An Efficient ΣΔ-STAP Detector for Radar Seeker using RPCA Post-processing

Adaptive detection of moving targets in sea clutter environment is considered as one of the crucial tasks for radar seekers. Due to the severe spreading of the sea clutter spectrum, the ability of space-time adaptive processing with sum and difference beams (ΣΔ-STAP) algorithms to suppress the sea clutter is very limited. This paper, investigated the low-rank property of the range-Doppler data matrix according to the eigenvalue distribution from the eigen spectrum, and proposed an efficient ΣΔ-STAP detector based on the robust principle component analysis (RPCA) algorithm to detect moving targets, which meets the low-rank matrix recovery conditions. The proposed algorithm first adopts ΣΔ-STAP algorithm to preprocess the sea clutter, then separates the sparse matrix of target component from the range-Doppler data matrix through the RPCA algorithm, and finally, effectively detects moving targets in the range-Doppler plane. Simulation results demonstrate the effectiveness and robustness of the proposed algorithm in the low signal-to-noise ratio scenarios.


Keywords:    Space-time adaptive processing,  low-rank matrix recovery,  principal component analysis,   accelerated proximal gradient 

In recent years, great development has been achieved in adaptive radar detection under Gaussian and compound-Gaussian clutter environment1-5. Many detectors have been used in radar signal processing as powerful tools of moving target detection, and have a constant false alarm rate (CFAR) property under homogenous data assumption, such as the generalized likelihood ratio test (GLRT)1-2, the adaptive matched filter (AMF)1, and the normalized adaptive matched filter (NAMF)5, and so on. However, the heterogeneous data will result in performance degradation of these detectors in a real environment, since they cannot estimate the clutter covariance matrix accordingly such that greatly mitigates their CFAR properties4. To overcome these problems, robust algorithm design in improving the detection performance is practical for space-time adaptive processing (STAP) techniques6-9.

Classical principle component analysis (PCA) has been widely used in data analysis and compression as one of the most popular tools10-11. PCA mainly studies the exact recovery problem from a corrupted low-rank data owing to small errors and noise, and provides the optimal estimation of the lower-dimensional subspace from the observed data. However, PCA cannot effectively deal with incomplete or missing real-world data under large corruption. Recently, a new theoretical framework, called the robust principle component analysis (RPCA), has been proposed for corrupted low-rank data recovery12-13, which can be applied in many engineering domains, such as background modeling, image processing, and face recognition, and so on. The variant of the Douglas-Rachford splitting method (VDRSM) is used to solve the recovery problem for object detection by exploiting the separable structure in both objective function and the constraint14. Principal component pursuit (PCP) is successfully applied to separate ground clutter and moving target in heterogeneous environments15. However, the velocity of moving target cannot be estimated due to the inexact extraction.

In this paper, authors find that the range-Doppler data matrix has low-rank property by analyzing its eigenvalue distribution, and consider to incorporate this property into STAP framework to further improve the performance of moving target detection. Therefore, authors extend the idea of the RPCA algorithm to space-time adaptive processing with sum and difference beams (ΣΔ-STAP)16, and propose an efficient ΣΔ-STAP detector based on the RPCA algorithm to detect moving targets, which meets the low-rank matrix recovery conditions. The proposed algorithm can accurately separate the sparse matrix of moving target from the range-Doppler data matrix after ΣΔ-STAP processing with the NAMF detector, which has the advantages in preserving the target signal during the process of clutter suppression. Simulation results show that the proposed algorithm greatly improves the performance of moving target detection in radar seeker, and also performs robustly in the case of low signal-to-noise ratio (SNR).

Authors consider a radar seeker with the sum and difference beams that transmits a sequence of M coherent pulses during the coherent processing interval (CPI) and samples the radar returns on sum-channel and delta-channel. For each pulse, it collects K temporal samples from each channel, where each temporal sample corresponds to a range cell. The entire received data can therefore be organized in a three-dimensional data cube denoted as X=[x, x 1 ,L, x K ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiwaiaab2 dacaqGBbGaamiEaiaacYcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGa aiilaiaadYeacaGGSaGaamiEamaaBaaaleaacaWGlbaabeaakiaab2 faaaa@42CA@  , where x 2M×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaCiEaiabgI GiolablkqiJoaaCaaaleqabaGaaGOmaiaad2eacqGHxdaTcaaIXaaa aaaa@400B@ represents the test data in the cell under test (CUT), x k 2M×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiEamaaBa aaleaacaWGRbaabeaakiabgIGiolablkqiJoaaCaaaleqabaGaaGOm aiaad2eacqGHxdaTcaaIXaaaaaaa@412B@ , k=1,2,,K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2 da9iaaigdacaGGSaGaaGOmaiaacYcacqWIVlctcaGGSaGaam4saaaa @3E21@ , denotes the secondary samples, and MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSOaHmkaaa@373D@ stands for the complex number field. Sum-channel data x Σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiEamaaBa aaleaacqqHJoWuaeqaaaaa@3A4D@ and delta-channel data x Δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8rrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxij=hbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeGabaa3ii aabIhadaWgaaWcbaGaeuiLdqeabeaaaaa@3A9D@ can be expressed as follows:

x Σ = [ x Σ1 ,, x Σm ,, x ΣM ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiEamaaBa aaleaacqqHJoWuaeqaaOGaeyypa0Jaai4waiaadIhadaWgaaWcbaGa eu4OdmLaaGymaaqabaGccaGGSaGaeS47IWKaaiilaiaadIhadaWgaa WcbaGaeu4OdmLaamyBaaqabaGccaGGSaGaeS47IWKaaiilaiaadIha daWgaaWcbaGaeu4OdmLaamytaaqabaGccaGGDbWaaWbaaSqabeaaca WGubaaaaaa@4F63@         (1)

x Δ = [ x Δ1 ,, x Δm ,, x ΔM ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiEamaaBa aaleaacqqHuoaraeqaaOGaaeypaiaacUfacaWG4bWaaSbaaSqaaiab fs5aejaaigdaaeqaaOGaaiilaiabl+UimjaacYcacaWG4bWaaSbaaS qaaiabfs5aejaad2gaaeqaaOGaaiilaiabl+UimjaacYcacaWG4bWa aSbaaSqaaiabfs5aejaad2eaaeqaaOGaaiyxamaaCaaaleqabaGaam ivaaaaaaa@4EA5@       (2)

where x Σm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacqqHJoWucaWGTbaabeaaaaa@3B41@ , x Δm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacqqHuoarcaWGTbaabeaaaaa@3B23@ are the mth elements of the sum-channel data and delta-channel data, respectively, m = 1, 2,...,M, and (.)T denotes the transpose. The test data at a range cell can be rearranged as

x=[ x Σ x Δ ]= [ x Σ1 ,, x Σm ,, x ΣM , x Δ1 ,, x Δm ,, x ΔM ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiEaiaab2 dadaWadaabaeqabaGaaeiEamaaBaaaleaacqqHJoWuaeqaaaGcbaGa aeiEamaaBaaaleaacqqHuoaraeqaaaaakiaawUfacaGLDbaacqGH9a qpcaGGBbGaamiEamaaBaaaleaacqqHJoWucaaIXaaabeaakiaacYca cqWIVlctcaGGSaGaamiEamaaBaaaleaacqqHJoWucaWGTbaabeaaki aacYcacqWIVlctcaGGSaGaamiEamaaBaaaleaacqqHJoWucaWGnbaa beaakiaacYcacaWG4bWaaSbaaSqaaiabfs5aejaaigdaaeqaaOGaai ilaiabl+UimjaacYcacaWG4bWaaSbaaSqaaiabfs5aejaad2gaaeqa aOGaaiilaiabl+UimjaacYcacaWG4bWaaSbaaSqaaiabfs5aejaad2 eaaeqaaOGaaiyxamaaCaaaleqabaGaamivaaaaaaa@6744@         (3)

In the line-of-sight (LOS) direction, the steering vector of the sum-channel and delta-channel can be written as
s=[ s Σ s Δ ]= [ s Σ1 ,, s Σm ,, s ΣM , s Δ1 ,, s Δm ,, s ΔM ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiEaiaab2 dadaWadaabaeqabaGaaeiEamaaBaaaleaacqqHJoWuaeqaaaGcbaGa aeiEamaaBaaaleaacqqHuoaraeqaaaaakiaawUfacaGLDbaacqGH9a qpcaGGBbGaamiEamaaBaaaleaacqqHJoWucaaIXaaabeaakiaacYca cqWIVlctcaGGSaGaamiEamaaBaaaleaacqqHJoWucaWGTbaabeaaki aacYcacqWIVlctcaGGSaGaamiEamaaBaaaleaacqqHJoWucaWGnbaa beaakiaacYcacaWG4bWaaSbaaSqaaiabfs5aejaaigdaaeqaaOGaai ilaiabl+UimjaacYcacaWG4bWaaSbaaSqaaiabfs5aejaad2gaaeqa aOGaaiilaiabl+UimjaacYcacaWG4bWaaSbaaSqaaiabfs5aejaad2 eaaeqaaOGaaiyxamaaCaaaleqabaGaamivaaaaaaa@6744@          =[ s Σ 0 ]= [ s Σ1 ,, s Σm ,, s ΣM , s Δ1 ,, s Δm ,, s ΔM ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiEaiaab2 dadaWadaabaeqabaGaaeiEamaaBaaaleaacqqHJoWuaeqaaaGcbaGa aeiEamaaBaaaleaacqqHuoaraeqaaaaakiaawUfacaGLDbaacqGH9a qpcaGGBbGaamiEamaaBaaaleaacqqHJoWucaaIXaaabeaakiaacYca cqWIVlctcaGGSaGaamiEamaaBaaaleaacqqHJoWucaWGTbaabeaaki aacYcacqWIVlctcaGGSaGaamiEamaaBaaaleaacqqHJoWucaWGnbaa beaakiaacYcacaWG4bWaaSbaaSqaaiabfs5aejaaigdaaeqaaOGaai ilaiabl+UimjaacYcacaWG4bWaaSbaaSqaaiabfs5aejaad2gaaeqa aOGaaiilaiabl+UimjaacYcacaWG4bWaaSbaaSqaaiabfs5aejaad2 eaaeqaaOGaaiyxamaaCaaaleqabaGaamivaaaaaaa@6744@         (4)
where s Σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8rrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxij=hbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqGZb WaaSbaaSqaaiabfo6atbqabaaaaa@39CF@ is the steering vector of the sum-channel, and s Δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaae4CamaaBa aaleaacqqHuoaraeqaaaaa@3A2A@ is the steering vector of the delta-channel. Note that the response of the delta-channel in a certain direction is usually null, so the steering vector s Δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaae4CamaaBa aaleaacqqHuoaraeqaaaaa@3A2A@ can be assumed to be zero.
Now we address the detection problem, which can be formulated in terms of the following binary hypothesis test

{ H 0 :x=c+n H 1 :x=d+c+n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe qaaiaabIeadaWgaaWcbaGaaeimaaqabaGccaqG6aGaaeiEaiaab2da caqGJbGaae4kaiaab6gaaeaacaqGibWaaSbaaSqaaiaabgdaaeqaaO GaaeOoaiaabIhacaqG9aGaaeizaiaabUcacaqGJbGaae4kaiaab6ga aaGaay5Eaaaaaa@47BB@         (5)

where x, c, n, and d are the test data, clutter, noise and signal, respectively. The signal d can be modeled as d=αs MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeizaiaab2 dacqaHXoqycaqGZbaaaa@3BDE@ , where s is the steering vector and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3941@ is the unknown complex amplitude.

Assuming that the clutter covariance matrix R is known, the NAMF detector is given by

Λ= | s H R -1 s | 2 ( s H R -1 s )( s H R -1 s ) ¤ H 0 H 1 η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeu4MdWKaae ypamaalaaabaWaaqWaaeaacaqGZbWaaWbaaSqabeaacaWGibaaaOGa aeOuamaaCaaaleqabaGaaeylaiaabgdaaaGccaqG4baacaGLhWUaay jcSdWaaWbaaSqabeaacaqGYaaaaaGcbaWaaeWaaeaacaqGZbWaaWba aSqabeaacaWGibaaaOGaaeOuamaaCaaaleqabaGaaeylaiaabgdaaa GccaqGZbaacaGLOaGaayzkaaWaaeWaaeaacaqG4bWaaWbaaSqabeaa caWGibaaaOGaaeOuamaaCaaaleqabaGaaeylaiaabgdaaaGccaqG4b aacaGLOaGaayzkaaaaamaaxadabaGaaiiPaaWcbaGaamisamaaBaaa meaacaaIWaaabeaaaSqaaiaadIeadaWgaaadbaGaaGymaaqabaaaaO Gaeq4TdGgaaa@57E0@        (6)

where ( · ) H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq WIpM+zaiaawIcacaGLPaaadaahaaWcbeqaaiaadIeaaaaaaa@3C95@ denotes the conjugate transpose, and η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeq4TdGgaaa@394E@ is the detection threshold9. Then, the test statistic of the NAMF detector is compared with a corresponding threshold to determine whether a target is present or not.
The NAMF detector needs to estimate the clutter covariance matrix R in Eqn. (6). However, the NAMF detector has a great loss in performance due to limited sample support, which results in inaccurate estimation of the clutter covariance matrix, especially in heterogeneous environment. In the conventional STAP algorithms, the clutter covariance matrix R can be obtained by the maximum likelihood (ML) estimator which makes use of secondary samples from adjacent range cells to estimate the unknown clutter covariance matrix6-7, and the sample covariance matrix R ˆ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabeOuayaaja Gaeyypa0ZaaSaaaeaacaaIXaaabaGaam4saaaadaaeWbqaaiaadIha caWG4bWaaWbaaSqabeaacaWGibaaaaqaaiaadUgacqGH9aqpcaaIXa aabaGaam4saaqdcqGHris5aaaa@43D4@ is estimated by

R ˆ = 1 K k = 1 K s s H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabeOuayaaja Gaeyypa0ZaaSaaaeaacaaIXaaabaGaam4saaaadaaeWbqaaiaadIha caWG4bWaaWbaaSqabeaacaWGibaaaaqaaiaadUgacqGH9aqpcaaIXa aabaGaam4saaqdcqGHris5aaaa@43D4@        (7)
The NAMF detector has excellent capability of sidelobe clutter suppression but at the cost of low target sensitivity. Therefore, it is not sensitive to the influence of any signal mismatch where the actual signal is not aligned with the presumed steering vector. Since the sidelobe clutter spreads severely in range-Doppler plane, the NAMF detector is exploited to preprocess the sidelobe clutter.

Recently, the RPCA algorithm is proposed that can accurately recover the low-rank component and the sparse component of observed data corrupted by large errors and noise, and has obvious advantage over classical PCA algorithm in the exact recovery problem. After STAP processing with the NAMF detector, we can get a range-Doppler spectrum image as a data matrix, where moving target has the sparsity in the range-Doppler domain while sea clutter forms a relatively low-rank property. Hence, the RPCA algorithm can be applied to detect moving targets in the range-Doppler plane by solving the convex optimization problem. Meanwhile, the recovery problem can be seen as a semi-definite programming (SDP) problem and solved by the accelerated proximal gradient (APG) algorithm13.

3.1   A. RPCA algorithm for STAP

After estimating the covariance matrix R ˆ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabeOuayaaja Gaeyypa0ZaaSaaaeaacaaIXaaabaGaam4saaaadaaeWbqaaiaadIha caWG4bWaaWbaaSqabeaacaWGibaaaaqaaiaadUgacqGH9aqpcaaIXa aabaGaam4saaqdcqGHris5aaaa@43D4@ in equation (7), we can obtain the range-Doppler data matrix D using the test statistic Λ from Eqn. (6). For example, the (k,i) entry of matrix D is calculated by employing the steering vector s at the k-th frequency to the data x of the i-th range cell. The detailed process about how to produce range-Doppler data in the ΣΔ-STAP can be found16,18. The new matrix D M×K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiraiabgI GiolablkqiJoaaCaaaleqabaGaamytaiabgEna0kaadUeaaaaaaa@3F2A@ formed by the range-Doppler data matrix can be decomposed into two matrices, named the low-rank matrix and the sparse matrix, which respectively correspond to the sea clutter component and the moving target one in our problem. Then the new matrix D has the form

D=L+S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiraiaab2 dacaqGmbGaae4kaiaabofaaaa@3B7C@        (8)

where L M×K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeitaiabgI GiolablkqiJoaaCaaaleqabaGaamytaiabgEna0kaadUeaaaaaaa@3F32@ is a low-rank matrix, and S M×K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaae4uaiabgI GiolablkqiJoaaCaaaleqabaGaamytaiabgEna0kaadUeaaaaaaa@3F39@ is a sparse matrix.
Taking the singular value decomposition (SVD) of D, we have

D=UΣ V H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiraiaab2 dacaqGvbGaeu4OdmLaaeOvamaaCaaaleqabaGaamisaaaaaaa@3D58@        (9)

where U M×M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeyvaiabgI GiolablkqiJoaaCaaaleqabaGaamytaiabgEna0kaad2eaaaaaaa@3F3D@ and V K×K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOvaiabgI GiolablkqiJoaaCaaaleqabaGaam4saiabgEna0kaadUeaaaaaaa@3F3A@ are the orthogonal matrices, and Σ M×K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeu4OdmLaey icI4SaeSOaHm6aaWbaaSqabeaacaWGnbGaey41aqRaam4saaaaaaa@3FE7@ is the diagonal matrix.
Assume that land S can be represented as follows

L= i=1 r σ i u i v i H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamitaiabg2 da9maaqahabaGaeq4Wdm3aaSbaaSqaaiaadMgaaeqaaOGaamyDamaa BaaaleaacaWGPbaabeaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaadk haa0GaeyyeIuoakiaadAhadaqhaaWcbaGaamyAaaqaaiaadIeaaaaa aa@473E@         (10)

S= i>r σ i u i v i H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4uaiabg2 da9maaqafabaGaeq4Wdm3aaSbaaSqaaiaadMgaaeqaaOGaamyDamaa BaaaleaacaWGPbaabeaakiaadAhadaqhaaWcbaGaamyAaaqaaiaadI eaaaaabaGaamyAaiabg6da+iaadkhaaeqaniabggHiLdaaaa@466D@        (11)

where rmin{M,K} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaiabgs MiJkGac2gacaGGPbGaaiOBaiaacUhacaWGnbGaaiilaiaadUeacaGG 9baaaa@4172@ denotes the rank of L as r=rank(L) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2 da9iaadkhacaWGHbGaamOBaiaadUgacaGGOaGaamitaiaacMcaaaa@3F89@ , u 1 ,, u r M×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaaIXaaabeaakiaacYcacqWIVlctcaGGSaGaamyDamaaBaaa leaacaWGYbaabeaakiabgIGiolablkqiJoaaCaaaleqabaGaamytai abgEna0kaaigdaaaaaaa@45AE@ and v 1 ,, v r M×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacaaIXaaabeaakiaacYcacqWIVlctcaGGSaGaamODamaaBaaa leaacaWGYbaabeaakiabgIGiolablkqiJoaaCaaaleqabaGaamytai abgEna0kaaigdaaaaaaa@45B0@ are two sets of the singular vectors of U and V, and σ 1 ,, σ r 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdaaeqaaOGaaiilaiabl+UimjaacYcacqaHdpWCdaWg aaWcbaGaamOCaaqabaGccqGHLjYScaaIWaaaaa@4314@ are the singular values of Σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeu4Odmfaaa@3926@ , respectively.
Then, to separate the low-rank matrix and the sparse matrix, we can solve the following convex optimization problem

min L,S L +δ S 1 s.t. D=L+S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyAaiaac6gaaSqaaiaadYeacaGGSaGaam4uaaqabaGcdaqb daqaaiaadYeaaiaawMa7caGLkWoadaWgaaWcbaGaey4fIOcabeaaki abgUcaRiabes7aKnaafmaabaGaam4uaaGaayzcSlaawQa7amaaBaaa leaacaaIXaaabeaakmaaBaaaleaaaeqaaOWaaubeaeqaleaaaeqane aaaaGccaqGZbGaaeOlaiaabshacaqGUaWaaubeaeqaleaaaeqaneaa aaGccaWGebGaeyypa0JaamitaiabgUcaRiaadofaaaa@5223@         (12)

where · MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaauWaaeaacq WIpM+zaiaawMa7caGLkWoadaWgaaWcbaGaey4fIOcabeaaaaa@3E54@ and · 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaauWaaeaacq WIpM+zaiaawMa7caGLkWoadaWgaaWcbaGaaGymaaqabaaaaa@3E20@ denote the nuclear norm and 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeS4eHW2aaS baaSqaaiaaigdaaeqaaaaa@37FE@ norm, respectively. δ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaey Opa4JaaGimaaaa@3B09@ is the weighted parameter for balancing that scales as 1/ N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca aIXaaabaWaaOaaaeaacaWGobaaleqaaaaaaaa@3961@ 12, and selection of the appropriate parameter is discussed in the later Section. Actually, the convex optimization algorithm addressed in Eqn. (12) is usually intractable in theory and practice. Instead of directly solving the Eqn. (12), we can solve the following dual problem equivalently:

min L,S L +δ S 1 + 1 2μ MLS F 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyAaiaac6gaaSqaaiaadYeacaGGSaGaam4uaaqabaGcdaqb daqaaiaadYeaaiaawMa7caGLkWoadaWgaaWcbaGaey4fIOcabeaaki abgUcaRiabes7aKnaafmaabaGaam4uaaGaayzcSlaawQa7amaaBaaa leaacaaIXaaabeaakiabgUcaRmaalaaabaGaaGymaaqaaiaaikdacq aH8oqBaaWaauWaaeaacaWGnbGaeyOeI0IaamitaiabgkHiTiaadofa aiaawMa7caGLkWoadaqhaaWcbaGaamOraaqaaiaaikdaaaaaaa@56F3@        (13)

where · F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaauWaaeaacq WIpM+zaiaawMa7caGLkWoadaWgaaWcbaGaamOraaqabaaaaa@3E30@ denotes the Frobenius norm, and μ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey Opa4JaaGimaaaa@3B1A@ is the penalty parameter for the violation of the linear constraint.
The APG algorithm is a fast algorithm to solve this dual problem, and its MATLAB codes are available online17.

3.2   Algorithm description

The procedure of the proposed algorithm is described in detail as follows. We first divide the complex matrix into two matrices according to its real and imaginary parts, and then use the APG algorithm to solve these two dual problems formed by real and imaginary matrices, respectively. After the separation of the real and imaginary parts of the sparse matrix, we unite them into the complex sparse matrix of moving target. Then, we can detect moving targets in the range-Doppler plane.

Now the procedures of the proposed algorithm are given as :
Step (1)    Use the ML estimator in Eqn (7) to estimate the clutter covariance matrix R ˆ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabeOuayaaja Gaeyypa0ZaaSaaaeaacaaIXaaabaGaam4saaaadaaeWbqaaiaadIha caWG4bWaaWbaaSqabeaacaWGibaaaaqaaiaadUgacqGH9aqpcaaIXa aabaGaam4saaqdcqGHris5aaaa@43D4@ from the secondary samples;
Step (2)    Generate a range-Doppler data matrix D by the NAMF detector in Eqn (6);
Step (3)    Divide the complex matrix D into real and imaginary matrices, D R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiramaaBa aaleaacaWGsbaabeaaaaa@396C@ , D I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeiramaaBa aaleaacaWGjbaabeaaaaa@3963@ ;
Step (4)     Use the APG algorithm to solve the two dual problems in Eqn (13), and obtain two sparse real matrices after low-rank and sparse matrices separation, S R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaae4uamaaBa aaleaacaWGsbaabeaaaaa@397B@ , S I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaae4uamaaBa aaleaacaWGjbaabeaaaaa@3972@ ;
Step (5)     Unite the real and imaginary matrices into the sparse complex matrix, S= S R +i* S I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbiqaaaMecaqGtb Gaeyypa0Jaae4uamaaBaaaleaacaWGsbaabeaakiabgUcaRiaadMga caGGQaGaae4uamaaBaaaleaacaWGjbaabeaaaaa@4024@ , and get the final moving target detection result.

Since the iterative process is involved in the APG algorithm, more computation is required in the proposed algorithm than that in the original ΣΔ-STAP detector. The details of computational performance regarding the APG algorithm are additionally discussed and compared17.


Authors used simulated sea clutter data to validate the performance of target detection in radar seekers. Simulation parameters are set as follows: PRF = 1 kHz, the platform velocity v=2000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2 da9iaaikdacaaIWaGaaGimaiaaicdaaaa@3AD1@ m/s, the number of pulses in a CPI M=32 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytaiabg2 da9iaaiodacaaIYaaaaa@3AF3@ , the number of range cells K=512 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9iaaiwdacaaIXaGaaGOmaaaa@3BAE@ , the antenna scanning angle is 2°, the shape parameter of the K-distributed sea clutter is set as ω=2.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdCNaey ypa0JaaGOmaiaac6cacaaI1aaaaa@3CA2@ , the clutter-to-noise ratio (CNR) is 50dB, and the root mean square (RMS) of sea clutter velocity dispersion σ sea =0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadohacaWGLbGaamyyaaqabaGccqGH9aqpcaaIWaGaaiOl aiaaiwdaaaa@3F94@ m/s. Meanwhile, there are two moving targets inserted into the simulated sea clutter data, and their parameters are listed in Table 1.


Table 1. Target parameters for numerical simulation


Firstly, we illustrate the low-rank matrix property of the range-Doppler data matrix. Fig. 1 shows the eigenvalue distribution of the observed range-Doppler data matrix. There are exactly 32 large eigenvalues corresponding to the inflexion of the eigenvalue distribution curve, and others are relatively small compared with 32 large eigenvalues. Therefore, it is reasonable that the proposed algorithm with sparse recovery can suppress the clutter properly in this scenario.




Figure 1. Eigenvalue distribution for the range-Doppler data matrix.


Then, we compare the performance of moving target detection in terms of different SNRs and weighted parameters according to the Monte Carlo simulations3. Figure 2 displays the test statistic of two moving targets with the parameter N for different SNRs, where N=1/ δ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2 da9maalyaabaGaaGymaaqaaiabes7aKnaaCaaaleqabaGaaGOmaaaa aaaaaa@3CDA@ ranges from 32 to 512. As the value of N becomes larger, the test statistic of moving target I keeps invariant at approximately the same rate for different SNRs in Fig. 2(a). While the case in Fig. 2(b) is different from that in Fig. 2(a), the test statistic of moving target II increases with N, and gradually to the extreme point 1. The reason of this phenomenon is that the strong clutter has influence on the result of targets detection.

Some moving targets may not be detected if N is too small ( δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiTdqgaaa@3947@ too large)15. Also, some clutter residue still exists, meaning that the setting of weighted parameter δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiTdqgaaa@3947@ is very important for the performance of moving target detection.




Figure 2. Performance of target detection with the proposed algorithm: (a) Simulation results of target I and (b) Simulation results of target II.


Hence, we can make a trade off between clutter suppression and target detection. However, in our simulation, the parameter N has little influence on moving target with positive Doppler frequency in range-Doppler area, and a bit influence on moving target with negative Doppler frequency. That is to say, the parameter N selection has little effect on the detection performance, so it is almost negligible, where δ=1/ 32 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacq aH0oazcqGH9aqpcaaIXaaabaWaaOaaaeaacaaIZaGaaGOmaaWcbeaa aaaaaa@3CB2@ ( N=min{M,L} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2 da9iGac2gacaGGPbGaaiOBaiaacUhacaWGnbGaaiilaiaadYeacaGG 9baaaa@40A0@ ) is selected in the simulation.

Next, the sea clutter range-Doppler spectrum with the scanning angle 2° in radar seeker is shown in Fig. 3(a), and the phenomenon analysis can be referred in18. Because the sea clutter spectrum severely spreads in a large part of the range-Doppler plane, two moving targets easily fall into strong clutter. In this case, radar seeker can difficultly detect moving targets. Meanwhile, we plot the curves of test statistic using the NAMF detector and the proposed algorithm in Figs. 3(b), 3(c) and 3(d). Figure 3(b) shows that the NAMF detector can detect two moving targets, but cannot effectively suppress the strong sea clutter, which may lead to high probability of false alarm and the failure of detecting moving targets with a small RCS. The proposed algorithm can also effectively detect two moving targets accompanied with clutter suppression from Figs. 3(c) and 3(d), and have a better performance in suppressing clutter than the NAMF detector. Therefore, the simulation results in Fig. 3 demonstrate the effectiveness and advantages of the proposed algorithm.






Figure 3. Simulation results with the NAMF detector and the proposed algorithm. (a) Range-Doppler spectrum for sea clutter data, (b) Test statistic of the NAMF detector, (c) Test statistic of target component with the proposed algorithm, and (d) Test statistic of clutter component with the proposed algorithm.


Finally, we compare two algorithms in terms of detection performance using 100 snapshots as the secondary samples with 200/ P fa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca aIYaGaaGimaiaaicdaaeaacaWGqbWaaSbaaSqaaiaadAgacaWGHbaa beaaaaaaaa@3CBA@ Monte Carlo trials. To reduce the computational burden, the probability of false alarm (Pfa) is set as P fa = 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGMbGaamyyaaqabaGccqGH9aqpcaaIXaGaaGimamaaCaaa leqabaGaeyOeI0IaaG4maaaaaaa@3ED0@ and the corresponding threshold is evaluated. For the sake of convenient simulation, the number of moving targets added in the simulated sea clutter is 20, where half of them are set as the parameters of target I listed in Table 1, and others are set as the parameters of target II. The probability of detection (Pd) is computed as the ratio between the number of detectable targets and the total number of targets. Figure 4 presents the performance of moving target detection. It can be clearly observed that the proposed algorithm outperforms the NAMF detector about 10 dB in improving the performance of moving target detection at P d =0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGKbaabeaakiabg2da9iaaicdacaGGUaGaaGynaaaa@3CC7@ .


Figure 4. Pd versus SNR with P fa = 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGMbGaamyyaaqabaGccqGH9aqpcaaIXaGaaGimamaaCaaa leqabaGaeyOeI0IaaG4maaaaaaa@3ED0@ .

In this paper, an efficient ΣΔ-STAP detector based on the low-rank matrix recovery for moving target detection was proposed in radar seeker. Compared with ΣΔ-STAP processing with NAMF detector, the proposed algorithm can compensate the deficiency of insufficient clutter suppression, and effectively detect moving targets in the range-Doppler plane. Meanwhile, the proposed algorithm can avoid the application of lots of homogeneous samples for STAP training, which is practical in application.

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Mr Zhao Lei

Mr Zhao Lei received the MS (circuits and systems) from Xidian University, Xi’an, China, in 2009. He is currently pursuing his PhD (Electrical engineering) at Beihang University, Beijing, China. His current research interests include: Space-time adaptive processing, synthetic aperture radar image interpretation, and digital image processing.

Dr Luo Xiling

Dr Luo Xiling received the BE and PhD from Beihang University, Beijing, China, in 1996 and 2003, respectively. He is presently working as an Associate Professor at the School of Electronics and Information Engineering, Beihang University, Beijing, China. He has published 20 research papers, co-edited one book. His research interests include air traffic management and radar data processing.

Dr Sun Jinping

Dr Sun Jinping received the MSc and PhD from Beihang University (BUAA), Beijing, China, in 1998 and 2001, respectively. He is currently a Professor with the School of Electronic and Information Engineering, BUAA. His research interests include : High-resolution radar signal processing, image understanding, and robust beamforming

Mr Lu Songtao

Mr Lu Songtao received his MS (Communication and information engineering) from Beihang University, Beijing, China, in 2011. He is currently pursuing the PhD in the Department of Electrical and Computer Engineering at Iowa State University, USA. His primary research interests include : Statistical signal processing, interference mitigation and wireless communications.