12] is given

ang (119860 + 1199081015840

119899) asymp arcsin

Im (1199081015840

119899)

1003816100381610038161003816119903119899

1003816100381610038161003816

asymp

Im (1199081015840

119899)

1003816100381610038161003816119903119899

1003816100381610038161003816

119899 = 0 119873 minus 1

(5)

22 SVR-Based Phase Unwrapping Process and FrequencyEstimation Algorithm In (5) there are two approximationsstill existing Based on the quadratic relation in (3) thispaper utilizes SVRrsquos excellent capability for learning unknownmodels to unwrap phases and estimate frequency We yield a

training set 119878119896= (119909

119896

119894 119910119896

119894) | 119894 = 1 119896 119909

119896

119894= 119894 minus 1 119910

119896

119894=

ang119903119894minus1

at time point 119896 (1 le 119896 le 119873) where ang119903119894minus1

denotes theestimation value of ang119903

119894minus1

At first we present a line119891119896(119909) = (w

119896sdot120601(119909)) + 119887

119896 where (sdot)

is an inner product operator and 120601(sdot) is a nonlinear mappingfrom low to high dimension feature space And also we defineits 120576-insensitive loss function as

119871 (119910119896

119894 119891119896(119909119896

119894)) =

10038161003816100381610038161003816119910119896

119894minus 119891119896(119909119896

119894)

10038161003816100381610038161003816120576

=

0

10038161003816100381610038161003816119910119896

119894minus 119891119896(119909119896

119894)

10038161003816100381610038161003816le 120576119896

10038161003816100381610038161003816119910119896

119894minus 119891119896(119909119896

119894)

10038161003816100381610038161003816minus 120576119896 else

(6)

where 120576119896is insensitive loss coefficient at time point 119896

Next we assume 119891119896(119909) insofar as for 120576

119896to completely fit

all elements of 119878119896 andwe denote119889119896

119894as the distance frompoint

(119909119896

119894 119910119896

119894) isin 119878119896to 119891119896(119909)

119889119896

119894=

10038161003816100381610038161003816(w119896sdot 120601 (119909119896

119894)) + 119887

119896minus 119910119896

119894

10038161003816100381610038161003816

radic1 +1003817100381710038171003817w119896

1003817100381710038171003817

2

le

120576119896

radic1 +1003817100381710038171003817w119896

1003817100381710038171003817

2

119894 = 1 119896

(7)

According to (7) we optimize 119891119896(119909) through maximizing

120576119896radic1 + w

1198962 that is minimizing w

1198962 Thereby SVR is

presented as

min 119869 (w119896 119887119896) =

1

2

1003817100381710038171003817w119896

1003817100381710038171003817

2

st 10038161003816100381610038161003816(w119896sdot 120601 (119909119896

119894)) + 119887

119896minus 119910119896

119894

10038161003816100381610038161003816le 120576119896 119894 = 1 119896

(8)

In fact fitting errors larger than 120576119896always exist By

introducing slack variables 120585119896119894 (120585lowast

)119896

119894ge 0 119894 = 1 119896 and

penalty factor 119862119896at time point 119896 (8) is converted into

min 119869 (w119896 119887119896 120585119896 120585lowast

119896) =

1

2

1003817100381710038171003817w119896

1003817100381710038171003817

2

+ 119862119896

119896

sum

119894=1

[120585119896

119894+ (120585lowast

)119896

119894]

st 119910119896

119894minus (w119896sdot 120601 (119909119896

119894)) minus 119887

119896le 120576119896+ 120585119896

119894

(w119896sdot 120601 (119909119896

119894)) + 119887

119896minus 119910119896

119894le 120576119896+ (120585lowast

)119896

119894

119894 = 1 119896

(9)

where 120585119896= [120585119896

1 120585119896

2 120585

119896

119896]

119879

120585lowast

119896= [(120585lowast

)119896

1 (120585lowast

)119896

2 (120585

lowast

)119896

119896]

119879

[sdot]119879 is a transpose operator and 119862 is a positive constant

to take compromise in SVRrsquos generalization capability andfitting errors which are denoted by the first and second itemsof 119869(w

119896 119887119896) respectively

Mathematical Problems in Engineering 3

Initialization

Set construction

Phase prediction

SVR

Phase unwrapping

Ultimate values

At time point k = 0 angr0 =

arg(r0) = arg[ [120601 + ang(1 + w0)

At time point 1 le k le N

xki = i minus 1 y

ki = angriminus1

Select a proper to satisfyangrk = arg(rk) + 2120587

120587

m isin

m isin

[angSVR rk minus angSVR rk + 120587)

At time point k = N

ang rk = [ [120572ki minus (120572

lowast)k

i xki )) 2

+ bk

=sum

Ni=1[120572Ni minus (120572lowast)

N

i]xki

120587fNd

fNr =

sumNi=1[ [120572

Ni minus (120572lowast)

N

ixki ))

2

2120587

minus (120572lowast)N

i

=

N

sumi=1

[

k

sumi=1

= [ [120572ki minus (120572

lowast)k

i xki x 1)) 2

+ bk

k

sumi=1

120572Ni ] + bN120601

N

Sk = xki y

ki )) |i = 1 middot middot middot k

Approximated function

Make use of SVR and derive

fk(x) = (wk middot 120601(x)) + bk

+

k 1+

k = N

k lt N

Figure 1 Recursive implementation of SVR-based algorithm

Equation (9) is a strict convex quadratic programming(QP) problem in optimal theories Then using Lagrangemultiplier method

w119896=

119896

sum

119894=1

[120572119896

119894minus (120572lowast

)119896

119894] 120601 (119909

119896

119894)

119896

sum

119894=1

[(120572lowast

)119896

119894minus 120572119896

119894] = 0

119862119896minus 120572119896

119894minus 120574119896

119894= 0 119894 = 1 119896

119862119896minus (120572lowast

)119896

119894minus (120574lowast

)119896

119894= 0 119894 = 1 119896

(10)

where 120572119896119894 (120572lowast

)119896

119894 120574119896

119894 and (120574lowast)119896

119894are the 119894th Lagrange multipli-

ers of 119910119896119894minus (w119896sdot 120601(119909119896

119894)) minus 119887119896le 120576119896+ 120585119896

119894 (w119896sdot 120601(119909119896

119894)) + 119887119896minus 119910119896

119894le

120576119896+ (120585lowast

)119896

119894 120585119894ge 0 and (120585lowast)119896

119894ge 0 respectively

Substituting (10) into (9) replacing (120601(119909119896119894) sdot 120601(119909

119896

119895)) with

quadratic kernel function 119870(119909119894 119909119895) = (119909

lowast

119894119909119895+ 1)2 in this

study and deriving the wolf dual problem of (9)

max119882(120572119896120572lowast

119896)

= minus

1

2

119896

sum

119894119895=1

[120572119896

119894minus (120572lowast

)119896

119894] [120572119896

119895minus (120572lowast

)119896

119895] (119909lowast

119894119909119895+ 1)

2

+

119896

sum

119894=1

119910119896

119894[120572119896

119894minus (120572lowast

)119896

119894] minus 120576119896

119896

sum

119894=1

[120572119896

119894+ (120572lowast

)119896

119894]

st119896

sum

119894=1

[120572119896

119894minus (120572lowast

)119896

119894] = 0

0 le 120572119896

119894 (120572

lowast

)119896

119894le 119862119896 119894 = 1 119896

(11)

where 120572119896= [120572119896

1 120572119896

2 120572

119896

119896]

119879

120572lowast119896= [(120572lowast

)119896

1 (120572lowast

)119896

2 (120572

lowast

)119896

119896]

119879

4 Mathematical Problems in Engineering

0 5 10 15 20 25 300

50

100

150

200

250

300

350Ph

ase

k

(a)

0 5 10 15 20 25 30

0

100

200

300

400

500

600

700

Phas

e

minus100

k

(b)

Without noiseActualSVR

DKLFK

Phas

e

0 5 10 15 20 25 30

0

200

400

600

800

1000

1200

1400

minus200

k

(c)

Figure 2 (a) Arbitrary phase unwrapping processes with119891119889= 01 119891

119903= 005 120601 = 0119873 = 32 and SNR = 8 dB (b)Arbitrary phase unwrapping

processes with 119891119889= 03 119891

119903= 005 120601 = 0119873 = 32 and SNR = 8 dB (c) Arbitrary phase unwrapping processes with 119891

119889= 01 119891

119903= 02 120601 =

0119873 = 32 and SNR = 8 dB

We obtain 120572119896 120572lowast119896through solving (11) Ultimately we get

the best approximation of (3) by using Karush-Kuhn-Tucker(KKT) conditions

119891119896(119909) = (w

119896sdot 120601 (119909)) + 119887

119896

=

119896

sum

119894=1

[120572119896

119894minus (120572lowast

)119896

119894] (119909lowast

119894119909119895+ 1)

2

+ 119887119896

(12)

where 119887119896= 119910119896

119894minussum119896

119895=1[120572119896

119895minus(120572lowast

)119896

119895](119909lowast

119894119909119895+ 1)2

minus120576119896 120572119896

119894isin (0 119862

119896)

or 119887119896= 119910119896

119894minussum119896

119895=1[120572119896

119895minus (120572lowast

)119896

119895](119909lowast

119894119909119895+ 1)2

+ 120576119896 (120572lowast

)119896

119894isin (0 119862

119896)

is bias at time point 119896Not only does (6) guarantee less fitting errors but it also

possesses a good generalization capability It means that after

learning previous unwrapped phases ang1199030 ang119903

119896minus1 (6) can

predict the variation trend of a phase efficiently for the nexttime point 119896 and deriveangSVR119903119896 = sum

119896

119894=1[120572119896

119894minus(120572lowast

)119896

119894](119909lowast

119894119896 + 1)

2

+

119887119896 The phase ang119903

119896is then unwrapped within a 2120587 interval

centered around angSVR119903119896 by adding multiples of plusmn2120587 to theprincipal value of ang119903

119896when the absolute difference between

angSVR119903119896 and the principal value of ang119903119896is greater than 120587

Namely we select a proper119898 to satisfy ang119903119896= arg(119903

119896)+2120587119898 isin

[angSVR119903119896 minus 120587 angSVR119903119896 + 120587) Till time point 119896 = 119873 we get 119891119873119889=

(sum119873

119894=1[120572119873

119894minus(120572lowast

)119873

119894]119909119873

119894)120587

119891119873

119903= (sum119873

119894=1[120572119873

119894minus(120572lowast

)119873

119894](119909119873

119894)

2

)2120587and

120601119873

= sum119873

119894=1[120572119873

119894minus (120572lowast

)119873

119894] + 119887119873as the ultimate estimation

values This algorithm can be implemented recursively intime as shown schematically in Figure 1

Mathematical Problems in Engineering 5

0 5 10

MSE

SNR (dB)minus5minus10

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

(a)

0 5 10 15 20SNR (dB)

MSE

10minus1

100

10minus2

10minus4

10minus5

10minus6

10minus7

10minus3

(b)

0 5 10 15 20SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus3

(c)

Figure 3 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 0 and119873 = 32 (b) MSE of IF with 119891

119889= 03 119891

119903= 005 120601 = 0 and119873 = 32 (c) MSE of

IF with 119891119889= 01 119891

119903= 02 120601 = 0 and119873 = 32

23 SVRrsquos Parameter Settings Setting SVRrsquos parameters is adifficult problem but has a pronounced impact on SVRrsquos per-formance for example insensitive loss coefficient 120576 penaltyfactor 119862 There is no complete theoretical basis or explicitclosed form Cross-validation is a widely used method untilnow but it is complex and time consuming In this study weformulate proper parameter values by understanding SVRrsquostheory and integrating a large number of references andexperiments

When 119896 is small imperfect phase unwrapping at aparticular time point can easily have an impact on thevariation trend of 119891(119909) In order to avoid this discrepancya new model must be constructed with the aptitude ofbetter generalization capability As 119896 increases its necessitydecreases contrarily which is due to the degrading impact of

improperly unwrapped phase Simple speaking the modelrsquosgeneralization capability is inversely proportional to the sizeof the set 119878

Intuitively insensitive loss coefficient 120576 is the verticalheight of 120576-tube The larger 120576 is the less support vectorsthere are Nevertheless too large 120576 will cause unfixable 119887Noted that if the predicted phase is in the vicinity of 120587rsquosodd times the estimation performance is deteriorated rapidlyfor its ambiguous phase characteristic Selecting a proper120576 can reduce this impact At the same time 120576 is inverselyproportional to SNR so insensitive loss coefficient at timepoint 119896 (1 le 119896 le 119873) is given by [14]

120576119896= 120591radic

1

SNRln 119896119896

119896 = 1 119873 (13)

6 Mathematical Problems in Engineering

0 2 4 6 8 10 12 14SNR (dB)

minus2minus4minus6

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus3

(a)

SNR (dB)0 2 4 6 8 10 12 14 16 18 20

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus3

(b)

0 5 10 15 20SNR (dB)

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus3

SVRHPFDK

LFKCRLB

(c)

Figure 4 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 0 and119873 = 32 (b) MSE of IFR with 119891

119889= 03 119891

119903= 005 120601 = 0 and119873 = 32 (c) MSE

of IFR with 119891119889= 01 119891

119903= 02 120601 = 0 and119873 = 32

where 120591 is a positive constant and 120591 = 08 is set in this studyafter experimental comparisons and validations and SNR isassumed to be known

Penalty factor 119862 controls the penalty degree of vectorsoutside the 120576-tube and determines SVRrsquos generalization capa-bility 119862 is directly proportional to the sample size and SNRAs (12) is a line in a two-dimension plane the first item oftarget function during solving SVRrsquos QP problem is directlyproportional to its slope so a very small 119862 will result in ahorizontal and underfitting 119862 Inspired by [14 15] penaltyfactor at time point 119896 (1 le 119896 le 119873) is given by

119862119896= 120575radicSNR 3radic119896max (100381610038161003816

1003816119892119896minus 120582120590119896

10038161003816100381610038161003816100381610038161003816119892119896+ 120582120590119896

1003816100381610038161003816)

119896 = 1 119873

(14)

where 119892119896= (1119896)sum

119896minus1

119899=0|119910119899|2

120590119896= radic(1119896)sum

119896minus1

119899=0(|119910119899|2

minus 119892119896)

2119910119899= ang119903119899minus1

is a section of the element of training set 119878119896 as

(13) SNR is assumed to be known 120575 120582 are positive constantsand 120575 = 01 120582 = 05 are set in this study by the same way as120591

3 Results and Analyses

We have compared the proposed algorithm entitled as SVRestimator with the other three the HPF estimator proposedin [6] the DK estimator proposed in [10] the LFK estimatorproposed in [12]

31 Phase Unwrapping Process Because HPF estimator doesnot need the phase unwrapping process Figure 2 illustrates

Mathematical Problems in Engineering 7M

SE 10minus4

10minus5

10minus6

10minus7

10minus3

10minus2

10minus1

SNR (dB)0 5 10minus5minus10

N = 8 N = 32

N = 64N = 16

Figure 5 MSE of IF of SVR estimator with 119891119889= 01 119891

119903= 005 120601 =

0

0 5 10SNR (dB)

minus5minus10

MSE

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus10

N = 8 N = 32

N = 64N = 16

Figure 6MSE of IFR of SVR estimator with119891119889= 01 119891

119903= 005 120601 =

0

the arbitrary phase unwrapping processes of the other threewhile 119891

119889= 01 119891

119903= 005 119891

119889= 03 119891

119903= 005 and 119891

119889=

01 119891119903= 02 The number of Monte Carlo experiments is

10000 and 120601 = 0 119873 = 32 and SNR = 8 dB It is shownthat SVR estimator can unwrap phase accurately whateverthe value of 119891

119889and 119891

119903is However errors emerge in DK and

LFK ones as 119891119889and 119891

119903increase

32 Estimation Performance Figures 2 3 and 4 illustrate theMSE curves of IF and IFR versus SNR respectively whereMSEs are defined as 119864[( 119891

119889minus 119891119889)

2

] 119864[(119891119903minus 119891119903)

2

] 119891119889119891119903are

the estimation values of 119891119889 119891119903 CRLBs are given by [12]

CRLB119891119889=

1

SNR6 (8119873 minus 11) (2119873 minus 1)

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB119891119903=

1

SNR360

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB120601=

1

SNR3 (3119873

2

minus 3119873 + 2)

2119873 (1198732+ 3119873 + 2)

(15)

It is shown that whether 119891119889= 01 119891

119903= 005 119891

119889=

03 119891119903= 005 or 119891

119889= 01 119891

119903= 02 SVR estimator is the

best one during both IF and IFR estimations all the whilealso MSE performances of IF and IFR totally decrease as 119891

119889

and119891119903increase but SVR estimator is the most robust and has

had a great advantage already when 119891119889= 03 119891

119903= 005 or

119891119889= 01 119891

119903= 02

33 Impact of the Sample Size119873 Everything is as in Figure 2other than 119891

119889= 01 119891

119903= 005 the MSE curves of IF and IFR

of SVR estimator versus SNR while119873 is 8 16 32 and 64 areplotted in Figures 5 and 6 respectively It is clear that as 119873increases MSE performances of IF and IFR of SVR estimatorare both improved

34 Impact of Initial Phase 120601 Everything is as in Figure 3except that 120601 = 04120587 08120587 the MSE curves of IF and IFRare plotted against SNR in Figures 7 and 8 respectivelyComparing with Figures 3(a) and 4(a) we can see that SVRestimator is immune to 120601 but the other three are not

35 Computational Complexity Because we translate SVRinto QP problem and need to search the minimums duringthe process we can not derive the explicit form of thecomputational complexity of SVR estimator So everythingis as in Figure 2 except that the number of Monte Carloexperiments is 100 SNR is 8 dB and 119891

119889= 01 119891

119903= 005

the consuming times are listed in Table 1 while119873 is 8 16 32and 64 respectively The running computer is ASUS-PC1111having Intel(R) Pentium 213GHz CPU and 200GB RAM

We can see that SVR estimatorrsquos consuming times aremore than the othersrsquo especially when 119873 becomes larger Asa matter of fact however it is acceptable and tolerable

4 Conclusions

Phase unwrapping process is a key point in phase-basedfrequency estimation of chirp signal Firstly we adopt SVRto learn the unwrapped phases at previous time pointspredict the variation trend of phase efficiently and derive theestimation value for the next time point Once acquired interms of relationship between absolute signal phase and timeseries we address a simple and effective frequency estimationalgorithmof chirp signalTheproposed algorithm completelyexhibits its advantages of higher estimation accuracy lowersensitivity of frequency and initial phase by sacrificing moreconsuming times

Because SVR predicts the curversquos variation trend merelyin terms of training set which consists of previous pointsrsquo

8 Mathematical Problems in Engineering

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

0 5 10SNR (dB)

minus5minus10

(a)

0 2 4 6 8 10

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

SNR (dB)minus6 minus4 minus2minus8minus10

(b)

Figure 7 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IF with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

0 2 4 6 8 10 12 14SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

minus6 minus4 minus2

(a)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

0 2 4 6 8 10 12 14SNR (dB)

minus6 minus4 minus2

(b)

Figure 8 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IFR with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

Table 1 Consuming time with different119873 (ms)

Algorithm 119873 = 8 119873 = 16 119873 = 32 119873 = 64

HPF 380 942 3721 10654DK 203 476 1098 3711LFK 299 784 2362 8643SVR 437 1107 4805 13074

values we even can estimate the frequency of chirp signalunder the nonGaussian condition by the same way

Stressing that the proposed algorithm learns training setand gets the approximate values of SVRrsquos insensitive losscoefficient 120576 and penalty factor 119862 as a next step thereforeimproving SVRrsquos parameter setting is an important researchpoint

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China (Grant no 61201380)

Mathematical Problems in Engineering 9

References

[1] J C Curlander andRNMcDonough Synthetic Aperture RadarSystems and Signal Processing Wiley New York NY USA 1992

[2] J Xu Y Peng and X Xia ldquoParametric autofocus of SARimagingmdashinherent accuracy limitations and realizationrdquo IEEETransactions on Geoscience and Remote Sensing vol 42 no 11pp 2397ndash2411 2004

[3] R Ramaswami K N Sivarajan and G H Sasaki OpticalNetworks A Practical Perspective Morgan Kaufmann SanMateo Calif USA 1998

[4] P OrsquoShea ldquoA new technique for instantaneous frequency rateestimationrdquo IEEE Signal Processing Letters vol 9 no 8 pp 251ndash252 2002

[5] PWang I Djurovic and J Yang ldquoGeneralized high-order phasefunction for parameter estimation of polynomial phase signalrdquoIEEE Transactions on Signal Processing vol 56 no 7 pp 3023ndash3028 2008

[6] P Wang H Li I Djurovic and B Himed ldquoPerformance ofinstantaneous frequency rate estimation using high-order phasefunctionrdquo IEEE Transactions on Signal Processing vol 58 no 4pp 2415ndash2421 2010

[7] M Benidir and A Ouldali ldquoPolynomial phase signal analysisbased on the polynomial derivatives decompositionsrdquo IEEETransactions on Signal Processing vol 47 no 7 pp 1954ndash19651999

[8] P OrsquoShea ldquoA fast algorithm for estimating the parameters of aquadratic FM signalrdquo IEEE Transactions on Signal Processingvol 52 no 2 pp 385ndash393 2004

[9] S Ma J Jiang and Q Meng ldquoA fast accurate and robustmethod for joint estimation of frequency and frequency raterdquoin Proceedings of the 2011 International Symposium on IntelligentSignal Processing and Communications Systems (ISPACS rsquo11) pp1ndash6 Chiang Mai Thailand December 2011

[10] P M Djuric and S M Kay ldquoParameter estimation of chirpsignalsrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 38 no 12 pp 2118ndash2126 1990

[11] H Fu and P-Y Kam ldquoPhase-based time-domain estimation ofthe frequency and phase of a single sinusoid in AWGNmdashtherole and applications of the additive observation phase noisemodelrdquo IEEE Transactions on Information Theory vol 59 no5 pp 3175ndash3188 2013

[12] Y Li H Fu and P Y Kam ldquoImproved approximate time-domain ML estimators of chirp signal parameters and theirperformance analysisrdquo IEEE Transactions on Signal Processingvol 57 no 4 pp 1260ndash1272 2009

[13] Y Li and P Y Kam ldquoImproved chirp parameter estimationusing signal recovery method approximate time-domain MLestimators of signal s and their performance analysisrdquo inProceedings of the IEEE 71st Vehicular Technology Conferencepp 1ndash5 Taipei Taiwan May 2010

[14] V Cherkassky and Y Ma ldquoSelection of meta-parameters forsupport vector regressionrdquo in Proceedings of the InternationalConference on Artificial Neural Networks pp 687ndash693 MadridSpain August 2002

[15] V Cherkassky X Shao F M Mulier and V N Vapnik ldquoModelcomplexity control for regression using VC generalizationboundsrdquo IEEE Transactions on Neural Networks vol 10 no 5pp 1075ndash1089 1999

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

2 The Proposed Algorithm

21 Signal Model The signal model used here is similar tothat in [10 12] The complex baseband chirp signal pollutedby noise is modelled as

119903119899= 119886119899119860 exp 119895 [120601 + 2120587119891

119889119899119879119904+ 2120587119891

119903(119899119879119904)2

] + 119908119899

119899 = 0 119873 minus 1

(1)

Here 119886119899is an independent symbol 119860 gt 0 119891

119889 119891119903isin

[minus05 05) and 120601 isin [minus120587 120587) are the amplitude deterministicbut unknown IF IFR and initial phase respectively 119879

119904is the

sampling period 119873 is the sample size 119908119899is an independent

complex additive white Gaussian noise (AWGN) with zeromean and variance 1205902 For the sake of simplicity we set119879

119904= 1

and rearrange (1)

119903119899= exp [119895 (120601 + 2120587119891

119889119899 + 2120587119891

1199031198992

)]

times 119860 + 119886lowast

119899exp [minus119895 (120601 + 2120587119891

119889119899 + 2120587119891

1199031198992

)]119908119899

= exp [119895 (120601 + 2120587119891119889119899 + 2120587119891

1199031198992

)] (119860 + 1199081015840

119899)

119899 = 0 119873 minus 1

(2)

where 1199081015840119899= 119886lowast

119899exp[minus119895(120601 + 2120587119891

119889119899 + 2120587119891

1199031198992

)]119908119899is still an

independent complex AWGN with zero mean and variance1205902 Absolute phase of 119903

119899is presented as

ang119903119899= 120601 + 2120587119891

119889119899 + 2120587119891

1199031198992

+ang (119860 + 1199081015840

119899) 119899 = 0 119873 minus 1

(3)

When SNR = 1198602

1205902 is large enough the approximate

model in [10] is expressed as

ang (119860 + 1199081015840

119899) asymp arctan

Im (1199081015840

119899)

119860 + Re (1199081015840119899)

asymp arctanIm (119908

1015840

119899)

119860

asymp

Im (1199081015840

119899)

119860

119899 = 0 119873 minus 1

(4)

A more reasonable model proposed in [12] is given

ang (119860 + 1199081015840

119899) asymp arcsin

Im (1199081015840

119899)

1003816100381610038161003816119903119899

1003816100381610038161003816

asymp

Im (1199081015840

119899)

1003816100381610038161003816119903119899

1003816100381610038161003816

119899 = 0 119873 minus 1

(5)

22 SVR-Based Phase Unwrapping Process and FrequencyEstimation Algorithm In (5) there are two approximationsstill existing Based on the quadratic relation in (3) thispaper utilizes SVRrsquos excellent capability for learning unknownmodels to unwrap phases and estimate frequency We yield a

training set 119878119896= (119909

119896

119894 119910119896

119894) | 119894 = 1 119896 119909

119896

119894= 119894 minus 1 119910

119896

119894=

ang119903119894minus1

at time point 119896 (1 le 119896 le 119873) where ang119903119894minus1

denotes theestimation value of ang119903

119894minus1

At first we present a line119891119896(119909) = (w

119896sdot120601(119909)) + 119887

119896 where (sdot)

is an inner product operator and 120601(sdot) is a nonlinear mappingfrom low to high dimension feature space And also we defineits 120576-insensitive loss function as

119871 (119910119896

119894 119891119896(119909119896

119894)) =

10038161003816100381610038161003816119910119896

119894minus 119891119896(119909119896

119894)

10038161003816100381610038161003816120576

=

0

10038161003816100381610038161003816119910119896

119894minus 119891119896(119909119896

119894)

10038161003816100381610038161003816le 120576119896

10038161003816100381610038161003816119910119896

119894minus 119891119896(119909119896

119894)

10038161003816100381610038161003816minus 120576119896 else

(6)

where 120576119896is insensitive loss coefficient at time point 119896

Next we assume 119891119896(119909) insofar as for 120576

119896to completely fit

all elements of 119878119896 andwe denote119889119896

119894as the distance frompoint

(119909119896

119894 119910119896

119894) isin 119878119896to 119891119896(119909)

119889119896

119894=

10038161003816100381610038161003816(w119896sdot 120601 (119909119896

119894)) + 119887

119896minus 119910119896

119894

10038161003816100381610038161003816

radic1 +1003817100381710038171003817w119896

1003817100381710038171003817

2

le

120576119896

radic1 +1003817100381710038171003817w119896

1003817100381710038171003817

2

119894 = 1 119896

(7)

According to (7) we optimize 119891119896(119909) through maximizing

120576119896radic1 + w

1198962 that is minimizing w

1198962 Thereby SVR is

presented as

min 119869 (w119896 119887119896) =

1

2

1003817100381710038171003817w119896

1003817100381710038171003817

2

st 10038161003816100381610038161003816(w119896sdot 120601 (119909119896

119894)) + 119887

119896minus 119910119896

119894

10038161003816100381610038161003816le 120576119896 119894 = 1 119896

(8)

In fact fitting errors larger than 120576119896always exist By

introducing slack variables 120585119896119894 (120585lowast

)119896

119894ge 0 119894 = 1 119896 and

penalty factor 119862119896at time point 119896 (8) is converted into

min 119869 (w119896 119887119896 120585119896 120585lowast

119896) =

1

2

1003817100381710038171003817w119896

1003817100381710038171003817

2

+ 119862119896

119896

sum

119894=1

[120585119896

119894+ (120585lowast

)119896

119894]

st 119910119896

119894minus (w119896sdot 120601 (119909119896

119894)) minus 119887

119896le 120576119896+ 120585119896

119894

(w119896sdot 120601 (119909119896

119894)) + 119887

119896minus 119910119896

119894le 120576119896+ (120585lowast

)119896

119894

119894 = 1 119896

(9)

where 120585119896= [120585119896

1 120585119896

2 120585

119896

119896]

119879

120585lowast

119896= [(120585lowast

)119896

1 (120585lowast

)119896

2 (120585

lowast

)119896

119896]

119879

[sdot]119879 is a transpose operator and 119862 is a positive constant

to take compromise in SVRrsquos generalization capability andfitting errors which are denoted by the first and second itemsof 119869(w

119896 119887119896) respectively

Mathematical Problems in Engineering 3

Initialization

Set construction

Phase prediction

SVR

Phase unwrapping

Ultimate values

At time point k = 0 angr0 =

arg(r0) = arg[ [120601 + ang(1 + w0)

At time point 1 le k le N

xki = i minus 1 y

ki = angriminus1

Select a proper to satisfyangrk = arg(rk) + 2120587

120587

m isin

m isin

[angSVR rk minus angSVR rk + 120587)

At time point k = N

ang rk = [ [120572ki minus (120572

lowast)k

i xki )) 2

+ bk

=sum

Ni=1[120572Ni minus (120572lowast)

N

i]xki

120587fNd

fNr =

sumNi=1[ [120572

Ni minus (120572lowast)

N

ixki ))

2

2120587

minus (120572lowast)N

i

=

N

sumi=1

[

k

sumi=1

= [ [120572ki minus (120572

lowast)k

i xki x 1)) 2

+ bk

k

sumi=1

120572Ni ] + bN120601

N

Sk = xki y

ki )) |i = 1 middot middot middot k

Approximated function

Make use of SVR and derive

fk(x) = (wk middot 120601(x)) + bk

+

k 1+

k = N

k lt N

Figure 1 Recursive implementation of SVR-based algorithm

Equation (9) is a strict convex quadratic programming(QP) problem in optimal theories Then using Lagrangemultiplier method

w119896=

119896

sum

119894=1

[120572119896

119894minus (120572lowast

)119896

119894] 120601 (119909

119896

119894)

119896

sum

119894=1

[(120572lowast

)119896

119894minus 120572119896

119894] = 0

119862119896minus 120572119896

119894minus 120574119896

119894= 0 119894 = 1 119896

119862119896minus (120572lowast

)119896

119894minus (120574lowast

)119896

119894= 0 119894 = 1 119896

(10)

where 120572119896119894 (120572lowast

)119896

119894 120574119896

119894 and (120574lowast)119896

119894are the 119894th Lagrange multipli-

ers of 119910119896119894minus (w119896sdot 120601(119909119896

119894)) minus 119887119896le 120576119896+ 120585119896

119894 (w119896sdot 120601(119909119896

119894)) + 119887119896minus 119910119896

119894le

120576119896+ (120585lowast

)119896

119894 120585119894ge 0 and (120585lowast)119896

119894ge 0 respectively

Substituting (10) into (9) replacing (120601(119909119896119894) sdot 120601(119909

119896

119895)) with

quadratic kernel function 119870(119909119894 119909119895) = (119909

lowast

119894119909119895+ 1)2 in this

study and deriving the wolf dual problem of (9)

max119882(120572119896120572lowast

119896)

= minus

1

2

119896

sum

119894119895=1

[120572119896

119894minus (120572lowast

)119896

119894] [120572119896

119895minus (120572lowast

)119896

119895] (119909lowast

119894119909119895+ 1)

2

+

119896

sum

119894=1

119910119896

119894[120572119896

119894minus (120572lowast

)119896

119894] minus 120576119896

119896

sum

119894=1

[120572119896

119894+ (120572lowast

)119896

119894]

st119896

sum

119894=1

[120572119896

119894minus (120572lowast

)119896

119894] = 0

0 le 120572119896

119894 (120572

lowast

)119896

119894le 119862119896 119894 = 1 119896

(11)

where 120572119896= [120572119896

1 120572119896

2 120572

119896

119896]

119879

120572lowast119896= [(120572lowast

)119896

1 (120572lowast

)119896

2 (120572

lowast

)119896

119896]

119879

4 Mathematical Problems in Engineering

0 5 10 15 20 25 300

50

100

150

200

250

300

350Ph

ase

k

(a)

0 5 10 15 20 25 30

0

100

200

300

400

500

600

700

Phas

e

minus100

k

(b)

Without noiseActualSVR

DKLFK

Phas

e

0 5 10 15 20 25 30

0

200

400

600

800

1000

1200

1400

minus200

k

(c)

Figure 2 (a) Arbitrary phase unwrapping processes with119891119889= 01 119891

119903= 005 120601 = 0119873 = 32 and SNR = 8 dB (b)Arbitrary phase unwrapping

processes with 119891119889= 03 119891

119903= 005 120601 = 0119873 = 32 and SNR = 8 dB (c) Arbitrary phase unwrapping processes with 119891

119889= 01 119891

119903= 02 120601 =

0119873 = 32 and SNR = 8 dB

We obtain 120572119896 120572lowast119896through solving (11) Ultimately we get

the best approximation of (3) by using Karush-Kuhn-Tucker(KKT) conditions

119891119896(119909) = (w

119896sdot 120601 (119909)) + 119887

119896

=

119896

sum

119894=1

[120572119896

119894minus (120572lowast

)119896

119894] (119909lowast

119894119909119895+ 1)

2

+ 119887119896

(12)

where 119887119896= 119910119896

119894minussum119896

119895=1[120572119896

119895minus(120572lowast

)119896

119895](119909lowast

119894119909119895+ 1)2

minus120576119896 120572119896

119894isin (0 119862

119896)

or 119887119896= 119910119896

119894minussum119896

119895=1[120572119896

119895minus (120572lowast

)119896

119895](119909lowast

119894119909119895+ 1)2

+ 120576119896 (120572lowast

)119896

119894isin (0 119862

119896)

is bias at time point 119896Not only does (6) guarantee less fitting errors but it also

possesses a good generalization capability It means that after

learning previous unwrapped phases ang1199030 ang119903

119896minus1 (6) can

predict the variation trend of a phase efficiently for the nexttime point 119896 and deriveangSVR119903119896 = sum

119896

119894=1[120572119896

119894minus(120572lowast

)119896

119894](119909lowast

119894119896 + 1)

2

+

119887119896 The phase ang119903

119896is then unwrapped within a 2120587 interval

centered around angSVR119903119896 by adding multiples of plusmn2120587 to theprincipal value of ang119903

119896when the absolute difference between

angSVR119903119896 and the principal value of ang119903119896is greater than 120587

Namely we select a proper119898 to satisfy ang119903119896= arg(119903

119896)+2120587119898 isin

[angSVR119903119896 minus 120587 angSVR119903119896 + 120587) Till time point 119896 = 119873 we get 119891119873119889=

(sum119873

119894=1[120572119873

119894minus(120572lowast

)119873

119894]119909119873

119894)120587

119891119873

119903= (sum119873

119894=1[120572119873

119894minus(120572lowast

)119873

119894](119909119873

119894)

2

)2120587and

120601119873

= sum119873

119894=1[120572119873

119894minus (120572lowast

)119873

119894] + 119887119873as the ultimate estimation

values This algorithm can be implemented recursively intime as shown schematically in Figure 1

Mathematical Problems in Engineering 5

0 5 10

MSE

SNR (dB)minus5minus10

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

(a)

0 5 10 15 20SNR (dB)

MSE

10minus1

100

10minus2

10minus4

10minus5

10minus6

10minus7

10minus3

(b)

0 5 10 15 20SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus3

(c)

Figure 3 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 0 and119873 = 32 (b) MSE of IF with 119891

119889= 03 119891

119903= 005 120601 = 0 and119873 = 32 (c) MSE of

IF with 119891119889= 01 119891

119903= 02 120601 = 0 and119873 = 32

23 SVRrsquos Parameter Settings Setting SVRrsquos parameters is adifficult problem but has a pronounced impact on SVRrsquos per-formance for example insensitive loss coefficient 120576 penaltyfactor 119862 There is no complete theoretical basis or explicitclosed form Cross-validation is a widely used method untilnow but it is complex and time consuming In this study weformulate proper parameter values by understanding SVRrsquostheory and integrating a large number of references andexperiments

When 119896 is small imperfect phase unwrapping at aparticular time point can easily have an impact on thevariation trend of 119891(119909) In order to avoid this discrepancya new model must be constructed with the aptitude ofbetter generalization capability As 119896 increases its necessitydecreases contrarily which is due to the degrading impact of

improperly unwrapped phase Simple speaking the modelrsquosgeneralization capability is inversely proportional to the sizeof the set 119878

Intuitively insensitive loss coefficient 120576 is the verticalheight of 120576-tube The larger 120576 is the less support vectorsthere are Nevertheless too large 120576 will cause unfixable 119887Noted that if the predicted phase is in the vicinity of 120587rsquosodd times the estimation performance is deteriorated rapidlyfor its ambiguous phase characteristic Selecting a proper120576 can reduce this impact At the same time 120576 is inverselyproportional to SNR so insensitive loss coefficient at timepoint 119896 (1 le 119896 le 119873) is given by [14]

120576119896= 120591radic

1

SNRln 119896119896

119896 = 1 119873 (13)

6 Mathematical Problems in Engineering

0 2 4 6 8 10 12 14SNR (dB)

minus2minus4minus6

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus3

(a)

SNR (dB)0 2 4 6 8 10 12 14 16 18 20

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus3

(b)

0 5 10 15 20SNR (dB)

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus3

SVRHPFDK

LFKCRLB

(c)

Figure 4 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 0 and119873 = 32 (b) MSE of IFR with 119891

119889= 03 119891

119903= 005 120601 = 0 and119873 = 32 (c) MSE

of IFR with 119891119889= 01 119891

119903= 02 120601 = 0 and119873 = 32

where 120591 is a positive constant and 120591 = 08 is set in this studyafter experimental comparisons and validations and SNR isassumed to be known

Penalty factor 119862 controls the penalty degree of vectorsoutside the 120576-tube and determines SVRrsquos generalization capa-bility 119862 is directly proportional to the sample size and SNRAs (12) is a line in a two-dimension plane the first item oftarget function during solving SVRrsquos QP problem is directlyproportional to its slope so a very small 119862 will result in ahorizontal and underfitting 119862 Inspired by [14 15] penaltyfactor at time point 119896 (1 le 119896 le 119873) is given by

119862119896= 120575radicSNR 3radic119896max (100381610038161003816

1003816119892119896minus 120582120590119896

10038161003816100381610038161003816100381610038161003816119892119896+ 120582120590119896

1003816100381610038161003816)

119896 = 1 119873

(14)

where 119892119896= (1119896)sum

119896minus1

119899=0|119910119899|2

120590119896= radic(1119896)sum

119896minus1

119899=0(|119910119899|2

minus 119892119896)

2119910119899= ang119903119899minus1

is a section of the element of training set 119878119896 as

(13) SNR is assumed to be known 120575 120582 are positive constantsand 120575 = 01 120582 = 05 are set in this study by the same way as120591

3 Results and Analyses

We have compared the proposed algorithm entitled as SVRestimator with the other three the HPF estimator proposedin [6] the DK estimator proposed in [10] the LFK estimatorproposed in [12]

31 Phase Unwrapping Process Because HPF estimator doesnot need the phase unwrapping process Figure 2 illustrates

Mathematical Problems in Engineering 7M

SE 10minus4

10minus5

10minus6

10minus7

10minus3

10minus2

10minus1

SNR (dB)0 5 10minus5minus10

N = 8 N = 32

N = 64N = 16

Figure 5 MSE of IF of SVR estimator with 119891119889= 01 119891

119903= 005 120601 =

0

0 5 10SNR (dB)

minus5minus10

MSE

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus10

N = 8 N = 32

N = 64N = 16

Figure 6MSE of IFR of SVR estimator with119891119889= 01 119891

119903= 005 120601 =

0

the arbitrary phase unwrapping processes of the other threewhile 119891

119889= 01 119891

119903= 005 119891

119889= 03 119891

119903= 005 and 119891

119889=

01 119891119903= 02 The number of Monte Carlo experiments is

10000 and 120601 = 0 119873 = 32 and SNR = 8 dB It is shownthat SVR estimator can unwrap phase accurately whateverthe value of 119891

119889and 119891

119903is However errors emerge in DK and

LFK ones as 119891119889and 119891

119903increase

32 Estimation Performance Figures 2 3 and 4 illustrate theMSE curves of IF and IFR versus SNR respectively whereMSEs are defined as 119864[( 119891

119889minus 119891119889)

2

] 119864[(119891119903minus 119891119903)

2

] 119891119889119891119903are

the estimation values of 119891119889 119891119903 CRLBs are given by [12]

CRLB119891119889=

1

SNR6 (8119873 minus 11) (2119873 minus 1)

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB119891119903=

1

SNR360

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB120601=

1

SNR3 (3119873

2

minus 3119873 + 2)

2119873 (1198732+ 3119873 + 2)

(15)

It is shown that whether 119891119889= 01 119891

119903= 005 119891

119889=

03 119891119903= 005 or 119891

119889= 01 119891

119903= 02 SVR estimator is the

best one during both IF and IFR estimations all the whilealso MSE performances of IF and IFR totally decrease as 119891

119889

and119891119903increase but SVR estimator is the most robust and has

had a great advantage already when 119891119889= 03 119891

119903= 005 or

119891119889= 01 119891

119903= 02

33 Impact of the Sample Size119873 Everything is as in Figure 2other than 119891

119889= 01 119891

119903= 005 the MSE curves of IF and IFR

of SVR estimator versus SNR while119873 is 8 16 32 and 64 areplotted in Figures 5 and 6 respectively It is clear that as 119873increases MSE performances of IF and IFR of SVR estimatorare both improved

34 Impact of Initial Phase 120601 Everything is as in Figure 3except that 120601 = 04120587 08120587 the MSE curves of IF and IFRare plotted against SNR in Figures 7 and 8 respectivelyComparing with Figures 3(a) and 4(a) we can see that SVRestimator is immune to 120601 but the other three are not

35 Computational Complexity Because we translate SVRinto QP problem and need to search the minimums duringthe process we can not derive the explicit form of thecomputational complexity of SVR estimator So everythingis as in Figure 2 except that the number of Monte Carloexperiments is 100 SNR is 8 dB and 119891

119889= 01 119891

119903= 005

the consuming times are listed in Table 1 while119873 is 8 16 32and 64 respectively The running computer is ASUS-PC1111having Intel(R) Pentium 213GHz CPU and 200GB RAM

We can see that SVR estimatorrsquos consuming times aremore than the othersrsquo especially when 119873 becomes larger Asa matter of fact however it is acceptable and tolerable

4 Conclusions

Phase unwrapping process is a key point in phase-basedfrequency estimation of chirp signal Firstly we adopt SVRto learn the unwrapped phases at previous time pointspredict the variation trend of phase efficiently and derive theestimation value for the next time point Once acquired interms of relationship between absolute signal phase and timeseries we address a simple and effective frequency estimationalgorithmof chirp signalTheproposed algorithm completelyexhibits its advantages of higher estimation accuracy lowersensitivity of frequency and initial phase by sacrificing moreconsuming times

Because SVR predicts the curversquos variation trend merelyin terms of training set which consists of previous pointsrsquo

8 Mathematical Problems in Engineering

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

0 5 10SNR (dB)

minus5minus10

(a)

0 2 4 6 8 10

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

SNR (dB)minus6 minus4 minus2minus8minus10

(b)

Figure 7 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IF with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

0 2 4 6 8 10 12 14SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

minus6 minus4 minus2

(a)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

0 2 4 6 8 10 12 14SNR (dB)

minus6 minus4 minus2

(b)

Figure 8 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IFR with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

Table 1 Consuming time with different119873 (ms)

Algorithm 119873 = 8 119873 = 16 119873 = 32 119873 = 64

HPF 380 942 3721 10654DK 203 476 1098 3711LFK 299 784 2362 8643SVR 437 1107 4805 13074

values we even can estimate the frequency of chirp signalunder the nonGaussian condition by the same way

Stressing that the proposed algorithm learns training setand gets the approximate values of SVRrsquos insensitive losscoefficient 120576 and penalty factor 119862 as a next step thereforeimproving SVRrsquos parameter setting is an important researchpoint

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China (Grant no 61201380)

Mathematical Problems in Engineering 9

References

[1] J C Curlander andRNMcDonough Synthetic Aperture RadarSystems and Signal Processing Wiley New York NY USA 1992

[2] J Xu Y Peng and X Xia ldquoParametric autofocus of SARimagingmdashinherent accuracy limitations and realizationrdquo IEEETransactions on Geoscience and Remote Sensing vol 42 no 11pp 2397ndash2411 2004

[3] R Ramaswami K N Sivarajan and G H Sasaki OpticalNetworks A Practical Perspective Morgan Kaufmann SanMateo Calif USA 1998

[4] P OrsquoShea ldquoA new technique for instantaneous frequency rateestimationrdquo IEEE Signal Processing Letters vol 9 no 8 pp 251ndash252 2002

[5] PWang I Djurovic and J Yang ldquoGeneralized high-order phasefunction for parameter estimation of polynomial phase signalrdquoIEEE Transactions on Signal Processing vol 56 no 7 pp 3023ndash3028 2008

[6] P Wang H Li I Djurovic and B Himed ldquoPerformance ofinstantaneous frequency rate estimation using high-order phasefunctionrdquo IEEE Transactions on Signal Processing vol 58 no 4pp 2415ndash2421 2010

[7] M Benidir and A Ouldali ldquoPolynomial phase signal analysisbased on the polynomial derivatives decompositionsrdquo IEEETransactions on Signal Processing vol 47 no 7 pp 1954ndash19651999

[8] P OrsquoShea ldquoA fast algorithm for estimating the parameters of aquadratic FM signalrdquo IEEE Transactions on Signal Processingvol 52 no 2 pp 385ndash393 2004

[9] S Ma J Jiang and Q Meng ldquoA fast accurate and robustmethod for joint estimation of frequency and frequency raterdquoin Proceedings of the 2011 International Symposium on IntelligentSignal Processing and Communications Systems (ISPACS rsquo11) pp1ndash6 Chiang Mai Thailand December 2011

[10] P M Djuric and S M Kay ldquoParameter estimation of chirpsignalsrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 38 no 12 pp 2118ndash2126 1990

[11] H Fu and P-Y Kam ldquoPhase-based time-domain estimation ofthe frequency and phase of a single sinusoid in AWGNmdashtherole and applications of the additive observation phase noisemodelrdquo IEEE Transactions on Information Theory vol 59 no5 pp 3175ndash3188 2013

[12] Y Li H Fu and P Y Kam ldquoImproved approximate time-domain ML estimators of chirp signal parameters and theirperformance analysisrdquo IEEE Transactions on Signal Processingvol 57 no 4 pp 1260ndash1272 2009

[13] Y Li and P Y Kam ldquoImproved chirp parameter estimationusing signal recovery method approximate time-domain MLestimators of signal s and their performance analysisrdquo inProceedings of the IEEE 71st Vehicular Technology Conferencepp 1ndash5 Taipei Taiwan May 2010

[14] V Cherkassky and Y Ma ldquoSelection of meta-parameters forsupport vector regressionrdquo in Proceedings of the InternationalConference on Artificial Neural Networks pp 687ndash693 MadridSpain August 2002

[15] V Cherkassky X Shao F M Mulier and V N Vapnik ldquoModelcomplexity control for regression using VC generalizationboundsrdquo IEEE Transactions on Neural Networks vol 10 no 5pp 1075ndash1089 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Initialization

Set construction

Phase prediction

SVR

Phase unwrapping

Ultimate values

At time point k = 0 angr0 =

arg(r0) = arg[ [120601 + ang(1 + w0)

At time point 1 le k le N

xki = i minus 1 y

ki = angriminus1

Select a proper to satisfyangrk = arg(rk) + 2120587

120587

m isin

m isin

[angSVR rk minus angSVR rk + 120587)

At time point k = N

ang rk = [ [120572ki minus (120572

lowast)k

i xki )) 2

+ bk

=sum

Ni=1[120572Ni minus (120572lowast)

N

i]xki

120587fNd

fNr =

sumNi=1[ [120572

Ni minus (120572lowast)

N

ixki ))

2

2120587

minus (120572lowast)N

i

=

N

sumi=1

[

k

sumi=1

= [ [120572ki minus (120572

lowast)k

i xki x 1)) 2

+ bk

k

sumi=1

120572Ni ] + bN120601

N

Sk = xki y

ki )) |i = 1 middot middot middot k

Approximated function

Make use of SVR and derive

fk(x) = (wk middot 120601(x)) + bk

+

k 1+

k = N

k lt N

Figure 1 Recursive implementation of SVR-based algorithm

Equation (9) is a strict convex quadratic programming(QP) problem in optimal theories Then using Lagrangemultiplier method

w119896=

119896

sum

119894=1

[120572119896

119894minus (120572lowast

)119896

119894] 120601 (119909

119896

119894)

119896

sum

119894=1

[(120572lowast

)119896

119894minus 120572119896

119894] = 0

119862119896minus 120572119896

119894minus 120574119896

119894= 0 119894 = 1 119896

119862119896minus (120572lowast

)119896

119894minus (120574lowast

)119896

119894= 0 119894 = 1 119896

(10)

where 120572119896119894 (120572lowast

)119896

119894 120574119896

119894 and (120574lowast)119896

119894are the 119894th Lagrange multipli-

ers of 119910119896119894minus (w119896sdot 120601(119909119896

119894)) minus 119887119896le 120576119896+ 120585119896

119894 (w119896sdot 120601(119909119896

119894)) + 119887119896minus 119910119896

119894le

120576119896+ (120585lowast

)119896

119894 120585119894ge 0 and (120585lowast)119896

119894ge 0 respectively

Substituting (10) into (9) replacing (120601(119909119896119894) sdot 120601(119909

119896

119895)) with

quadratic kernel function 119870(119909119894 119909119895) = (119909

lowast

119894119909119895+ 1)2 in this

study and deriving the wolf dual problem of (9)

max119882(120572119896120572lowast

119896)

= minus

1

2

119896

sum

119894119895=1

[120572119896

119894minus (120572lowast

)119896

119894] [120572119896

119895minus (120572lowast

)119896

119895] (119909lowast

119894119909119895+ 1)

2

+

119896

sum

119894=1

119910119896

119894[120572119896

119894minus (120572lowast

)119896

119894] minus 120576119896

119896

sum

119894=1

[120572119896

119894+ (120572lowast

)119896

119894]

st119896

sum

119894=1

[120572119896

119894minus (120572lowast

)119896

119894] = 0

0 le 120572119896

119894 (120572

lowast

)119896

119894le 119862119896 119894 = 1 119896

(11)

where 120572119896= [120572119896

1 120572119896

2 120572

119896

119896]

119879

120572lowast119896= [(120572lowast

)119896

1 (120572lowast

)119896

2 (120572

lowast

)119896

119896]

119879

4 Mathematical Problems in Engineering

0 5 10 15 20 25 300

50

100

150

200

250

300

350Ph

ase

k

(a)

0 5 10 15 20 25 30

0

100

200

300

400

500

600

700

Phas

e

minus100

k

(b)

Without noiseActualSVR

DKLFK

Phas

e

0 5 10 15 20 25 30

0

200

400

600

800

1000

1200

1400

minus200

k

(c)

Figure 2 (a) Arbitrary phase unwrapping processes with119891119889= 01 119891

119903= 005 120601 = 0119873 = 32 and SNR = 8 dB (b)Arbitrary phase unwrapping

processes with 119891119889= 03 119891

119903= 005 120601 = 0119873 = 32 and SNR = 8 dB (c) Arbitrary phase unwrapping processes with 119891

119889= 01 119891

119903= 02 120601 =

0119873 = 32 and SNR = 8 dB

We obtain 120572119896 120572lowast119896through solving (11) Ultimately we get

the best approximation of (3) by using Karush-Kuhn-Tucker(KKT) conditions

119891119896(119909) = (w

119896sdot 120601 (119909)) + 119887

119896

=

119896

sum

119894=1

[120572119896

119894minus (120572lowast

)119896

119894] (119909lowast

119894119909119895+ 1)

2

+ 119887119896

(12)

where 119887119896= 119910119896

119894minussum119896

119895=1[120572119896

119895minus(120572lowast

)119896

119895](119909lowast

119894119909119895+ 1)2

minus120576119896 120572119896

119894isin (0 119862

119896)

or 119887119896= 119910119896

119894minussum119896

119895=1[120572119896

119895minus (120572lowast

)119896

119895](119909lowast

119894119909119895+ 1)2

+ 120576119896 (120572lowast

)119896

119894isin (0 119862

119896)

is bias at time point 119896Not only does (6) guarantee less fitting errors but it also

possesses a good generalization capability It means that after

learning previous unwrapped phases ang1199030 ang119903

119896minus1 (6) can

predict the variation trend of a phase efficiently for the nexttime point 119896 and deriveangSVR119903119896 = sum

119896

119894=1[120572119896

119894minus(120572lowast

)119896

119894](119909lowast

119894119896 + 1)

2

+

119887119896 The phase ang119903

119896is then unwrapped within a 2120587 interval

centered around angSVR119903119896 by adding multiples of plusmn2120587 to theprincipal value of ang119903

119896when the absolute difference between

angSVR119903119896 and the principal value of ang119903119896is greater than 120587

Namely we select a proper119898 to satisfy ang119903119896= arg(119903

119896)+2120587119898 isin

[angSVR119903119896 minus 120587 angSVR119903119896 + 120587) Till time point 119896 = 119873 we get 119891119873119889=

(sum119873

119894=1[120572119873

119894minus(120572lowast

)119873

119894]119909119873

119894)120587

119891119873

119903= (sum119873

119894=1[120572119873

119894minus(120572lowast

)119873

119894](119909119873

119894)

2

)2120587and

120601119873

= sum119873

119894=1[120572119873

119894minus (120572lowast

)119873

119894] + 119887119873as the ultimate estimation

values This algorithm can be implemented recursively intime as shown schematically in Figure 1

Mathematical Problems in Engineering 5

0 5 10

MSE

SNR (dB)minus5minus10

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

(a)

0 5 10 15 20SNR (dB)

MSE

10minus1

100

10minus2

10minus4

10minus5

10minus6

10minus7

10minus3

(b)

0 5 10 15 20SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus3

(c)

Figure 3 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 0 and119873 = 32 (b) MSE of IF with 119891

119889= 03 119891

119903= 005 120601 = 0 and119873 = 32 (c) MSE of

IF with 119891119889= 01 119891

119903= 02 120601 = 0 and119873 = 32

23 SVRrsquos Parameter Settings Setting SVRrsquos parameters is adifficult problem but has a pronounced impact on SVRrsquos per-formance for example insensitive loss coefficient 120576 penaltyfactor 119862 There is no complete theoretical basis or explicitclosed form Cross-validation is a widely used method untilnow but it is complex and time consuming In this study weformulate proper parameter values by understanding SVRrsquostheory and integrating a large number of references andexperiments

When 119896 is small imperfect phase unwrapping at aparticular time point can easily have an impact on thevariation trend of 119891(119909) In order to avoid this discrepancya new model must be constructed with the aptitude ofbetter generalization capability As 119896 increases its necessitydecreases contrarily which is due to the degrading impact of

improperly unwrapped phase Simple speaking the modelrsquosgeneralization capability is inversely proportional to the sizeof the set 119878

Intuitively insensitive loss coefficient 120576 is the verticalheight of 120576-tube The larger 120576 is the less support vectorsthere are Nevertheless too large 120576 will cause unfixable 119887Noted that if the predicted phase is in the vicinity of 120587rsquosodd times the estimation performance is deteriorated rapidlyfor its ambiguous phase characteristic Selecting a proper120576 can reduce this impact At the same time 120576 is inverselyproportional to SNR so insensitive loss coefficient at timepoint 119896 (1 le 119896 le 119873) is given by [14]

120576119896= 120591radic

1

SNRln 119896119896

119896 = 1 119873 (13)

6 Mathematical Problems in Engineering

0 2 4 6 8 10 12 14SNR (dB)

minus2minus4minus6

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus3

(a)

SNR (dB)0 2 4 6 8 10 12 14 16 18 20

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus3

(b)

0 5 10 15 20SNR (dB)

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus3

SVRHPFDK

LFKCRLB

(c)

Figure 4 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 0 and119873 = 32 (b) MSE of IFR with 119891

119889= 03 119891

119903= 005 120601 = 0 and119873 = 32 (c) MSE

of IFR with 119891119889= 01 119891

119903= 02 120601 = 0 and119873 = 32

where 120591 is a positive constant and 120591 = 08 is set in this studyafter experimental comparisons and validations and SNR isassumed to be known

Penalty factor 119862 controls the penalty degree of vectorsoutside the 120576-tube and determines SVRrsquos generalization capa-bility 119862 is directly proportional to the sample size and SNRAs (12) is a line in a two-dimension plane the first item oftarget function during solving SVRrsquos QP problem is directlyproportional to its slope so a very small 119862 will result in ahorizontal and underfitting 119862 Inspired by [14 15] penaltyfactor at time point 119896 (1 le 119896 le 119873) is given by

119862119896= 120575radicSNR 3radic119896max (100381610038161003816

1003816119892119896minus 120582120590119896

10038161003816100381610038161003816100381610038161003816119892119896+ 120582120590119896

1003816100381610038161003816)

119896 = 1 119873

(14)

where 119892119896= (1119896)sum

119896minus1

119899=0|119910119899|2

120590119896= radic(1119896)sum

119896minus1

119899=0(|119910119899|2

minus 119892119896)

2119910119899= ang119903119899minus1

is a section of the element of training set 119878119896 as

(13) SNR is assumed to be known 120575 120582 are positive constantsand 120575 = 01 120582 = 05 are set in this study by the same way as120591

3 Results and Analyses

We have compared the proposed algorithm entitled as SVRestimator with the other three the HPF estimator proposedin [6] the DK estimator proposed in [10] the LFK estimatorproposed in [12]

31 Phase Unwrapping Process Because HPF estimator doesnot need the phase unwrapping process Figure 2 illustrates

Mathematical Problems in Engineering 7M

SE 10minus4

10minus5

10minus6

10minus7

10minus3

10minus2

10minus1

SNR (dB)0 5 10minus5minus10

N = 8 N = 32

N = 64N = 16

Figure 5 MSE of IF of SVR estimator with 119891119889= 01 119891

119903= 005 120601 =

0

0 5 10SNR (dB)

minus5minus10

MSE

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus10

N = 8 N = 32

N = 64N = 16

Figure 6MSE of IFR of SVR estimator with119891119889= 01 119891

119903= 005 120601 =

0

the arbitrary phase unwrapping processes of the other threewhile 119891

119889= 01 119891

119903= 005 119891

119889= 03 119891

119903= 005 and 119891

119889=

01 119891119903= 02 The number of Monte Carlo experiments is

10000 and 120601 = 0 119873 = 32 and SNR = 8 dB It is shownthat SVR estimator can unwrap phase accurately whateverthe value of 119891

119889and 119891

119903is However errors emerge in DK and

LFK ones as 119891119889and 119891

119903increase

32 Estimation Performance Figures 2 3 and 4 illustrate theMSE curves of IF and IFR versus SNR respectively whereMSEs are defined as 119864[( 119891

119889minus 119891119889)

2

] 119864[(119891119903minus 119891119903)

2

] 119891119889119891119903are

the estimation values of 119891119889 119891119903 CRLBs are given by [12]

CRLB119891119889=

1

SNR6 (8119873 minus 11) (2119873 minus 1)

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB119891119903=

1

SNR360

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB120601=

1

SNR3 (3119873

2

minus 3119873 + 2)

2119873 (1198732+ 3119873 + 2)

(15)

It is shown that whether 119891119889= 01 119891

119903= 005 119891

119889=

03 119891119903= 005 or 119891

119889= 01 119891

119903= 02 SVR estimator is the

best one during both IF and IFR estimations all the whilealso MSE performances of IF and IFR totally decrease as 119891

119889

and119891119903increase but SVR estimator is the most robust and has

had a great advantage already when 119891119889= 03 119891

119903= 005 or

119891119889= 01 119891

119903= 02

33 Impact of the Sample Size119873 Everything is as in Figure 2other than 119891

119889= 01 119891

119903= 005 the MSE curves of IF and IFR

of SVR estimator versus SNR while119873 is 8 16 32 and 64 areplotted in Figures 5 and 6 respectively It is clear that as 119873increases MSE performances of IF and IFR of SVR estimatorare both improved

34 Impact of Initial Phase 120601 Everything is as in Figure 3except that 120601 = 04120587 08120587 the MSE curves of IF and IFRare plotted against SNR in Figures 7 and 8 respectivelyComparing with Figures 3(a) and 4(a) we can see that SVRestimator is immune to 120601 but the other three are not

35 Computational Complexity Because we translate SVRinto QP problem and need to search the minimums duringthe process we can not derive the explicit form of thecomputational complexity of SVR estimator So everythingis as in Figure 2 except that the number of Monte Carloexperiments is 100 SNR is 8 dB and 119891

119889= 01 119891

119903= 005

the consuming times are listed in Table 1 while119873 is 8 16 32and 64 respectively The running computer is ASUS-PC1111having Intel(R) Pentium 213GHz CPU and 200GB RAM

We can see that SVR estimatorrsquos consuming times aremore than the othersrsquo especially when 119873 becomes larger Asa matter of fact however it is acceptable and tolerable

4 Conclusions

Phase unwrapping process is a key point in phase-basedfrequency estimation of chirp signal Firstly we adopt SVRto learn the unwrapped phases at previous time pointspredict the variation trend of phase efficiently and derive theestimation value for the next time point Once acquired interms of relationship between absolute signal phase and timeseries we address a simple and effective frequency estimationalgorithmof chirp signalTheproposed algorithm completelyexhibits its advantages of higher estimation accuracy lowersensitivity of frequency and initial phase by sacrificing moreconsuming times

Because SVR predicts the curversquos variation trend merelyin terms of training set which consists of previous pointsrsquo

8 Mathematical Problems in Engineering

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

0 5 10SNR (dB)

minus5minus10

(a)

0 2 4 6 8 10

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

SNR (dB)minus6 minus4 minus2minus8minus10

(b)

Figure 7 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IF with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

0 2 4 6 8 10 12 14SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

minus6 minus4 minus2

(a)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

0 2 4 6 8 10 12 14SNR (dB)

minus6 minus4 minus2

(b)

Figure 8 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IFR with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

Table 1 Consuming time with different119873 (ms)

Algorithm 119873 = 8 119873 = 16 119873 = 32 119873 = 64

HPF 380 942 3721 10654DK 203 476 1098 3711LFK 299 784 2362 8643SVR 437 1107 4805 13074

values we even can estimate the frequency of chirp signalunder the nonGaussian condition by the same way

Stressing that the proposed algorithm learns training setand gets the approximate values of SVRrsquos insensitive losscoefficient 120576 and penalty factor 119862 as a next step thereforeimproving SVRrsquos parameter setting is an important researchpoint

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China (Grant no 61201380)

Mathematical Problems in Engineering 9

References

[1] J C Curlander andRNMcDonough Synthetic Aperture RadarSystems and Signal Processing Wiley New York NY USA 1992

[2] J Xu Y Peng and X Xia ldquoParametric autofocus of SARimagingmdashinherent accuracy limitations and realizationrdquo IEEETransactions on Geoscience and Remote Sensing vol 42 no 11pp 2397ndash2411 2004

[3] R Ramaswami K N Sivarajan and G H Sasaki OpticalNetworks A Practical Perspective Morgan Kaufmann SanMateo Calif USA 1998

[4] P OrsquoShea ldquoA new technique for instantaneous frequency rateestimationrdquo IEEE Signal Processing Letters vol 9 no 8 pp 251ndash252 2002

[5] PWang I Djurovic and J Yang ldquoGeneralized high-order phasefunction for parameter estimation of polynomial phase signalrdquoIEEE Transactions on Signal Processing vol 56 no 7 pp 3023ndash3028 2008

[6] P Wang H Li I Djurovic and B Himed ldquoPerformance ofinstantaneous frequency rate estimation using high-order phasefunctionrdquo IEEE Transactions on Signal Processing vol 58 no 4pp 2415ndash2421 2010

[7] M Benidir and A Ouldali ldquoPolynomial phase signal analysisbased on the polynomial derivatives decompositionsrdquo IEEETransactions on Signal Processing vol 47 no 7 pp 1954ndash19651999

[8] P OrsquoShea ldquoA fast algorithm for estimating the parameters of aquadratic FM signalrdquo IEEE Transactions on Signal Processingvol 52 no 2 pp 385ndash393 2004

[9] S Ma J Jiang and Q Meng ldquoA fast accurate and robustmethod for joint estimation of frequency and frequency raterdquoin Proceedings of the 2011 International Symposium on IntelligentSignal Processing and Communications Systems (ISPACS rsquo11) pp1ndash6 Chiang Mai Thailand December 2011

[10] P M Djuric and S M Kay ldquoParameter estimation of chirpsignalsrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 38 no 12 pp 2118ndash2126 1990

[11] H Fu and P-Y Kam ldquoPhase-based time-domain estimation ofthe frequency and phase of a single sinusoid in AWGNmdashtherole and applications of the additive observation phase noisemodelrdquo IEEE Transactions on Information Theory vol 59 no5 pp 3175ndash3188 2013

[12] Y Li H Fu and P Y Kam ldquoImproved approximate time-domain ML estimators of chirp signal parameters and theirperformance analysisrdquo IEEE Transactions on Signal Processingvol 57 no 4 pp 1260ndash1272 2009

[13] Y Li and P Y Kam ldquoImproved chirp parameter estimationusing signal recovery method approximate time-domain MLestimators of signal s and their performance analysisrdquo inProceedings of the IEEE 71st Vehicular Technology Conferencepp 1ndash5 Taipei Taiwan May 2010

[14] V Cherkassky and Y Ma ldquoSelection of meta-parameters forsupport vector regressionrdquo in Proceedings of the InternationalConference on Artificial Neural Networks pp 687ndash693 MadridSpain August 2002

[15] V Cherkassky X Shao F M Mulier and V N Vapnik ldquoModelcomplexity control for regression using VC generalizationboundsrdquo IEEE Transactions on Neural Networks vol 10 no 5pp 1075ndash1089 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

0 5 10 15 20 25 300

50

100

150

200

250

300

350Ph

ase

k

(a)

0 5 10 15 20 25 30

0

100

200

300

400

500

600

700

Phas

e

minus100

k

(b)

Without noiseActualSVR

DKLFK

Phas

e

0 5 10 15 20 25 30

0

200

400

600

800

1000

1200

1400

minus200

k

(c)

Figure 2 (a) Arbitrary phase unwrapping processes with119891119889= 01 119891

119903= 005 120601 = 0119873 = 32 and SNR = 8 dB (b)Arbitrary phase unwrapping

processes with 119891119889= 03 119891

119903= 005 120601 = 0119873 = 32 and SNR = 8 dB (c) Arbitrary phase unwrapping processes with 119891

119889= 01 119891

119903= 02 120601 =

0119873 = 32 and SNR = 8 dB

We obtain 120572119896 120572lowast119896through solving (11) Ultimately we get

the best approximation of (3) by using Karush-Kuhn-Tucker(KKT) conditions

119891119896(119909) = (w

119896sdot 120601 (119909)) + 119887

119896

=

119896

sum

119894=1

[120572119896

119894minus (120572lowast

)119896

119894] (119909lowast

119894119909119895+ 1)

2

+ 119887119896

(12)

where 119887119896= 119910119896

119894minussum119896

119895=1[120572119896

119895minus(120572lowast

)119896

119895](119909lowast

119894119909119895+ 1)2

minus120576119896 120572119896

119894isin (0 119862

119896)

or 119887119896= 119910119896

119894minussum119896

119895=1[120572119896

119895minus (120572lowast

)119896

119895](119909lowast

119894119909119895+ 1)2

+ 120576119896 (120572lowast

)119896

119894isin (0 119862

119896)

is bias at time point 119896Not only does (6) guarantee less fitting errors but it also

possesses a good generalization capability It means that after

learning previous unwrapped phases ang1199030 ang119903

119896minus1 (6) can

predict the variation trend of a phase efficiently for the nexttime point 119896 and deriveangSVR119903119896 = sum

119896

119894=1[120572119896

119894minus(120572lowast

)119896

119894](119909lowast

119894119896 + 1)

2

+

119887119896 The phase ang119903

119896is then unwrapped within a 2120587 interval

centered around angSVR119903119896 by adding multiples of plusmn2120587 to theprincipal value of ang119903

119896when the absolute difference between

angSVR119903119896 and the principal value of ang119903119896is greater than 120587

Namely we select a proper119898 to satisfy ang119903119896= arg(119903

119896)+2120587119898 isin

[angSVR119903119896 minus 120587 angSVR119903119896 + 120587) Till time point 119896 = 119873 we get 119891119873119889=

(sum119873

119894=1[120572119873

119894minus(120572lowast

)119873

119894]119909119873

119894)120587

119891119873

119903= (sum119873

119894=1[120572119873

119894minus(120572lowast

)119873

119894](119909119873

119894)

2

)2120587and

120601119873

= sum119873

119894=1[120572119873

119894minus (120572lowast

)119873

119894] + 119887119873as the ultimate estimation

values This algorithm can be implemented recursively intime as shown schematically in Figure 1

Mathematical Problems in Engineering 5

0 5 10

MSE

SNR (dB)minus5minus10

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

(a)

0 5 10 15 20SNR (dB)

MSE

10minus1

100

10minus2

10minus4

10minus5

10minus6

10minus7

10minus3

(b)

0 5 10 15 20SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus3

(c)

Figure 3 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 0 and119873 = 32 (b) MSE of IF with 119891

119889= 03 119891

119903= 005 120601 = 0 and119873 = 32 (c) MSE of

IF with 119891119889= 01 119891

119903= 02 120601 = 0 and119873 = 32

23 SVRrsquos Parameter Settings Setting SVRrsquos parameters is adifficult problem but has a pronounced impact on SVRrsquos per-formance for example insensitive loss coefficient 120576 penaltyfactor 119862 There is no complete theoretical basis or explicitclosed form Cross-validation is a widely used method untilnow but it is complex and time consuming In this study weformulate proper parameter values by understanding SVRrsquostheory and integrating a large number of references andexperiments

When 119896 is small imperfect phase unwrapping at aparticular time point can easily have an impact on thevariation trend of 119891(119909) In order to avoid this discrepancya new model must be constructed with the aptitude ofbetter generalization capability As 119896 increases its necessitydecreases contrarily which is due to the degrading impact of

improperly unwrapped phase Simple speaking the modelrsquosgeneralization capability is inversely proportional to the sizeof the set 119878

Intuitively insensitive loss coefficient 120576 is the verticalheight of 120576-tube The larger 120576 is the less support vectorsthere are Nevertheless too large 120576 will cause unfixable 119887Noted that if the predicted phase is in the vicinity of 120587rsquosodd times the estimation performance is deteriorated rapidlyfor its ambiguous phase characteristic Selecting a proper120576 can reduce this impact At the same time 120576 is inverselyproportional to SNR so insensitive loss coefficient at timepoint 119896 (1 le 119896 le 119873) is given by [14]

120576119896= 120591radic

1

SNRln 119896119896

119896 = 1 119873 (13)

6 Mathematical Problems in Engineering

0 2 4 6 8 10 12 14SNR (dB)

minus2minus4minus6

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus3

(a)

SNR (dB)0 2 4 6 8 10 12 14 16 18 20

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus3

(b)

0 5 10 15 20SNR (dB)

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus3

SVRHPFDK

LFKCRLB

(c)

Figure 4 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 0 and119873 = 32 (b) MSE of IFR with 119891

119889= 03 119891

119903= 005 120601 = 0 and119873 = 32 (c) MSE

of IFR with 119891119889= 01 119891

119903= 02 120601 = 0 and119873 = 32

where 120591 is a positive constant and 120591 = 08 is set in this studyafter experimental comparisons and validations and SNR isassumed to be known

Penalty factor 119862 controls the penalty degree of vectorsoutside the 120576-tube and determines SVRrsquos generalization capa-bility 119862 is directly proportional to the sample size and SNRAs (12) is a line in a two-dimension plane the first item oftarget function during solving SVRrsquos QP problem is directlyproportional to its slope so a very small 119862 will result in ahorizontal and underfitting 119862 Inspired by [14 15] penaltyfactor at time point 119896 (1 le 119896 le 119873) is given by

119862119896= 120575radicSNR 3radic119896max (100381610038161003816

1003816119892119896minus 120582120590119896

10038161003816100381610038161003816100381610038161003816119892119896+ 120582120590119896

1003816100381610038161003816)

119896 = 1 119873

(14)

where 119892119896= (1119896)sum

119896minus1

119899=0|119910119899|2

120590119896= radic(1119896)sum

119896minus1

119899=0(|119910119899|2

minus 119892119896)

2119910119899= ang119903119899minus1

is a section of the element of training set 119878119896 as

(13) SNR is assumed to be known 120575 120582 are positive constantsand 120575 = 01 120582 = 05 are set in this study by the same way as120591

3 Results and Analyses

We have compared the proposed algorithm entitled as SVRestimator with the other three the HPF estimator proposedin [6] the DK estimator proposed in [10] the LFK estimatorproposed in [12]

31 Phase Unwrapping Process Because HPF estimator doesnot need the phase unwrapping process Figure 2 illustrates

Mathematical Problems in Engineering 7M

SE 10minus4

10minus5

10minus6

10minus7

10minus3

10minus2

10minus1

SNR (dB)0 5 10minus5minus10

N = 8 N = 32

N = 64N = 16

Figure 5 MSE of IF of SVR estimator with 119891119889= 01 119891

119903= 005 120601 =

0

0 5 10SNR (dB)

minus5minus10

MSE

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus10

N = 8 N = 32

N = 64N = 16

Figure 6MSE of IFR of SVR estimator with119891119889= 01 119891

119903= 005 120601 =

0

the arbitrary phase unwrapping processes of the other threewhile 119891

119889= 01 119891

119903= 005 119891

119889= 03 119891

119903= 005 and 119891

119889=

01 119891119903= 02 The number of Monte Carlo experiments is

10000 and 120601 = 0 119873 = 32 and SNR = 8 dB It is shownthat SVR estimator can unwrap phase accurately whateverthe value of 119891

119889and 119891

119903is However errors emerge in DK and

LFK ones as 119891119889and 119891

119903increase

32 Estimation Performance Figures 2 3 and 4 illustrate theMSE curves of IF and IFR versus SNR respectively whereMSEs are defined as 119864[( 119891

119889minus 119891119889)

2

] 119864[(119891119903minus 119891119903)

2

] 119891119889119891119903are

the estimation values of 119891119889 119891119903 CRLBs are given by [12]

CRLB119891119889=

1

SNR6 (8119873 minus 11) (2119873 minus 1)

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB119891119903=

1

SNR360

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB120601=

1

SNR3 (3119873

2

minus 3119873 + 2)

2119873 (1198732+ 3119873 + 2)

(15)

It is shown that whether 119891119889= 01 119891

119903= 005 119891

119889=

03 119891119903= 005 or 119891

119889= 01 119891

119903= 02 SVR estimator is the

best one during both IF and IFR estimations all the whilealso MSE performances of IF and IFR totally decrease as 119891

119889

and119891119903increase but SVR estimator is the most robust and has

had a great advantage already when 119891119889= 03 119891

119903= 005 or

119891119889= 01 119891

119903= 02

33 Impact of the Sample Size119873 Everything is as in Figure 2other than 119891

119889= 01 119891

119903= 005 the MSE curves of IF and IFR

of SVR estimator versus SNR while119873 is 8 16 32 and 64 areplotted in Figures 5 and 6 respectively It is clear that as 119873increases MSE performances of IF and IFR of SVR estimatorare both improved

34 Impact of Initial Phase 120601 Everything is as in Figure 3except that 120601 = 04120587 08120587 the MSE curves of IF and IFRare plotted against SNR in Figures 7 and 8 respectivelyComparing with Figures 3(a) and 4(a) we can see that SVRestimator is immune to 120601 but the other three are not

35 Computational Complexity Because we translate SVRinto QP problem and need to search the minimums duringthe process we can not derive the explicit form of thecomputational complexity of SVR estimator So everythingis as in Figure 2 except that the number of Monte Carloexperiments is 100 SNR is 8 dB and 119891

119889= 01 119891

119903= 005

the consuming times are listed in Table 1 while119873 is 8 16 32and 64 respectively The running computer is ASUS-PC1111having Intel(R) Pentium 213GHz CPU and 200GB RAM

We can see that SVR estimatorrsquos consuming times aremore than the othersrsquo especially when 119873 becomes larger Asa matter of fact however it is acceptable and tolerable

4 Conclusions

Phase unwrapping process is a key point in phase-basedfrequency estimation of chirp signal Firstly we adopt SVRto learn the unwrapped phases at previous time pointspredict the variation trend of phase efficiently and derive theestimation value for the next time point Once acquired interms of relationship between absolute signal phase and timeseries we address a simple and effective frequency estimationalgorithmof chirp signalTheproposed algorithm completelyexhibits its advantages of higher estimation accuracy lowersensitivity of frequency and initial phase by sacrificing moreconsuming times

Because SVR predicts the curversquos variation trend merelyin terms of training set which consists of previous pointsrsquo

8 Mathematical Problems in Engineering

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

0 5 10SNR (dB)

minus5minus10

(a)

0 2 4 6 8 10

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

SNR (dB)minus6 minus4 minus2minus8minus10

(b)

Figure 7 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IF with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

0 2 4 6 8 10 12 14SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

minus6 minus4 minus2

(a)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

0 2 4 6 8 10 12 14SNR (dB)

minus6 minus4 minus2

(b)

Figure 8 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IFR with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

Table 1 Consuming time with different119873 (ms)

Algorithm 119873 = 8 119873 = 16 119873 = 32 119873 = 64

HPF 380 942 3721 10654DK 203 476 1098 3711LFK 299 784 2362 8643SVR 437 1107 4805 13074

values we even can estimate the frequency of chirp signalunder the nonGaussian condition by the same way

Stressing that the proposed algorithm learns training setand gets the approximate values of SVRrsquos insensitive losscoefficient 120576 and penalty factor 119862 as a next step thereforeimproving SVRrsquos parameter setting is an important researchpoint

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China (Grant no 61201380)

Mathematical Problems in Engineering 9

References

[1] J C Curlander andRNMcDonough Synthetic Aperture RadarSystems and Signal Processing Wiley New York NY USA 1992

[2] J Xu Y Peng and X Xia ldquoParametric autofocus of SARimagingmdashinherent accuracy limitations and realizationrdquo IEEETransactions on Geoscience and Remote Sensing vol 42 no 11pp 2397ndash2411 2004

[3] R Ramaswami K N Sivarajan and G H Sasaki OpticalNetworks A Practical Perspective Morgan Kaufmann SanMateo Calif USA 1998

[4] P OrsquoShea ldquoA new technique for instantaneous frequency rateestimationrdquo IEEE Signal Processing Letters vol 9 no 8 pp 251ndash252 2002

[5] PWang I Djurovic and J Yang ldquoGeneralized high-order phasefunction for parameter estimation of polynomial phase signalrdquoIEEE Transactions on Signal Processing vol 56 no 7 pp 3023ndash3028 2008

[6] P Wang H Li I Djurovic and B Himed ldquoPerformance ofinstantaneous frequency rate estimation using high-order phasefunctionrdquo IEEE Transactions on Signal Processing vol 58 no 4pp 2415ndash2421 2010

[7] M Benidir and A Ouldali ldquoPolynomial phase signal analysisbased on the polynomial derivatives decompositionsrdquo IEEETransactions on Signal Processing vol 47 no 7 pp 1954ndash19651999

[8] P OrsquoShea ldquoA fast algorithm for estimating the parameters of aquadratic FM signalrdquo IEEE Transactions on Signal Processingvol 52 no 2 pp 385ndash393 2004

[9] S Ma J Jiang and Q Meng ldquoA fast accurate and robustmethod for joint estimation of frequency and frequency raterdquoin Proceedings of the 2011 International Symposium on IntelligentSignal Processing and Communications Systems (ISPACS rsquo11) pp1ndash6 Chiang Mai Thailand December 2011

[10] P M Djuric and S M Kay ldquoParameter estimation of chirpsignalsrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 38 no 12 pp 2118ndash2126 1990

[11] H Fu and P-Y Kam ldquoPhase-based time-domain estimation ofthe frequency and phase of a single sinusoid in AWGNmdashtherole and applications of the additive observation phase noisemodelrdquo IEEE Transactions on Information Theory vol 59 no5 pp 3175ndash3188 2013

[12] Y Li H Fu and P Y Kam ldquoImproved approximate time-domain ML estimators of chirp signal parameters and theirperformance analysisrdquo IEEE Transactions on Signal Processingvol 57 no 4 pp 1260ndash1272 2009

[13] Y Li and P Y Kam ldquoImproved chirp parameter estimationusing signal recovery method approximate time-domain MLestimators of signal s and their performance analysisrdquo inProceedings of the IEEE 71st Vehicular Technology Conferencepp 1ndash5 Taipei Taiwan May 2010

[14] V Cherkassky and Y Ma ldquoSelection of meta-parameters forsupport vector regressionrdquo in Proceedings of the InternationalConference on Artificial Neural Networks pp 687ndash693 MadridSpain August 2002

[15] V Cherkassky X Shao F M Mulier and V N Vapnik ldquoModelcomplexity control for regression using VC generalizationboundsrdquo IEEE Transactions on Neural Networks vol 10 no 5pp 1075ndash1089 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0 5 10

MSE

SNR (dB)minus5minus10

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

(a)

0 5 10 15 20SNR (dB)

MSE

10minus1

100

10minus2

10minus4

10minus5

10minus6

10minus7

10minus3

(b)

0 5 10 15 20SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus3

(c)

Figure 3 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 0 and119873 = 32 (b) MSE of IF with 119891

119889= 03 119891

119903= 005 120601 = 0 and119873 = 32 (c) MSE of

IF with 119891119889= 01 119891

119903= 02 120601 = 0 and119873 = 32

23 SVRrsquos Parameter Settings Setting SVRrsquos parameters is adifficult problem but has a pronounced impact on SVRrsquos per-formance for example insensitive loss coefficient 120576 penaltyfactor 119862 There is no complete theoretical basis or explicitclosed form Cross-validation is a widely used method untilnow but it is complex and time consuming In this study weformulate proper parameter values by understanding SVRrsquostheory and integrating a large number of references andexperiments

When 119896 is small imperfect phase unwrapping at aparticular time point can easily have an impact on thevariation trend of 119891(119909) In order to avoid this discrepancya new model must be constructed with the aptitude ofbetter generalization capability As 119896 increases its necessitydecreases contrarily which is due to the degrading impact of

improperly unwrapped phase Simple speaking the modelrsquosgeneralization capability is inversely proportional to the sizeof the set 119878

Intuitively insensitive loss coefficient 120576 is the verticalheight of 120576-tube The larger 120576 is the less support vectorsthere are Nevertheless too large 120576 will cause unfixable 119887Noted that if the predicted phase is in the vicinity of 120587rsquosodd times the estimation performance is deteriorated rapidlyfor its ambiguous phase characteristic Selecting a proper120576 can reduce this impact At the same time 120576 is inverselyproportional to SNR so insensitive loss coefficient at timepoint 119896 (1 le 119896 le 119873) is given by [14]

120576119896= 120591radic

1

SNRln 119896119896

119896 = 1 119873 (13)

6 Mathematical Problems in Engineering

0 2 4 6 8 10 12 14SNR (dB)

minus2minus4minus6

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus3

(a)

SNR (dB)0 2 4 6 8 10 12 14 16 18 20

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus3

(b)

0 5 10 15 20SNR (dB)

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus3

SVRHPFDK

LFKCRLB

(c)

Figure 4 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 0 and119873 = 32 (b) MSE of IFR with 119891

119889= 03 119891

119903= 005 120601 = 0 and119873 = 32 (c) MSE

of IFR with 119891119889= 01 119891

119903= 02 120601 = 0 and119873 = 32

where 120591 is a positive constant and 120591 = 08 is set in this studyafter experimental comparisons and validations and SNR isassumed to be known

Penalty factor 119862 controls the penalty degree of vectorsoutside the 120576-tube and determines SVRrsquos generalization capa-bility 119862 is directly proportional to the sample size and SNRAs (12) is a line in a two-dimension plane the first item oftarget function during solving SVRrsquos QP problem is directlyproportional to its slope so a very small 119862 will result in ahorizontal and underfitting 119862 Inspired by [14 15] penaltyfactor at time point 119896 (1 le 119896 le 119873) is given by

119862119896= 120575radicSNR 3radic119896max (100381610038161003816

1003816119892119896minus 120582120590119896

10038161003816100381610038161003816100381610038161003816119892119896+ 120582120590119896

1003816100381610038161003816)

119896 = 1 119873

(14)

where 119892119896= (1119896)sum

119896minus1

119899=0|119910119899|2

120590119896= radic(1119896)sum

119896minus1

119899=0(|119910119899|2

minus 119892119896)

2119910119899= ang119903119899minus1

is a section of the element of training set 119878119896 as

(13) SNR is assumed to be known 120575 120582 are positive constantsand 120575 = 01 120582 = 05 are set in this study by the same way as120591

3 Results and Analyses

We have compared the proposed algorithm entitled as SVRestimator with the other three the HPF estimator proposedin [6] the DK estimator proposed in [10] the LFK estimatorproposed in [12]

31 Phase Unwrapping Process Because HPF estimator doesnot need the phase unwrapping process Figure 2 illustrates

Mathematical Problems in Engineering 7M

SE 10minus4

10minus5

10minus6

10minus7

10minus3

10minus2

10minus1

SNR (dB)0 5 10minus5minus10

N = 8 N = 32

N = 64N = 16

Figure 5 MSE of IF of SVR estimator with 119891119889= 01 119891

119903= 005 120601 =

0

0 5 10SNR (dB)

minus5minus10

MSE

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus10

N = 8 N = 32

N = 64N = 16

Figure 6MSE of IFR of SVR estimator with119891119889= 01 119891

119903= 005 120601 =

0

the arbitrary phase unwrapping processes of the other threewhile 119891

119889= 01 119891

119903= 005 119891

119889= 03 119891

119903= 005 and 119891

119889=

01 119891119903= 02 The number of Monte Carlo experiments is

10000 and 120601 = 0 119873 = 32 and SNR = 8 dB It is shownthat SVR estimator can unwrap phase accurately whateverthe value of 119891

119889and 119891

119903is However errors emerge in DK and

LFK ones as 119891119889and 119891

119903increase

32 Estimation Performance Figures 2 3 and 4 illustrate theMSE curves of IF and IFR versus SNR respectively whereMSEs are defined as 119864[( 119891

119889minus 119891119889)

2

] 119864[(119891119903minus 119891119903)

2

] 119891119889119891119903are

the estimation values of 119891119889 119891119903 CRLBs are given by [12]

CRLB119891119889=

1

SNR6 (8119873 minus 11) (2119873 minus 1)

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB119891119903=

1

SNR360

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB120601=

1

SNR3 (3119873

2

minus 3119873 + 2)

2119873 (1198732+ 3119873 + 2)

(15)

It is shown that whether 119891119889= 01 119891

119903= 005 119891

119889=

03 119891119903= 005 or 119891

119889= 01 119891

119903= 02 SVR estimator is the

best one during both IF and IFR estimations all the whilealso MSE performances of IF and IFR totally decrease as 119891

119889

and119891119903increase but SVR estimator is the most robust and has

had a great advantage already when 119891119889= 03 119891

119903= 005 or

119891119889= 01 119891

119903= 02

33 Impact of the Sample Size119873 Everything is as in Figure 2other than 119891

119889= 01 119891

119903= 005 the MSE curves of IF and IFR

of SVR estimator versus SNR while119873 is 8 16 32 and 64 areplotted in Figures 5 and 6 respectively It is clear that as 119873increases MSE performances of IF and IFR of SVR estimatorare both improved

34 Impact of Initial Phase 120601 Everything is as in Figure 3except that 120601 = 04120587 08120587 the MSE curves of IF and IFRare plotted against SNR in Figures 7 and 8 respectivelyComparing with Figures 3(a) and 4(a) we can see that SVRestimator is immune to 120601 but the other three are not

35 Computational Complexity Because we translate SVRinto QP problem and need to search the minimums duringthe process we can not derive the explicit form of thecomputational complexity of SVR estimator So everythingis as in Figure 2 except that the number of Monte Carloexperiments is 100 SNR is 8 dB and 119891

119889= 01 119891

119903= 005

the consuming times are listed in Table 1 while119873 is 8 16 32and 64 respectively The running computer is ASUS-PC1111having Intel(R) Pentium 213GHz CPU and 200GB RAM

We can see that SVR estimatorrsquos consuming times aremore than the othersrsquo especially when 119873 becomes larger Asa matter of fact however it is acceptable and tolerable

4 Conclusions

Phase unwrapping process is a key point in phase-basedfrequency estimation of chirp signal Firstly we adopt SVRto learn the unwrapped phases at previous time pointspredict the variation trend of phase efficiently and derive theestimation value for the next time point Once acquired interms of relationship between absolute signal phase and timeseries we address a simple and effective frequency estimationalgorithmof chirp signalTheproposed algorithm completelyexhibits its advantages of higher estimation accuracy lowersensitivity of frequency and initial phase by sacrificing moreconsuming times

Because SVR predicts the curversquos variation trend merelyin terms of training set which consists of previous pointsrsquo

8 Mathematical Problems in Engineering

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

0 5 10SNR (dB)

minus5minus10

(a)

0 2 4 6 8 10

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

SNR (dB)minus6 minus4 minus2minus8minus10

(b)

Figure 7 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IF with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

0 2 4 6 8 10 12 14SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

minus6 minus4 minus2

(a)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

0 2 4 6 8 10 12 14SNR (dB)

minus6 minus4 minus2

(b)

Figure 8 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IFR with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

Table 1 Consuming time with different119873 (ms)

Algorithm 119873 = 8 119873 = 16 119873 = 32 119873 = 64

HPF 380 942 3721 10654DK 203 476 1098 3711LFK 299 784 2362 8643SVR 437 1107 4805 13074

values we even can estimate the frequency of chirp signalunder the nonGaussian condition by the same way

Stressing that the proposed algorithm learns training setand gets the approximate values of SVRrsquos insensitive losscoefficient 120576 and penalty factor 119862 as a next step thereforeimproving SVRrsquos parameter setting is an important researchpoint

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China (Grant no 61201380)

Mathematical Problems in Engineering 9

References

[1] J C Curlander andRNMcDonough Synthetic Aperture RadarSystems and Signal Processing Wiley New York NY USA 1992

[2] J Xu Y Peng and X Xia ldquoParametric autofocus of SARimagingmdashinherent accuracy limitations and realizationrdquo IEEETransactions on Geoscience and Remote Sensing vol 42 no 11pp 2397ndash2411 2004

[3] R Ramaswami K N Sivarajan and G H Sasaki OpticalNetworks A Practical Perspective Morgan Kaufmann SanMateo Calif USA 1998

[4] P OrsquoShea ldquoA new technique for instantaneous frequency rateestimationrdquo IEEE Signal Processing Letters vol 9 no 8 pp 251ndash252 2002

[5] PWang I Djurovic and J Yang ldquoGeneralized high-order phasefunction for parameter estimation of polynomial phase signalrdquoIEEE Transactions on Signal Processing vol 56 no 7 pp 3023ndash3028 2008

[6] P Wang H Li I Djurovic and B Himed ldquoPerformance ofinstantaneous frequency rate estimation using high-order phasefunctionrdquo IEEE Transactions on Signal Processing vol 58 no 4pp 2415ndash2421 2010

[7] M Benidir and A Ouldali ldquoPolynomial phase signal analysisbased on the polynomial derivatives decompositionsrdquo IEEETransactions on Signal Processing vol 47 no 7 pp 1954ndash19651999

[8] P OrsquoShea ldquoA fast algorithm for estimating the parameters of aquadratic FM signalrdquo IEEE Transactions on Signal Processingvol 52 no 2 pp 385ndash393 2004

[9] S Ma J Jiang and Q Meng ldquoA fast accurate and robustmethod for joint estimation of frequency and frequency raterdquoin Proceedings of the 2011 International Symposium on IntelligentSignal Processing and Communications Systems (ISPACS rsquo11) pp1ndash6 Chiang Mai Thailand December 2011

[10] P M Djuric and S M Kay ldquoParameter estimation of chirpsignalsrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 38 no 12 pp 2118ndash2126 1990

[11] H Fu and P-Y Kam ldquoPhase-based time-domain estimation ofthe frequency and phase of a single sinusoid in AWGNmdashtherole and applications of the additive observation phase noisemodelrdquo IEEE Transactions on Information Theory vol 59 no5 pp 3175ndash3188 2013

[12] Y Li H Fu and P Y Kam ldquoImproved approximate time-domain ML estimators of chirp signal parameters and theirperformance analysisrdquo IEEE Transactions on Signal Processingvol 57 no 4 pp 1260ndash1272 2009

[13] Y Li and P Y Kam ldquoImproved chirp parameter estimationusing signal recovery method approximate time-domain MLestimators of signal s and their performance analysisrdquo inProceedings of the IEEE 71st Vehicular Technology Conferencepp 1ndash5 Taipei Taiwan May 2010

[14] V Cherkassky and Y Ma ldquoSelection of meta-parameters forsupport vector regressionrdquo in Proceedings of the InternationalConference on Artificial Neural Networks pp 687ndash693 MadridSpain August 2002

[15] V Cherkassky X Shao F M Mulier and V N Vapnik ldquoModelcomplexity control for regression using VC generalizationboundsrdquo IEEE Transactions on Neural Networks vol 10 no 5pp 1075ndash1089 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 2 4 6 8 10 12 14SNR (dB)

minus2minus4minus6

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus3

(a)

SNR (dB)0 2 4 6 8 10 12 14 16 18 20

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus3

(b)

0 5 10 15 20SNR (dB)

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus3

SVRHPFDK

LFKCRLB

(c)

Figure 4 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 0 and119873 = 32 (b) MSE of IFR with 119891

119889= 03 119891

119903= 005 120601 = 0 and119873 = 32 (c) MSE

of IFR with 119891119889= 01 119891

119903= 02 120601 = 0 and119873 = 32

where 120591 is a positive constant and 120591 = 08 is set in this studyafter experimental comparisons and validations and SNR isassumed to be known

Penalty factor 119862 controls the penalty degree of vectorsoutside the 120576-tube and determines SVRrsquos generalization capa-bility 119862 is directly proportional to the sample size and SNRAs (12) is a line in a two-dimension plane the first item oftarget function during solving SVRrsquos QP problem is directlyproportional to its slope so a very small 119862 will result in ahorizontal and underfitting 119862 Inspired by [14 15] penaltyfactor at time point 119896 (1 le 119896 le 119873) is given by

119862119896= 120575radicSNR 3radic119896max (100381610038161003816

1003816119892119896minus 120582120590119896

10038161003816100381610038161003816100381610038161003816119892119896+ 120582120590119896

1003816100381610038161003816)

119896 = 1 119873

(14)

where 119892119896= (1119896)sum

119896minus1

119899=0|119910119899|2

120590119896= radic(1119896)sum

119896minus1

119899=0(|119910119899|2

minus 119892119896)

2119910119899= ang119903119899minus1

is a section of the element of training set 119878119896 as

(13) SNR is assumed to be known 120575 120582 are positive constantsand 120575 = 01 120582 = 05 are set in this study by the same way as120591

3 Results and Analyses

We have compared the proposed algorithm entitled as SVRestimator with the other three the HPF estimator proposedin [6] the DK estimator proposed in [10] the LFK estimatorproposed in [12]

31 Phase Unwrapping Process Because HPF estimator doesnot need the phase unwrapping process Figure 2 illustrates

Mathematical Problems in Engineering 7M

SE 10minus4

10minus5

10minus6

10minus7

10minus3

10minus2

10minus1

SNR (dB)0 5 10minus5minus10

N = 8 N = 32

N = 64N = 16

Figure 5 MSE of IF of SVR estimator with 119891119889= 01 119891

119903= 005 120601 =

0

0 5 10SNR (dB)

minus5minus10

MSE

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus10

N = 8 N = 32

N = 64N = 16

Figure 6MSE of IFR of SVR estimator with119891119889= 01 119891

119903= 005 120601 =

0

the arbitrary phase unwrapping processes of the other threewhile 119891

119889= 01 119891

119903= 005 119891

119889= 03 119891

119903= 005 and 119891

119889=

01 119891119903= 02 The number of Monte Carlo experiments is

10000 and 120601 = 0 119873 = 32 and SNR = 8 dB It is shownthat SVR estimator can unwrap phase accurately whateverthe value of 119891

119889and 119891

119903is However errors emerge in DK and

LFK ones as 119891119889and 119891

119903increase

32 Estimation Performance Figures 2 3 and 4 illustrate theMSE curves of IF and IFR versus SNR respectively whereMSEs are defined as 119864[( 119891

119889minus 119891119889)

2

] 119864[(119891119903minus 119891119903)

2

] 119891119889119891119903are

the estimation values of 119891119889 119891119903 CRLBs are given by [12]

CRLB119891119889=

1

SNR6 (8119873 minus 11) (2119873 minus 1)

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB119891119903=

1

SNR360

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB120601=

1

SNR3 (3119873

2

minus 3119873 + 2)

2119873 (1198732+ 3119873 + 2)

(15)

It is shown that whether 119891119889= 01 119891

119903= 005 119891

119889=

03 119891119903= 005 or 119891

119889= 01 119891

119903= 02 SVR estimator is the

best one during both IF and IFR estimations all the whilealso MSE performances of IF and IFR totally decrease as 119891

119889

and119891119903increase but SVR estimator is the most robust and has

had a great advantage already when 119891119889= 03 119891

119903= 005 or

119891119889= 01 119891

119903= 02

33 Impact of the Sample Size119873 Everything is as in Figure 2other than 119891

119889= 01 119891

119903= 005 the MSE curves of IF and IFR

of SVR estimator versus SNR while119873 is 8 16 32 and 64 areplotted in Figures 5 and 6 respectively It is clear that as 119873increases MSE performances of IF and IFR of SVR estimatorare both improved

34 Impact of Initial Phase 120601 Everything is as in Figure 3except that 120601 = 04120587 08120587 the MSE curves of IF and IFRare plotted against SNR in Figures 7 and 8 respectivelyComparing with Figures 3(a) and 4(a) we can see that SVRestimator is immune to 120601 but the other three are not

35 Computational Complexity Because we translate SVRinto QP problem and need to search the minimums duringthe process we can not derive the explicit form of thecomputational complexity of SVR estimator So everythingis as in Figure 2 except that the number of Monte Carloexperiments is 100 SNR is 8 dB and 119891

119889= 01 119891

119903= 005

the consuming times are listed in Table 1 while119873 is 8 16 32and 64 respectively The running computer is ASUS-PC1111having Intel(R) Pentium 213GHz CPU and 200GB RAM

We can see that SVR estimatorrsquos consuming times aremore than the othersrsquo especially when 119873 becomes larger Asa matter of fact however it is acceptable and tolerable

4 Conclusions

Phase unwrapping process is a key point in phase-basedfrequency estimation of chirp signal Firstly we adopt SVRto learn the unwrapped phases at previous time pointspredict the variation trend of phase efficiently and derive theestimation value for the next time point Once acquired interms of relationship between absolute signal phase and timeseries we address a simple and effective frequency estimationalgorithmof chirp signalTheproposed algorithm completelyexhibits its advantages of higher estimation accuracy lowersensitivity of frequency and initial phase by sacrificing moreconsuming times

Because SVR predicts the curversquos variation trend merelyin terms of training set which consists of previous pointsrsquo

8 Mathematical Problems in Engineering

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

0 5 10SNR (dB)

minus5minus10

(a)

0 2 4 6 8 10

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

SNR (dB)minus6 minus4 minus2minus8minus10

(b)

Figure 7 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IF with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

0 2 4 6 8 10 12 14SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

minus6 minus4 minus2

(a)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

0 2 4 6 8 10 12 14SNR (dB)

minus6 minus4 minus2

(b)

Figure 8 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IFR with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

Table 1 Consuming time with different119873 (ms)

Algorithm 119873 = 8 119873 = 16 119873 = 32 119873 = 64

HPF 380 942 3721 10654DK 203 476 1098 3711LFK 299 784 2362 8643SVR 437 1107 4805 13074

values we even can estimate the frequency of chirp signalunder the nonGaussian condition by the same way

Stressing that the proposed algorithm learns training setand gets the approximate values of SVRrsquos insensitive losscoefficient 120576 and penalty factor 119862 as a next step thereforeimproving SVRrsquos parameter setting is an important researchpoint

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China (Grant no 61201380)

Mathematical Problems in Engineering 9

References

[1] J C Curlander andRNMcDonough Synthetic Aperture RadarSystems and Signal Processing Wiley New York NY USA 1992

[2] J Xu Y Peng and X Xia ldquoParametric autofocus of SARimagingmdashinherent accuracy limitations and realizationrdquo IEEETransactions on Geoscience and Remote Sensing vol 42 no 11pp 2397ndash2411 2004

[3] R Ramaswami K N Sivarajan and G H Sasaki OpticalNetworks A Practical Perspective Morgan Kaufmann SanMateo Calif USA 1998

[4] P OrsquoShea ldquoA new technique for instantaneous frequency rateestimationrdquo IEEE Signal Processing Letters vol 9 no 8 pp 251ndash252 2002

[5] PWang I Djurovic and J Yang ldquoGeneralized high-order phasefunction for parameter estimation of polynomial phase signalrdquoIEEE Transactions on Signal Processing vol 56 no 7 pp 3023ndash3028 2008

[6] P Wang H Li I Djurovic and B Himed ldquoPerformance ofinstantaneous frequency rate estimation using high-order phasefunctionrdquo IEEE Transactions on Signal Processing vol 58 no 4pp 2415ndash2421 2010

[7] M Benidir and A Ouldali ldquoPolynomial phase signal analysisbased on the polynomial derivatives decompositionsrdquo IEEETransactions on Signal Processing vol 47 no 7 pp 1954ndash19651999

[8] P OrsquoShea ldquoA fast algorithm for estimating the parameters of aquadratic FM signalrdquo IEEE Transactions on Signal Processingvol 52 no 2 pp 385ndash393 2004

[9] S Ma J Jiang and Q Meng ldquoA fast accurate and robustmethod for joint estimation of frequency and frequency raterdquoin Proceedings of the 2011 International Symposium on IntelligentSignal Processing and Communications Systems (ISPACS rsquo11) pp1ndash6 Chiang Mai Thailand December 2011

[10] P M Djuric and S M Kay ldquoParameter estimation of chirpsignalsrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 38 no 12 pp 2118ndash2126 1990

[11] H Fu and P-Y Kam ldquoPhase-based time-domain estimation ofthe frequency and phase of a single sinusoid in AWGNmdashtherole and applications of the additive observation phase noisemodelrdquo IEEE Transactions on Information Theory vol 59 no5 pp 3175ndash3188 2013

[12] Y Li H Fu and P Y Kam ldquoImproved approximate time-domain ML estimators of chirp signal parameters and theirperformance analysisrdquo IEEE Transactions on Signal Processingvol 57 no 4 pp 1260ndash1272 2009

[13] Y Li and P Y Kam ldquoImproved chirp parameter estimationusing signal recovery method approximate time-domain MLestimators of signal s and their performance analysisrdquo inProceedings of the IEEE 71st Vehicular Technology Conferencepp 1ndash5 Taipei Taiwan May 2010

[14] V Cherkassky and Y Ma ldquoSelection of meta-parameters forsupport vector regressionrdquo in Proceedings of the InternationalConference on Artificial Neural Networks pp 687ndash693 MadridSpain August 2002

[15] V Cherkassky X Shao F M Mulier and V N Vapnik ldquoModelcomplexity control for regression using VC generalizationboundsrdquo IEEE Transactions on Neural Networks vol 10 no 5pp 1075ndash1089 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7M

SE 10minus4

10minus5

10minus6

10minus7

10minus3

10minus2

10minus1

SNR (dB)0 5 10minus5minus10

N = 8 N = 32

N = 64N = 16

Figure 5 MSE of IF of SVR estimator with 119891119889= 01 119891

119903= 005 120601 =

0

0 5 10SNR (dB)

minus5minus10

MSE

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus2

10minus10

N = 8 N = 32

N = 64N = 16

Figure 6MSE of IFR of SVR estimator with119891119889= 01 119891

119903= 005 120601 =

0

the arbitrary phase unwrapping processes of the other threewhile 119891

119889= 01 119891

119903= 005 119891

119889= 03 119891

119903= 005 and 119891

119889=

01 119891119903= 02 The number of Monte Carlo experiments is

10000 and 120601 = 0 119873 = 32 and SNR = 8 dB It is shownthat SVR estimator can unwrap phase accurately whateverthe value of 119891

119889and 119891

119903is However errors emerge in DK and

LFK ones as 119891119889and 119891

119903increase

32 Estimation Performance Figures 2 3 and 4 illustrate theMSE curves of IF and IFR versus SNR respectively whereMSEs are defined as 119864[( 119891

119889minus 119891119889)

2

] 119864[(119891119903minus 119891119903)

2

] 119891119889119891119903are

the estimation values of 119891119889 119891119903 CRLBs are given by [12]

CRLB119891119889=

1

SNR6 (8119873 minus 11) (2119873 minus 1)

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB119891119903=

1

SNR360

(2120587)2

119873(119873 minus 1) (1198733+ 1198732minus 4119873 minus 4)

CRLB120601=

1

SNR3 (3119873

2

minus 3119873 + 2)

2119873 (1198732+ 3119873 + 2)

(15)

It is shown that whether 119891119889= 01 119891

119903= 005 119891

119889=

03 119891119903= 005 or 119891

119889= 01 119891

119903= 02 SVR estimator is the

best one during both IF and IFR estimations all the whilealso MSE performances of IF and IFR totally decrease as 119891

119889

and119891119903increase but SVR estimator is the most robust and has

had a great advantage already when 119891119889= 03 119891

119903= 005 or

119891119889= 01 119891

119903= 02

33 Impact of the Sample Size119873 Everything is as in Figure 2other than 119891

119889= 01 119891

119903= 005 the MSE curves of IF and IFR

of SVR estimator versus SNR while119873 is 8 16 32 and 64 areplotted in Figures 5 and 6 respectively It is clear that as 119873increases MSE performances of IF and IFR of SVR estimatorare both improved

34 Impact of Initial Phase 120601 Everything is as in Figure 3except that 120601 = 04120587 08120587 the MSE curves of IF and IFRare plotted against SNR in Figures 7 and 8 respectivelyComparing with Figures 3(a) and 4(a) we can see that SVRestimator is immune to 120601 but the other three are not

35 Computational Complexity Because we translate SVRinto QP problem and need to search the minimums duringthe process we can not derive the explicit form of thecomputational complexity of SVR estimator So everythingis as in Figure 2 except that the number of Monte Carloexperiments is 100 SNR is 8 dB and 119891

119889= 01 119891

119903= 005

the consuming times are listed in Table 1 while119873 is 8 16 32and 64 respectively The running computer is ASUS-PC1111having Intel(R) Pentium 213GHz CPU and 200GB RAM

We can see that SVR estimatorrsquos consuming times aremore than the othersrsquo especially when 119873 becomes larger Asa matter of fact however it is acceptable and tolerable

4 Conclusions

Phase unwrapping process is a key point in phase-basedfrequency estimation of chirp signal Firstly we adopt SVRto learn the unwrapped phases at previous time pointspredict the variation trend of phase efficiently and derive theestimation value for the next time point Once acquired interms of relationship between absolute signal phase and timeseries we address a simple and effective frequency estimationalgorithmof chirp signalTheproposed algorithm completelyexhibits its advantages of higher estimation accuracy lowersensitivity of frequency and initial phase by sacrificing moreconsuming times

Because SVR predicts the curversquos variation trend merelyin terms of training set which consists of previous pointsrsquo

8 Mathematical Problems in Engineering

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

0 5 10SNR (dB)

minus5minus10

(a)

0 2 4 6 8 10

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

SNR (dB)minus6 minus4 minus2minus8minus10

(b)

Figure 7 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IF with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

0 2 4 6 8 10 12 14SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

minus6 minus4 minus2

(a)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

0 2 4 6 8 10 12 14SNR (dB)

minus6 minus4 minus2

(b)

Figure 8 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IFR with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

Table 1 Consuming time with different119873 (ms)

Algorithm 119873 = 8 119873 = 16 119873 = 32 119873 = 64

HPF 380 942 3721 10654DK 203 476 1098 3711LFK 299 784 2362 8643SVR 437 1107 4805 13074

values we even can estimate the frequency of chirp signalunder the nonGaussian condition by the same way

Stressing that the proposed algorithm learns training setand gets the approximate values of SVRrsquos insensitive losscoefficient 120576 and penalty factor 119862 as a next step thereforeimproving SVRrsquos parameter setting is an important researchpoint

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China (Grant no 61201380)

Mathematical Problems in Engineering 9

References

[1] J C Curlander andRNMcDonough Synthetic Aperture RadarSystems and Signal Processing Wiley New York NY USA 1992

[2] J Xu Y Peng and X Xia ldquoParametric autofocus of SARimagingmdashinherent accuracy limitations and realizationrdquo IEEETransactions on Geoscience and Remote Sensing vol 42 no 11pp 2397ndash2411 2004

[3] R Ramaswami K N Sivarajan and G H Sasaki OpticalNetworks A Practical Perspective Morgan Kaufmann SanMateo Calif USA 1998

[4] P OrsquoShea ldquoA new technique for instantaneous frequency rateestimationrdquo IEEE Signal Processing Letters vol 9 no 8 pp 251ndash252 2002

[5] PWang I Djurovic and J Yang ldquoGeneralized high-order phasefunction for parameter estimation of polynomial phase signalrdquoIEEE Transactions on Signal Processing vol 56 no 7 pp 3023ndash3028 2008

[6] P Wang H Li I Djurovic and B Himed ldquoPerformance ofinstantaneous frequency rate estimation using high-order phasefunctionrdquo IEEE Transactions on Signal Processing vol 58 no 4pp 2415ndash2421 2010

[7] M Benidir and A Ouldali ldquoPolynomial phase signal analysisbased on the polynomial derivatives decompositionsrdquo IEEETransactions on Signal Processing vol 47 no 7 pp 1954ndash19651999

[8] P OrsquoShea ldquoA fast algorithm for estimating the parameters of aquadratic FM signalrdquo IEEE Transactions on Signal Processingvol 52 no 2 pp 385ndash393 2004

[9] S Ma J Jiang and Q Meng ldquoA fast accurate and robustmethod for joint estimation of frequency and frequency raterdquoin Proceedings of the 2011 International Symposium on IntelligentSignal Processing and Communications Systems (ISPACS rsquo11) pp1ndash6 Chiang Mai Thailand December 2011

[10] P M Djuric and S M Kay ldquoParameter estimation of chirpsignalsrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 38 no 12 pp 2118ndash2126 1990

[11] H Fu and P-Y Kam ldquoPhase-based time-domain estimation ofthe frequency and phase of a single sinusoid in AWGNmdashtherole and applications of the additive observation phase noisemodelrdquo IEEE Transactions on Information Theory vol 59 no5 pp 3175ndash3188 2013

[12] Y Li H Fu and P Y Kam ldquoImproved approximate time-domain ML estimators of chirp signal parameters and theirperformance analysisrdquo IEEE Transactions on Signal Processingvol 57 no 4 pp 1260ndash1272 2009

[13] Y Li and P Y Kam ldquoImproved chirp parameter estimationusing signal recovery method approximate time-domain MLestimators of signal s and their performance analysisrdquo inProceedings of the IEEE 71st Vehicular Technology Conferencepp 1ndash5 Taipei Taiwan May 2010

[14] V Cherkassky and Y Ma ldquoSelection of meta-parameters forsupport vector regressionrdquo in Proceedings of the InternationalConference on Artificial Neural Networks pp 687ndash693 MadridSpain August 2002

[15] V Cherkassky X Shao F M Mulier and V N Vapnik ldquoModelcomplexity control for regression using VC generalizationboundsrdquo IEEE Transactions on Neural Networks vol 10 no 5pp 1075ndash1089 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

0 5 10SNR (dB)

minus5minus10

(a)

0 2 4 6 8 10

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus6

10minus3

SNR (dB)minus6 minus4 minus2minus8minus10

(b)

Figure 7 (a) MSE of IF with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IF with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

0 2 4 6 8 10 12 14SNR (dB)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

minus6 minus4 minus2

(a)

SVRHPFDK

LFKCRLB

MSE

10minus1

10minus2

10minus4

10minus5

10minus7

10minus8

10minus6

10minus3

0 2 4 6 8 10 12 14SNR (dB)

minus6 minus4 minus2

(b)

Figure 8 (a) MSE of IFR with 119891119889= 01 119891

119903= 005 120601 = 04120587 and119873 = 32 (b) MSE of IFR with 119891

119889= 01 119891

119903= 005 120601 = 08120587 and119873 = 32

Table 1 Consuming time with different119873 (ms)

Algorithm 119873 = 8 119873 = 16 119873 = 32 119873 = 64

HPF 380 942 3721 10654DK 203 476 1098 3711LFK 299 784 2362 8643SVR 437 1107 4805 13074

values we even can estimate the frequency of chirp signalunder the nonGaussian condition by the same way

Stressing that the proposed algorithm learns training setand gets the approximate values of SVRrsquos insensitive losscoefficient 120576 and penalty factor 119862 as a next step thereforeimproving SVRrsquos parameter setting is an important researchpoint

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China (Grant no 61201380)

Mathematical Problems in Engineering 9

References

[1] J C Curlander andRNMcDonough Synthetic Aperture RadarSystems and Signal Processing Wiley New York NY USA 1992

[2] J Xu Y Peng and X Xia ldquoParametric autofocus of SARimagingmdashinherent accuracy limitations and realizationrdquo IEEETransactions on Geoscience and Remote Sensing vol 42 no 11pp 2397ndash2411 2004

[3] R Ramaswami K N Sivarajan and G H Sasaki OpticalNetworks A Practical Perspective Morgan Kaufmann SanMateo Calif USA 1998

[4] P OrsquoShea ldquoA new technique for instantaneous frequency rateestimationrdquo IEEE Signal Processing Letters vol 9 no 8 pp 251ndash252 2002

[5] PWang I Djurovic and J Yang ldquoGeneralized high-order phasefunction for parameter estimation of polynomial phase signalrdquoIEEE Transactions on Signal Processing vol 56 no 7 pp 3023ndash3028 2008

[6] P Wang H Li I Djurovic and B Himed ldquoPerformance ofinstantaneous frequency rate estimation using high-order phasefunctionrdquo IEEE Transactions on Signal Processing vol 58 no 4pp 2415ndash2421 2010

[7] M Benidir and A Ouldali ldquoPolynomial phase signal analysisbased on the polynomial derivatives decompositionsrdquo IEEETransactions on Signal Processing vol 47 no 7 pp 1954ndash19651999

[8] P OrsquoShea ldquoA fast algorithm for estimating the parameters of aquadratic FM signalrdquo IEEE Transactions on Signal Processingvol 52 no 2 pp 385ndash393 2004

[9] S Ma J Jiang and Q Meng ldquoA fast accurate and robustmethod for joint estimation of frequency and frequency raterdquoin Proceedings of the 2011 International Symposium on IntelligentSignal Processing and Communications Systems (ISPACS rsquo11) pp1ndash6 Chiang Mai Thailand December 2011

[10] P M Djuric and S M Kay ldquoParameter estimation of chirpsignalsrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 38 no 12 pp 2118ndash2126 1990

[11] H Fu and P-Y Kam ldquoPhase-based time-domain estimation ofthe frequency and phase of a single sinusoid in AWGNmdashtherole and applications of the additive observation phase noisemodelrdquo IEEE Transactions on Information Theory vol 59 no5 pp 3175ndash3188 2013

[12] Y Li H Fu and P Y Kam ldquoImproved approximate time-domain ML estimators of chirp signal parameters and theirperformance analysisrdquo IEEE Transactions on Signal Processingvol 57 no 4 pp 1260ndash1272 2009

[13] Y Li and P Y Kam ldquoImproved chirp parameter estimationusing signal recovery method approximate time-domain MLestimators of signal s and their performance analysisrdquo inProceedings of the IEEE 71st Vehicular Technology Conferencepp 1ndash5 Taipei Taiwan May 2010

[14] V Cherkassky and Y Ma ldquoSelection of meta-parameters forsupport vector regressionrdquo in Proceedings of the InternationalConference on Artificial Neural Networks pp 687ndash693 MadridSpain August 2002

[15] V Cherkassky X Shao F M Mulier and V N Vapnik ldquoModelcomplexity control for regression using VC generalizationboundsrdquo IEEE Transactions on Neural Networks vol 10 no 5pp 1075ndash1089 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

References

[1] J C Curlander andRNMcDonough Synthetic Aperture RadarSystems and Signal Processing Wiley New York NY USA 1992

[2] J Xu Y Peng and X Xia ldquoParametric autofocus of SARimagingmdashinherent accuracy limitations and realizationrdquo IEEETransactions on Geoscience and Remote Sensing vol 42 no 11pp 2397ndash2411 2004

[3] R Ramaswami K N Sivarajan and G H Sasaki OpticalNetworks A Practical Perspective Morgan Kaufmann SanMateo Calif USA 1998

[4] P OrsquoShea ldquoA new technique for instantaneous frequency rateestimationrdquo IEEE Signal Processing Letters vol 9 no 8 pp 251ndash252 2002

[5] PWang I Djurovic and J Yang ldquoGeneralized high-order phasefunction for parameter estimation of polynomial phase signalrdquoIEEE Transactions on Signal Processing vol 56 no 7 pp 3023ndash3028 2008

[6] P Wang H Li I Djurovic and B Himed ldquoPerformance ofinstantaneous frequency rate estimation using high-order phasefunctionrdquo IEEE Transactions on Signal Processing vol 58 no 4pp 2415ndash2421 2010

[7] M Benidir and A Ouldali ldquoPolynomial phase signal analysisbased on the polynomial derivatives decompositionsrdquo IEEETransactions on Signal Processing vol 47 no 7 pp 1954ndash19651999

[8] P OrsquoShea ldquoA fast algorithm for estimating the parameters of aquadratic FM signalrdquo IEEE Transactions on Signal Processingvol 52 no 2 pp 385ndash393 2004

[9] S Ma J Jiang and Q Meng ldquoA fast accurate and robustmethod for joint estimation of frequency and frequency raterdquoin Proceedings of the 2011 International Symposium on IntelligentSignal Processing and Communications Systems (ISPACS rsquo11) pp1ndash6 Chiang Mai Thailand December 2011

[10] P M Djuric and S M Kay ldquoParameter estimation of chirpsignalsrdquo IEEE Transactions on Acoustics Speech and SignalProcessing vol 38 no 12 pp 2118ndash2126 1990

[11] H Fu and P-Y Kam ldquoPhase-based time-domain estimation ofthe frequency and phase of a single sinusoid in AWGNmdashtherole and applications of the additive observation phase noisemodelrdquo IEEE Transactions on Information Theory vol 59 no5 pp 3175ndash3188 2013

[12] Y Li H Fu and P Y Kam ldquoImproved approximate time-domain ML estimators of chirp signal parameters and theirperformance analysisrdquo IEEE Transactions on Signal Processingvol 57 no 4 pp 1260ndash1272 2009

[13] Y Li and P Y Kam ldquoImproved chirp parameter estimationusing signal recovery method approximate time-domain MLestimators of signal s and their performance analysisrdquo inProceedings of the IEEE 71st Vehicular Technology Conferencepp 1ndash5 Taipei Taiwan May 2010

[14] V Cherkassky and Y Ma ldquoSelection of meta-parameters forsupport vector regressionrdquo in Proceedings of the InternationalConference on Artificial Neural Networks pp 687ndash693 MadridSpain August 2002

[15] V Cherkassky X Shao F M Mulier and V N Vapnik ldquoModelcomplexity control for regression using VC generalizationboundsrdquo IEEE Transactions on Neural Networks vol 10 no 5pp 1075ndash1089 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of