# MIMO SAR Imaging for Wide-Swath Based on Compressed Sensing

Authors: Feng Liu; Shanxiang Mu; Wanghan Lv
Journal: Mathematical Problems in Engineering
 (2017)

---

## Abstract

To reduce the amount of data to be stored and software/hardware complexity and suppress range ambiguity, a novel MIMO SAR imaging based on compressed sensing is proposed under the condition of wide-swath imaging. Random phase orthogonal waveform (RPOW) is designed for MIMO SAR based on compressed sensing (CS). Echo model of sparse array in range and compressive sampling is reconstructed with CS theory. Resolution in range imaging is improved by using the techniques of digital beamforming (DBF) in transmit. Zero-point technique based on CS is proposed with DBF in receive and the range ambiguity is suppressed effectively. Comprehensive numerical simulation examples are performed. Its validity and practicality are validated by simulations.


---

## Body

## 1. Introduction

Synthetic aperture radar (SAR) has been widely used in remote sensing imaging technology. However, current single-antenna SAR systems have become more sophisticated and cannot fulfill the increasing demands of future remote sensing in high-resolution and wide-swath (HRWS) imaging performance [1]. On one hand, a small antenna length in azimuth is needed to achieve a fine azimuth resolution. On the other hand, the imaging of large swaths requires low pulse repetition frequencies (PRFs) to avoid range ambiguities, and azimuth ambiguities will appear if the PRF is too low for the chosen azimuth antenna length. In order to suppress range ambiguity, LCMV [2] and MUSIC [3] algorithms of DBF are introduced. So current single-antenna SAR systems cannot provide simultaneously high-resolution and wide-swath imaging. One solution is the displaced phase center antenna (DPCA) technique whose receive antennas are located in along-track direction [4], but the potential drawback is that sampling is nonuniform in azimuth constantly. Another solution is to use multiaperture in receive to gather additional information [5]. The digital beamforming (DBF) on receive techniques is employed.In recent years, the multiple-input and multiple-output (MIMO) technique is applied to the SAR system of multiple antennas in range [1, 6]. Orthogonal waveforms are simultaneously transmitted by antennas in range and echoes are isolated from different transmit signals with different weighting coefficients. Using DBF technology resolution in range is improved. The challenge of large amount of data is the common character of the above methods. To solve this problem, compressed sensing technology is used in SAR imaging [7–9]. The effect of sampling rate and the channel capacity on imaging system are analyzed in a given measurement matrix and SNR in [10]. In literature [11], a new CS-SAR imaging method is proposed that can be applied to high-quality and high-resolution imaging under sub-Nyquist rate sampling, while saving the computational cost substantially both in time and in memory.In addition to solving the problem of large amount of data, waveform design is the key technology to achieve MIMO SAR imaging [1, 12, 13]. Signal waveform should have enough time-bandwidth products and less peak power. According to the requirement of signal waveform, a modified genetic algorithm (GA) is proposed to numerically search optimal frequency firing order for discrete frequency-coding waveform (DFCW) in [14], and linear frequency modulation hybrid coding (LFMHC) waveform is proposed to enable MIMO SAR to operate efficiently in practical applications [15], but the methods can increase the bandwidth of RF system. A random OFDM-LFM waveform is proposed in [1] and has good performance in MIMO SAR imaging.In this paper we proposed a random phase OFDM-LFM waveform whose orthogonality is used to separate the echo signals and beamforming. The proposed waveform has only changed the initial frequency and phase of the LFM signal, so the complexity of the transmitted signal is reduced to meet the requirements of MIMO SAR imaging. Using this waveform, a new MIMO SAR imaging for sparse receive array in range based on CS is proposed. The quantity of receive antennas is reduced whose spacing between the array does not have to satisfy a limit of less thanλ/2 where λ is the transmit wavelength. According to the CS theory, the received echo signals can be directly sampled randomly, and the sampling frequency does not need to meet the Nyquist sampling frequency, which can reduce the sampling data. Zero-point technique based on CS is proposed with DBF in receive to suppress range ambiguity. The rest of this paper is organized as follows. In Section 2, a random phase orthogonal waveform for MIMO SAR is designed. Then compressed sensing theory is introduced in Section 3. In Section 4, illustration and implementation of the signal model of MIMO SAR in range based on CS are proposed. In Section 5, the numerical simulation results are presented. Finally, conclusions are reported in Section 6.


## 2. Random Phase Orthogonal Waveform Design

Waveform design is considered in this section. Waveform design is the key technology to MIMO SAR imaging. Because the linear frequency modulation (LFM) signal has a large time-bandwidth product and it is also the main waveform used by the conventional SAR system, the main research in this paper is based on the LFM signal.Consider that transmit signals of MIMO SAR ares(t)=[s1t,s2t,…,sm(t),…,sM(t)]T, and themth transmit signal can be expressed by (1)smt=recttCmUmt,where(2)rectt=1TP,0≤t≤TP0,othersis a rectangular pulse signal, and Cm=exp⁡(jφm) shows themth phase encoding signal where φm=(2π/L)am is random phase,L is the number of available phases for phase encoding, and am is an element in sequence of 0,1,…,L-1. Um(t)=exp⁡j2πfmt+(1/2)Kmt2 ismth LFM signal of random encoding waveform, where fm is the initial frequency of the LFM signal, Km=bmB/TP is the modulation slope, B is pulse signal bandwidth, and bm is a pseudorandom number.As an example, Figure1 is a schematic diagram of eight waveforms whose color represents different random phases. Figure 1(a) expresses that the phase and modulation slope are random, and Figures 1(b) and 1(c) are the simplified waveforms whose modulation slope is consistent, where modulation slope is up in Figure 1(b) and down in Figure 1(c). In practical applications, the waveforms shown in Figure 1(a) are commonly used in signal processing to matched filter (MF), and the waveforms shown in Figures 1(b) and 1(c) are used for the Stretch processing system. Compared with the matched filter processing, the Stretch processing system only requires one reference signal which uses the waveforms shown in Figure 1(b) or Figure 1(c). The analysis of signal model and simulation mainly uses the waveform shown in Figure 1(b).Figure 1
Schematic diagram of random phase orthogonal waveform.
(a) (b) (c)Figure2 is a frequency domain result of eight signals after Stretch processing in one receive channel. The parameters are B=10 MHz, Tp=10 μs, the starting frequencies fm=[50,70,0,20,40,10,60,30]×106 Hz, and phase φm=2π/8×[5,1,4,0,2,3,7,6]. It can be seen that the echoes are separated after Stretch processing using the random phase orthogonal waveforms. Analyzed by ambiguity function [1, 15], this waveform has a satisfactory ambiguity function performance in range resolution and Doppler frequency resolution.Figure 2
Output of eight signals in frequency domain after Stretch processing in one receive channel.
(a) (b) (c) (d) (e) (f) (g) (h)


## 3. Compressed Sensing Theory

Compressed sensing theory is mainly used for sparse signal restoration. As sparse signal representation has greater flexibility in matching structure in the signal, it can be used in SAR imaging. Assume that vectorx=x1,x2,…,xNT and transform based matrix Ψ=ψ1,ψ2,…,ψN, the signal x on transform based matrix Ψ can be expressed by(3)x=∑n=1NsnΨn=As,where s=s1,s2,…,sNT is weight coefficient vector of N×1 dimensions. The signal x is a K-sparse signal if there are K significant coefficients in S where K is the sparse level, while others are nearly zeros.Linear measurements of sparse signalx can be expressed as(4)y=Φx,where Φ=φ1φ2⋯φM is measurement matrix.Put (3) into (4) and we can get (5)y=Φx=ΦΨs,(6)y=Ψs,where D=ΦΨ is recovery matrix.In the above transformation process, it is possible to reconstruct the original signalx from the observation signal y by the reconstruction algorithm when the restricted isometry property (RIP) is met [16]. Therefore, the design of the measurement matrix Φ is very important. There are many measurement matrixes that have been used, such as the Hadamard matrix, Gaussian random matrix, sparse random matrix, and part of the Fourier matrix.


## 4. The Signal Model of MIMO SAR in Range Based on CS

The range ambiguity is existing because MIMO SAR antenna patterns are wider than swath, and the external signals can be received with useful echoes. The echoes of MIMO SAR in range can be used by DBF that the antenna main lobe is pointed to the direction of the desired swath and nulls in the direction of interferers by null steering techniques. Thereby range ambiguity can be suppressed while desired targets can be imaged. Figure3 is schematic diagram of range ambiguity suppression using null steering techniques.Figure 3
Schematic diagram of range ambiguity suppression using null steering techniques.


### 4.1. Echo Model in One Receive Channel

Echo in one receive channel is the superposition of all reflected signals. Using RPOW signals as shown in Figure1(b), themth transmit signal in nth receive channel in the receiving array is assumed as(7)sr=ej2πfct-τejφmej2πfsmt-τ+1/2Krt-τ2,where fsm is starting frequency of random phase orthogonal waveform and τ is delay time. The expression of the reference signal is(8)sref=ej2πfcte-jφmejπKrt2.The signal of difference frequency mixing after Stretch can be written as(9)sif=srt·sref∗t=ej2π-fcτ+1/2Krτ2ej2πfsmt-τe-j2πKrτt.Radar echo signal is sparse in Fourier transform domain from (5). Therefore, it can be processed and recovered by compressed sensing theory. Diagram of sparse receive array and signal sampling is shown in Figure 4, where Nr indicates the sampling point in range and Na denotes the sampling point in azimuth.Figure 4
Diagram of sparse receive array and signal sampling.When the sample numbernc is large enough, after discretization, a band limited analog signal can be recovered by(10)xrt=∑nc=-Nc/2Nc/2-1xncsin⁡πt-ncT/Tπt-ncT/T,where x[nc] is sample sequence of original signal xs which meets the Nyquist frequency and T is sampling period. The original signal is restored from Mc random samples of Nc samples. Sampled sequence s(Δtm) can be expressed by(11)sΔtm=∑nc=1NcxncTesin⁡πΔtm-ncTe/TeπΔtm-ncTe/Te,where Δtm is relative time interval ofmth random sample, 1≤m≤Mc, and Te is random sampling interval which can be considered the same as T.The relationship between the random sample values and sample values satisfying the Nyquist sampling rate can be expressed as [17](12)sΔt1sΔt2⋮sΔtM=ϕ11ϕ12⋯ϕ1Ncϕ21ϕ22⋯ϕ2Nc⋮⋮⋱⋮ϕM1ϕM2⋯ϕMcNcsTes2Te⋮sNcTe.Using vector representation we have(13)s=Φx,where Φ is measure matrix and ϕmcnc=sin⁡[π(Δtm-ncTe)/Te]/(π(Δtm-ncTe)/Te).Transformation matrix is(14)Ψ=1e-iΔϕB1e-i2ΔϕB1e-iN-1ΔϕB11e-iΔϕB2e-i2ΔϕB2e-iN-1ΔϕB2⋮⋮⋱⋮1e-iΔϕBM⋯e-iN-1ΔϕBM.Multibeam can also be restored by this transformation matrix in DBF.To solve the problem of sparse signal representation, various methods have been proposed, such as basis pursuit [18, 19] and orthogonal matching pursuit [20, 21]. This paper takes orthogonal matching pursuit (OMP) algorithm to recover the original signal.


### 4.2. DBF in Transmit

Suppose that the transmit array is a uniform linear array, the transmit signals can be expressed by(15)stt=vtTθts1t,s2t,…,sMtT,where vtT(θt)=[1,e-j2π/λd0sinθt,…,e-j2π/λM-1d0sinθt]T is transmit array response vector and θt is the incidence angle in the transmit array response vector. The receive signals can be written as(16)xrt=vrθrstt-τd=vrθrvtTθts1t-τd,s2t-τd,…,sMt-τdT,where vr(θr)=[1,e-j(2π/λ)d0sinθr,…,e-j(2π/λ)(N-1)d0sinθr]T is the receive array response vector and τd represents the time which the signal takes to travel the transmitter-target-receiver distance.With Stretch process and band pass filter, the echo expressed in (9) can be performed with DBF. With DBF in nth receive channel, the output can be expressed by(17)FBb=∑m=0M-1ej2πd0/λm-1sin⁡ϕbe-j2πd0/λm-1sin⁡θt·ej2πfsmt-τ-fcτe-j2πKrτte-j2π/λn-1d0sinθr,where ϕb is beam direction. When ϕb=θt, the output can be expressed as(18)FBθtb=∑m=0M-1e-j2πfcτej2πfsmt-τe-j2πKrτte-j2π/λn-1d0sinθr=e-j2πfcτe-j2πKrτte-j2π/λn-1d0sinθr∑m=0M-1ej2πfsmt-τ.Now we consider the expression (9). Each received signal is processed by demodulation. The expression in frequency domain is(19)Sifsncrnf=e-j2πd0/λm-1sin⁡θt·sin⁡cTp2πf+Krτe-j2πfcτe-j2πfsmτ.Because of the correspondence between the frequency and time, the expression (19) can be written as(20)sifsncrnt=SifsncrnfKr=sin⁡cTp2πf+KrτKr=sin⁡cTp2πt+τ.After symmetric processing of y-axis, we can get(21)fliplrsifsncrnt=sin⁡cTp2πt-τ.Multiplying the formulae (18) and (21), the new expression is(22)HFBθtb=FBθtb·fliplrsifsncrnt=e-j2πfcτe-j2πKrτtejM-1/2fst-τe-j2π/λn-1d0sinθrsin⁡M/2fst-τsin⁡1/2fst-τ·sin⁡cTp2πt-τ,where(23)τ=2Rtslow,R0C≈2CR0+Vtslow22R0and R0 is the closest slant range to the target.Considering the envelope, the second addend in (23) can be ignored; that is, τ≈2R0/C, and the expression (22) is written as(24)HFBϕtb=e-j2π/λn-1d0sinθre-j2πfcτ+KrτtejM-1/2fst-τsin⁡N/2fst-2R0/Csin⁡1/2fst-2R0/C·sin⁡cTp2πt-2R0C.Taking the absolute value of (24), the expression is(25)HFBϕtb=sin⁡M/2fst-2R0/Csin⁡1/2fst-2R0/C·sin⁡cTp2πt-2R0C.Envelope of the first factor in right side of (25) is sin⁡c function shape, and the −4 dB width is ΔR1=C/(2Mfs). The total resolution after the product is ΔRtotal≈ΔR1=C/(2MB). Thus, resolution of random phase orthogonal waveform is M times resolution of single LFM signal where M is the number of transmit waveforms.Figure5 is a schematic diagram of output with RPOW signal. The number of transmit channels is eight, and the target range is set to 10000 m. Figure 5(a) shows the resolution of single LFM signal with Stretch processing. Figure 5(b) is output of RPOW signal in one receiving channel after the digital beamforming which is expressed in (18), and Figure 5(c) is waveform processed by multiplying which is expressed in (25). So resolution is improved by eight times which is equal to the number of transmit signals. Figure 5(d) represents output comparison of single LFM signal with random phase orthogonal waveform. Two targets are set to 9999 m and 10001 m. As shown in Figure 5(d), the two targets are distinguished easily with random phase orthogonal waveform.Figure 5
Resolution comparison of single LFM signal with RPOW signal.
(a)
Output of single LFM (b)
Output of RPOW signal with DBF (c)
The final output of RPOW signal (d)
Resolution comparison of single LFM and RPOW signal


### 4.3. Echo Model of Random Sparse Array

Figure6 shows the uniform linear array where the number of receive array elements is N=8. The array space arrangements satisfy d0≤λ/2. In the figure below shown in Figure 6, the red elements indicate the randomly selected 3 elements, whose numbers are 1, 3, and 6, respectively. The incident angle of target signal u(t) is θ. The phase difference in space is Δϕ=2π/λd0sinθ and the phase difference in array is ΔϕB=2π/λd0sinϕB. Thekth beam-pointing is ϕBk, and the phase compensation value provided by the digital beamforming processor shall be ΔϕBk=2π/λd0sinϕBk. The antenna pattern function of uniform linear array can be expressed as(26)Fk=∑i=0N-1ejiΔϕ-ΔϕBk.Figure 6
Schematic diagram of sparse array in receive.For receiving DBF, assume that the direction of arrival of the desired signal isθ1, and the arrival directions of the interferences are θ2,…,θk,…,θK. The steering vector of the receive signals can be expressed as(27)asin⁡θk=1,ej2πdsin⁡θk/λ,…,ej2πN-1dsin⁡θk/λT.The echo signal of receive array can be represented as a matrix by(28)st=∑k=1Kuktasin⁡θk.In the real scene, echoes can be from multiple directions in receiving window because of range ambiguity. In order to estimate direction of arrival, we decompose the entire space from −90°to 90° intoΓ parts; then we obtain transformation matrix:(29)A=asin⁡θ1,asin⁡θ2,…,asin⁡θГ.The echo signal of the receiving array can be expressed as(30)St=AUt,where U(t)=[0,0,…,u1t,0,…,0,…,uKt,0,…,0]T. It is clear that Ut is sparse and has a few nonzero elements. Therefore, according to CS theory, the received signal S(t) can be recovered accurately by using a CS reconstruction algorithm.The compressed vectorYt can be expressed with measurement matrix Φ as(31)Yt=ΦSt=ΦAUt=ΨUt.Therefore, the reconstruction of the received signal becomes an optimal estimation of (32)(32)minU⌢0s.t.Y=ΨU⌢.After the projection coefficient vectorU⌢ is estimated from the compressed vector Yt, the received signal S⌢(t) can be reconstructed, as shown in the following equation:(33)S⌢t=AU⌢t.The signal can be reconstructed using orthogonal matching pursuit algorithm in this paper.After obtaining the reconstructed echoes using compressed sensing reconstruction methods, using LCMV beamforming algorithm [2, 22], the adaptive digital beamforming is implemented that the antenna main lobe is pointed to the direction of the desired signal and nulls in the direction of interferers. Thereby range ambiguity can be suppressed while desired targets can be imaged.


### 4.4. MIMO SAR Imaging Process Based on CS

In order to reduce the amount of data, the received signal of Stretch process is compressive sampled from the random equivalent sampling method and is restored with CS. Then DBF is implemented in transmit in receive channel. So the range resolution can be improved byM times. With DBF in transmit of sparse array, the null steering techniques based on CS is put forward to suppress echo outside of swath in order to range ambiguity suppression. Under normal circumstances, the impact of adjacent ambiguity region with imaging swath is considered because the ambiguity echo power in adjacent ambiguity region generally accounts for more than 80% of all ambiguity echo power. For the far ambiguity region, the antenna sidelobe gain is low and the echo power is less due to the far distance. MIMO SAR imaging process based on CS is shown in Figure 7.Figure 7
MIMO SAR imaging process based on CS.


## 4.1. Echo Model in One Receive Channel

Echo in one receive channel is the superposition of all reflected signals. Using RPOW signals as shown in Figure1(b), themth transmit signal in nth receive channel in the receiving array is assumed as(7)sr=ej2πfct-τejφmej2πfsmt-τ+1/2Krt-τ2,where fsm is starting frequency of random phase orthogonal waveform and τ is delay time. The expression of the reference signal is(8)sref=ej2πfcte-jφmejπKrt2.The signal of difference frequency mixing after Stretch can be written as(9)sif=srt·sref∗t=ej2π-fcτ+1/2Krτ2ej2πfsmt-τe-j2πKrτt.Radar echo signal is sparse in Fourier transform domain from (5). Therefore, it can be processed and recovered by compressed sensing theory. Diagram of sparse receive array and signal sampling is shown in Figure 4, where Nr indicates the sampling point in range and Na denotes the sampling point in azimuth.Figure 4
Diagram of sparse receive array and signal sampling.When the sample numbernc is large enough, after discretization, a band limited analog signal can be recovered by(10)xrt=∑nc=-Nc/2Nc/2-1xncsin⁡πt-ncT/Tπt-ncT/T,where x[nc] is sample sequence of original signal xs which meets the Nyquist frequency and T is sampling period. The original signal is restored from Mc random samples of Nc samples. Sampled sequence s(Δtm) can be expressed by(11)sΔtm=∑nc=1NcxncTesin⁡πΔtm-ncTe/TeπΔtm-ncTe/Te,where Δtm is relative time interval ofmth random sample, 1≤m≤Mc, and Te is random sampling interval which can be considered the same as T.The relationship between the random sample values and sample values satisfying the Nyquist sampling rate can be expressed as [17](12)sΔt1sΔt2⋮sΔtM=ϕ11ϕ12⋯ϕ1Ncϕ21ϕ22⋯ϕ2Nc⋮⋮⋱⋮ϕM1ϕM2⋯ϕMcNcsTes2Te⋮sNcTe.Using vector representation we have(13)s=Φx,where Φ is measure matrix and ϕmcnc=sin⁡[π(Δtm-ncTe)/Te]/(π(Δtm-ncTe)/Te).Transformation matrix is(14)Ψ=1e-iΔϕB1e-i2ΔϕB1e-iN-1ΔϕB11e-iΔϕB2e-i2ΔϕB2e-iN-1ΔϕB2⋮⋮⋱⋮1e-iΔϕBM⋯e-iN-1ΔϕBM.Multibeam can also be restored by this transformation matrix in DBF.To solve the problem of sparse signal representation, various methods have been proposed, such as basis pursuit [18, 19] and orthogonal matching pursuit [20, 21]. This paper takes orthogonal matching pursuit (OMP) algorithm to recover the original signal.


## 4.2. DBF in Transmit

Suppose that the transmit array is a uniform linear array, the transmit signals can be expressed by(15)stt=vtTθts1t,s2t,…,sMtT,where vtT(θt)=[1,e-j2π/λd0sinθt,…,e-j2π/λM-1d0sinθt]T is transmit array response vector and θt is the incidence angle in the transmit array response vector. The receive signals can be written as(16)xrt=vrθrstt-τd=vrθrvtTθts1t-τd,s2t-τd,…,sMt-τdT,where vr(θr)=[1,e-j(2π/λ)d0sinθr,…,e-j(2π/λ)(N-1)d0sinθr]T is the receive array response vector and τd represents the time which the signal takes to travel the transmitter-target-receiver distance.With Stretch process and band pass filter, the echo expressed in (9) can be performed with DBF. With DBF in nth receive channel, the output can be expressed by(17)FBb=∑m=0M-1ej2πd0/λm-1sin⁡ϕbe-j2πd0/λm-1sin⁡θt·ej2πfsmt-τ-fcτe-j2πKrτte-j2π/λn-1d0sinθr,where ϕb is beam direction. When ϕb=θt, the output can be expressed as(18)FBθtb=∑m=0M-1e-j2πfcτej2πfsmt-τe-j2πKrτte-j2π/λn-1d0sinθr=e-j2πfcτe-j2πKrτte-j2π/λn-1d0sinθr∑m=0M-1ej2πfsmt-τ.Now we consider the expression (9). Each received signal is processed by demodulation. The expression in frequency domain is(19)Sifsncrnf=e-j2πd0/λm-1sin⁡θt·sin⁡cTp2πf+Krτe-j2πfcτe-j2πfsmτ.Because of the correspondence between the frequency and time, the expression (19) can be written as(20)sifsncrnt=SifsncrnfKr=sin⁡cTp2πf+KrτKr=sin⁡cTp2πt+τ.After symmetric processing of y-axis, we can get(21)fliplrsifsncrnt=sin⁡cTp2πt-τ.Multiplying the formulae (18) and (21), the new expression is(22)HFBθtb=FBθtb·fliplrsifsncrnt=e-j2πfcτe-j2πKrτtejM-1/2fst-τe-j2π/λn-1d0sinθrsin⁡M/2fst-τsin⁡1/2fst-τ·sin⁡cTp2πt-τ,where(23)τ=2Rtslow,R0C≈2CR0+Vtslow22R0and R0 is the closest slant range to the target.Considering the envelope, the second addend in (23) can be ignored; that is, τ≈2R0/C, and the expression (22) is written as(24)HFBϕtb=e-j2π/λn-1d0sinθre-j2πfcτ+KrτtejM-1/2fst-τsin⁡N/2fst-2R0/Csin⁡1/2fst-2R0/C·sin⁡cTp2πt-2R0C.Taking the absolute value of (24), the expression is(25)HFBϕtb=sin⁡M/2fst-2R0/Csin⁡1/2fst-2R0/C·sin⁡cTp2πt-2R0C.Envelope of the first factor in right side of (25) is sin⁡c function shape, and the −4 dB width is ΔR1=C/(2Mfs). The total resolution after the product is ΔRtotal≈ΔR1=C/(2MB). Thus, resolution of random phase orthogonal waveform is M times resolution of single LFM signal where M is the number of transmit waveforms.Figure5 is a schematic diagram of output with RPOW signal. The number of transmit channels is eight, and the target range is set to 10000 m. Figure 5(a) shows the resolution of single LFM signal with Stretch processing. Figure 5(b) is output of RPOW signal in one receiving channel after the digital beamforming which is expressed in (18), and Figure 5(c) is waveform processed by multiplying which is expressed in (25). So resolution is improved by eight times which is equal to the number of transmit signals. Figure 5(d) represents output comparison of single LFM signal with random phase orthogonal waveform. Two targets are set to 9999 m and 10001 m. As shown in Figure 5(d), the two targets are distinguished easily with random phase orthogonal waveform.Figure 5
Resolution comparison of single LFM signal with RPOW signal.
(a)
Output of single LFM (b)
Output of RPOW signal with DBF (c)
The final output of RPOW signal (d)
Resolution comparison of single LFM and RPOW signal


## 4.3. Echo Model of Random Sparse Array

Figure6 shows the uniform linear array where the number of receive array elements is N=8. The array space arrangements satisfy d0≤λ/2. In the figure below shown in Figure 6, the red elements indicate the randomly selected 3 elements, whose numbers are 1, 3, and 6, respectively. The incident angle of target signal u(t) is θ. The phase difference in space is Δϕ=2π/λd0sinθ and the phase difference in array is ΔϕB=2π/λd0sinϕB. Thekth beam-pointing is ϕBk, and the phase compensation value provided by the digital beamforming processor shall be ΔϕBk=2π/λd0sinϕBk. The antenna pattern function of uniform linear array can be expressed as(26)Fk=∑i=0N-1ejiΔϕ-ΔϕBk.Figure 6
Schematic diagram of sparse array in receive.For receiving DBF, assume that the direction of arrival of the desired signal isθ1, and the arrival directions of the interferences are θ2,…,θk,…,θK. The steering vector of the receive signals can be expressed as(27)asin⁡θk=1,ej2πdsin⁡θk/λ,…,ej2πN-1dsin⁡θk/λT.The echo signal of receive array can be represented as a matrix by(28)st=∑k=1Kuktasin⁡θk.In the real scene, echoes can be from multiple directions in receiving window because of range ambiguity. In order to estimate direction of arrival, we decompose the entire space from −90°to 90° intoΓ parts; then we obtain transformation matrix:(29)A=asin⁡θ1,asin⁡θ2,…,asin⁡θГ.The echo signal of the receiving array can be expressed as(30)St=AUt,where U(t)=[0,0,…,u1t,0,…,0,…,uKt,0,…,0]T. It is clear that Ut is sparse and has a few nonzero elements. Therefore, according to CS theory, the received signal S(t) can be recovered accurately by using a CS reconstruction algorithm.The compressed vectorYt can be expressed with measurement matrix Φ as(31)Yt=ΦSt=ΦAUt=ΨUt.Therefore, the reconstruction of the received signal becomes an optimal estimation of (32)(32)minU⌢0s.t.Y=ΨU⌢.After the projection coefficient vectorU⌢ is estimated from the compressed vector Yt, the received signal S⌢(t) can be reconstructed, as shown in the following equation:(33)S⌢t=AU⌢t.The signal can be reconstructed using orthogonal matching pursuit algorithm in this paper.After obtaining the reconstructed echoes using compressed sensing reconstruction methods, using LCMV beamforming algorithm [2, 22], the adaptive digital beamforming is implemented that the antenna main lobe is pointed to the direction of the desired signal and nulls in the direction of interferers. Thereby range ambiguity can be suppressed while desired targets can be imaged.


## 4.4. MIMO SAR Imaging Process Based on CS

In order to reduce the amount of data, the received signal of Stretch process is compressive sampled from the random equivalent sampling method and is restored with CS. Then DBF is implemented in transmit in receive channel. So the range resolution can be improved byM times. With DBF in transmit of sparse array, the null steering techniques based on CS is put forward to suppress echo outside of swath in order to range ambiguity suppression. Under normal circumstances, the impact of adjacent ambiguity region with imaging swath is considered because the ambiguity echo power in adjacent ambiguity region generally accounts for more than 80% of all ambiguity echo power. For the far ambiguity region, the antenna sidelobe gain is low and the echo power is less due to the far distance. MIMO SAR imaging process based on CS is shown in Figure 7.Figure 7
MIMO SAR imaging process based on CS.


## 5. Simulation Analysis

### 5.1. Signal Recovery from the Random Sampling Method in Range Based on CS

In this subsection, some simulation results under different situations are provided. Assume that there are four targets, and the parameters of four RPOW signals are as follows: the starting frequenciesfsm= [0,20,10,30]×106 Hz, B=10 MHz, Tp=10 μs, and sparse ratio is 0.37. Figure 8 shows the simulation results. It is shown that the amount of data is significantly reduced with compressive sampling, and the frequency spectrum of the targets can be displayed correctly after the restoration compared with the original signal.Figure 8
Simulation results of image in range based on CS. (a) shows the original echo of four targets and the echo with compressive sampling of one transmit signal. (b) shows the spectrum of original echo and the echo with compressive sampling, and (c) shows the spectrum of reconstruction with CS method.
(a) (b) (c)Secondly, the influence of SNR is evaluated with CS method. The parameters are the same as in Figure8. Figure 9 shows the spectrum of reconstruction compared with the original signal. It is shown that the spectrum of reconstruction can display correctly the targets when SNR>10 dB, but when SNR<10dB, such as SNR=5 dB, the spectrum of reconstruction shows that the target will be lost or false targets appear.Figure 9
The spectrum of reconstruction in different SNR. (a) shows that the reconstruction in the case of SNR = 20 dB, (b) shows SNR = 15 dB, (c) shows SNR = 10 dB, and (d) shows SNR = 5 dB.
(a)
SNR = 20 dB (b)
SNR = 15 dB (c)
SNR = 10 dB (d)
SNR = 5 dBNow let us consider the influence of transmit number. Figure10 shows the reconstruction spectrum of the RPOW signals from five to eight. The starting frequencies are randomly chosen from [0,10,20,30,40,50,60,70]×106 Hz. Other parameters are as the same as in Figure 7. The recovered signal can display the position of the spectrum correctly, which shows that the increase of the number of signals will not affect the recovery of the data.Figure 10
The reconstruction spectrum of different numbers of RPOW signals. (a) shows that the number is 5, (b) shows 6, (c) shows 7, and (d) shows 8.
(a)
M = 5 (b)
M = 6 (c)
M = 7 (d)
M = 8Then we consider the Monte Carlo simulation. SNR and sparse ratio are two important indexes in the data recovery of MIMO SAR based on compressed sensing. When SNR or sparse ratio is low, the original signal may not be able to recover correctly. In order to analyze the application of MIMO SAR data recovery based on compressed sensing, we give the following simulation analysis. The carrier frequency is 1 GHz,fsm=[0,20,10,30]×106 Hz, B=10 MHz, Tp=10 μs, SNR is from −5 dB to 20 dB, and sparse ratio is from 0.2 to 1. We do Monte Carlo analysis 200 times and give the results of successful probability in Figure 11. It can be seen that successful probability of reconstruction is more than 95% when the sparse ratio is 0.4 and SNR is greater than 10 dB. At the same SNR, the higher the sparse ratio, the more successful the probability. Under the condition of the same sparse ratio, the higher the SNR, the more successful the probability. Due to the fact that random sampling requires the signal above the noise level, and in the case of small signal, the sparse coefficient is submerged in the noise, so it cannot be well estimated. Small signal will be lost and the probability of success is reduced. Therefore, this method is not suitable for small signal.Figure 11
Successful probability with different SNR and sparse ratio.


### 5.2. DBF Algorithm of Zero-Pointing Technology Based on CS

To verify the correctness of the algorithm in the cases of sparse array, we choose 30 elements sparsely from 100 elements. Set SNR 5 dB and 10 dB, INR 10 dB and 40 dB, and the directions of range ambiguity are set to −10° and 10°. The angular space within the swath is discretized into 720 subspaces. Figure12 shows the beam patterns with LCMV algorithm in different SNR and INR. It is shown that when SNR is lower than 10 dB, the interference can be suppressed, and the greater the interference, the greater the depth. But the precision is not ideal. When the SNR is greater than 10 dB, and the number of array elements is reduced from 100 to 30, the data recovery after beamforming, the performance of the proposed algorithm is almost the same as that of the full array. The main reason is that, in the case of low SNR, the signal is relatively low to noise level, and the signal cannot be successfully restored.Figure 12
The beam patterns with different SNR and INR. (a) shows reconstruction in the case of SNR = 5 dB, INR = 10 dB, (b) shows SNR = 5 dB, INR = 40 dB, (c) shows SNR = 10 dB, INR = 10 dB, and (d) shows SNR = 10 dB, INR = 40 dB.
(a)
SNR = 5 dB, INR = 10 dB (b)
SNR = 5 dB, INR = 40 dB (c)
SNR = 10 dB, INR = 10 dB (d)
SNR = 10 dB, INR = 40 dB


### 5.3. MIMO SAR Imaging Simulation Based on CS

Imaging simulations are also performed to evaluate the algorithm based on CS. Figure13 shows imaging of HRWS SAR and MIMO SAR based on CS. The parameters are as follows: carrier frequency fc=10GHz, platform altitude is 600 km, platform velocity is 7640 m/s, pulse repeated frequency is 1908 Hz, minimum slant range is 558 km, maximum slant range is 727 km, and SNR = 10 dB, and we apply RPOW waveform to MIMO SAR as shown in Figure 1(b). In order to reduce the computational complexity, number of subswathes is set to 2, and there are two point targets locating at the positions of (600 km, 0) and (685 km, 0). Figure 13 shows the comparative imaging results. In Figure 13(b), sparse ratio of receive array is set to 0.375, and sampling rate is 0.4 which means the data is 0.15 times of original sample points. As shown in Figure 13, resolution is improved by 8 times in range and the range ambiguity suppression is better.Figure 13
Imaging of HRWS SAR and MIMO SAR based on CS.
(a)
Imaging of HRWS SAR (b)
Imaging of MIMO SAR based on CS


## 5.1. Signal Recovery from the Random Sampling Method in Range Based on CS

In this subsection, some simulation results under different situations are provided. Assume that there are four targets, and the parameters of four RPOW signals are as follows: the starting frequenciesfsm= [0,20,10,30]×106 Hz, B=10 MHz, Tp=10 μs, and sparse ratio is 0.37. Figure 8 shows the simulation results. It is shown that the amount of data is significantly reduced with compressive sampling, and the frequency spectrum of the targets can be displayed correctly after the restoration compared with the original signal.Figure 8
Simulation results of image in range based on CS. (a) shows the original echo of four targets and the echo with compressive sampling of one transmit signal. (b) shows the spectrum of original echo and the echo with compressive sampling, and (c) shows the spectrum of reconstruction with CS method.
(a) (b) (c)Secondly, the influence of SNR is evaluated with CS method. The parameters are the same as in Figure8. Figure 9 shows the spectrum of reconstruction compared with the original signal. It is shown that the spectrum of reconstruction can display correctly the targets when SNR>10 dB, but when SNR<10dB, such as SNR=5 dB, the spectrum of reconstruction shows that the target will be lost or false targets appear.Figure 9
The spectrum of reconstruction in different SNR. (a) shows that the reconstruction in the case of SNR = 20 dB, (b) shows SNR = 15 dB, (c) shows SNR = 10 dB, and (d) shows SNR = 5 dB.
(a)
SNR = 20 dB (b)
SNR = 15 dB (c)
SNR = 10 dB (d)
SNR = 5 dBNow let us consider the influence of transmit number. Figure10 shows the reconstruction spectrum of the RPOW signals from five to eight. The starting frequencies are randomly chosen from [0,10,20,30,40,50,60,70]×106 Hz. Other parameters are as the same as in Figure 7. The recovered signal can display the position of the spectrum correctly, which shows that the increase of the number of signals will not affect the recovery of the data.Figure 10
The reconstruction spectrum of different numbers of RPOW signals. (a) shows that the number is 5, (b) shows 6, (c) shows 7, and (d) shows 8.
(a)
M = 5 (b)
M = 6 (c)
M = 7 (d)
M = 8Then we consider the Monte Carlo simulation. SNR and sparse ratio are two important indexes in the data recovery of MIMO SAR based on compressed sensing. When SNR or sparse ratio is low, the original signal may not be able to recover correctly. In order to analyze the application of MIMO SAR data recovery based on compressed sensing, we give the following simulation analysis. The carrier frequency is 1 GHz,fsm=[0,20,10,30]×106 Hz, B=10 MHz, Tp=10 μs, SNR is from −5 dB to 20 dB, and sparse ratio is from 0.2 to 1. We do Monte Carlo analysis 200 times and give the results of successful probability in Figure 11. It can be seen that successful probability of reconstruction is more than 95% when the sparse ratio is 0.4 and SNR is greater than 10 dB. At the same SNR, the higher the sparse ratio, the more successful the probability. Under the condition of the same sparse ratio, the higher the SNR, the more successful the probability. Due to the fact that random sampling requires the signal above the noise level, and in the case of small signal, the sparse coefficient is submerged in the noise, so it cannot be well estimated. Small signal will be lost and the probability of success is reduced. Therefore, this method is not suitable for small signal.Figure 11
Successful probability with different SNR and sparse ratio.


## 5.2. DBF Algorithm of Zero-Pointing Technology Based on CS

To verify the correctness of the algorithm in the cases of sparse array, we choose 30 elements sparsely from 100 elements. Set SNR 5 dB and 10 dB, INR 10 dB and 40 dB, and the directions of range ambiguity are set to −10° and 10°. The angular space within the swath is discretized into 720 subspaces. Figure12 shows the beam patterns with LCMV algorithm in different SNR and INR. It is shown that when SNR is lower than 10 dB, the interference can be suppressed, and the greater the interference, the greater the depth. But the precision is not ideal. When the SNR is greater than 10 dB, and the number of array elements is reduced from 100 to 30, the data recovery after beamforming, the performance of the proposed algorithm is almost the same as that of the full array. The main reason is that, in the case of low SNR, the signal is relatively low to noise level, and the signal cannot be successfully restored.Figure 12
The beam patterns with different SNR and INR. (a) shows reconstruction in the case of SNR = 5 dB, INR = 10 dB, (b) shows SNR = 5 dB, INR = 40 dB, (c) shows SNR = 10 dB, INR = 10 dB, and (d) shows SNR = 10 dB, INR = 40 dB.
(a)
SNR = 5 dB, INR = 10 dB (b)
SNR = 5 dB, INR = 40 dB (c)
SNR = 10 dB, INR = 10 dB (d)
SNR = 10 dB, INR = 40 dB


## 5.3. MIMO SAR Imaging Simulation Based on CS

Imaging simulations are also performed to evaluate the algorithm based on CS. Figure13 shows imaging of HRWS SAR and MIMO SAR based on CS. The parameters are as follows: carrier frequency fc=10GHz, platform altitude is 600 km, platform velocity is 7640 m/s, pulse repeated frequency is 1908 Hz, minimum slant range is 558 km, maximum slant range is 727 km, and SNR = 10 dB, and we apply RPOW waveform to MIMO SAR as shown in Figure 1(b). In order to reduce the computational complexity, number of subswathes is set to 2, and there are two point targets locating at the positions of (600 km, 0) and (685 km, 0). Figure 13 shows the comparative imaging results. In Figure 13(b), sparse ratio of receive array is set to 0.375, and sampling rate is 0.4 which means the data is 0.15 times of original sample points. As shown in Figure 13, resolution is improved by 8 times in range and the range ambiguity suppression is better.Figure 13
Imaging of HRWS SAR and MIMO SAR based on CS.
(a)
Imaging of HRWS SAR (b)
Imaging of MIMO SAR based on CS


## 6. Conclusion

A MIMO SAR imaging based on compressed sensing is proposed in this paper to reduce the amount of data and suppress the range ambiguity. Random phase orthogonal waveform is designed for MIMO SAR based on compressed sensing. Using DBF technology in transmit and receive, the echoes of RPOW enable range resolution improved and the range ambiguity is suppressed by zero-point technology based on CS. However, due to the characteristics of signal recovery based CS, it is required that the signal be sparse on a projection basis. In the case of small signal, the sparse coefficient is submerged in the noise, and the recovery of signal can be distorted. Therefore, this method has a certain scope of application.



---

*Source: 9310865-2017-02-21.xml*