Wideband MIMO Radar Waveform Design for Multiple ... - IEEE Xplore

IEEE SENSORS JOURNAL, VOL. 16, NO. 23, DECEMBER 1, 2016

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Wideband MIMO Radar Waveform Design for Multiple Target Imaging Hongwei Liu, Member, IEEE, Xu Wang, Bo Jiu, Member, IEEE, Junkun Yan, Meng Wu, and Zheng Bao Senior Member, IEEE Abstract— This paper focuses on the design of wideband multiple-input multiple-output (MIMO) radar waveform and its application to multi-target imaging. In a multiple targets scenario, the imaging performances are affected by the transmit beampattern and the power spectral density (PSD) illuminating on targets. Taking into account these factors, we have established an unconstrained optimization model based on jointly minimizing the matching errors of the beampattern and PSDs. To solve the large-scale optimization problem, a conjugate gradient-based wideband MIMO waveform design method is proposed. For the purpose of improving the computational efficiency, the Chirp-Z transform is employed in evaluating the discrete frequency spectrum. Numerical results show that the proposed method can approximately realize the desired multibeam and PSDs. By utilizing a sparse representation method, the designed waveform with randomly distributed PSDs can be applied to multitarget imaging. Index Terms— MIMO radar, waveform design, wideband multibeam, radar imaging, sparse representation.

I. I NTRODUCTION ULTIPLE-INPUT MULTIPLE-OUTPUT (MIMO) radar, a new technology in radar systems, has become a hot research topic [1]–[17]. MIMO radar is superior to phased array radar in many aspects, such as its higher degree of freedom (DOF), better parameter identification capability [18], and the flexibility in transmit beampattern design [2]–[19]. Compared with narrowband radar, wideband radar has better range resolution and is able to obtain more target information. Meanwhile, wideband radar has more reliable detection and tracking performance and is not sensitive to active or passive interference [20]. Therefore, wideband MIMO radar is an important research field. In the scene of multi-target tracking or imaging, the loads placed on the radar system are heavy. Phased-array radar adopts the conventional time-division multibeam which focuses the energy on one beam and switches the beam to the interesting directions one by one. However, long time observation for a single target is not acceptable in a multi-function

M

Manuscript received March 18, 2016; revised June 14, 2016 and August 5, 2016; accepted August 16, 2016. Date of publication August 31, 2016; date of current version November 4, 2016. This work was supported in part by the National Science Fund for Distinguished Young Scholars under Grant 61525105, and in part by the National Natural Science Foundation of China under Grant 61271291 and Grant 61671351. The associate editor coordinating the review of this paper and approving it for publication was Dr. Francis P. Hindle. (Corresponding author: Bo Jiu.) H. Liu, B. Jiu, J. Yan, M. Wu, and Z. Bao are with the National Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071, China (e-mail: [email protected]). X. Wang is with the Xi’an Electronic Engineering Research Institute, Xi’an 710100, China. Digital Object Identifier 10.1109/JSEN.2016.2604844

radar system. Unlike the approach in phased-array radar, the simultaneous multibeam technique [10]–[14], [21]–[25] in MIMO radar does not decrease the pulse repetition frequency (PRF) and the dwell time on each target. Then, the tracking accuracy and the resolution of the radar image can be guaranteed. Furthermore, the availability of resources can also be improved by allocating the needed powers for multiple beams. Therefore, the multibeam technique in wideband MIMO radar can be applied to multi-target imaging. MIMO radar waveform design has received considerable attention in the past few years. These existing design approaches can be classified into four categories. The first exploits waveform design to extract the information of targets and improve target identification and classification, based on maximizing MI (mutual information) or minimizing MSE (mean square error) [5]–[7]. The second focuses on optimizing the range, angular, and Doppler resolution based on radar ambiguity function [8]–[10]. In the third category, the beampattern is optimized by minimizing the beampattern matching error or maximum SINR [11]–[14], [21]–[26]. The last one emphasizes the correlation properties, based on the merits of auto-correlation level and cross-correlation level [15], [16]. At present, MIMO radar waveform design algorithms for realizing multibeam mainly consist of two-stage methods and single-stage ones. For a two-stage method, the optimal correlation matrices (R) are firstly obtained by solving beampattern matching models [11]–[13], and then an optimal/acceptable waveform is solved [14]–[22]. In the single-stage method, the transmit waveform is designed without the prior R [23]–[25]. An iterative quasi-newton-based waveform design method is proposed in [23] to realize multiple beams and suppress the auto/cross-correlation sidelobes in the directions of interest. By utilizing a weight matrix, a set of independent finitealphabet constant envelope waveforms are generated and pre-processed. The beam-shape method proposed in [25] shapes the transmit beampattern by optimizing the waveform matrix and the auxiliary variables alternatively. This method is applicable to several kinds of signals, such as low-rank, discrete-phase or low-PAR. Due to the chromatic dispersion in wideband radar systems, the steering vectors at different frequencies are different. As a result, the wideband MIMO radar waveform design is more complicated than the narrowband one. In [27], the optimal power spectral density matrix is designed to generate the desired beampattern, but the constant-modulus signal has not been synthesized. The wideband beampattern formation via iterative technique (WBFIT) proposed in [28] is an efficient

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TABLE I N OTATIONS

Fig. 1.

The transmit array of a MIMO radar system.

method to realize the desired spatial-frequency beampattern. However, the desired spatial-frequency beampattern in the target direction is distributed in one continuous subband, which may lead to bandwidth loss and decrease the range resolution. Based on the practical demand for multi-target imaging, three issues should be considered in waveform design: (i) On account of the enough signal-to-noise ratio (SNR), the designed beampattern need to forming a mainlobe in each target direction [Therefore, the signal in the target direction is described as mainlobe signal in the following]; (ii) The power spectrums (or power spectral densities, PSDs) illuminating on targets should be designed to guarantee the range resolution; (iii) The signals arriving at different targets should be approximately orthogonal in order to reduce the mutual interferences among different echoes. Aiming at the above issues, a wideband MIMO radar waveform design for multi-target imaging is proposed in this paper. The main contributions of this paper can be summarized as follows: 1. An unconstrained optimization model is established to jointly design the beampattern and the PSDs in the target directions. To decrease the mutual interferences of target echoes without the degeneration of range resolution, the PSDs are randomly distributed over multiple non-overlapping subbands. 2. To improve the computational efficiency, the Chirp-Z transform (CZT) is employed in the calculation of the spatial frequency spectrum. 3. By utilizing a sparse representation method, the designed waveform can be applied to the simultaneous imaging of multiple targets. The rest of the paper is organized as follows. Section II gives the problem formulation. In section III, the wideband MIMO radar waveform design is proposed. The signal processing method for multi-target imaging is introduced in Section IV. Several designed waveforms with different desired PSDs and the corresponding results of imaging are shown in Section V. Conclusions are drawn in Section VI. Notation: Bold uppercase letters and lowercase letters denote matrices and vectors, respectively. The notations a, (·)∗ , and (·) H denote the transpose, the conjugate, and the conjugate transpose, respectively. See Table I for other notations used throughout this paper. II. S IGNAL M ODEL Consider a MIMO radar system with a uniform linear array (ULA) as shown in Fig.1. There are M transmit antennas with inter-element spacing d. The signal transmitted by the

mth antenna can be written as sm (t) = re{x m (t)e j 2π f c t }, 0 ≤ t ≤ T p

(1)

where x m (t) is the baseband signal, f c is the carrier frequency, and T p is the pulse duration. Suppose that the bandwidth of x m (t) is B. The baseband signal can be sampled as x ml = x m (t)|t =(l−1)Ts , l = 1, · · · , L, where Ts = 1/B is the sampling interval and L = T p /Ts denotes the code length. Let X = [x1 , · · · , x M ]T denote the transmit waveform matrix, where xm = [x m1 , · · · , x m L ]T is the signal transmitted by the mth antenna. The discrete Fourier transform (DFT) of x ml , l = 1, · · · , L can be given by ym (n) =

L

x ml e− j

2π L

(l−1)n

T = xm fn,

l=1

n = −L/2, · · · , L/2 − 1

(2)

where f n = [1, e− j 2πn/L , · · · , e− j 2πn(L−1)/L ]T . Then, the discrete frequency spectrums of the waveform X can be written as yn = [y1 (n), · · · , y M (n)]T = Xf n , n = −L/2, · · · , L/2 − 1

(3)

Defining the transmit steering vector at frequency f c + n B/L as an (θ ) = [1, e j 2π( f c+n B/L)

d sin θ c

, · · · , e j 2π( f c+n B/L)

(M−1)d sin θ c

]T , (4)

the discrete frequency spectrums of the signal at angle θ can be obtained by (see [28]) z n (θ ) = anT (θ )yn = anT (θ )Xf n , n = −L/2, · · · , L/2 − 1.

(5)

LIU et al.: WIDEBAND MIMO RADAR WAVEFORM DESIGN FOR MULTIPLE TARGET IMAGING

min φ,α

K k=1

I L/2−1 2 μ wk [G(θk , φ)/L − αg(θk )/L] + ςin Pn (θ i , φ) − αpin L 2

Pn (θ ) = |z n (θ )|2 /L = anT (θ )Xf n f nH X H a∗n (θ )/L

(6)

The transmit beampattern is the power distribution in space [11], [13] and can be denoted as the summation of the discrete power spectrum at angle θ

=

1 L 1 L

L/2−1

anT (θ )Xf n f nH X H a∗n (θ ) =

n=−L/2

L/2−1

Pn (θ ).

n=−L/2

(7) To maximum transmit efficiency, power amplifiers are presently operating at saturation and do not allow linear changes in amplitude [29]. As a result, the waveform should be designed with a constant modulus, i.e., |x ml | = 1. III. W IDEBAND MIMO R ADAR WAVEFORM D ESIGN As mentioned in Section I, the waveform design for multitarget imaging concerns the following three aspects: the transmit beampattern, the range resolution and the orthogonality of the mainlobe signals. The transmit beampattern determines the power illuminating on each target and thus influences the SNR of the echoes. The PSD of the mainlobe signal determines the range resolution and the orthogonality of the echoes. Therefore, the waveform is designed by jointly minimizing the matching errors of the transmit beampattern and the PSDs. Under the constant-modulus constraint, the waveform can be denoted as x ml = e j ϕml , where ϕml is the phase of x ml . Defining the phase matrix as φ = [ϕ 1 , · · · , ϕ L ], where ϕ l = [ϕ1l , · · · , ϕ Ml ]T , the waveform matrix can be written as X = exp( j φ). Substituting X = exp( j φ) into (6) and (7), the transmit beampattern and the PSD at angle θ can be rewritten as (8) and (9), respectively

Pn (θ, φ) =

L/2−1

L/2−1

pin pin˜ = 0, i = i˜

(11)

n=−L/2

Therefore, the effective frequency bands of the desired PSDs are non-overlapping in this paper.

It is worth noting that α in (10) can be optimally determined with a given φ. Therefore, with a fixed waveform matrix X = exp( j φ), the optimal α can be expressed as αopt = σ −1 [

K

wk g(θk )G(θk , φ)

k=1

+ μL

I L/2−1

ςin pin Pn (θ i , φ)]

(12)

i=1 n=−L/2 K

σ =

k=1

wk g (θk ) + μL 2

L/2−1 I

ςin pin

(13)

i=1 n=−L/2

Based on (12), the waveform design problem (10) can be transformed into a problem w.r.t. φ as follows min{F(φ) φ

|anT (θ ) exp( j φ)f n |2

n=−L/2 T |an (θ ) exp( j φ)f n |2 /L.

(8)

=

I . Let g(θ ) Suppose that there are I targets located at {θ i }i=1 L/2−1 denote the desired beampattern and { pin }n=−L/2 be the desired

PSD of the signal at angle θ i , i = 1, . . . , I . Then, the waveform design problem can be constructed as (10) at the top of the page, where θk is the discrete azimuth angle, α ≥ 0 is the scaling factor, and μ ≥ 0 is the weight for the second term.

L/2−1 K I 1 μ 2 w [e (φ)] + ςin [qin (φ)]2 } k k L2 L

(14)

ek (φ) = G(θk , φ) − αopt g(θk )

(15)

k=1

(9)

Under this condition, the same scaling factor α is used for the desired beampattern αg(θ i ) and the desired PSD αpin . To realize the orthogonality of the echoes from different L/2−1 targets, { pin }n=−L/2 are set to be orthogonal with each other, e.g.,

B. The Waveform Design Method

A. Problem Formulation

1 G(θ, φ) = L

K are the weights for different azimuths, Additionally, {wk }k=1 L/2−1 and {ςin }n=−L/2 denote the weights for different frequencies. The first and the second terms in (10) denote the matching errors of the beampattern and the matching errors of the PSDs, respectively. The expression G(θk , φ)/L can be seen as the average power spectrum at angle θk . Due to the relationship of the beampattern and the PSD in (7), the desired PSD L/2−1 L/2−1 pin = g(θ i ). { pin }n=−L/2 should satisfy the condition n=−L/2

|z n (θ )|2

n=−L/2 L/2−1

(10)

i=1 n=−L/2

Following the frequency spectrums z n (θ ), the power spectrum of the signal at angle θ can be formulated as

G(θ ) =

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i=1 n=−L/2

where

qin (φ) = Pn (θ i , φ) − αopt pin .

(16)

As seen from (14), the cost function is a fourth-order trigonometric polynomial w.r.t. φ. Because the cost function in (14) is nonconvex w.r.t. φ, the global optimum may not be guaranteed.

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TABLE II

C. The CZT-Based Wideband MIMO Waveform Design

WAVEFORM D ESIGN P ROCEDURE BASED ON THE C ONJUGATE G RADIENT A LGORITHM

As seen from Table II and equations (15)-(17), the main computational cost of the waveform design is dominated by

k ,φ) , Pn (θ i , φ) and ∂ Pn (∂φθ i ,φ) . Furthremore, the G(θk , φ), ∂G(θ ∂φ discrete frequency spectrums {anT (θk )Xf n } play an important role in these equations. Based on (5), the frequency spectrum of the signal at θ can be written as

z n (θ ) = anT (θ )Xf n = anT (θ )yn =

M−1

(n)

ym (n)e j 2πmu ,

(18)

m=0 sin θ where u (n) = c/( fdc +n B/L) is the normalized spatial frequency corresponding to the signal frequency fc + n B/L, n = −L/2, · · · , L/2 − 1. To avoid spatial ambiguity (grating lobes), the spacing between adjacent antennas is chosen as c [28]. d = 2( fc +B/2) In this paper, the discrete spatial angles are determined as follows

However, the performance of the local optimum for a fourthorder trigonometric polynomial is illustrated in [23], i.e., the local minimizer is a 1/2-approximation of the global minimum. Though the problem (14) is nonconvex, the numerical optimization method can be used to optimize the phase matrix. As a result of wideband signal, the dimensionality of the Hessian matrix or the approximate Hessian matrix may be too large to compute and store. Then, the second-order derivativesbased method (e.g., the Newton or quasi-Newton method) may not be applicable for the problem (14). The conjugate gradient method need not compute and save the second-order derivatives. Furthermore, it overcomes the steepest descent method’s shortcoming of slow convergence. Consequently, it is widely used to solve large-scale optimization problems [30]. Due to the moderate computational complexity and convergence rate, the conjugate gradient method is adopted in this paper. According to (12), (15), and (16), the gradient of the objective function in (14) can be expressed by ∇φ =

K 2 ∂ G(θk , φ) wk eˆk (φ) L2 ∂φ k=1

+

L/2−1 I 2μ ∂ Pn (θ i , φ) ςin qîn (φ) L ∂φ

(17)

i=1 n=−L/2

−K /2 + k − 1 ), k = 1, · · · , K K

(L/2)

uk

=

1 k−1 d sin θk =− + , k = 1, · · · , K , c/( f c + B/2) 2 K (20)

Moreover, the normalized spatial frequency at f c + n B/L can be further expressed as d sin θk 1 k −1 = n (− + ), c/( f c + n B/L) 2 K k = 1, · · · , K

u (n) k =

where n = u (n) k

f c +n B/L f c +B/2 ,

i=1 n=−L/2

The derivation of (17) is given in Appendix A. The waveform design procedure based on the conjugate gradient algorithm [30] is outlined in Table II.

(21)

n = −L/2, · · · , L/2 − 1. Therefore,

is uniformly sampled in the interval [−n /2, n /2]. Based on the sampled criterion (19), the frequency spectrum z n (θ ) in (18) can be sampled as z n (θk ) = anT (θk )Xf n . Additionally, the transmit steering vector at the frequency f c + n B/L can be further written as an (θk ) = [1, e j 2π( f c+n B/L)

d sin θk c

1 k−1 K )

eˆk (φ) = [ek (φ) − ε/σ g(θk )] qîn (φ) = [qin (φ) − ε/σ pin ] L/2−1 K I ε= wk ek (φ)g(θk ) + μL ςin qin (φ) pin .

(19)

As a result, the normalized spatial frequency at f c + B/2 can be sampled uniformly in [−1/2, 1/2]

= [1, e j 2πn (−2+

where

k=1

θk = arcsin(2

, · · · , e j 2π( f c+n B/L)

, · · · , e j 2πn (M−1)(−2 + 1

(M−1)d sin θk c

k−1 K )

]T

]T (22)

Then, the transmit steering matrix at the frequency f c + n B/L can be defined as (23), shown at the top of the next page. The spatial frequency spectrum can be formulated as zn = [z n (θ1 ), · · · , z n (θ K )]T = AnT yn .

(24)

From (23) and (24), it can be seen that zn is the frequency spectrum of yn and is uniformly sampled in the interval [−n /2, n /2]. Based on digital signal processing theory, the CZT can be used for rapidly computing the frequency spectrums that are uniformly distributed in a circular arc.


⎡

1

⎢ ⎢ An = [an (θ1 ), · · · , an (θ K )] = ⎢ ⎢ ⎣

e j 2πn .. . e j 2πn

z n (θk ) = = =

M−1

m=0 M−1 m=0 M−1

M−1

(n)

ym (n)e j 2πmu k = ym (n)e− j πn m e j

ym (n)e j 2πn m(− 2 + 1

k−1 K )

m(k−1)

ym (n)Hn−m Wnm(k−1) , k = 1, · · · , K

(25)

m=0 2π n

where Hn = e j πn and Wn = e j K . Equation (25) satisfies the form of the CZT. Therefore, the computational efficiency of the objective function in (14) and its gradient can be improved by using the CZT (the derivation is given in Appendix B). The wideband MIMO waveform design algorithm proposed in this paper is named the CZT-based Wideband MIMO Waveform Design algorithm (CZT-WWD).

Inverse synthetic aperture radar (ISAR) imaging is a useful tool for target identification and classification [32]–[34]. The CZT-WWD algorithm can realize wideband multiple beam illumination on multiple targets. The dwell time on each target is no decrease, which guarantees the cross-resolution of the radar image. In addition, enough down-range resolution is achieved by illuminating wideband signal on each target. With the combination of the multiple beams technique and the nonoverlapping distribution of the desired PSDs, radar imaging for multiple targets can be implemented simultaneously.

.. .

(−K /2+1)(M−1) K

1

··· .. .

e j 2πn .. .

· · · e j 2πn

(K /2−1) K

(K /2−1)(M−1) K

I denote the azimuth angles of I targets. Based Let {θ i }i=1

on (18), the mainlobe signal at angle θ i can be formulated as

T = F LH [a−L/2 (θ i )Xf −L/2 , . . . , a TL/2−1 (θ i )Xf L/2−1 ]T

(26) √ where F L = [f −L/2 , · · · , f L/2−1 ]T / L is the FFT matrix with size L × L. The signal reflected from the i th target can be formulated in the frequency domain as follows [35]

r(θ i ) = F L R [st ar (θ i ) ∗ γ i

= [F L R |L st ar (θ i )] [F L R |L T γ i ]

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(27)

where γ i ∈ C L T ×1 is the scattering coefficient of the i th target and L R = L T + L − 1. Additionally, F L R |L and F L R |L T denote the first L and L T columns of the FFT matrix F L R respectively.

(23)

Therefore, F L R |L st ar (θ i ) and F L R |L T γ i denote the L R -point FFT of st ar (θ i ) and γ i , respectively. It should be noted that L T is the estimated size of the target and is generally larger than the real size. The scattering coefficient γ i is referred to as the high resolution range profile (HRRP). The reflected signals are received by an array consisting of M˜ antennas with the inter-element spacing d. Then, the output of the wideband beamformer [36] directing to i th the target can be expressed in the frequency domain as follows

(28) vi = M˜ F L R |L st ar (θ i ) F L R |L T γ i + ni where vi ∈ C L R ×1 is the output of the i th beamformer and ni involves the interference reflected from other targets and noise. By extracting the data located in the effective frequency band from vi , the effective measured data with size L¯ R can be obtained as follows

v¯ i = M˜ F¯ L R |L st ar (θ i ) F¯ L R |L T γ i + n¯ i = Gi γ i + n¯ i ˜ ag(F¯ L R |L st ar (θ i ))F¯ L R |L T Gi = Mdi

(29)

where n¯ i is the interference and noise data located in the effective frequency band. F¯ L R |L and F¯ L R |L T are the submatrices extracted from F L R |L and F L R |L T by the approach the same as that of v¯ i . In general, strong scattering centers are sparse in the complete range bins of the HRRP γ i , whereas signals from weak scattering centers contribute little to the HRRP [32]–[34]. Similar to the sparse representation model in [32], the problem of recovering the sparse vector γ i can be formulated as follows min ||¯vi − Gi γ i ||22 + λ||γ i ||1 γi

A. Range Profile Estimation

e j 2πn

(−K /2+1) K

···

IV. M ULTIPLE TARGET I MAGING

st ar (θ i )

e j 2πn

(−K /2)(M−1) K

m=0 2π n K

1

(−K /2) K

In fact, based on (18), the spatial frequency spectrum z n (θk ) in (24) can be further denoted as

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(30)

where λ is the regularization parameter that is related to the SNR and the sparsity of γ i [32]–[34]. Some methods in [32] and [37]–[39] can be used to determine λ. The problem (30) is a convex l1 -norm optimization problem that can be solved by various convex optimization toolboxes, such as CVX [40]. B. Target Imaging Compared with conventional ISAR imaging [32]–[34], [40], the echo model of multi-target imaging differs only in the signal itself. Therefore, the conventional motion compensation technique [41]–[44] can be applied to multi-target imaging. Without loss of generality, we assume that the motion compensation has been performed. In the ISAR model, the scattering coefficients of the i th target can be expressed as [33] i = i F∗Na ,

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Fig. 2.

The spatial and frequency properties. The power spectrums at (a) −40°, (b) 0°, (c) 40°, (d) beampattern, and (e) beampattern error.

Fig. 3.

The spatial-frequency power distribution.

Fig. 5.

The desired PSDs { pin }.

of i denotes the scattering coefficient in the nth pulse. Similar to extracting v¯ i from vi in (29), the effective measured data in the Na pulses can be written as ¯ i = Gi i + N ¯ i = Gi i F∗N + N ¯i V a

(31)

¯ i = [n(1) , · · · , n¯ (Na ) ], ¯ i = [v¯ (1) , · · · , v¯ (Na ) ] and N where V i i i i (n a ) a) and n ¯ denoting the effective measured data and with v¯ (n i i the noise term in the n a th pulse, respectively.

In the ISAR imagery i , strong scattering centers are usually sparse [32]–[34]. Based on the sparse representation, the problem of recovering i is stated as follows

I . (a) Auto-correlation. Fig. 4. Correlation properties of the signals at { θ i }i=1 (b) Cross-correlation.

ˆ i = arg min ||V ¯ i − Gi i F∗N ||2F + η||i ||1 a i

¯ i FTN − Gi i ||2F + η||i ||1 = arg min ||V a i

where i ∈ C L T ×Na is the two-dimensional ISAR image of the i th target and Na is the pulse number. The nth column

(32)

where η is the regularization parameter. The problem (32) is a convex l1 -norm optimization problem. Due to the


Fig. 10. Fig. 6.

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I . Auto-correlation properties of the signals at { θ i }i=1


V. N UMERICAL E XAMPLES

Fig. 7.

I . Auto-correlation properties of the signals at { θ i }i=1

Consider a MIMO radar system comprising M = 16 transmit antennas and M˜ = 16 receive antennas. The system parameters are set as follows: carrier frequency f c = 2G H z, pulse width T p = 5.12μs, and bandwidth B = 200M H z. The spacing between adjacent antennas is d = c/2( f c + B/2). The code length of the transmitted signals is L = T p B = 1024. The discrete azimuth angles are sampled according to (19) with K = 100. Assume that I = 3 targets are located at θ 1 = −40°, θ 2 = 0° and θ 3 = 40°. This section consists of two parts, the wideband waveform design and the multi-target imaging. Several numerical examples are presented in the two parts to illustrate the merits of the proposed CZT-WWD algorithm and the performance of its application in multi-target imaging. The simulations are implemented on a 2.7 GHz PC with 4 GB of RAM. A. Wideband Waveform Design for MIMO Radar Suppose that the desired power ratio of the three targets is 1:1:1. Then, the desired beampattern can be set as gi (θ ), θ ∈ i , i = 1, · · · , I (33) g(θ ) = 0, other wi se

Fig. 8.

Fig. 9.

The desired PSDs { pin }.


independence of the columns in ωi , the problem (32) can be decomposed into many subproblems related to the columns of i . Each subproblem can be solved by CVX [40].

where gi (θ ) is the traditional beam pointing at θ i and i is the mainlobe region of gi (θ ). In this paper, gi (θ ) is weighted by a -40dB Chebyshev window. To suppress the sidelobe K are set as follows at −90° and 90°, the weights {wk }k=1 5, k = 1, · · · , 4, K − 3, · · · , K wk = 1, other s. The weight μ can create a tradeoff between the performances of the beampattern and the power spectrum, and it is empirically set to 1. Without specification, the weights {ςin } are set to 1. The termination threshold and maximum iteration number in Table II are set as εth = 10−3 and pmax = 100. The phase matrix is initialized to random values following a uniform distribution over [0, 2π]. To evaluate the performance of the designed beampattern, the peak sidelobe level (PSL) and average sidelobe level (ASL) of the beampattern G(θ ) are defined as G(θk ) P S L = max θk ∈side M 2 G(θk ) 1 AS L = K side M2 θk ∈side

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Fig. 11.

Range profile estimation. (a) Waveform 1. (b) Waveform 2. (c) Waveform 3.

where side is the sidelobe region (i.e. the complementary set I ), K side is the quantity of the discrete angles in the of {i }i=1 sidelobe region, M 2 is the peak level of G(θ ) with respect to X=1 M×1 1TN×1 . The desired PSDs { pin } are assigned under the condition (11). In order to analyze the influence of { pin } on the imaging performance, several desired PSDs { pin } are considered in this paper. 1) Waveform 1: The desired PSDs { pin } are distributed in three continuous bands without overlapping: ⎧ 1, i f − 512 ≤ n ≤ −173, i = 1 ⎪ ⎪ ⎪ ⎨1, i f − 172 ≤ n ≤ 167, i = 2 (34) pin = ⎪ 1, i f 168 ≤ n ≤ 507, i = 3 ⎪ ⎪ ⎩ 0, else The dashed lines in Fig. 2(a)-(c) show the desired PSDs { pin }. The dotted line shown in Fig. 2(d) is the desired beampattern defined in (33). Based on g(θ ) and { pin }, the waveform X is optimized by using CZT-WWD. The solid lines in Fig. 2(a)-(c) show the PSDs of the mainlobe signals I . The solid line in Fig. 2(d) is the beampattern at {θ i }i=1 designed by CZT-WWD. The performances of WBFIT are also shown in Fig. 2. Herein, the desired 2-D beampattern of I and zero for other azimuths, WBFIT is set as (34) for {θ i }i=1 which is identical to the approach in [19]. The ASL and the PSL of the beampatterns in Fig.2(d) are given in Table III. The beampattern errors (between the designed one and the desired one) normalized by the peak of the desired beampattern are shown in Fig.2(e). The ASL of WBFIT is better than that of CZT-WWD. However, the CZT-WWD algorithm has merits in the PSL and the beampattern error. Furthmore, the spectrums designed by CZT-WWD are close to the desired ones. Fig. 3 shows the spatial-frequency power distribution of the waveform X designed by CZT-WWD. Three mainlobes are formed in the desired frequency bands. The auto-/cross correlation properties of the signals st ar (θ i ) are shown in Fig. 4. The auto-correlation peak sidelobe level (APSL) is approximately −14.5 dB. The peak cross-correlation level (PCCL) is approximately −22 dB. 2) Waveform 2: The desired PSDs { pin } are periodically distributed over the non-overlapping bands with the interval 48B/L and the width 16B/L. In Fig. 5, the PSDs designed

Fig. 12.

Reconstruction error as a function of λ.

Fig. 13.

Scattering mode.

by CZT-WWD are close to the desired ones. The designed beampattern (not shown) is similar to the beampattern of waveform 1. The performances of the waveform designed by WBFIT are identical to those in Fig. 2 and are not shown. In Fig.6, three mainlobes are periodically distributed across the non-overlapping bands. The auto-correlation properties of the mainlobe signals st ar (θ i ) are shown in Fig. 7. Compared with the result in Fig. 4(a), the mainlobe width in Fig. 7 is noticeably decreased. However, the APSL is approximately −4 dB, which is much higher than that in Fig. 4(a). The crosscorrelation properties are similar to those of Fig. 4(b) and are not shown. In fact, it is the periodical distribution of { pin } that leads to the grating lobes in Fig. 7. To suppress the APSL, the desired PSD { pin } with aperiodic distribution is considered in the following example.


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Fig. 14. Imaging results with the proposed approach. (a) The result based on waveform 1. (b) The result based on waveform 2. (c) The result based on waveform 3. TABLE III ASL AND PSL ( D B)

3) Waveform 3: The desired PSDs { pin } are randomly distributed over the non-overlapping bands with the width 16B/L. As shown in Fig. 8, the PSDs designed CZT-WWD are close to the desired PSDs { pin }. Fig. 9 shows the spatialfrequency power distribution which is identical to the desired beampattern and PSDs. The auto-correlation properties of st ar (θ i ) are shown in Fig. 10. The mainlobe width in Fig. 10 is close to that in Fig. 7. The APSL is approximately −10 dB, which is a compromise between those in Fig. 4 and Fig. 7. The cross-correlation properties are similar to those of examples 1 and 2. To illustrate the efficiency of the CZT in the algorithm, the execution time of designing the three waveforms by using the CZT fore-and-aft is listed in Table IV. The measurement shows that the execution time of CZT-WWD is roughly one-third of that without CZT. In addition, the execution time of WBFIT in the above three example is also given. It can be seen that WBFIT is more efficient than CZT-WWD. The WBFIT in [28] adopt the 2D (i.e., spatial-frequency) beampattern matching model and can effectively realize the desired 2D beampattern. However, for target imaging, we only concern the PSDs in the target directions. The 2D beampattern model in WBFIT may limit the DOF of the waveform design for target imaging. The results in Fig.2 and Table III indicate the advantages of CZT-WWD in beampattern and spectrums design. The CZT-WWD can also be used in the background of narrow band interference. With the prior information of narrow band interferences, the desired PSDs can be set to skip over the frequency bands of interferences. As a result, the polluted data in the band of frequency can be neglected to reduce the influence of the interference. B. Multiple Target Imaging The results of range profile estimation and target imaging using the designed waveforms are given in this subsection.

TABLE IV E XECUTION T IME (S ECONDS )

Fig. 11 shows the results of range profile estimation. In the simulation, we suppose that a target is located at −40° and that the dominant scattering centers are sparse. The length of the range profile is 128. The SNR of the simulated data is 15 dB. With λ determined by the approach in [38], the estimated results are obtained by solving (30). In Fig. 11, the estimated HRRP corresponding to waveform 1 is a failure, whereas the result of waveform 3 is slightly better than that of waveform 2. These results can be explained by the auto-correlation properties shown in the subsection V-A. The sidelobe levels (except for the 0th lag) reflect the correlations of the basis matrix which influence the reconstruction performance. To analyze the reconstruction performances, the normalized reconstruction error is defined as N R E = ||γ i − γˆ i ||/||γ i ||, where γˆ i is the estimated scattering coefficient vector. Fig. 12 shows the N R E as a function of λ, averaged over 1000 Monte Carlo tests. In the interval λ ∈ [10−2 , 1], the reconstruction errors change little with λ, and the N R E of waveform 3 is the smallest one. Fig. 13 is the scattering model of the Yak-42 airplane, which is obtained from the data set recorded by an ISAR experimental system. The simulated data with SNR = 10d B is generated by using the scattering coefficients and the relative position of the scattering centers in Fig. 13. There are 256 pulses in the simulated data. The PRF is 400 Hz. The range resolution and the Doppler resolution are 0.75 m and 1.56 Hz, respectively. It is supposed that the target shown in Fig. 13 is located at −40°. By solving (32) with η determined via the approach in [38], the estimated results are obtained. The imaging results and the reconstruction errors are shown in Figs. 14 - 15. Compared with the scattering model, the imaging result for waveform 1 loses too much information and the result for waveform 2 creates many false scattering centers. The imaging result for waveform 3 recovers the scattering information efficiently.

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Fig. 15.

Reconstruction error of the image. (a) Waveform 1. (b) Waveform 2. (c) Waveform 3.

∂ Pn (θ i , φ) ∂φ

The reconstructed error energy ratio is defined as ||i − ˆ i is the estimated ISAR image. ˆ i ||2 /||i ||2 , where F F The reconstructed error energy ratios corresponding to Fig. 15(a)-(c) are 80.2%, 32.31% and 5.36%, respectively. The reconstruction performances of the signals at 0° and 40° are similar to those in Figs. 14-15 and are not shown in this paper.

= 2L −1 I m{[a∗n (θ i )anT (θ i )Xf n f nH ] (X∗ )} ∂αopt = σ −1 ∂φ +σ

K

wk g(θk )

k=1

−1

μL

(36)

∂ G(θk , φ) ∂φ

L/2−1 I i=1 n=−L/2

∂ P(θ i , φ) ςin pin ∂φ

(37)

Therefore, the gradient of the objective function in (14) can be expressed by equation K 2 ∂ G(θk , φ) ∇φ = 2 wk eˆk (φ) L ∂φ k=1

L/2−1 I 2μ ∂ P(θ i , φ) + ςin qîn (φ) L ∂φ

(38)

i=1 n=−L/2

Where eˆk (φ) = [ek (φ) − ε/σ g(θk )] and qîn (φ) = [qin (φ) − ε/σ pin ]. A PPENDIX B A PPLY CZT TO F(φ) AND ∇φ

A PPENDIX A T HE G RADIENT OF F(φ) According to (12), (15), and (16), the gradient of ek (φ) and qin (φ) can be written as ∂αopt ∂ G(θk , φ) ∂ek (φ) = − g(θk ) ∂φ ∂φ ∂φ

The objective function F(φ) is composed of {anT (θk )Xf n }, which can be computed by using the CZT. The gradient of F(φ) can be rewritten as ∇φ K 2 ∂ G(θk , φ) = 2 wk eˆk (φ) L ∂φ

∂αopt ∂ P(θ i , φ) ∂qin (φ) = − pin ∂φ ∂φ ∂φ

k=1 I L/2−1

2μ + L

where L/2−1 ∂ G(θk , φ) = L −1 {[an (θk )anH (θk )X∗ f ∗n f nT ] ( j X) ∂φ

=

+ [a∗n (θ i )anT (θ i )Xf n f nH ] (− j X∗ )}

VI. C ONCLUSION In this paper, a wideband MIMO radar waveform design algorithm is proposed for multi-target imaging. The algorithm can jointly optimize the transmit beampattern and the PSDs of the mainlobe signals. By employing the CZT to calculate the spatial frequency spectrum, the computational cost is efficiently decreased. Several waveforms with different desired PSDs are designed to illustrate the validity of the proposed algorithm. The simulation results show that the designed waveform can approximately realize the desired beampattern and the desired PSDs. The waveform with randomly distributed PSDs is applicable to multi-target imaging. By utilizing the sparse representation method, the scattering information can be efficiently recovered. In future work, the PSD optimization will be further exploited to improve the estimation accuracy.

= L −1 {[an (θ i )anH (θ i )X∗ f ∗n f nT ] ( j X)

n=−L/2 ∗ + [an (θk )anT (θk )Xf n f nH ] (− j X∗ )} L/2−1 2L −1 I m{[a∗n (θk )anT (θk )Xf n f nH ] (X∗ )} n=−L/2

(35)

=

i=1 n=−L/2

∂ P(θ i , φ) ςin qîn (φ) ∂φ

L/2−1 K 4 I m{[ a∗n (θk )wk eˆk (φ)anT (θk )Xf n f nH ] (X∗ )} L3 n=−L/2 k=1

+

L/2−1 I 4μ I m{[ a∗n (θ i )ςin qîn (φ)anT (θ i )Xf n f nH ] (X∗ )} 2 L n=−L/2 i=1

(39)


The first term in (39) can be further expressed as T1 (φ) =

L/2−1 4 I m{[ (A∗n AnT Xf n )f nH ] (X∗ )} 3 L n=−L/2

=

L/2−1 4 I m{[ (A∗n κ n )f nH ] (X∗ )} 3 L

(40)

n=−L/2

where = diag{[w1 eˆ1 (φ), · · · , w K eˆ K (φ)]} and κ n = AnT Xf n . Similar to (24), AnT Xf n and A∗n κ n in (40) can be computed by using the CZT. R EFERENCES [1] A. M. Haimovich, R. S. Blum, and L. J. Cimini, “MIMO radar with widely separated antennas,” IEEE Signal Process Mag., vol. 25, no. 1, pp. 116–129, Jan. 2008. [2] E. Fishler, A. Haimovich, R. S. Blum, L. J. Cimini, D. Chizhik, and R. A. Valenzuela, “Spatial diversity in radars—Models and detection performance,” IEEE Trans. Signal Process, vol. 54, no. 3, pp. 823–838, Mar. 2006. [3] J. Li and P. Stoica, “MIMO radar with colocated antennas,” IEEE Signal Process Mag., vol. 24, no. 5, pp. 106–114, Sep. 2007. [4] D. W. Bliss and K. W. Forsythe, “Multiple-input multiple-output (MIMO) radar and imaging: Degrees of freedom and resolution,” in Proc. 37th Asilomar Conf. Signals, Syst., Comput., vol. 1, Nov. 2003, pp. 54–59. [5] Y. Yang and R. S. Blum, “MIMO radar waveform design based on mutual information and minimum mean-square error estimation,” IEEE Trans. Aerosp. Electron. Syst., vol. 43, no. 1, pp. 330–343, Jan. 2007. [6] Y. Yang and R. S. Blum, “Minimax robust MIMO radar waveform design,” IEEE J. Sel. Topics Signal Process, vol. 1, no. 1, pp. 147–155, Jun. 2007. [7] M. M. Naghsh, M. Modarres-Hashemi, S. ShahbazPanahi, M. Soltanalian, and P. Stoica, “Unified optimization framework for multi-static radar code design using information-theoretic criteria,” IEEE Trans. Signal Process, vol. 61, no. 21, pp. 5401–5416, Nov. 2013. [8] G. S. Antonio, D. R. Fuhrmann, and F. C. Robey, “MIMO radar ambiguity functions,” IEEE J. Sel. Topics Signal Process, vol. 1, no. 1, pp. 167–177, Jun. 2007. [9] C.-Y. Chen and P. P. Vaidyanathan, “MIMO radar ambiguity properties and optimization using frequency-hopping waveforms,” IEEE Trans. Signal Process, vol. 56, no. 12, pp. 5926–5936, Dec. 2008. [10] J. Li, P. Stoica, and X. Zheng, “Signal synthesis and receiver design for MIMO radar imaging,” IEEE Trans. Signal Process, vol. 56, no. 8, pp. 3959–3968, Aug. 2008. [11] P. Stoica, J. Li, and Y. Xie, “On probing signal design for MIMO radar,” IEEE Trans. Signal Process, vol. 55, no. 8, pp. 4151–4161, Aug. 2007. [12] D. R. Fuhrmann and G. S. Antonio, “Transmit beamforming for MIMO radar systems using signal cross-correlation,” IEEE Trans. Aerosp. Electron. Syst., vol. 44, no. 1, pp. 171–186, Jan. 2008. [13] S. Ahmed, J. S. Thompson, Y. R. Petillot, and B. Mulgrew, “Unconstrained synthesis of covariance matrix for MIMO radar transmit beampattern,” IEEE Trans. Signal Process, vol. 59, no. 8, pp. 3837–3849, Aug. 2011. [14] P. Stoica, J. Li, and X. Zhu, “Waveform synthesis for diversity-based transmit beampattern design,” IEEE Trans. Signal Process, vol. 56, no. 6, pp. 2593–2598, Jun. 2008. [15] B. Liu, Z. He, J. Zeng, and B. Y. Liu, “Polyphase orthogonal code design for MIMO radar systems,” in Proc. Int. Conf. Radar, Oct. 2006, pp. 1–4. [16] H. He, P. Stoica, and J. Li, “Designing unimodular sequence sets with good correlations—including an application to MIMO radar,” IEEE Trans. Signal Process, vol. 57, no. 11, pp. 4391–4405, Nov. 2009. [17] L. Xu, J. Li, and P. Stoica, “Radar imaging via adaptive MIMO techniques,” presented at the 14th Eur. Signal Process Conf., Florence, Italy, Sep. 2006. [18] J. Li, P. Stoica, L. Xu, and W. Roberts, “On parameter identifiability of MIMO radar,” IEEE Signal Process Lett., vol. 14, no. 12, pp. 968–971, Dec. 2007.

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[19] D. J. Rabideau and P. A. Parker, “Ubiquitous MIMO multifunction digital array radar and the role of time-energy management in radar,” Lincoln Lab., Massachusetts Inst. Technol., Cambridge, MA, USA, Tech. Rep. PR-DAR-4, Mar. 2004. [20] Y. D. Shirman, S. P. Leshchenko, and V. M. Orlenko, “Advantages and problems of wideband radar,” in Proc. Intern. Radar Conf., Adelaide, SA, Australia, 2003, pp. 15–21. [21] B. Friedlander, “On transmit beamforming for MIMO radar,” IEEE Trans. Aerosp. Electron. Syst., vol. 48, no. 4, pp. 3376–3388, Oct. 2012. [22] S. Ahmed, J. S. Thompson, Y. R. Petillot, and B. Mulgrew, “Finite alphabet constant-envelope waveform design for MIMO radar,” IEEE Trans. Signal Process, vol. 59, no. 11, pp. 5326–5337, Nov. 2011. [23] Y. C. Wang, X. Wang, H. Liu, and Z. Q. Luo, “On the design of constant modulus probing signals for MIMO radar,” IEEE Trans. Signal Process., vol. 60, no. 8, pp. 4432–4438, Aug. 2012. [24] S. Ahmed and M.-S. Alouini, “MIMO radar transmit beampattern design without synthesising the covariance matrix,” IEEE Trans. Signal Process., vol. 62, no. 9, pp. 2278–2289, May 2014. [25] M. Soltanalian, H. Hu, and P. Stoica, “Single-stage transmit beamforming design for MIMO radar,” Signal Process., vol. 102, no. 9, pp. 132–138, Sep. 2014. [26] S. Ahmed and M.-S. Alouini, “MIMO-Radar waveform covariance matrix for high SINR and low side-lobe levels,” IEEE Trans. Signal Process., vol. 62, no. 8, pp. 2056–2065, Apr. 2014. [27] G. S. Antonio and D. R. Fuhrmann, “Beampattern synthesis for wideband MIMO radar systems,” in Proc. 1st IEEE Int. Workshop Comput. Adv. Multi-Sensor Adapt. Process., Puerto Vallarta, Mexico, Dec. 2005, pp. 105–108. [28] H. He, P. Stoica, and J. Li, “Wideband MIMO systems: Signal design for transmit beampattern synthesis,” IEEE Trans. Signal Process, vol. 59, no. 2, pp. 618–628, Feb. 2011. [29] N. Levanon and E. Mozeson, Radar Signals. Hoboken, NJ, USA: Wiley, 2004. [30] W. Sun and Y.-X. Yuan, Optimization Theory and Methods—Nonlinear Programming. New York, NY, USA: Springer, 2006. [31] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [32] L. Zhang, Z. J. Qiao, M. Xing, Y. Li, and Z. Bao, “Highresolution ISAR imaging with sparse stepped-frequency waveforms,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 11, pp. 4630–4651, Nov. 2011. [33] L. Zhang, Z.-J. Qiao, M.-D. Xing, J.-L. Sheng, R. Guo, and Z. Bao, “High-resolution ISAR imaging by exploiting sparse apertures,” IEEE Trans. Antennas Propag., vol. 60, no. 2, pp. 997–1008, Feb. 2012. [34] L. Zhang, M. D. Xing, C.-W. Qiu, J. Li, and Z. Bao, “Achieving higher resolution ISAR imaging with limited pulses via compressed sampling,” IEEE Geosci. Remote Sens. Lett., vol. 6, no. 3, pp. 567–571, Jul. 2009. [35] D. Brandwood, Fourier Transforms in Radar and Signal Processing. Norwood, MA, USA: Artech House, 2003. [36] H. L. Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory. New York, NY, USA: Wiley, 2002. [37] S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process., vol. 56, no. 6, pp. 2346–2356, Jun. 2008. [38] S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput., vol. 20, no. 1, pp. 33–61, 1999. [39] S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process., vol. 57, no. 7, pp. 2479–2493, Jul. 2009. [40] M. Grant, S. Boyd, and Y. Ye. (2009). CVX: Matlab Software for Disciplined Convex Programming. [Online]. Available: http://www.stanford.edu/~boyd/cvx [41] C. C. Chen and H. C. Andrews, “Target-motion-induced radar imaging,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-16, no. 1, pp. 2–14, Jan. 1980. [42] W. Ye, T. S. Yeo, and Z. Bao, “Weighted least-squares estimation of phase errors for SAR/ISAR autofocus,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 5, pp. 2487–2494, Sep. 1999. [43] L. Xi, L. Guosui, and J. Ni, “Autofocusing of ISAR images based on entropy minimization,” IEEE Trans. Aerosp. Electron. Syst., vol. 35, no. 4, pp. 1240–1251, Oct. 1999. [44] J. Wang and D. Kasilingam, “Global range alignment for ISAR,” IEEE Trans. Aerosp. Electron. Syst., vol. 39, no. 1, pp. 351–357, Jan. 2003.

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Hongwei Liu (M’00) received the M.Eng. and Ph.D. degrees in electronic engineering from Xidian University, Xi’an, China, in 1995 and 1999, respectively. He was a Visiting Scholar with the Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA, from 2001 to 2002. He is currently the Director and a Professor with the National Laboratory of Radar Signal Processing, Xidian University. His research interests are radar signal processing, radar automatic target recognition, adaptive signal processing, and cognitive radar.

Xu Wang received the B.S. and Ph.D. degrees in electronic engineering from Xidian University, Xi’an, China, in 2009 and 2014, respectively. Since 2015, he has been with Xi’an Electronic Engineering Research Institute, Xi’an. His major research interests are radar signal processing, cognitive radar, and radar waveform design.

Bo Jiu (M’13) received the B.S., M.S., and Ph.D. degrees from Xidian University, Xi’an, China, in 2003, 2006, and 2009, respectively, all in electronic engineering. He is currently an Associate Professor with the National Laboratory of Radar Signal Processing, Xidian University. His research interests are radar signal processing, radar automatic target recognition, radar imaging, and cognitive radar.

Junkun Yan (M’16) was born in Sichuan, China, in 1987. He received the B.S. and Ph.D. degrees in electronic engineering from Xidian University, Xi’an, China, in 2009 and 2015, respectively. He is currently pursuing the Ph.D. degree with the National Laboratory of Radar Signal Processing, Xidian University. His research interests include adaptive signal processing, target tracking, and cognitive radar.

Meng Wu received the B.Sc. degree in electronic and information engineering and the Ph.D. degree in signal and information processing from Xidian University, Xi’an, China, in 2007 and 2015, respectively. Her main research interests are parameters estimation of MIMO radar and adaptive algorithms for array signal processing.

Zheng Bao (M’80–SM’90) was born in Jiangsu, China. He is currently a Professor with Xidian University and the Chairman of the Academic Board of the National Key Laboratory of Radar Signal Processing. He is a member of the Chinese Academy of Sciences. He has authored or coauthored over six books and published over 300 papers. His research fields include space-time adaptive processing, radar imaging, automatic target recognition, and over-thehorizon radar signal processing.