IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 14, NO. 6, JUNE 2017
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Joint Cancelation of Autocorrelation Sidelobe and Cross Correlation in MIMO-SAR Lilong Qin, Sergiy A. Vorobyov, and Zhen Dong
Abstract— Waveform separation based on matched filtering leads to autocorrelation sidelobe and cross correlation, which deteriorate the performance of multiple-input multiple-output synthetic aperture radar (MIMO-SAR). This letter investigates the performance of a waveform-separation approach employing an extended space-time coding (STC) scheme for MIMO-SAR. Using the autocorrelation property of multiphase complementary codes, we propose a novel STC scheme to effectively cancel out both the autocorrelation sidelobe and cross correlation. Using theoretical analysis also confirmed by simulations, we show that the proposed scheme decreases the sidelobe ratio while increasing the signal-to-noise ratio, leading to high-quality. Index Terms— Multiphase complementary (MPC) codes, multiple-input multiple-output (MIMO) radar, space-time coding (STC), synthetic aperture radar (SAR).
I. I NTRODUCTION UE to waveform and spatial diversity, multiple-input multiple-output synthetic aperture radar (MIMO-SAR) is an attractive option for radar imaging. First, MIMO-SAR can be used to achieve wide-swath imaging [1], [2]. Second, it can be used to achieve high resolution in the range direction [3]. Third, the extra phase centers in an MIMO radar system [4] lead to new capabilities for fading and scintillation mitigation, jammer suppression [5], additional and longer baselines for ground moving target indication, and along-track interferometry implementation [6]. It is well known, however, that in MIMO radar, the orthogonality of waveforms is a key issue determining radar system performance [7]. Waveforms with perfect orthogonality for arbitrary time delay do not exist [6]. If the cross-correlation level is high, the MIMO-SAR imaging performance suffers from significant degradation. Thus, waveform orthogonality optimization for MIMO radar is an important but difficult problem [8], [9]. Indeed, not only low cross-correlation level, but also other factors, such as the autocorrelation sidelobe
D
Manuscript received November 24, 2015; revised March 14, 2016, July 24, 2016, November 12, 2016, and March 9, 2017; accepted March 22, 2017. Date of publication April 25, 2017; date of current version May 19, 2017. This work was supported by the National Natural Science Foundation of China under Project 61101178. (Corresponding author: Lilong Qin.) L. Qin is with the School of Electronic Science and Engineering, Institute of Space Electronics and Information Technology, National University of Defense Technology, Changsha 410073, China, and also with the Department of Signal Processing and Acoustics, Aalto University, FI-00076 Espoo, Finland (e-mail:
[email protected]). S. A. Vorobyov is with the Department of Signal Processing and Acoustics, Aalto University, FI-00076 Espoo, Finland (e-mail:
[email protected]). Z. Dong is with the School of Electronic Science and Engineering, Institute of Space Electronics and Information Technology, National University of Defense Technology, Changsha 410073, China. Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2017.2688122
level, range and Doppler resolutions, and interference rejection ability, should be considered while optimizing the waveforms. In some publications [10], [11], space-time coding (STC) techniques are used to cancel out cross correlation in MIMO radar. However, the autocorrelation sidelobe level is still high after using conventional STC, which decreases the contrast ratio of the radar image and directly influences the accuracy of the detection results of weak targets [12]. In this letter, we develop a novel STC scheme in combination with the multiphase complementary (MPC) coding technique. In particular, we exploit the Autocorrelation Function (ACF) property of the MPC pair to achieve the high-quality imaging for a distributed scene, i.e., both the autocorrelation sidelobe and cross correlation are effectively canceled out. II. BACKGROUND AND P ROBLEM F ORMULATION Consider an MIMO radar system equipped with N subarrays along the azimuth direction (along-track direction) [13], [14]. At the slow time η = kT , i.e., during the kth pulse repetition interval (PRI), the nth transmit (Tx) subarray emits the signal sn,k (τ ), where τ represents the fast time. In the receiving stage, echoes arising from the simultaneous scene illumination are received by M different receive (Rx) subarrays forming multiple receiving phase centers in each PRI. If the cross section of the radar target is assumed to remain constant during the whole period, then the received echo at the mth Rx subarray at the kth PRI can be expressed in the frequency domain as [15] T Rm,k ( fr ) = hm ( fr )sk ( fr )e j (K −1)υ + n m,k ( fr )
(1)
where fr denotes the range frequency, sk ( fr ) = [S1,k ( fr ), S2,k ( fr ), . . . , S N,k ( fr )]T denotes the frequency domain form of the Tx waveform of size N × 1, and (·)T represents the transpose. Here, hm ( fr ) = T where Hn,m ( fr ) [H1,m ( fr ), H2,m ( fr ), . . . , H N,m ( fr )] , denotes the channel frequency response between the nth transmitter and the mth receiver, n m,k ( fr ) stands for radar system additive noise in the kth PRI, which is assumed to be Gaussian and white, and υ is the Doppler-induced differential phase shift caused by the movement of the radar platform.1 The echo received from the first PRI to the K th PRI is represented as it T T T rm ( fr ) = hm ( fr )S( fr ) + nm ( fr )
(2)
1 In (1), the Doppler shift during receiving the pulse is neglected. For airborne side-looking SAR, the Doppler value of the target is usually much smaller than the Doppler tolerance of the waveform. Note, however, that the performance of pulse compression can be significantly deteriorated by the effect of this Doppler shift if the Doppler value of the target is large.
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where rm ( fr ) = [Rm,1 ( fr ) Rm,2 ( fr ) · · · Rm,K ( fr )]T , S( fr ) = [ s1 ( fr ) · · · s K ( fr )e j (K −1)υ ], and nm ( fr ) = [ n m,1 ( fr ) · · · n m,K ( fr ) ]T . The cross correlation can be canceled out by multiplying by a decoding matrix D ( fr ), which can be expressed as T ( fr )D( fr ) rTD,m ( fr ) = rm T T = hm ( fr )S( fr )D( fr ) + nm ( fr )D( fr ).
(3)
One of the commonly used STC schemes is as follows. The transmitted waveforms are encoded by coefficients an,k ’s, so that in the kth PRI, the nth transmitter emits the signal an,k sn (τ ). In this case, we have ˜ [s( fr ) · · · s( fr )e j (K −1)υ ] S( fr ) = A
(4)
˜ = [a1 , . . . , a K ], ak = where s( fr ) = [ S1 ( fr ) · · · SN ( fr ) ], A T [a1,k , . . . , a N,k ] , and stands for the Hadamard product. Hence, the decoding matrix should be constructed as
˜ T sH ( fr ) · · · sH ( fr )e− j (K −1)υ ] D( fr ) = D
(5)
˜ is constructed by decoding coefficients bn,k ’s (the where D value of bn,k will be specified later) and (·)H represents the conjugate transpose. Then, the received signal after decoding processing can be written as T T ˜D ˜ T ) E( fr )] + n˜ m rTD,m ( fr ) = hm ( fr )[(A ( fr )
(6)
is the spectral density matrix of size N × N where E( fr ) = with E i, j ( fr ) denoting the power (i = j ) or cross-spectral density (i = j ) between waveforms si (τ ) and s j (τ ). ˜D ˜ T is a diagonal matrix of size N × N, after decoding, If A only the diagonal elements of the spectral density matrix E( fr ) will remain, and the cross-spectral density can be canceled out. Hence, the waveforms will be completely separated by STC processing regardless of whether the initial waveforms are orthogonal, which simplifies the waveform design. There is, however, another problem in this conventional STC processing arising from the waveform itself. After STC processing, the autocorrelation sidelobe cannot be reduced, which becomes a serious issue, especially for distributed targets. In fact, almost all autocorrelation sidelobe levels of waveforms cannot satisfy the strict imaging requirements. There are several modified STC schemes that have been used to cancel out the autocorrelation sidelobe. The Alamouti codes with complementary pairs have been used in [16] to cancel out the autocorrelation sidelobe in the context of radar polarization. However, only parts of the cross correlation are canceled out. In [17], the autocorrelation sidelobe and cross correlation are perfectly canceled out by setting the extended waveform matrix as S1 ( fr ) S1 ( fr ) S3 ( fr ) S3 ( fr ) S( fr ) = (7) S2 ( fr ) −S2 ( fr ) S4 ( fr ) −S4 ( fr ) ssH
where {S1 ( fr ), S3 ( fr )} and {S2 ( fr ), S4 ( fr )} are two Golay complementary pairs and the decoding matrix is given as D( fr ) = SH ( fr ). However, this autocorrelation sidelobe and cross-correlation cancelation is achieved at the high cost of increasing the number of accumulated pulses. Moreover, these schemes are only well suited for double-input stationary radar systems. In Section III, we propose an STC waveform scheme for MIMO-SAR aiming to overcome the drawbacks of conventional STC processing.
III. N OVEL STC WAVEFORM S CHEME A. Case of Two Transmitting Subarrays Consider that the whole phased array antenna is divided into left and right subarrays along the azimuth direction, and two beams generated by these two subarrays illuminate the same area on the ground. The radar platform flies at a speed V along the azimuth direction. In the signal transmitting stage, at the slow time η = kT , the two subarrays transmit waveforms s1 (τ ) and s2 (τ ), respectively. Then, at the next slow time η = (k + 1)T , the waveforms s2∗ (−τ ) and −s1∗ (−τ ) are transmitted, where (·)∗ represents the complex conjugate. The echoes received by the mth Rx subarray (the impulse response of the system) in two transmitting periods can be expressed as rm,1 (η) rm,1 (η) rm (τ, η) = exp − j 2π f c s1 τ − c c rm,2 (η) rm,2 (η) + exp − j 2π f c s2 τ − c c (8) +n m (τ, η) ) ) r (η (η r m,1 m,1 rm (τ, η ) = exp − j 2π f c s2∗ −τ + c c rm,2 (η ) ∗ rm,2 (η ) − exp − j 2π f c s1 −τ + c c (9) +n¯ m (τ, η) where n m (τ, η) and n¯ m (τ, η) denote radar system noise, which is assumed to be Gaussian and white. Moreover (m − 1)d 2 2 2 rm,n (η) = Rc + V η + V (n − 1)d 2 + Rc2 + V 2 η + V (m + n − 2)d 2 ≈ 2 Rc2 + V 2 η + (10) 2V is the transmission path between the nth transmitter and the mth receiver, where Rc is the range of closest approach2 and d represents the distance between elements of the subarrays. The approximation comes from the well-known displaced phase center principle [1], and it is reasonable in the case of Rc d. Transforming (8) into the 2-D frequency domain, we obtain
R m ( fr , f a ) = Hm,1( fr , fa )S1 ( fr ) + Hm,2( fr , fa )S2 ( fr ) + Nm,1 ( fr , fa ) (11) where fa denotes the Doppler frequency and Hm,n ( fr , fa ) is the Fourier transform of the term exp{− j 2π( f c + fr )(rm,n (η)/c)}, which can be regarded as the channel response. Using the principle of the stationary phase method [18], we obtain −j4π Rc ( fr + f c ) D( fr , fa ) Hm,n ( fr , f a ) = exp c (m + n − 2)d × exp j π f a (12) V
where D( fr , f a ) = 1 − (c2 f a2 /4V 2 fr + f c )2 )1/2 denotes the range migration of 2-D frequency domain. Note that η = 2 The range of closest approach is a minimum range when the zero-Doppler line across the target [18].
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η + T ; hence, the Fourier transform of the channel response is Hm,n ( fr , fa ) e j 2π f a T . Moreover, similar to the derivation of (8), the 2-D frequency domain expression of (9) can be obtained directly according to the time-shifting property of the Fourier transform as
R m ( fr , f a ) = Hm,1 ( fr , f a )S2∗ ( fr )e j 2π f a T − Hm,2( fr , fa ) (13) ×S1∗ ( fr )e j 2π f a T + Nm,2 ( fr , fa ). Grouping together (11) and (13), we obtain
T rm ( fr , fa ) = Hm,1( fr , f a ) Hm,2 ( fr , fa ) S ( f ) S2∗ ( fr )e j 2π f a T T ( fr , fa ) (14) × 1 r +nm S2 ( fr ) −S1∗ ( fr )e j 2π f a T T( f , f ) = [ N where nm m,1 ( f r , f a ) Nm,2 ( f r , f a ) ]. Similar to r a the processing of the 2×2 Alamouti code [13], the decoding matrix can be constructed as S1∗ ( fr ) S2∗ ( fr ) D( fr , f a ) = . (15) S2 ( fr )e− j 2π f a T −S1 ( fr )e− j 2π f a T
According to (3), the decoded signals can, therefore, be obtained as T rTD,m ( fr , f a ) = rm ( fr , fa ) · D( fr , f a )
= Hm,1 ( fr , f a )Hm,2 ( fr , fa ) ⎤ ⎡ 2 2 |Si ( fr )| 0 ⎥ ⎢ ⎥ ⎢ i=1 ⎥ ×⎢ ⎥ ⎢ 2 ⎣ 2⎦ 0 |Si ( fr )| i=1
+ ¯n Tm ( fr , fa ).
(16)
Remark 1: After decoding, two equivalent phase center outputs are obtained. Compared with the amplitude of the matched-filter output in conventional MIMO radar (which is Hm,n ( fr , fa ) · |Sn ( fr )|2 ), the amplitude of the nth (n = 1, 2) phase center output is doubled, that is, Hm,n ( fr , fa ) · 2 2 |S i=1 i ( f r )| . Hence, the power will be increased by a factor of 4. However, the power of the noise output after STC processing is approximated as 2i=1 |Si ( fr ) · Nm,i ( fr , fa )|2 . Consequently, the signal-to-noise ratio (SNR) will increase by 3 dB when compared with the conventional matchedfilter output |Si ( fr ) · Nm ( fr , f a )|2 . Thus, SNR improvement is another major benefit of the proposed scheme. Generally speaking, the benefit is obtained by the coherent integration of two successive PRIs. Every equivalent phase center output in (16) can be used for SAR imaging. Inverse transforming the nth output into the time domain, we obtain 2 −1 2 R˜ m,n (τ, f a ) = F fr Hm,n ( fr , f a ) |Si ( fr )|
Fig. 1. Transmitted waveforms during the 2K PRIs. In the odd PRI, the nth subarray transmits waveforms a1,k s1 (τ ), . . . , a N,k s2N (τ ). Then, in the even ∗ PRI, the waveforms a1,k s2∗ (−τ ), . . . , −a N,k s2N −1 (−τ ) are transmitted. For the transmitters to perform at optimum, coefficients an,k ’s should be chosen from the set {−1, 1}.
time, and rrd ( f a ) = Rc + (λ2 Rc fa2 /8V 2 ) denotes range cell migration. To cancel out the autocorrelation sidelobe, the property of the MPC pair can be employed. Specifically, an MPC pair is a pair of phase-coded waveforms whose autocorrelation functions are complementary [19] 0, τ = 0 ac f 1 (τ ) + ac f 2 (τ ) = (18) ξ, τ = 0 where ξ is a constant. Hence, if the waveforms s1 (τ ) and s2 (τ ) are an MPC pair, (17) can be rewritten as rrd ( f a ) 4π f c Rc ˜ Rm,n (τ, fa ) = ξ · δ τ − exp − j c c 2 (m + n − 2)d fa fa + × exp j π (19) Ka V where δ (·) is the impulse function. Equation (19) shows that the output energy is concentrated on the correct position (τ = 0), and no energy leaks into the sidelobe region (τ = 0). Hence, the autocorrelation sidelobe and cross correlation are completely canceled out in the sidelobe position. B. Case of General MIMO Configuration The proposed scheme can be extended to an MIMO configuration with 2N subarrays, where N is an arbitrarily constant. The waveforms transmitted during the 2K PRIs are shown in Fig. 1. Note that every dashed box in Fig. 1 denotes a proposed scheme encoded by the coefficient an,k , and {sn (τ ), sn+1 (τ )|n = 1, 3, . . . , 2N − 1} are N different MPC pairs. The echoes received by the mth Rx subarray in 2K transmitting periods can be expressed as
T ( fr , f a ) = Hm,1 ( fr , f a ) · · · Hm,2N ( fr , f a ) rm ⎤ ⎡ a1,1 S2∗ ( fr )e j 2π f a T ··· a1,1 S1 ( fr ) ∗ j 2π f a T ⎢ · · ·⎥ × ⎣a1,1 S2 ( fr ) −a1,1 S1 ( fr )e ⎦ .. .. .. . . .
i=1
rrd ( f a ) 4π f c Rc = ac f i τ − ×exp − j c c i=1 2 f (m + n − 2) d fa × exp j π a + (17) Ka V
T + nm ( fr , f a ) .
2
where ac f i (·) denotes the ACF of the waveform si (τ ), K a = (2V 2 /λRc ) is the chirp rate for the slow
⎡ ⎢ ⎢ ⎢ ⎣
(20)
The decoding matrix D( fr , fa ) is constructed as d1,1 S1∗ ( fr ) d1,1 S2 ( fr )e− j 2π f a T .. .
d1,1 S2∗ ( fr ) −d1,1 S1 ( fr )e−j 2π f a T .. .
⎤ ··· · · ·⎥ ⎥ .. ⎥ .⎦
d1,K S2 ( fr )e− j 2π f a (2K −1)T d1,K S1 ( fr )e− j 2π f a (2K −1)T · · · (21)
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Fig. 2. Compression outputs of MIMO and conventional/windowing weighted STC radars. The integrated sidelobe ratio is ISLR = 10×log10 (( pt − pm / pt )), where pt and pm are the total power and the main lobe power of the compression output, respectively. The main lobe is specified by the null-to-null width
and the code (subpulse) width for up/downchirp and phase-coded waveforms, respectively. The peak sidelobe ratio is defined as PSLR = 20 × log10 ((rs /rm )), where rs and rm are the peak intensity of the most prominent sidelobe and the peak intensity of the main lobe, respectively. The parameter ρ denotes the resolution defined as the half power (3 dB) width. (1.a) and (2.a) MIMO output. (1.b) and (2.b) STC output. (1.c) and (2.c) Windowing weighted STC output.
˜D ˜ T is a where dn,k denotes the decoding coefficient. If A ˜ ˜ diagonal matrix, where A and D are constructed with the coefficients an,k ’s and bn,k ’s, where A and D are constructed with the coefficients an,k ’s and bn,k ’s, autocorrelation sidelobe and cross correlation are completely canceled out. IV. S IMULATION R ESULTS Our simulations are carried out as follows. The range of closest approach is 14.142 km. The optimum spatial filter is assumed, i.e., the antenna pattern ideally covers the angular range of interest and suppresses the signals from the sidelobe of the antenna pattern. The channel is assumed to cause only transmission delay, i.e., effects, such as multipath and fading, are neglected. The additive white Gaussian noise is added, and the SNR is set to 20 dB. The range sampling rate is 100 MHz, and the time duration of the transmitted waveforms is 20.48 μs. The radar platform flies at an altitude of 10 km at a speed of 200 m/s, the PRF is 1000 Hz, and the carrier frequency is 10 GHz. First, the outputs of the conventional MIMO radar processing [6] and conventional STC processing [11] are shown in Fig. 2. Orthogonal up/downchirp and orthogonal phasecoded waveforms are employed in Fig. 2(1.a) and (1.b) and (2.a) and (2.b), respectively. The bandwidth of the up/downchirp is 50 MHz, and 1024 orthogonal codes (subpulses) are used for the phase-coded waveforms. Although most of the echo energy is compressed into the correct position, regardless of what types of waveforms are transmitted, the ISLR of the MIMO radar, when compared with the output of STC processing, increases significantly due to the cross correlation. The simulation result verifies the conclusion that the cross correlation is completely canceled out
by the STC-MIMO system, but the waveform autocorrelation sidelobe remains after the pulse compression. The imaging requirement of SAR should satisfy ISLR < −17 dB and PSLR < −20 dB [18]. After conventional STC processing, however, ISLR ≈ −10 dB and PSLR ≈ −13 dB according to Fig. 2(1.b). In Fig. 2(2.b), the PSLR can satisfy the above requirement, but ISLR = −1.69 dB. Hence, the exorbitant sidelobe level after conventional STC processing cannot satisfy the imaging requirement. To decrease the sidelobe level, windowing techniques are widely used in SAR imaging [18]. In Fig. 2(1.c) and (2.c), the Kaiser window with the parameter β = 2.5 is employed. The windowing technique can significantly decrease the sidelobe level for the up/downchirps. However, this is achieved at the cost of decreasing resolution (because the windowing method narrows the effective signal bandwidth). In addition, the windowing technique is invalid for phase-coded waveforms. Second, the performance of the proposed STC scheme in combination with the MPC coding technique is given in Fig. 3. It can be seen that both the waveform autocorrelation sidelobe and cross correlation are jointly canceled out by the proposed STC scheme, and the ISLR is decreased. The sidelobe level of the output satisfies the imaging requirement. Finally, the performance improvement of the imaging for the scene with 200 pixels in the range direction is shown in Fig. 4. To display the sidelobe levels, the SAR image with only 128 pixels in the range direction is placed in the middle of the scene, and the remaining pixel values in the scene are set to zero. The top image in Fig. 4 shows the output of the conventional STC scheme when the phase-coded waveforms are employed. Because the autocorrelation sidelobe leads to a high ISLR, a severe degradation of the image quality and
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are some other potential benefits in MPC-based STC SAR imaging, such as SNR improvement and the simplification of waveform design. The new scheme provides a better choice for SAR applications. R EFERENCES
Fig. 3. Compression outputs of MPC-based STC scheme. The ISLR is decreased significantly.
Fig. 4. Simulation of MIMO-SAR imaging. (a) Output image after the conventional STC processing. (b) Output image after the proposed STC processing. (c) Range profiles of the azimuth averaged power for the original scene (solid line), the difference between the first image and the original scene (dotted line), and the difference between the second image and the original scene (dashed line).
contrast can be seen. The contrast ratio of the image decreases significantly, and some gray spots can be seen outside the region of the SAR image. Thus, the improvement in the imaging performance by the conventional STC is not sufficient. The second image of Fig. 4 shows the output of the proposed MPC-based STC scheme. It can be seen that the scene is well reconstructed, and there is no energy leakage outside of the SAR image. For a quantitative analysis, three power profiles have been added at the bottom of Fig. 4. The dotted and dashed lines show the differences of the azimuth averaged power between the two images and the original scene. Because of the cancelation of the autocorrelation sidelobe, there is an approximately 10-dB improvement between the conventional and the proposed STC scheme. V. C ONCLUSION This letter presents a novel MPC-based STC scheme for MIMO-SAR. Autocorrelation sidelobe and cross-correlation cancelation is the main advantage of this scheme. There
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