School of Information Science and Engineering,Yan Shan University,Qinhuangdao 066004,China

SAR imaging method based on coprime sampling and nested sparse sampling

Hongyin Shi*and Baojing Jia

School of Information Science and Engineering,Yan Shan University,Qinhuangdao 066004,China

As the signal bandwidth and the number of channels increase,the synthetic aperture radar(SAR)imaging system produces huge amount of data according to the Shannon-Nyquist theorem,causing a huge burden for data transmission.This paper concerns the coprime sampling and nested sparse sampling, which are proposed recently but have never been applied to real world for target detection,and proposes a novel way which utilizes these new sub-Nyquist sampling structures for SAR sampling in azimuth and reconstructs the data of SAR sampling by compressive sensing(CS).Both the simulated and real data are processed to test the algorithm,and the results indicate the way which combines these new undersampling structures and CS is able to achieve the SAR imaging effectively with much less data than regularly ways required.Finally,the influenc of a little sampling jitter to SAR imaging is analyzed by theoretical analysis and experimental analysis,and then it concludes a little sampling jitter have no effect on image quality of SAR.

synthetic aperture radar(SAR)imaging,compressive sensing,coprime sampling,nested sparse sampling.

1.Introduction

In synthetic aperture radar(SAR),the classical sampling basic is the Shannon-Nyquist theorem.However,the traditional SAR data acquisition increases the scale of SAR system forobtainingmoreandmoredata.Thus someproblems such as data processing and transmission in real time would occur[1].For example,for high-resolution spaceborne SAR image,the data would several gigabytes,and requires a satellite-earth datalink reach of over Gbps data rate,and such high data rate is difficul to achieve.Because the required number of frequencies is usually very large with the traditional sampling theory,the application of stepped-frequency waveform is also limited in SAR.It becomes a central objective in signal processing to reduce the amount of data of SAR sampling without loss of meaningful information.Fortunately,in recent years,an explosion of theoretical and computational methods have been developed primarily to study how to recovery the information using a fewer samples.This has given rise to the possibility of reducing the number of measurements by down samplingthe data,which automaticallygives rise to underdetermined systems.

A new approach to undersampling using coprime sampling and nested sparse sampling is provided by[2,3].The authors has already provedthat these two new sub-Nyquist samplingalgorithmscanachieveenhanceddegreesoffreedom(DOF)in the coarray domain for applications such as direction of arrival(DOA)estimation and spectrum estimation[4,5].The number of DOF is an important criteria because more DOF mean more sources can be resolved by the system.In order to exploit these increased DOF of the coarray,the authors has developed a new algorithm which can achieves the maximum identifiabilit among all DOA estimation methods using 2qth-order cumulants[6]. In[7],Chen et al.utilized the nested sparse sampling and coprime sampling to estimate the quadrature phase shift keying(QPSK)signal’s autocorrelationandpowerspectral density(PSD)with a set of sparse samples,and Wu et al. provedthat the mainlobeof PSD becamenarrowerthanthe original signal as the sampling intervals increased.In[8], Wu et al.applied the coprime sampling and nested sparse sampling to ultra wideband(UWB)radar sensor networks (RSN),andprovedthatbothofthesesub-Nyquistsampling algorithms can work better than the uniform undersampling does,especially when the signal quality is poor.In this paper,coprime sampling and nested sparse sampling will be utilized for SAR sampling in azimuth.These sampling strategies can drastically reduce more redundancy data of SAR sampling than the Shannon-Nyquist theorem does.

Compressive sensing(CS)[9,10]provides us a new point of view,which could only use much less samplesto perfectly recover the original signal at a high compression ratio.CS theory has been explored for a wide range of radar imaging applications.In order to acquire superresolution imaging under limited bandwidth,a novel CS imaging framework[11]has been proposed for the purposeofimprovingtheimagingperformanceof steppedfrequency SAR.In[12],to meet the requirements of the processing power and reduce the storage,a framework which is 2D SAR imaging scheme based on CS is adopted to solve the ground reflectvity.In[13],in order to mitigate azimuthambiguities,the authorsutilized CS to spaceborne stripmap SAR images.In addition,various imaging methods based on CS were proposed for different SAR modes including spotlight SAR[14],inverse SAR[15],tomography SAR[16]and random frequency SAR[17].

In this paper,coprime sampling and nested sparse samplingwill be respectivelyutilized for SAR samplingin azimuth.Because these new sub-Nyquist sampling structures are only applied to SAR in azimuth,in order to demonstrate the validity of these methods,CS is used to pulse compression of SAR in azimuth,and the matching filte is still used to pulse compression of SAR in range.

The paper is organized as follows.Section 2 and Section 3 brieflreviews the coprime sampling and nested sparse sampling respectively.Section 4 describes SAR imaging based on CS in detail.Section 5 presents the results of both simulated and real data to validate the performance of the proposed methods,and Section 6 analyses whether the jitter of sampling will influenc the image quality of SAR.Section 7 concludes the paper with future work remarked.

2.Coprime sampling

The coprime sampling is a non-uniform sampling,which involves two sets of uniformly spaced samplers[3].Fig.1 shows this structure with sample spacing AT and BT respectively,whereAandBare coprime,1/Tis the Nyquist rate.S1(t)andS2(t)come from the firs and the second sampler.

Fig.1 Coprime sampling

Assume a coprime difference set as

Because the set includesX(k1,k2)and?X(k1,k2),its DOF is more thanmn,though there arem+nphysical sample points at most.

For the example whereA=4,B=3,m=3 andn=4,the elements of the full set are–9,–8,×,–6,–5,–4,–3,–2,–1,0,1,2,3,4,5,6,×,8,9.

Although there are 17 different freedoms here,there are“holes”in the set indicated by×.Namely,the elements–7 and 7 are missing.If we setAandBlarger,there will be more“holes”in the set.

For coprime sampling,the average sampling rate is

It can be noticed the average sampling rate of coprime sampling is much smaller than the conventional Nyquist sampling rate of 1/T.

3.Nested sparse sampling

The nested array was introduced in[5]as an effective approach to array processing with enhanced degrees of freedom[2].It also can be seen as a non-uniform sampling, using two different samplers in each period as shown in Fig.2.

Fig.2 Nested sparse sampling

The two level nested array is the simplest form with the level 1 samples at theN1locations and the level 2 samples at theN2locations.At the firs periodthe cross-differences are define as

The range ofkis?[(N1+1)N2?1]≤k≤[(N1+ 1)N2?1],and the missing elements are[(N1+1),(N1+ 1),...,(N2?1)(N1+1)].

For example,whenN1=2 andN2=3 in Fig.2,the cross-differenceskare 1,2,4,5,7,8,with 3,6 missing.

However,it should be noticed that the self differences can cover all of the missing differences,such as

Thus it can be demonstratedthat with a two-level nested array,we can attain 2N2(N1+1)?1 freedoms in the coarray using onlyN1+N2elements.

For nested sparse sampling,the average sampling rate is

It can be noticed the average sampling rate of nested sparse sampling is smaller than the conventional Nyquist sampling rate of 1/T.In a nested array the self differences are not entirely contained in the cross-differences,so additionalfreedomsaregenerated.Comparedwiththearraysof coprime sampling,the nested array indeed gives us a more effective way to increase the DOF.However,adjacent elements are spaced farther apart in co-prime arrays,which can be used to reduce mutual coupling between elements [4].

4.SAR imaging based on CS

4.1CS theory

CS is a new way in information representation introduced in recent years.In the framework of CS,sampling below the Nyquist rate may also reconstruct the signal correctly when some property of the signal is guaranteed.For a signalXwith length ofN,if there is a basisΨN×Nand it canbeexpressedasXN×1=ΨN×NSN×1,thesignalcan be projected ontoΨ,whereSis a coefficien vector andΨi=(Ψ1,Ψ2···ΨN)is aN×Nmatrix.Ifa coefficien vectorShasK(K＜＜N)nonzerovalues,Xis knownas aK-sparse signal.Then consider the measurement matrixΦM×NwithM＜N.IfXis aK-Sparse signal simultaneously,the measurement process can be described as

whereA=ΦΨis the sensing matrix.When the matrixAsatisfie the restricted isometry property and measurement timesMsatisfieM≥o(Klg4N),the signalXcan be recovered exactly fromyby solving anl1minimization problem taking by

whereεlimits the amount of noise.Currently there are many reconstruction methods to solve(6),such as orthogonal matching pursuit(OMP),basis pursuit(BP)and regularized orthogonal matching pursuit(ROMP).

4.2SAR imaging model

Suppose SAR transmits a linear frequency modulated (LFM)signal as follows:

whereA0is the amplitude,τis the fast time,Tris the time width of the chirp pulse,f0is the carrier frequency.

For a point reflecto at rangeR,the echo is

whereRis the instantaneous distance between SAR and target,andcis the speed of light.

After dechirp processing and compressing pulse in range,(8)can be written as

whereA=A0pr[τ?2R(t)/c],pr(τ)is the envelope of compression pulse andλis the wavelength of signal.

ThenSrwill beseparatelysampledbythecoprimesampling and the nested sparse sampling in azimuth.

WhenR0＞＞vtn,RCM can be ignored,so the pulse compression in azimuth can be expressed as

whereg(k,n)is the reference function,Xkis the target coordinate in azimuth,vis the platform velocity.

Equation(10)can be expressed in other form as

Then the CS model of SAR imaging can be designed by (12),and we should construct a measurement matrixΦ.It is provedthat therandommatrixperformswell,suchas the Gaussian or Bernoulli matrix.For simplicity,we choose the randommatrix asΦ.Now the CS model of SAR imaging can be expressed as

In this paper,the echo will be sampled by the new sub-Nyquist sampling structures,so we chooseMequalsN. According to(13),for one range cell,Sa(N×1)can be recovered by the convex optimization as follows:

In this paper,OMP is used to solve this problem directly.Loop through all the range cells,then the azimuthcompressed signalSa(N×L)is obtained.

5.Imaging results

5.1Simulated data

Inthissection,inorderto illustratetheperformanceofproposed methods,some simulation results will be showed to valuate the performanceof the proposedsampling scheme. The simulation parameters are shown in Table 1.

Table 1 Simulation radar parameters

Fig.3 shows the comparison of the results of simulation data obtained via different sampling methods,from which it can be noted that the proposed method showed in Fig.3(c)has the best performance.

Fig.3 Comparison of imaging results

5.2Real data

In order to illustrate the performance of proposed method, a real SAR imaging experiment is carried out as follows. The SAR experiment data are taken from RADARSAT-1. The experiment parameters are shown in Table 2.The imaged targets are four ships in sea.The total number of the samples of SAR data is 1 500×1 350.

Table 2 Experiment parameters

In this experiment,we separately sample the SAR data in azimuth by two methods.For the firs method,the data are selected by coprime sampling with sample spacingA=7,B=9 andA=17,B=19.It means that only 25%and 10%of raw data are selected.For the second method,the data are selected by nested sparse sampling withN1=6,N2=11 andN1=15,N2=23. It means that only 25%and 10%of raw data are selected. The simulation results are shown in Fig.4.

Fig.4 Imaging results of real data via different sampling methods

Fig.4(c)and Fig.4(d)show the CS reconstruction result of the firs method;Fig.4(e)and Fig.4(f)show the CS reconstruction result of the second method.It can be seen that the proposed methods are able to accurately reconstruct the targets with only part of raw data,and even show an improvement in terms of image quality with the sidelobe suppressed.

From the results of real data obtained via different sampling methods,it can be noted that the second method has better performance than the firs method,especially with 10%of the raw data.It is because that nested sparse samplingcan producemuchmore freedomsthan coprimesampling,when they have the same sample rate.

6.Influenc of sampling jitter

6.1Theoretical analysis

The speed of traditional SAR platform is uniform,however,in some special situations,it is difficul to remain in the state of uniform motion,such as the missile-borne SAR,whose speed even has a higherfluctuation When the SAR platform produces motion error,some of sampling points in azimuth are not according to coprime sampling or nested sparse sampling.

As thenestedsparsesamplingandthecoprimesampling are similar,we will use the coprime sampling to state theoretical analysis.For the coprime sampling,the number of freedoms is taken by(2),when one sampling pointAi(0≤i＜m?1)in Fig.1 has deviation with the coprime sampling,the coprime difference set can be expressed by

whereXdis the offset ofAi.

For the example introduced in Section 2,we seti=0,Xd=1,the elements of the full set are×,–8,×,–6,–5,–4,×,–2,–1,0,1,2,×,4,5,6,×,8,×.When we seti=1,the elements of the full set become–9,–8,–7,–6,–5,–4,–3,–2,–1,0,1,2,3,4,5,6,7,8,9.

Comparing with the number of freedoms with no jitter, there are 13 and 19 freedoms in the two examples respectively introduced above.When the sampling points have jitter,the number of freedoms will change.

6.2Experimental analysis

In this experiment,the part of data of SAR sampling have jitter,and then the influenc of sampling jitter to SAR imaging will be analyzed.In order to clearly observe the influenc for image quality,we only choose one of ships shown in Fig.4 as the experiment data.Because a lot of sampling jitter can be equivalent to random sampling,weonly analyze whether a little jitter will influenc the image quality of SAR.

For the array of coprime sampling,a little jitter can not result in a high fluctuatio of freedoms.Comparing Fig.5(a)with Fig.5(b),and Fig.5(c)with Fig.5(d),althoughthereare slight changesin SAR imagingbeforeand after havingsamplingjitter,it can be noticedthese changes have no effect on image quality of SAR.

Fig.5 Analysis of sampling jitter to SAR imaging by CS based on different sampling methods

7.Conclusions

In this paper,a novel way that coprime sampling and nested sparse sampling combine with CS is proposed to SAR imaging.Theproposedmethodscan achievethe SAR imaging effectively with only 25%or even 10%of the raw data,and a little samplingjitter have no effect onthe image quality of SAR.We have demonstratedthe validity and the stablity of the methods by processing both simulated and real data.

As for future work,in order to reduce much more storage and computation burden for SAR,it is worthwhile to apply these new undersampling methods for the range direction of SAR,or both the range direction and the azimuth direction of SAR.Many other things remain to be explored,such as,higher dimensional structure of the coprime sampling and the nested sparse sampling is worthwhile to be utilized for SAR sampling.

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Biographies

Hongyin Shiwas born in 1976.He received his Ph.D.degree from Beihang University in 2009. Now,he is an associate professor in Yanshan University.His main research interests include SAR imaging and moving target detection

E-mail:shihy@ysu.edu.cn

Baojing Jiawas born in 1988.She received her B.S.degree in Electrical&Information Engineering from Inner Mongolia Normal University,Huhehot, China,in 2011.She has been working toward her M.S.degree in the School of Information Science and Engineering,Yanshan University,since 2012. Her main research interests include radar imaging and radar countermeasure.

E-mail:jiabaojing@126.com

10.1109/JSEE.2015.00134

Manuscript received November 15,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61571388;U1233109).