a novel interference detection method of stap based on

Research ArticleA Novel Interference Detection Method of STAP Based onSimplified TT Transform

QiangWang Yongshun Zhang Hanwei Liu and Yiduo Guo

Air and Missile Defense College Air Force Engineering University Xirsquoan Shaanxi 710051 China

Correspondence should be addressed to Qiang Wang 1019611183qqcom

Received 20 June 2017 Accepted 26 October 2017 Published 21 November 2017

Academic Editor Fazal M Mahomed

Copyright copy 2017 Qiang Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Training samples contaminated by target-like signals is one of the major reasons for inhomogeneous clutter environment In suchenvironment clutter covariance matrix in STAP (space-time adaptive processing) is estimated inaccurately which finally leads todetection performance reduction In terms of this problem a STAP interference detection method based on simplified TT (time-time) transform is proposed in this letter Considering the sparse physical property of clutter in the space-time plane data on eachrange cell is first converted into a discrete slow time series Then the expression of simplified TT transform about sample datais derived step by step Thirdly the energy of each training sample is focalized and extracted by simplified TT transform fromenergy-variant difference between the unpolluted and polluted stage and the physical significance of discarding the contaminatedsamples is analyzed Lastly the contaminated samples are picked out in light of the simplified TT transform-spectrum differenceThe result onMonte Carlo simulation indicates that when training samples are contaminated by large power target-like signals theproposed method is more effective in getting rid of the contaminated samples reduces the computational complexity significantlyand promotes the target detection performance compared with the method of GIP (generalized inner product)

1 Introduction

The homogeneous clutter environment is the prerequisiteof STAP to suppress groundsea clutter and detect targetseffectively [1ndash3] In terms of the classical STAP [4 5]obtaining the CCM (clutter covariance matrix) is importantCCM is estimated by the training samples that have the IID(independent and identically distributed) feature with theCUT (cell under test) In general the number of IID trainingsamples needs to be twice the DOF (degrees of freedom)of the system in order to get the approximate optimalperformance [6] This can be ensured in the homogeneousclutter environment However in the actual condition theheterogeneity of clutter affects the IID relationship betweenthe training samples and the CUT Furthermore the greatreduction in the number of satisfactory samples leads toinaccurate CCM estimation Finally the target detectionperformance of STAP is obviously deteriorated [7ndash11]

Training samples containing target-like signals namelythe samples contaminated by the target-like signals is oneof the major factors that cause clutter heterogeneity [12 13]Before calculating the CCM the interference detection ofeach sample is required because the existing contaminatedsamples have a bad influence on the estimated accuracyof CCM [14] Hence it is important to find a more effec-tive method to solve the interference detection problemAt present GIP (generalized inner product) is a commonmethod which builds the test statistics by calculating theinversion of CCM to pick out the contaminated samples [15]GIP is available when fewer target-like signals are containedor the jamming intensity of the contained ones is smallerNevertheless in the condition of target-like signals with bigjamming intensity for example when JNR (jamming noiseratio) is 20 dB greater than SNR (signal-to-noise ratio) thereis no possibility of picking out the contaminated samplesdue to the obvious inaccuracy in CCM estimation and the

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 1424835 9 pageshttpsdoiorg10115520171424835

2 Mathematical Problems in Engineering

dramatic fluctuation in test statistics Meanwhile the CCMestimation and its inversion are required in the processingof GIP which lead to the heavy computational complexity ofSTAP and go against efficient target detection In addition SR(sparse recovery) is another solution Recently consideringthe sparse physical property of clutter in space-time planeSR is applied in STAP [6 16] As for SR-STAP the space-time spectrum of CUT is directly estimated to calculate theCCM which avoids the contamination problem of trainingsamples However high computational complexity has arisenfrom sparse grid partition and clutter suppression perfor-mance deterioration caused by big CCM estimation error hasappeared when the isolated interference signals exist in theCUT

Based on the above analysis TT (time-time) transformis considered to solve the interference detection problemof the training samples in this paper [17ndash19] TT transformwas proposed by Pinnegar and Mansinha [20 21] as a newtransform in 2003 which came from the inverse Fourier formof the time-frequency analysis S transform [22ndash25] One-dimensional time series is expressed as a two-dimensionaltime-time series by TT transform which is good for observ-ing the local features of the signal [26] An important featureof TT transform is that the main energy of the signal can befocalized in the main diagonal position [27] In this letter thesignal energy in the main diagonal position is only extractedto realize the rejection of the contaminated samples and thereduction of the computational cost The method is calledsimplified TT transformWhen the training samples are con-taminated by some target-like signals with bigger jammingintensity the energy of these samples varies greatly betweenthe unpolluted stage and the polluted stageTherefore a kindof interference detection method based on simplified TTtransform is put forward from the viewpoint of time-domainenergy in this letter Firstly the data on each training sampleis converted into one-dimensional discrete slow time seriesrespectively Then the transform-spectrum energy of eachseries is extracted to separate the contaminated samples andthe clean samples in the simplified TT transform domainLastly the polluted training samples are rejectedThrough theabove treatment the target detection performance on STAPis improved and the total computational cost is reduced in theheterogeneous clutter environment

2 Signal Model of STAP

Assuming the side-looking antenna array is 119872 times 119873 in theairborne radar this array is shown as the 119873 uniform lineararrays via the column equivalent synthesis 119870 pulses arecontained in a CPI (coherent processing interval) and thenumber of the observed range cells is 119876 The data samplingprocess about STAP can be described by the element-pulse-range domain and then data collection is composed of 119873 times119870 times 119876 sampling points Each range cell is a matrix of119873 times 119870If the matrix is converted into a vector of 119873119870 times 1 whichcorresponds with the slow time (pulse domain) a slow timesequence of each range cell is obtained Its specific form isdenoted in Figure 1

Range domain

Range sampling

Element dom

ain

Pulse domainA slow time data sequence

Figure 1 Data sampling process of STAP

As for Figure 1 the data matrix about the 119902th (1 2 119876)range cell is expressed as

X119902 = ((

11990911990211 11990911990221 sdot sdot sdot 119909119902119873111990911990212 11990911990222 sdot sdot sdot 1199091199021198732 d1199091199021119870 1199091199022119870 sdot sdot sdot 119909119902119873119870

))

(1)

By vector conversation (1) has the following form

vec (X119902) = (11990911990211 1199091199021198731 11990911990212 1199091199021198732 119909119902119873119870) (2)

vec(X119902) is the sampling data of the 119902th (1 2 119876)range cell in the different elements and different pulsesConsidering the sparse physical property of clutter in space-time plane vec(X119902) is discretized by the sampling intervalof different elements in the same pulse Supposing g119873times119870 isa discrete time sequence g119873times119870 is one-to-one mapping withvec(X119902) Then the relevant discrete slow time data sequenceof vec(X119902) can be expressed as119891 (g vec (X119902)) = (119909 (1198921) 119909 (1198922) 119909 (119892119873times119870)) (3)

If 119891(g vec(X119902)) is replaced with W119902 and the CUT isindicated by 1199020 the data in the CUT is denoted as

W1199020 = 119873119888sum119894=1

1205881199020119888119894 s1199020119888 (119908119905119894 119908119904119894) + 1205881199020119905 s1199020119905 (120594119905 120594119904) + n1199020 (4)

where119873119888 is the number of clutter patches1205881199020119888119894 and s1199020119888 (119908119905119894 119908119904119894)are the reflected coefficient of the 119894th clutter patch and itsspace-time steering vector respectively119908119905119894 and119908119904119894 show thenormalized Doppler frequency of the 119894th clutter patch and itsnormalized spatial frequency respectively 1205881199020119905 and s1199020119905 (120594119905 120594119904)are the reflected coefficient of the target in the CUT and itsspace-time steering vector respectively 120594119905 and 120594119904 show thenormalized Doppler frequency of the target in the CUT andits normalized spatial frequency respectively n1199020 is the noise

The data about each training cell is defined as

W119902 = 119873119888sum119894=1

120588119902119888119894s119902119888 (119908119905119894 119908119904119894) + (120572119902 minus 1) 120590119902119892120577119902119892 (120594119892119905 120594119892119904)+ n119902 (5)

Mathematical Problems in Engineering 3

where the value range of 119902 is 0 lt 119902 le 119876 which is unequalto 1199020 the polluted situation about each sample is reflected by120572119902 when 120572119902 is equal to 1 the samples are unpolluted when 120572119902is bigger than 1 the samples are polluted 120590119902119892 and 120577119902119892(120594119892119905 120594119892119904)denote the reflected coefficient of the target-like signal and itsspace-time steering vector respectively 120594119892119905 and 120594119892119904 show thenormalized Doppler frequency of the target-like signal andits normalized spatial frequency respectively n119902 is the noise

Combined with the above signal model two kinds ofinterference detection methods are discussed In the fol-lowing analysis and comparison the interference detectionmethods of STAP using GIP and simplified TT transformare simply called GIP-STAP and simplified TT-STAP respec-tively

3 Interference Detection Method of GIP-STAP

Detecting the contaminated training samples W119902 by GIP-STAP the test covariance matrix R119902 is firstly constructed torealize the prewhitening processing about W119902 and get 1006704W119902Then the test statistics of each training sample are calculatedin order to pick out the heterogeneous samples caused bythe target-like signals The test statistics about GIP-STAP aredefined as

120574119902 = 1006704WH1199021006704W119902 = (WH

119902 Rminus12119902 ) (Rminus12119902 W119902) = WH

119902 Rminus1119902 W119902 (6)

According to (6) before the test statistics about GIP-STAP are obtained calculating the prior covariance matrixR119902 is firstly required The estimation error of R119902 directlyaffects the performance of detecting the polluted samples Ifthere are many training samples contaminated by the target-like signals with big jamming intensity the error of the teststatistics increases seriously which leads to the failure ofrejecting the polluted samplesMeanwhile the computationalcomplexity is obviously sharpened because of estimating theCCM and calculating its inversion

In terms of the above problem the following part adoptsthe simplified TT transform to get rid of the polluted samples

from the viewpoint of energy-variant difference between theunpolluted stage and the polluted stage

4 Interference Detection Method ofSimplified TT-STAP

41 Simplified TT Transform Before the definition of simpli-fied TT transform is given TT transform is firstly introducedConcerning the continuous signal 120583(119905) its TT transform isexpressed as

TT (119905 120591)= int+infinminusinfin

int+infinminusinfin

120583 (120599) 10038161003816100381610038161198911003816100381610038161003816120576radic2120587119890minus[119891(119905minus120599)radic2120576]21198901198942120587119891(120599minus120591)119889120599 119889119891= int+infinminusinfin

120583 (120599) 119889120599int+infinminusinfin

10038161003816100381610038161198911003816100381610038161003816120576radic2120587119890minus[119891(120599minus119905)radic2120576]21198901198942120587119891(120599minus120591)119889119891(7)

where 119905 and 120591 denote time 119891 indicates frequency(|119891|120576radic2120587)119890minus[119891(119905minus120599)radic2120576]2 means the Gaussian windowfunction 120576 is the regulatory factor of window scale

Assuming 120578 and ] are equal to 120599minus119905 and 120599minus120591 respectively(7) is arranged as



10038161003816100381610038161198911003816100381610038161003816120576radic2120587119890minus[119891120578radic2120576]21198901198942120587119891]119889119891 (8)

where

int+infinminusinfin

10038161003816100381610038161198911003816100381610038161003816120576radic2120587119890minus[119891120578radic2120576]21198901198942120587119891]119889119891=

1205761205782radic 2120587 minus 21205871205762]1205783 119890minus21205872]212057621205782119890119903119891119894 (radic2120576120587]120578 ) 120578 = 0minusradic 21205762120587]4 120578 = 0

(9)

where 119890119903119891119894(∙) is the imaginary error functionCombining (9) with (8) TT transform is

TT (119905 120591) = int+infinminusinfin

120583 (120599) [ 120576(120599 minus 119905)2radic 2120587 minus 21205871205762 (120599 minus 120591)(120599 minus 119905)3 119890minus21205872(120599minus120591)21205762(120599minus119905)2119890119903119891119894 (radic2120576120587 (120599 minus 120591)120599 minus 119905 )]119889120599 120599 = 119905int+infinminusinfin

minus120583 (120599)radic 21205762120587 (120599 minus 120591)4 119889120599 120599 = 119905 (10)

We simplify (10) to


120583 (120599)(120599 minus 119905)2 [120576radic 2120587 minus 21205871205762 (120599 minus 120591)120599 minus 119905 119890minus21205872(120599minus120591)21205762(120599minus119905)2119890119903119891119894 (radic2120576120587 (120599 minus 120591)120599 minus 119905 )]119889120599 120599 = 119905int+infinminusinfin

minus 120583 (120599)120576 (120599 minus 120591)2radic 2120587119889120599 120599 = 119905 (11)


Table 1 Analysis on computational complexity

Methods CCM estimation Matrix inversion Simplified TT transformGIP-STAP 119874(119876 (119873119870)2) 119874 ((119873119870)3) mdashSimplified TT-STAP mdash mdash 119874 (119873119870119876)Hence

120575 = 120599 minus 120591120599 minus 119905 120585 (120575) = 120576radic 2120587 minus 21205871205762120575119890minus2120587212057621205752119890119903119891119894 (radic2120576120587120575) (12)

Furthermore (11) is denoted as


120583 (120599)(120599 minus 119905)2 120585 (120575) 119889120599 120599 = 119905int+infinminusinfin


In a physical way (13) is explained in such a way that thefunction 120585(120575) is rapidly convergent to 0 because of 120575 gt 1Meanwhile considering that time is nonnegative the mainenergy about 120585(120575) is contained in the region of |120575| le 1 (0 le120591 le 119905) For 120591 = 119905 the main energy has been maximallyreflected which indicates that the main energy about thesignal 120583(119905) is presented in the main diagonal position ofTT transform domain Summing up the above analysesextracting the diagonal feature of TT transform spectrumabout 120583(119905) can obtain the main energy of 120583(119905) in the timedomain

In terms of 120591 = 119905 TT(119905 120591) transform is changed intoa simplified form that is called simplified TT transformCombined with (13) the simplified TT transform is definedas


120583 (120599)(120599 minus 119905)2 120585 (1) 119889120599 120599 = 119905 (14)

42 Interference Detection Method Since the discrete datais often adopted in the processing of STAP convertingsimplified TT transform into the discrete form is necessaryDiscretization about (14) is expressed as

TT (119896119879 119896119879) = 119875minus1sum119895=0

120585 (1) 120583 (119895119879)(119895119879 minus 119896119879)2 119895119879 = 119896119879 (15)

where 119879 is the time sampling interval 119875 is the number ofsampling points 119896 = 0 1 119875 minus 1

Combining (4) with (5) the dimension of a range cell is119873119870 times 1 So the number of sampling points is 119873119870 Discretesimplified TT transform ofW119902 has the following form

TTstap (119896119879 119896119879) = 119873119870minus1sum119895=0

120585 (1) W119902 (119895119879)(119895119879 minus 119896119879)2 119895119879 = 119896119879 (16)

where 0 lt 119902 le 119876

Based on the physical property analysis about TT trans-form in Section 41 the energy of the signal in the timedomain is mainly focused in the diagonal position of the TTtransform which is regarded as the simplified TT transformspectrum of the signal When the clutter presents the hetero-geneity caused by the polluted training samples the energyabout training samples produces a valid change Meanwhilethe energy difference between the polluted samples andthe unpolluted samples increases By means of this featureextracting the energy of training samples using (16) canrealize the effective rejection of the training samples thatcontain the target-like signals

5 Computational Complexity Analyses

In STAP the computation load is one of the most importantstandards to judge the performance of interference detectionmethods Smaller computation loadmeans better applicationin actual environments Estimating the CCM and gettingits inversion are the two main steps of aggravating thecomputational complexity in STAP Before comparing theperformance of GIP-STAP with that of simplified TT-STAPthe computational load analysis is firstly denoted in Table 1

The computational load about simplified TT-STAPreduces rapidly because CCM estimation and its inversionare not required in the jamming detection Using simpli-fied TT-STAP the total computational load descends from119874(119876(119873119870)2) + 119874((119873119870)3) to 119874(119873119870119876) which promotes theprocessing of STAP obviously

6 Simulations

In order to compare GIP-STAP with simplified TT-STAP interms of detection performance several simulation resultshave been presented in this partThemain parameters are setin Table 2

Based on the parameters in Table 2 the simulation resultsare presented as follows

61 Comparison on the Test Statistics Concerning detectionabout the contaminated samples the test statistics of eachtraining sample are plotted in Figures 2 and 3

Considering the 150th sample asCUTand its both sides asa protecting unit these cells are ignored in the experimentsThus the test statistics results about 237 training samplesare presented in the two figures Comparing Figure 2 withFigure 3 when four target-like signals with big jammingintensity are contained in the training samples the precisionof estimating CCM by GIP-STAP decreases evidently whichleads to the large fluctuation about the statistics value anddetecting the polluted samples invalidly As for simplified


Table 2 Parameter setting

Parameter ValueRadar array 10 times 10Velocity of plane 140msHeight of plane 6000mNumber of pulses 12PRF (pulse repetition frequency) 24348HzElement distance 0115mWavelength 023mCNR 60 dBNumber of training samples 240Position of CUT 150SNR 10 dBJNR 30 dBPosition of the contaminated samples 30 90 210 230Number of Monte Carlo simulations 1000

0

1

2

3

4

5

6

7

Test

stat

istic

s of G

IP-S

TAP

50 100 150 200 2500Training samples

times10minus17

Figure 2 Test statistics of GIP-STAP

TT-STAP it avoids CCM estimation and the four contami-nated samples can be clearly seen which is good for realizingthe polluted samples rejection

For ease of the physical explanation the energy varianceof each polluted sample between the polluted stage and theunpolluted stage is analyzed as follows

62 Simplified TT Transform Spectrum Simplified TT trans-form spectrums of the four training samples are respectivelydenoted in Figures 4 5 6 and 7

According to Figures 4ndash7 simplified TT transform spec-trums change obviously between the unpolluted stage andthe polluted stage In terms of the same cell the color ofthe simplified TT transform spectrum deepens from theunpolluted stage to the polluted stage which has a directrelationwith the energy distribution of the sampleMoreoverthe power of the simplified TT transform spectrum in eachsample is increased by at least an order of magnitude

0

05

1

15

2

25

3

35

Test

stat

istic

s of s

impl

ified

TT-

STA

P


times10minus8

Figure 3 Test statistics of simplified TT-STAP

Hence it is ensured that simplified TT-STAP can detectthe contaminated samples effectively from the viewpoint ofenergy

63 Comparison on the Output Power Combined with thesame simulation data the output power of STAP is simulatedbetween the unpolluted stage and the polluted stage in thispart Meanwhile comparison on the output power usingGIP-STAP and simplified TT-STAP is made The results aredenoted in Figures 8 and 9

Analyzing Figures 8 and 9 there are two points we canget (1) in terms of Figure 8 when the contaminated trainingsamples exist estimating the CCM is affected and the error ofthe adaptive weight increases Furthermore the output powerdeteriorates seriously after STAP and the target in the CUTcannot be obtained (2) as for Figure 9 the prior covariancematrix estimation is required about GIP-STAP If there aresome target-like signals with big interference intensity in thetraining samples the error of estimating the CCM exists andthe polluted samples are not rejected effectively So the outputpower is not improved by GIP-STAP and the real target isdetected invalidly Aiming at simplified TT-STAP it analyzesthe training samples from the viewpoint of energyThematrixerror has no influence on jamming detection and then thefour polluted samples are eliminated and the real target isobtained

64 Comparison on the Detection Performance The finalpurpose of STAP is to detect the target in a strong clutterenvironment In this part simulations about IF (improvedfactor) the gap between primary peak of the output powerand its secondary peak the output SCR (signal clutter ratio)and PD (probability of detection) are made The results arepresented in Figures 10 11 12 and 13

As for GIP-STAP in Figure 10 the deep null is formednot only in the main clutter region but also in the Dopplerfrequency of the real target because of the contaminated

6 Mathematical Problems in EngineeringD

iscre

te-ti

me s

erie

s n2

100120

604020

80

806040 100 12020Discrete-time series n1

12108642

14times10

minus9

(a) Uncontaminated simplified TT transform spectrum

Disc

rete

-tim

e ser

ies n

2

20406080

100120

40 60 80 100 12020Discrete-time series n1

123456times10

minus7

(b) Contaminated simplified TT transform spectrum

Figure 4 Energy distribution of the 30th range cell


Disc

rete

-tim

e ser

ies n

2

100120

4020

8060

24681012times10

minus8


Disc

rete

-tim

e ser

ies n

220406080

100120


1234567times10

minus7



Disc

rete

-tim

e ser

ies n

2

20406080

100120

12345


times10minus8


Disc

rete

-tim

e ser

ies n

2

20406080

100120

12345


times10minus7



Disc

rete

-tim

e ser

ies n

2

20406080

100120


1

2

3

4

times10minus9


Disc

rete

-tim

e ser

ies n

2

20406080

100120


123456times10

minus7



samples rejected invalidly And then the target detectionperformance decreases obviously However the four con-taminated samples are all rejected by simplified TT-STAPEstimating the CCM is not affected and the deep null is only

formed in the main clutter region Furthermore the cluttersuppression performance is obviously improved

In terms of Figure 11 the gap between the primary peakand secondary peak reflects the target detection performance


UncontaminatedContaminated

minus60

minus50

minus40

minus30

minus20

minus10

0

Out

put p

ower

(dB)

120 140 160 180 200100Range cell

Figure 8 Output power changing with range cell before jammingdetection

GIP-STAPSimplified TT-STAP

minus80

minus70

minus60

minus50

minus40

minus30

minus20

Out

put p

ower

(dB)

120 140 160 180 200100Range cell

Figure 9 Comparison on the output power after jamming detec-tion

after interference detectionThe target is more easily detectedif the gap is bigger With the increase of INSCR the gapis promoted which means the target detection performanceis improved gradually Meanwhile in the same INSCRsimplifiedTT-STAP is superior toGIP-STAPWhen INSCR issmaller the performances of the two methods are equivalentbecause the signal is weak As the signal becomes strongerthe performance of GIP-STAP is improved but is still inferiorto the simplified TT-STAP

In Figure 12 OUTSCR and INSCR of CUThave a positivecorrelation In the same INSCR the OUTSCR improvementof simplified TT-STAP is more obvious than that of GIP-STAP Since the target-like signals are rejected ineffectively

Simplified TT-STAPGIP-STAP

45

50

55

60

65

70

75

80

85

IF (d

B)

0 05minus05Normalized Doppler

Figure 10 Relation between IF and normalized Doppler frequency


minus55 minus50 minus45 minus40 minus35 minus30 minus25 minus20minus60INSCR (dB)

0

5

10

15

20

25Pr

imar

y an

d se

cond

ary

peak

s rat

io (d

B)

Figure 11 Difference between primary peak and secondary peak ofthe output power

using GIP-STAP the error of the adaptive weight increaseswhich leads to a disappointing OUTSCR improvement

Based on the output SCR the relation between PD andINSCR is given in Figure 13 using CFAR In the same falsealarm probability PD is promoted gradually with increasesin INSCR Since estimating the CCM is inaccurate by GIP-STAP PDofGIP-STAP ismore inferior than that of simplifiedTT-STAP with the same INSCR

7 Conclusions

The inhomogeneous clutter environment appearswhen train-ing samples are contaminated by some target-like signals withbig jamming intensity In order to eliminate the bad influence



0

5

10

15

20

25

30

OU

TSC

R (d

B)


Figure 12 Relation between INSCR and OUTSCR


0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f det

ectio

n


Figure 13 Relation between INSCR and PD

of such environment an interference detectionmethod basedon simplified TT transform is proposed from the viewpointof energy in time domain Combined with the sparse physicalproperty of clutter in the space-time plane the training sam-ples are firstly converted into discrete slow time sequencesThen the reason of getting rid of the polluted samplesinvalidly is analyzedThirdly the formula about simplifiedTTtransform is derived based on TT transform and the physicalexplanation of rejecting the polluted samples by simplifiedTT-STAP is given Fourthly the computational complexityof each method is discussed At last the performances onpicking out the polluted samples and detecting the realtarget are verified Compared with GIP-STAP the proposedmethod is more effective which avoids the influence of thepolluted samples on the adaptive weight and reduces the

computational load of STAP Furthermore simplified TT-STAP has a better practical value and theoretical researchsignificance

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant 61501501

References

[1] Z Wang Y Wang K Duan andW Xie ldquoSubspace-augmentedclutter suppression technique for STAP radarrdquo IEEE Geoscienceand Remote Sensing Letters vol 13 no 3 pp 462ndash466 2016

[2] W Wang Z Chen X Li and B Wang ldquoSpace time adaptiveprocessing algorithm for multiple-input-multiple-output radarbased onNystrommethodrdquo IETRadar SonarampNavigation vol10 no 3 pp 459ndash467 2016

[3] ZGaoH Tao S Zhu and J Zhao ldquoL1-regularised joint iterativeoptimisation space-time adaptive processing algorithmrdquo IETRadar Sonar amp Navigation vol 10 no 3 pp 435ndash441 2016

[4] L E Brennan and L S Reed ldquoTheory of adaptive radarrdquo IEEETransactions on Aerospace and Electronic Systems vol 9 no 2pp 237ndash252 1973

[5] T Wang Y Zhao T Lai and J Wang ldquoPerformance analysisof polarization-space-time adaptive processing for airbornepolarization array multiple-input multiple-output radarrdquo ActaPhysica Sinica vol 66 no 4 pp 048401(1)ndash048401(9) 2017

[6] Z Ma X Wang Y Liu and H Meng ldquoAn overview on sparserecovery-based STAPrdquo Journal of Radars vol 3 no 2 pp 217ndash228 2014

[7] Y Tong T Wang and J Wu ldquoImproving EFA-STAP perfor-mance using persymmetric covariancematrix estimationrdquo IEEETransactions on Aerospace and Electronic Systems vol 51 no 2pp 924ndash936 2015

[8] Y Wu T Wang J Wu and J Duan ldquoRobust training samplesselection algorithm based on spectral similarity for space-time adaptive processing in heterogeneous interference envi-ronmentsrdquo IET Radar SonarampNavigation vol 9 no 7 pp 778ndash782 2015

[9] Y Wu T Wang J Wu and J Duan ldquoTraining sample selectionfor space-time adaptive processing in heterogeneous environ-mentsrdquo IEEE Geoscience and Remote Sensing Letters vol 12 no4 pp 691ndash695 2015

[10] J H Bang W L Melvin and A D Lanterman ldquoModel-basedclutter cancellation based on enhanced knowledge-aided para-metric covariance estimationrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 51 no 1 pp 154ndash166 2015

[11] CHao S Gazor DOrlandoG Foglia and J Yang ldquoParametricspace-time detection and range estimation of a small targetrdquoIET Radar Sonar amp Navigation vol 9 no 2 pp 221ndash231 2015

[12] L B Fertig ldquoAnalytical expressions for space-time adaptive pro-cessing (STAP) performancerdquo IEEE Transactions on Aerospaceand Electronic Systems vol 51 no 1 pp 442ndash453 2015


[13] W Zhang Z He J Li and C Li ldquoBeamspace reduced-dimension space-time adaptive processing for multiple-inputmultiple-output radar based on maximum cross-correlationenergyrdquo IET Radar Sonar amp Navigation vol 9 no 7 pp 772ndash777 2015

[14] K Duan W Xie Y Wang W Liu and F Gao ldquoA deterministicauto-regressive STAP approach for nonhomogenerous cluttersuppressionrdquo Multidimensional Systems and Signal ProcessingAn International Journal vol 27 no 1 pp 105ndash119 2016

[15] T Wang and Y Zhao ldquoKnowledge-aided non-homogeneoussamples detection method for airborne MIMO radar STAPrdquo XiTong Gong Cheng Yu Dian Zi Ji Shu vol 37 no 10 pp 2260ndash2265 2015

[16] ZWang K DuanW Xie and YWang ldquoA joint sparse recoverySTAP method based on SA-MUSICrdquo Tien Tzu Hsueh PaoActaElectronica Sinica vol 43 no 5 pp 846ndash853 2015

[17] S Suja and J Jerome ldquoPattern recognition of power signal dis-turbances using S Transform and TT Transformrdquo InternationalJournal of Electrical Power amp Energy Systems vol 32 no 1 pp37ndash53 2010

[18] J Ma and Q Li ldquoSurface Wave Suppression with Joint STransform and TT Transformrdquo Procedia Earth and PlanetaryScience vol 3 pp 246ndash252 2011

[19] H Shareef A Mohamed and A A Ibrahim ldquoIdentificationof voltage sag source location using S and TT transformeddisturbance powerrdquo Journal of Central South University vol 20no 1 pp 83ndash97 2013

[20] C R Pinnegar and L Mansinha ldquoA method of time-timeanalysis The TT-transformrdquo Digital Signal Processing vol 13no 4 pp 588ndash603 2003

[21] C R Pinnegar ldquoGeneralizing the TT-transformrdquo Digital SignalProcessing vol 19 no 1 pp 144ndash152 2009

[22] G-Z ShaoG P Tsoflias andC-J Li ldquoDetection of near-surfacecavities by generalized S-transform of Rayleigh wavesrdquo Journalof Applied Geophysics vol 129 pp 53ndash65 2016

[23] M V Reddy and R Sodhi ldquoA rule-based S-Transform andAdaBoost based approach for power quality assessmentrdquo Elec-tric Power Systems Research vol 134 pp 66ndash79 2016

[24] S Zhang P Li L Zhang H LiW Jiang and Y Hu ldquoModified Stransform and ELM algorithms and their applications in powerquality analysisrdquo Neurocomputing vol 185 pp 231ndash241 2016

[25] Z Huang J Zhang T Zhao and Y Sun ldquoSynchrosqueezingS-transform and its application in seismic spectral decomposi-tionrdquo IEEE Transactions on Geoscience and Remote Sensing vol54 no 2 pp 817ndash825 2016

[26] Y Qin and L Tian ldquoPattern recognition and time location ofpower quality disturbances using TT-transformrdquo in Proceedingsof the 2010 International Conference on Intelligent System Designand Engineering Application ISDEA 2010 pp 53ndash56 ChinaOctober 2010

[27] C Simon M Schimmel and J J Danobeitia ldquoOn the TT-transform and its diagonal elementsrdquo IEEE Transactions onSignal Processing vol 56 no 11 pp 5709ndash5713 2008

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


dramatic fluctuation in test statistics Meanwhile the CCMestimation and its inversion are required in the processingof GIP which lead to the heavy computational complexity ofSTAP and go against efficient target detection In addition SR(sparse recovery) is another solution Recently consideringthe sparse physical property of clutter in space-time planeSR is applied in STAP [6 16] As for SR-STAP the space-time spectrum of CUT is directly estimated to calculate theCCM which avoids the contamination problem of trainingsamples However high computational complexity has arisenfrom sparse grid partition and clutter suppression perfor-mance deterioration caused by big CCM estimation error hasappeared when the isolated interference signals exist in theCUT

Based on the above analysis TT (time-time) transformis considered to solve the interference detection problemof the training samples in this paper [17ndash19] TT transformwas proposed by Pinnegar and Mansinha [20 21] as a newtransform in 2003 which came from the inverse Fourier formof the time-frequency analysis S transform [22ndash25] One-dimensional time series is expressed as a two-dimensionaltime-time series by TT transform which is good for observ-ing the local features of the signal [26] An important featureof TT transform is that the main energy of the signal can befocalized in the main diagonal position [27] In this letter thesignal energy in the main diagonal position is only extractedto realize the rejection of the contaminated samples and thereduction of the computational cost The method is calledsimplified TT transformWhen the training samples are con-taminated by some target-like signals with bigger jammingintensity the energy of these samples varies greatly betweenthe unpolluted stage and the polluted stageTherefore a kindof interference detection method based on simplified TTtransform is put forward from the viewpoint of time-domainenergy in this letter Firstly the data on each training sampleis converted into one-dimensional discrete slow time seriesrespectively Then the transform-spectrum energy of eachseries is extracted to separate the contaminated samples andthe clean samples in the simplified TT transform domainLastly the polluted training samples are rejectedThrough theabove treatment the target detection performance on STAPis improved and the total computational cost is reduced in theheterogeneous clutter environment

2 Signal Model of STAP

Assuming the side-looking antenna array is 119872 times 119873 in theairborne radar this array is shown as the 119873 uniform lineararrays via the column equivalent synthesis 119870 pulses arecontained in a CPI (coherent processing interval) and thenumber of the observed range cells is 119876 The data samplingprocess about STAP can be described by the element-pulse-range domain and then data collection is composed of 119873 times119870 times 119876 sampling points Each range cell is a matrix of119873 times 119870If the matrix is converted into a vector of 119873119870 times 1 whichcorresponds with the slow time (pulse domain) a slow timesequence of each range cell is obtained Its specific form isdenoted in Figure 1

Range domain

Range sampling

Element dom

ain

Pulse domainA slow time data sequence

Figure 1 Data sampling process of STAP

As for Figure 1 the data matrix about the 119902th (1 2 119876)range cell is expressed as

X119902 = ((

11990911990211 11990911990221 sdot sdot sdot 119909119902119873111990911990212 11990911990222 sdot sdot sdot 1199091199021198732 d1199091199021119870 1199091199022119870 sdot sdot sdot 119909119902119873119870

))

(1)

By vector conversation (1) has the following form

vec (X119902) = (11990911990211 1199091199021198731 11990911990212 1199091199021198732 119909119902119873119870) (2)

vec(X119902) is the sampling data of the 119902th (1 2 119876)range cell in the different elements and different pulsesConsidering the sparse physical property of clutter in space-time plane vec(X119902) is discretized by the sampling intervalof different elements in the same pulse Supposing g119873times119870 isa discrete time sequence g119873times119870 is one-to-one mapping withvec(X119902) Then the relevant discrete slow time data sequenceof vec(X119902) can be expressed as119891 (g vec (X119902)) = (119909 (1198921) 119909 (1198922) 119909 (119892119873times119870)) (3)

If 119891(g vec(X119902)) is replaced with W119902 and the CUT isindicated by 1199020 the data in the CUT is denoted as

W1199020 = 119873119888sum119894=1

1205881199020119888119894 s1199020119888 (119908119905119894 119908119904119894) + 1205881199020119905 s1199020119905 (120594119905 120594119904) + n1199020 (4)

where119873119888 is the number of clutter patches1205881199020119888119894 and s1199020119888 (119908119905119894 119908119904119894)are the reflected coefficient of the 119894th clutter patch and itsspace-time steering vector respectively119908119905119894 and119908119904119894 show thenormalized Doppler frequency of the 119894th clutter patch and itsnormalized spatial frequency respectively 1205881199020119905 and s1199020119905 (120594119905 120594119904)are the reflected coefficient of the target in the CUT and itsspace-time steering vector respectively 120594119905 and 120594119904 show thenormalized Doppler frequency of the target in the CUT andits normalized spatial frequency respectively n1199020 is the noise

The data about each training cell is defined as

W119902 = 119873119888sum119894=1

120588119902119888119894s119902119888 (119908119905119894 119908119904119894) + (120572119902 minus 1) 120590119902119892120577119902119892 (120594119892119905 120594119892119904)+ n119902 (5)






120574119902 = 1006704WH1199021006704W119902 = (WH


119902 Rminus1119902 W119902 (6)







int+infinminusinfin









where

int+infinminusinfin



(9)





We simplify (10) to















120583 (120599)(120599 minus 119905)2 120585 (1) 119889120599 120599 = 119905 (14)


TT (119896119879 119896119879) = 119875minus1sum119895=0

120585 (1) 120583 (119895119879)(119895119879 minus 119896119879)2 119895119879 = 119896119879 (15)



TTstap (119896119879 119896119879) = 119873119870minus1sum119895=0

120585 (1) W119902 (119895119879)(119895119879 minus 119896119879)2 119895119879 = 119896119879 (16)

where 0 lt 119902 le 119876





6 Simulations








0

1

2

3

4

5

6

7

Test

stat

istic

s of G

IP-S

TAP


times10minus17






0

05

1

15

2

25

3

35

Test

stat

istic

s of s

impl

ified

TT-

STA

P


times10minus8








iscre

te-ti

me s

erie

s n2

100120

604020

80


12108642

14times10

minus9


Disc

rete

-tim

e ser

ies n

2

20406080

100120


123456times10

minus7




Disc

rete

-tim

e ser

ies n

2

100120

4020

8060

24681012times10

minus8


Disc

rete

-tim

e ser

ies n

220406080

100120


1234567times10

minus7



Disc

rete

-tim

e ser

ies n

2

20406080

100120

12345


times10minus8


Disc

rete

-tim

e ser

ies n

2

20406080

100120

12345


times10minus7



Disc

rete

-tim

e ser

ies n

2

20406080

100120


1

2

3

4

times10minus9


Disc

rete

-tim

e ser

ies n

2

20406080

100120


123456times10

minus7








minus60

minus50

minus40

minus30

minus20

minus10

0

Out

put p

ower

(dB)

120 140 160 180 200100Range cell



minus80

minus70

minus60

minus50

minus40

minus30

minus20

Out

put p

ower

(dB)

120 140 160 180 200100Range cell





45

50

55

60

65

70

75

80

85

IF (d

B)





0

5

10

15

20

25Pr

imar

y an

d se

cond

ary

peak

s rat

io (d

B)




7 Conclusions




0

5

10

15

20

25

30

OU

TSC

R (d

B)




0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f det

ectio

n







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









120574119902 = 1006704WH1199021006704W119902 = (WH


119902 Rminus1119902 W119902 (6)







int+infinminusinfin









where

int+infinminusinfin



(9)





We simplify (10) to















120583 (120599)(120599 minus 119905)2 120585 (1) 119889120599 120599 = 119905 (14)


TT (119896119879 119896119879) = 119875minus1sum119895=0

120585 (1) 120583 (119895119879)(119895119879 minus 119896119879)2 119895119879 = 119896119879 (15)



TTstap (119896119879 119896119879) = 119873119870minus1sum119895=0

120585 (1) W119902 (119895119879)(119895119879 minus 119896119879)2 119895119879 = 119896119879 (16)

where 0 lt 119902 le 119876





6 Simulations








0

1

2

3

4

5

6

7

Test

stat

istic

s of G

IP-S

TAP


times10minus17






0

05

1

15

2

25

3

35

Test

stat

istic

s of s

impl

ified

TT-

STA

P


times10minus8








iscre

te-ti

me s

erie

s n2

100120

604020

80


12108642

14times10

minus9


Disc

rete

-tim

e ser

ies n

2

20406080

100120


123456times10

minus7




Disc

rete

-tim

e ser

ies n

2

100120

4020

8060

24681012times10

minus8


Disc

rete

-tim

e ser

ies n

220406080

100120


1234567times10

minus7



Disc

rete

-tim

e ser

ies n

2

20406080

100120

12345


times10minus8


Disc

rete

-tim

e ser

ies n

2

20406080

100120

12345


times10minus7



Disc

rete

-tim

e ser

ies n

2

20406080

100120


1

2

3

4

times10minus9


Disc

rete

-tim

e ser

ies n

2

20406080

100120


123456times10

minus7








minus60

minus50

minus40

minus30

minus20

minus10

0

Out

put p

ower

(dB)

120 140 160 180 200100Range cell



minus80

minus70

minus60

minus50

minus40

minus30

minus20

Out

put p

ower

(dB)

120 140 160 180 200100Range cell





45

50

55

60

65

70

75

80

85

IF (d

B)





0

5

10

15

20

25Pr

imar

y an

d se

cond

ary

peak

s rat

io (d

B)




7 Conclusions




0

5

10

15

20

25

30

OU

TSC

R (d

B)




0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f det

ectio

n







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of















120583 (120599)(120599 minus 119905)2 120585 (1) 119889120599 120599 = 119905 (14)


TT (119896119879 119896119879) = 119875minus1sum119895=0

120585 (1) 120583 (119895119879)(119895119879 minus 119896119879)2 119895119879 = 119896119879 (15)



TTstap (119896119879 119896119879) = 119873119870minus1sum119895=0

120585 (1) W119902 (119895119879)(119895119879 minus 119896119879)2 119895119879 = 119896119879 (16)

where 0 lt 119902 le 119876





6 Simulations








0

1

2

3

4

5

6

7

Test

stat

istic

s of G

IP-S

TAP


times10minus17






0

05

1

15

2

25

3

35

Test

stat

istic

s of s

impl

ified

TT-

STA

P


times10minus8








iscre

te-ti

me s

erie

s n2

100120

604020

80


12108642

14times10

minus9


Disc

rete

-tim

e ser

ies n

2

20406080

100120


123456times10

minus7




Disc

rete

-tim

e ser

ies n

2

100120

4020

8060

24681012times10

minus8


Disc

rete

-tim

e ser

ies n

220406080

100120


1234567times10

minus7



Disc

rete

-tim

e ser

ies n

2

20406080

100120

12345


times10minus8


Disc

rete

-tim

e ser

ies n

2

20406080

100120

12345


times10minus7



Disc

rete

-tim

e ser

ies n

2

20406080

100120


1

2

3

4

times10minus9


Disc

rete

-tim

e ser

ies n

2

20406080

100120


123456times10

minus7








minus60

minus50

minus40

minus30

minus20

minus10

0

Out

put p

ower

(dB)

120 140 160 180 200100Range cell



minus80

minus70

minus60

minus50

minus40

minus30

minus20

Out

put p

ower

(dB)

120 140 160 180 200100Range cell





45

50

55

60

65

70

75

80

85

IF (d

B)





0

5

10

15

20

25Pr

imar

y an

d se

cond

ary

peak

s rat

io (d

B)




7 Conclusions




0

5

10

15

20

25

30

OU

TSC

R (d

B)




0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f det

ectio

n







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







0

1

2

3

4

5

6

7

Test

stat

istic

s of G

IP-S

TAP


times10minus17






0

05

1

15

2

25

3

35

Test

stat

istic

s of s

impl

ified

TT-

STA

P


times10minus8








iscre

te-ti

me s

erie

s n2

100120

604020

80


12108642

14times10

minus9


Disc

rete

-tim

e ser

ies n

2

20406080

100120


123456times10

minus7




Disc

rete

-tim

e ser

ies n

2

100120

4020

8060

24681012times10

minus8


Disc

rete

-tim

e ser

ies n

220406080

100120


1234567times10

minus7



Disc

rete

-tim

e ser

ies n

2

20406080

100120

12345


times10minus8


Disc

rete

-tim

e ser

ies n

2

20406080

100120

12345


times10minus7



Disc

rete

-tim

e ser

ies n

2

20406080

100120


1

2

3

4

times10minus9


Disc

rete

-tim

e ser

ies n

2

20406080

100120


123456times10

minus7








minus60

minus50

minus40

minus30

minus20

minus10

0

Out

put p

ower

(dB)

120 140 160 180 200100Range cell



minus80

minus70

minus60

minus50

minus40

minus30

minus20

Out

put p

ower

(dB)

120 140 160 180 200100Range cell





45

50

55

60

65

70

75

80

85

IF (d

B)





0

5

10

15

20

25Pr

imar

y an

d se

cond

ary

peak

s rat

io (d

B)




7 Conclusions




0

5

10

15

20

25

30

OU

TSC

R (d

B)




0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f det

ectio

n







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





iscre

te-ti

me s

erie

s n2

100120

604020

80


12108642

14times10

minus9


Disc

rete

-tim

e ser

ies n

2

20406080

100120


123456times10

minus7




Disc

rete

-tim

e ser

ies n

2

100120

4020

8060

24681012times10

minus8


Disc

rete

-tim

e ser

ies n

220406080

100120


1234567times10

minus7



Disc

rete

-tim

e ser

ies n

2

20406080

100120

12345


times10minus8


Disc

rete

-tim

e ser

ies n

2

20406080

100120

12345


times10minus7



Disc

rete

-tim

e ser

ies n

2

20406080

100120


1

2

3

4

times10minus9


Disc

rete

-tim

e ser

ies n

2

20406080

100120


123456times10

minus7








minus60

minus50

minus40

minus30

minus20

minus10

0

Out

put p

ower

(dB)

120 140 160 180 200100Range cell



minus80

minus70

minus60

minus50

minus40

minus30

minus20

Out

put p

ower

(dB)

120 140 160 180 200100Range cell





45

50

55

60

65

70

75

80

85

IF (d

B)





0

5

10

15

20

25Pr

imar

y an

d se

cond

ary

peak

s rat

io (d

B)




7 Conclusions




0

5

10

15

20

25

30

OU

TSC

R (d

B)




0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f det

ectio

n







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






minus60

minus50

minus40

minus30

minus20

minus10

0

Out

put p

ower

(dB)

120 140 160 180 200100Range cell



minus80

minus70

minus60

minus50

minus40

minus30

minus20

Out

put p

ower

(dB)

120 140 160 180 200100Range cell





45

50

55

60

65

70

75

80

85

IF (d

B)





0

5

10

15

20

25Pr

imar

y an

d se

cond

ary

peak

s rat

io (d

B)




7 Conclusions




0

5

10

15

20

25

30

OU

TSC

R (d

B)




0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f det

ectio

n







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






0

5

10

15

20

25

30

OU

TSC

R (d

B)




0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty o

f det

ectio

n







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




a novel interference detection method of stap based on

Documents