Open access peer-reviewed chapter

Space-Time Transmit-Receive Design for Colocated MIMO Radar

By Guolong Cui, Xianxiang Yu and Lingjiang Kong

Submitted: June 12th 2017Reviewed: October 25th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.71946

Downloaded: 382

Abstract

This chapter deals with the design of multiple input multiple-output (MIMO) radar space-time transmit code (STTC) and space-time receive filter (STRF) to enhance moving targets detection in the presence of signal-dependent interferences, where we assume that some knowledge of target and clutter statistics are available for MIMO radar system according to a cognitive paradigm by using a site-specific (possible dynamic) environment database. Thus, an iterative sequential optimization algorithm with ensuring the convergence is proposed to maximize the signal to interference plus noise ratio (SINR) under the similarity and constant modulus constraints on the probing waveform. In particular, each iteration of the proposed algorithm requires to solve the hidden convex problems. The computational complexity is linear with the number of iterations and polynomial with the sizes of the STTW and the STRF. Finally, the gain and the computation time of the proposed algorithm also compared with the available methods are evaluated.

Keywords

  • multiple input multiple output (MIMO)
  • space-time transmit code (STTC)
  • space-time receive filter (STRF)
  • signal-dependent interferences
  • signal to interference plus noise ratio (SINR)

1. Introduction

Multiple-input multiple-output (MIMO) radar emits multiple probing signals via its transmit antennas, which provides the greater flexibility for the design of the whole radar system, and boosts the development of more sophisticated signal processing algorithms [1]. On the basis of the configurations of transmitter/receiver antennas, MIMO radar systems can be classified into two categories: widely distributed [2, 3] and colocated [4, 5]. The former has different angles of view on the target owing to widely separated antennas, and this feature can be used to improve the performance of target detection and angle estimation, as well as the capabilities of target identification and classification [6]. The latter shares the same aspect angle of the target by using tightly spaced antennas. However, colocated MIMO radar exploits the waveform diversity to form a long virtual array, thus providing better results concerning spatial resolution, target localization, and the interference rejection, as well as obtaining the degrees of freedom for the design of transmit beam pattern [1, 7, 8].

Recently, colocated MIMO radar waveform design is a hot and challenging topic and has received significant attention. In general, these works can be divided into two categories. The first category focuses on the fast-time waveforms design exploiting some a priori information. In particular, in [6], by using the a priori knowledge of target power spectral density, the minimax robust waveforms are designed based on the rules of the mutual information (MI) and minimum mean-square error (MMSE). In [9], MIMO waveforms for the case of an extended target are devised based on the maximization of signal-to-interference plus-noise ratio (SINR) through a gradient-based algorithm assuming the knowledge of both the target and signal-dependent clutter statistics. In [10], by considering MMSE as figure of merit, MIMO radar waveforms are synthesized under signal-dependent clutter. The join design of the transmit waveform and the receive filter is addressed for improving the extended target delectability in the presence of signal-dependent clutter, by employing a cycle iteration algorithm with ensuring convergence [11]. In [12], by designing the transmit waveform and the receive filter, two sequential optimization algorithms are proposed to maximize SINR subject to the constant modulus and similarity constraints. Based on the rule of the worst-case output SINR in the presence of unknown target angle, the robust joint design of transmit waveform and the receive filter is considered [13]. Some more works can be found in [7, 8, 14, 15].

The second category addresses the MIMO radar space-(slow) time code design for moving target scenarios. In particular, in [16], MIMO radar slow-time code shares the ability of improving the resolution in angle-Doppler images and obtaining enhanced moving target detection performance. In [17], the signal-dependent interference is alleviated by the space-time coding framework based on a beamspace space-time adaptive processing (STAP). In [18], based on the max-min SINR optimization criteria, the time-division beamforming signal is designed for a multiple target scenario. For a moving point-like target detection, based on the worst case SINR over the actual and signal-dependent clutter statistics, the robust joint design of the space-time transmit code (STTC) satisfying the energy and similarity constraints and the space-time receive filter (STRF) is addressed in [19].

This chapter handles the joint design of the STTC and STRF with the aim of enhancing the moving target detectability under signal-dependent interferences and white Gaussian noise. Unlike [19, 20], some knowledge of target and clutter statistics is assumed to be available. In particular, the SINR is considered as figure of merit to maximize subject to a constant modulus constraint on the transmit signal in addition to a similarity constraint. To deal with the resulting nonconvex design problem, an iterative algorithm ensuring convergence is proposed. Each iteration of the proposed algorithm involves the solution of hidden convex problems. Specifically, both a convex problem with closed-form solution and a set of fractional programming problems, which can be globally solved through the Dinkelback’s algorithm, are solved. The resulting computational complexity is linear with the number of iterations and polynomial with the sizes of the STTC and the STRF.

The remainder of the chapter is organized as follows. In Section 2, the system model is formalized. In Section 3, the constrained optimization problem under constant modulus and similarity constraints is formulated. In Section 4, the new optimization algorithm is presented. In Section 5, the performance of the new procedure is evaluated. Finally, in Section 6, concluding remarks and possible future research tracks are provided.

2. System model

We focus on a colocated narrow band MIMO radar system consisting of NTtransmitters antennas and NRreceivers. Each transmitter emits a slow-time phase-coded coherent train with Kpulses. Let sk=s1ks2ksNTkTCNT, k=1,2,,Kdenote the transmitted space code vector at the kth transmission interval, where sntkdenotes the kth transmitted phase-code pulse of the ntth transmitting antenna, for nt=1,2,,NT, Tstands for the transpose, and CNis the set of N-dimensional vectors of complex numbers. At each receiver, the received waveform is downconverted to baseband, undergoes a pulse matched filtering operation, and then is sampled. Hence, the observations of the kth slow-time sample for a far-field moving target at the azimuth angle θ0can be expressed as [21]

xk=α0ej2πk1vd0Aθ0sk+dk+vk,E1

where

  • α0is a complex parameter taking into account the target radar cross section (RCS), channel propagation effects, and other terms involved into the radar range equation.

  • vd0denotes the normalized target Doppler frequency, which is related to the radial velocity vrvia the equation vd0=2vrT/λwith λbeing the carrier wavelength and Tbeing the pulse repetition time (PRT).

  • Aθ=arθatθ, in which atθand arθdenote the transmit spatial steering vector and the receive spatial steering vector at the azimuth angle θ, respectively, and and are the conjugate and the conjugate transpose operators, respectively. In particular, for the uniform linear arrays (ULAs), they are given by

atθ=1NT1ej2πdTλsinθej2πdTλNT1sinθT,E2
arθ=1NR1ej2πdRλsinθej2πdRλNR1sinθTE3

with dTand dRbeing the array interelement spacing of the transmitter and the receiver, respectively.

  • dkCNR,k=1,2,,K, considering Msignal-dependent uncorrelated point-like interfering scatterers. Specifically, as shown in Figure 1, the angle space is discretized as Θ=01L×2πL+1. For the mth interfering source located at the range-azimuth bin rmlm, rm01K1, lm01L, the received interfering vector dkcan be expressed as the superposition of the returns from Minterference sources, i.e.,

dk=m=1Mρmej2πvdmk1Aθmskrm,0rmk1,E4

with ρm, vdm, and θm, respectively, the complex amplitude, the normalized Doppler frequency, and the look angle, given by θm=2πL+1lm, of the mth interferences. Furthermore, Mis nominally equal to K×L+1.

  • vkCNR,k=1,2,,Kdenotes additive noise, modeled as independent and identically distributed (i.i.d.) complex circular zero-mean Gaussian random vector, i.e., vkCN0σ2INR, where INRdenotes NR×NR-dimensional identity matrix.

Figure 1.

Range-azimuth bins (the target of interest is represented by the red (solid) circle).

Let x=xT1xTKT, s=sT1sTKT, d=dT1dTKT, and v=vT1vTKT. Then, Eq. (1) can be expressed in a compact form as

x=α0Âvd0θ0s+d+v,E5

where

Âvdθ0=DiagpvdAθ0E6

with pvd=1ej2πvdej2πK1vdTbeing the temporal steering vector, denotes the Kronecker product, and Diagdenotes the diagonal matrix formed by the entries of the vector argument. Additionally, we assume that the noise vector vis a zero-mean circular complex Gaussian random vector with covariance matrix Σv=Evv=σv2INRK. Finally, interference vector dcan be expressed as

d=m=1MρmPrmÂvdmθms,E7

where Prmis given by

Prm=JrmINR,E8

in which Jrdenotes the shift matrix [23], whose k1k2th entry is defined as1,

Jrk1k2=1k1k2=r0k1k2r,E9

r01K1and k1k212K2. In particular, we assume that ρm, m=1,2,,M, and α0are a zero-mean uncorrelated random variables with, respectively, σm2=Eρm2and σ02=Eα02. As to the normalized Doppler frequency of the interfering signals, we model vdmas a random variable uniformly distributed around a mean Doppler frequency v¯dm, i.e.,

vdmUv¯dmεm2v¯dm+εm2,m1,2,,ME10

where εmaccounts for the uncertainty on vdm. Basing on the previous assumptions, the interference vector dhas zero mean and covariance matrix

Σds=Edd=m=1MJrmAθmssΞmJrmAθm,E11

where

Ξm=σm2Φεmv¯dmϒt,E12

in which

Φεmv¯dmk1k2=ej2πv¯dmk1k2sinπεmk1k2πεmk1k2,k1k212K2,E13

and ϒt=1t1tTwith 1t=111Tbeing the NT×1vector, and Edenote the Hadamard product and the statistical expectation, respectively. This expression, for the covariance matrix Σds, follows from the results obtained in ([19], Appendix 1).

Inspection of (11) and (12) reveals that the interference covariance matrix Σdsrequires the knowledge of θmand σm2as well as v¯dmand εm, for m=1,2,,M. These information can be obtained according to a cognitive paradigm [22, 23, 24] through exploiting a site-specific (possible dynamic) environment database, which involves a geographical information system (GIS), digital terrain maps, previous scans, tracking files, clutter models (in terms of electromagnetic reflectivity and spectral density), and meteorological information.

3. Problem formulation

This section formulates the joint design problem of the STTC and STRF based on the maximization of the output SINR considering practical constraints.

3.1. Output SINR

Letting the observations xbe processed via the STRF wCNRK, the SINR ρ̂swat the output of the receiver can be expressed as

ρ̂sw=α0wÂvd0θ0s2Ewd2+Ewv2=σ02wÂvd0θ0ssÂvd0θ0wwΣdsw+σv2ww,E14

where we exploit

Ewd2=wEddwE15

and

Ewv2=wEvvwE16

and assume w0and the independence between the disturbance and the noise random processes.

In particular, the numerator in (14) denotes the useful energy at the output of the STRF, wΣdswand σv2wwrepresent the clutter energy and noise energy, respectively, at the output of w. Observe that the clutter energy wΣdswfunctionally relies on the STTC wand the STRF sthrough Σdsas well as the useful energy. Furthermore, we note that the objective function ρ̂swrequires that the exact angle θ0and normalized Doppler frequency vd0are known. However, from a practical point of view, the explicit knowledge of θ0and vd0cannot be available. To circumvent this drawback, the averaged SINR defined as ρsw=Eρ̂swas figure of merit is exploited. More specifically, we suppose that vd0and θ0are independent random variables uniformly distributed around a mean Doppler frequency v¯d0and a mean azimuth θ¯0, respectively, i.e., vd0Uv¯d0ε02v¯d0+ε02, θ0Uθ¯0ϑ02θ¯0+ϑ02, where means “distribute” and Urepresents uniform distribution and ε0and ϑ0accounts for the uncertainty on vd0and θ0, respectively. Interestingly, after some algebraic manipulations, the objective function ρswshares the following two equivalent expressions,

where

ΓS=σ02EDiagpvdAθ0SDiagpvdAθ0E17
ΣdvS=m=1MJrmAθmSΞmJrmAθm+σv2INRKE18
ΘW=σ02EDiagpvdAθ0WDiagpvdAθ0E19
Σ¯dvW=m=1MJrmAθmWΞ¯mJrmAθm+σv2trWINTKE,E20

While S=ssHKNTand W=wwHKNR, Ξmis given by (12), E denotes the energy of s, Ξ¯m=σm2Ψεmv¯dmϒr, Ψεmv¯dmk1k2=Φεmv¯dmk1k2, k1k212K2and ϒr=1r1rTwith 1r=111TCNR, and trdenotes the trace of square matrix. These expressions follow from the results obtained in ([19], Appendix 3).

Note that ΓSand ΘWcan be rewritten in block matrix form, i.e.,

ΓS=σ02Γm1m2K×KE21
ΘW=σ02Θi1i2K×KE22

where Γm1m2CNR×NRand Θi1i2CNT×NTcan be computed by (38) and (46) respectively, m1m2i1i212K4, as shown in Appendix A.

3.2. Constant modulus and similarity constraints

In practical applications, the designed STTC is enforced to be unimodular (i.e., constant modulus) since the nonlinear property of radar amplifiers [24, 25]. To this end, we limit the modulus of each element of the code sas a constant. Precisely, the ith element siof scan be written as

si=1NTKejφi,i=1,2,,NTK,E23

with φidenoting the phase of si. Furthermore, Kdifferent similarity constraints are enforced on the NTtransmitting waveforms, namely

sks0kξk,k=1,2,,K,E24

where s0kCNTis the reference code vector at the kth transmission interval, ξkis a real parameter ruling the extent of the similarity, and xdenotes the infinite norm.

Without loss of generality, we assume the same similarity parameter ξ0(i.e., ξ0=ξ1==ξK) [12, 26, 28, 29, 30] on the sought STTC. Thus, Eq. (24) can be written as ss0ξ0, where s0=s0T1s0TKTis the reference code vector. Several reasons are presented to show the motivation to exploit the similarity constraints on radar codes. Actually, an arbitrary optimization of SINR via designing an STTC does not offer any kind of control on the shape of the resulting designed waveforms. Specifically, an pure optimization of the SINR can cause signals sharing high peak sidelobe levels and, in general, with an undesired ambiguity function feature. To this end, by exploiting the similarity constraint, when s0possesses suitable properties, such as low peak sidelobe levels, and reasonable Doppler resolutions, the designed STTC can enjoy some of the good ambiguity function feature of s0. In other words, the similarity constraint compromises the performance between SINR improvement and suitable waveform features [31].

3.3. Design problem

Summarizing, the joint design of the STTC and the STRF can be formulated as the following constrained optimization problem:

P1maxs,wρsws.t.sks0kξk,k=1,2,,K,si=1NTK,i=1,2,,NTK,w2=1,E25

where and , respectively, represent the modulus and the Euclidean norm. Without loss of generality, we add the constraint w2=1. P1is a NP-hard problem [12, 28] whose optimal solution cannot be found in polynomial time. Next, we develop a new iterative algorithm to offer high-quality solution to the NP-hard problem (25).

4. STTC and STRF design procedure

This section focuses on the design of an iterative algorithm ensuring convergence properties, which is capable of offering high-quality solutions to the NP-hard problem P1by sequentially improving the SINR. In particular, we exploit the pattern search framework to cyclically optimize the design variables ws1s2sNTK.

4.1. STRF optimization

In this subsection, we deal with the STRF optimization for a fixed STTC s. Specifically, we handle the optimization problem

PwmaxwwΓsswwΣdvssws.t.w2=1.E26

We observe that the optimal solution woto Pwis the maximum eigenvector of the matrix

Σdvss1Γss,

i.e., to a generalized eigenvector of the matrices Γssand Σdvsscorresponding to the maximum generalized eigenvalue. Thus, a closed-form solution to Pwcan be obtained by normalizing wo.

4.2. STTC optimization

This subsection is devoted to the optimization of the STTC under a fixed STRF. Precisely, each code element in sis sequentially optimized under the fixed remaining NTK1elements. Performing some algebraic manipulations to similarity constraints [26], the optimization problem Ps¯iwith respect to the ith STTC variable, i=1,,NTK, is written by,

Ps¯imaxs¯isΘwwssΣ¯dvwwss.t.args¯iγiγi+δ,s¯i=1NTK,E27

where s=s1s2si1s¯isi+1sKNTT, γi=args0iarccos1ξ2/2, δ=2arccos1ξ2/2, ξ=NTKξ0with 0ξ2, and s0iis the ith element of s0. Notice that for ξ=0, the code sis equal to the reference code s0, whereas the similarity constraint would become the constant modulus constraint with ξ=2.

Remark: This procedure by resorting to pattern search framework offers a new strategy to address the code design problem under a fixed filter. In addition, this STTC optimization problem can be efficiently but approximatively settled by semidefinite relaxation (SDR) and randomization procedure with the computational complexity of ONTK3.5+OLNTK2, where Lis the number of randomization trials. However, the SDR technique usually shares a huge computational complexity, especially in large dimension NTK, thus limiting its applications in real-time systems; moreover, the existing approach also needs the reasonable selection of L. On the other hand, it is shown that a higher quality solution can be further obtained via a sequential iteration optimization algorithm, which is capable of monotonically increasing the SINR value and achieving a stationary point of the formulated NP-hard problem [27].

Next, we focus on the proposed iteration algorithm to solve problem (27) in a polynomial time. In particular, performing some algebraic manipulations to the objective function in (27), Ps¯ican be equivalently rewritten as a fractional programming optimization problem by the following proposition.

Proposition 4.1 The problem Ps¯iis equivalent to

maxs¯ia1,is¯i+a3,ib1,is¯i+b3,is.t.s¯i=1NTKe,φγiγi+δ,E28

where

a3,i=a0,iNTK+a2,i,b3,i=b0,iNTK+b2,i,E29

and ak,i,bk,iare constants for k=0,1,2, xdenotes the real part of x.

Proof. See Appendix B.

Problem (28) is solvable [32] since the objective function is continuous with b1,is¯i+b3,i>0and the constraint is a compact set (closed and bounded set of C). Thus, we consider the following parametric problem [32],

maxs¯iϱμ=a1,is¯i+a3,iμb1,is¯i+b3,is.t.s¯i=1NTKe,φγiγi+δ.E30

After some simple manipulations, problem (30) can be rewritten as

maxs¯icis¯is.t.s¯i=1NTKe,φγiγi+δ,E31

where ci=a1,iμb1,iand the constant a3,iμb3,ido not affect the optimal value.

Interestingly, problem (31) shares a closed-form solution whose phase φis given by,

φ=φci,φciγiγi+δ,

where φciis the phase of ci; otherwise, the optimal solution φis given by,

φ=γi+δcosφci+γi+δcosφci+γiγicosφci+γi+δ<cosφci+γi.E32

We observe that problems (28) and (30) are relevant in each other via Lemma 2.1 of [32]. Specifically, we can find a solution to problem (28) by obtaining a solution of the equation ϱμ=0concerning s¯i. To this end, the Dinkelbach-type procedure [32, 33] summarized in Algorithm 1 is introduced to solve problem (27).

Algorithm 1. Dinkelbach-type algorithm for solving Ps¯i

Input: a1,i, a3,i, b1,i, b3,i, γiand δ;

Output: An optimal solution ŝito Ps¯i;

  1. Randomly generate s¯i,0within the feasible sets;

  2. Compute μ1=a1,is¯i,0+a3,ib1,is¯i,0+b3,iand let k1;

  3. Find the optimal solution s¯i,kby solving problem (30),

  4. If ϱμk=0, then s¯i,kis an optimal solution of Ps¯iwith optimal value μkand stop. Otherwise, go to step 5;

  5. Let μk=a1,is¯i,k+a3,ib1,is¯i,k+b3,iand kk+1; Then go to step 2.

Algorithm 1 sharing a linear convergence rate [34] is needed to handle the problem (30) in each iteration. The objective value of the generated sequence of points has a monotonic convergence property, and the optimal value of (28) can be achieved eventually. We set the exit condition ϱμ=0, actually, which can be replaced by ϱμς, with ςbeing a prescribed accuracy.

4.3. Transmit-receive system design

This subsection reports the iteration optimization procedure for the STTC and STRF in Algorithm 2. In particular, Algorithm 2 guarantees that the SINR monotonically increases2. Furthermore, we need to point out that the maximum block improvement (MBI) [24] framework could be used to ensure the convergence to a stationary point of problem P1.

The global computation consume of the Algorithm 2 is linear to the number of iterations and polynomial with the sizes of the STTC and the STRF. More specifically, each iteration of the proposed algorithm involves the computational cost associated with the solution to problems (26) and Ps¯i, for i=1,2,,NTK. The former requires to solve the generalized eigenvalue decomposition with the order of ONRK3(see [35], p. 500). Similarly, the latter is linear to polynomial with the size of the STTC, while each iteration needs the solution of a generalized fractional programming problem with the computational complexity of ONTK2. We need to point out that SOA2, based on the SDR and randomization method, can also be used to the solution of problem (25). However, it cannot guarantee the convergence to a stationary point due to the use of randomized approximations. Moreover, from computational complexity, each iteration of SOA2 has the order of ONRK3+ ONTK3.5+ OLNTK2, whereas Algorithm 2 is ONRK3+ ONTK3.

Algorithm 2. Algorithm for the joint STTC sand STRF wdesign

Input: θ¯0, ϑ0, s0, ξ, σm,rm,v¯dm,εm, for m=0,1,,M, and θp, for p=1,2,,M;

Output: An optimal solution swto P1;

  1. Construct γm,δ,m=1,2,,NTKexploiting s0;

  2. For n=0and initialize sn=s0;

  3. Compute w0=wo0wo0and ρ0=ρs0w0;

  4. nn+1and i=0;

  5. Compute Σ¯dvwnwnand Θwnwnby (20) and (22), respectively;

  6. ii+1;

  7. Compute ak,iand bk,iby (50) and (51), k=0,1,2, respectively;

  8. Find a3,iand b3,iby (29);

  9. Exploit Algorithm 1 to update siby maximizing the problem (27);

  10. If i=NTK, output sn=s1s2sKNTT. Otherwise, return to step 7;

  11. Compute Σdvsnsnand Γsnsnby (18) and (21), respectively;

  12. Find the generalized eigenvector wonof matrices Γsnsnand Σdvsnsncorresponding to the maximum generalized eigenvalue;

  13. Compute wn=wonwonand ρn=ρsnwn;

  14. If ρnρn1κ, where κis a user selected parameter to control convergence, output s=snand w=wn; Otherwise, repeat step 5 until convergence.

5. Numerical results

This section focuses on assessing the capability of the proposed algorithm for designing optimized STTC and STRF in signal-dependent interference for both a nonuniform and an uniform point-like clutter environment. In particular, for both scenarios, we consider an L-band radar with operating frequency fc=1.4GHz, which is equipped with an ULA of NT=4transmit elements and NR=8receive elements under an interelement spacing dt=dr=λ/2. We set the code length K=13for each transmitter and the orthogonal linear frequency modulation (LFM3) is used as the reference waveform s0[12] with the ntkth entry of the reference S0given by,

S0ntk=expj2πntk1/NTexpk12/NTKNTE33

where nt=1,2,,NTand k=1,2,,K. Hence, the reference code is derived as s0=vecS0. Moreover, we assume the target located at range-azimuth bin of interest (0,0) with power σ02=10dB. In addition, we set a mean azimuth θ¯0=0with azimuth uncertainty ϑ/2=1, and a normalized mean Doppler frequency v¯d0=0.4with Doppler uncertainty ε0/2=0.04for the presence of target. We set the noise variance to σv2=0dB. Finally, the exit condition4 ς=103for Algorithms 1 and 2 is κ=103, i.e.,

ρnρn1103.E34

All simulations are performed using Matlab 2010a version, running on a standard PC (with a 3.3 GHz Core i5 CPU and 8 GB RAM).

5.1. Nonuniform point-like clutter environment

This subsection focuses on a scenario where three disturbances, respectively, are located at the spatial angles θ1=55°,θ2=20°,θ3=40°, with corresponding range bins ri=0,i=1,2,3and powers σ12=30dB, σ22=28dB, σ32=25dB. Moreover, we suppose v¯d1=0.35,v¯d2=0.15,v¯d3=0.25,εm/2=0.04,m=1,2,3for the presence of the disturbances.

For comparison purpose, we also perform simulations for the SOA2 with constant modulus and similarity constraints as well as the algorithm in [19] with energy constraint (i.e., s2=1), respectively. In particular, Figure 2 shows the SINR versus the iteration number for different ξby also comparing the results obtained via Algorithm 2 and SOA2 considering L = 100 and exploiting the CVX toolbox [36] to handle the semidefinite programming (SDP) involved in SOA2. The results exhibit that the SINR values achieved using Algorithm 2 and SOA2 increase as the iteration number increases. In addition, the SINR increases as ξincreases owing to the higher degrees of freedom available at the design stage. Precisely, Algorithm 2 is superior to SOA2 for ξ=0.1,0.5,1.3. It is interesting to note that Algorithm 2 and SOA2 share almost the same SINR for ξ=2, whereas both obtain lower SINR than the case considering energy constraint. Finally, it is worth pointing out that a loss of SINR caused by constant constraint can be observed since the gap of SINR between ξ=2and energy constraint is about 1 dB.

Figure 2.

The SINR behavior versus iteration number assuming a target with − π / 180 ≤ θ 0 ≤ π / 180 , 0.36 ≤ v d 0 ≤ 0.44 for ξ = 0.1 , 0.5 , 1.3 , 2 , s 0 as the initial point.

Table 1 reports the achieved SINR values, iterations number, and global computation time of Algorithm 2 and SOA2 supposing a target with π/180θ0π/180, 0.36vd00.44for ξ=0.1,0.5,1.3,2and setting the same exit condition for SOA2. We observe that Algorithm 2 and SOA2 both converge very fast. Additionally, Algorithm 2 is superior to SOA2 concerning the achieved SINR value for ξ=0.1,0.5,1.3and concerning the required computational cost for ξ=0.1,0.5,1.3,2.

Algorithm 2SOA2
ξSINRnTimeSINRnTime
0.12.720.32362.334.0145
0.55.260.89423.533.9688
1.38.3121.71756.345.3498
28.8131.81028.879.3621

Table 1.

SINR values (in dB), iterations number, and global computation time (in seconds) of Algorithm 2 and SOA2 assuming a target with π/180θ0π/180, 0.36vd00.44for ξ=0.1,0.5,1.3,2, s0as the initial point.

In the following, the joint frequency and azimuth behavior of STTC and STRF are considered corresponding to ξ=2supposing π/180θ0π/180, 0.36vd00.44for different iteration numbers, by using the contour map of the slow-time cross ambiguity function (CAF) [19],

gnsnwnrvθ=wnPrÂ(vθ)sn2,E35

where Âvθand Prare obtained by exploiting Eqs. (6) and (8), respectively. Figure 3 plots the contour map of the Doppler-azimuth plane of CAF at r=0versus the iteration number n=01415for Algorithm 2. As expected, the lower and lower values in the regions of (highlighted by black ellipses) θ1=55°and 0.39v0.31, θ2=20°and 0.19v0.11, and θ3=40°and 0.21v0.29are achieved, with the increase of n. Thus, it is worth pointing out that the proposed algorithm can suitably shape the CAF to resist interferences.

Figure 3.

Doppler-azimuth plane of CAF at r = 0 for ξ = 2 of Algorithm 2 for n = 0 1 4 15 assuming a target with − π / 180 ≤ θ 0 ≤ π / 180 , 0.36 ≤ v d 0 ≤ 0.44 (black ellipses represent the locations of three interference sources), s 0 as the initial point.

For the uniform distribution, we define both standard deviations σvd0and σθ0of target Doppler and azimuth as, respectively,

σvd0=ε0/12,σθ0=ϑ0/12.

Figure 4 shows the SINR behaviors versus the standard deviations σvd0(Figure 4a) and σθ0(Figure 4b) supposing θ¯0=0°, v¯d0=0.4, respectively. Our curves highlight that the proposed algorithm can further improve SINR gain in comparison with SOA2 for ξ=0.1,0.5,1.3. We also observe that the higher σvd0and σθ0and the lower SINR can be obtained due to the larger inaccuracies on the knowledge of Doppler and azimuth of the actual target. Finally, we need to point out that the proposed design procedure still has the better robustness against a large uncertain set in comparison with SOA2.

Figure 4.

The SINR behaviors versus the standard deviations σ v d 0 (Figure 4a) and σ θ 0 (Figure 4b) of Doppler and azimuth of target with θ ¯ 0 = 0 ° , v ¯ d 0 = 0.4 considering ξ = 0.1 , 0.5 , 1.3 , 2 , respectively, s 0 as the initial point.

5.2. Uniform clutter environment

This subsection focuses on a scenario where we consider a homogeneous range-azimuth ground clutter interfering with the range-azimuth bin of interest (0,0). Specifically, for each range-azimuth ground clutter bin, a clutter to noise ratio (CNR) of 25 dB and a normalized Doppler frequency v¯=0with Doppler uncertainty ε/2=0.04are considered. We suppose M=50range-azimuth ground clutter bins located within the azimuth angular sector π/2π/2. Moreover, we set the range ring ri=0for all range-azimuth ground clutter bins.

In Figure 5, we show the SINR of Algorithm 2 and SOA2 for ξ=0.1,0.5,1.3,2supposing a target π/180θ0π/180, 0.36vd00.44. The SINR values increases both for Algorithm 2 and SOA2 with the increasing iteration number n. Furthermore, we observe the higher ξ, the better SINR values reflecting the larger and larger feasible set. Interestingly, Algorithm 2 significantly outperforms SOA2 for all the considered ξ, except for ξ=2where they both achieve the same SINR value. In particular, we see that the gap between ξ=2and energy constraint is about 1.1 dB because of the introduction of constant modulus constraint. We also observe that in this scenario, Algorithm 2 needs a higher number of iterations to achieve convergence compared with that in Figure 2. For instance, for ξ=0.1, Algorithm 2 converges with about 12 iterations in Figure 5, whereas in Figure 2 after about 2 iterations.

Figure 5.

The SINR behavior versus iteration number assuming a target with − π / 180 ≤ θ 0 ≤ π / 180 , 0.36 ≤ v d 0 ≤ 0.44 in uniform clutter environment for ξ = 0.1 , 0.5 , 1.3 , 2 , s 0 as the initial point.

In Table 2, we summarize the SINR values, iterations number, and the global computation time of Algorithm 2 and SOA2. In particular, Algorithm 2 shows a lower computational time for ξ=0.1,2. Furthermore, it is observed that the gains of 2.3 and 3 dB are achieved using Algorithm 2 with a slightly higher computational cost for ξ=0.5,1.3, respectively.

Algorithm 2SOA2
ξSINRnTimeSINRnTime
0.12.3122.63251.734.3007
0.54.96813.01832.6811.4606
1.38.012021.48755.01014.5547
28.88215.86998.82435.0873

Table 2.

SINR values (in dB), iterations number, and global computation time (in seconds) of Algorithm 2 and SOA2 assuming a target with π/180θ0π/180, 0.36vd00.44in uniform clutter environment for ξ=0.1,0.5,1.3,2, s0as the initial point.

Figure 6 shows the joint frequency and azimuth behavior of STTC and STRF concerning CAF. Specifically, the contour map of the Doppler-azimuth plane of CAF at r=0against the iteration number (n=0103082) considering ξ=2for Algorithm 2 is plotted. We observe that gnsnwnrvθobtains lower and lower values in the region of π/2θπ/2, 0.04v0.04(highlighted by black rectangles) with the increase of iteration number n. This performance behavior highlights that the proposed algorithm of joint design STTC and STRF possesses the ability of sequentially refining the shape of the CAF to achieve better and better clutter suppression levels.

Figure 6.

Doppler-azimuth plane of CAF at r = 0 for ξ = 2 of Algorithm 2 for n = 0 10 30 82 assuming a target with − π / 180 ≤ θ 0 ≤ π / 180 , 0.36 ≤ v d 0 ≤ 0.44 in uniform clutter environment (black rectangles represent the locations of uniform clutter), s 0 as the initial point of Algorithm 2 and SOA2.

Figure 7 plots the SINR versus the standard deviations σvd0(Figure 7a) and σθ0(Figure 7b) of Doppler and azimuth of target with θ¯0=0°, v¯d0=0.4, respectively. Again, we see that Algorithm 2 obtains a higher SINR gain than SOA2 for ξ=0.1,0.5,1.3, whereas they both fulfill the near same gain at ξ=2. Interestingly, we also observe that a decreasing trend in gain with the increase in standard deviation. This is reasonable due to that the larger standard deviation results in the larger uncertainty on the knowledge of target.

Figure 7.

The SINR behaviors versus the standard deviations σ v d 0 (a) and σ θ 0 () of Doppler and azimuth of target with θ ¯ 0 = 0 ° , v ¯ d 0 = 0.4 , respectively, s 0 as the initial point of Algorithm 2 and SOA2.

6. Conclusions

This chapter has considered the joint STTC and STRF design for MIMO radar under signal-dependent interference. We focus on a narrow band colocated MIMO radar with a moving point-like target considering imprecise a prior knowledge including Doppler and azimuth. Summarizing,

  • We have devised an iterative algorithm to maximize the SINR accounting for both a similarity constraint and constant modulus requirements on the probing waveform. Each iteration of the algorithm requires the solution of hidden convex problems. The consequent computational complexity is linear with the number of iterations and polynomial with the sizes of the STTC and the STRF.

  • We have assessed the performance of the proposed iteration algorithm through numerical simulations. The results have manifested that the larger the similarity parameter (i.e., the weaker the similarity constraint), the larger the output SINR due to the expanded feasible set. Moreover, we observed that the devised iteration procedure can provide a monotonic improvement of SINR and ensuring convergence to a stationary point, which possesses excellent superiority in computation complexity and performance gain compared with the related SOA2. The numerical examples also have revealed the capability of the developed procedure to sequentially refine the shape of the CAF both in nonuniform point-like clutter environment and uniform clutter environment.

Possible future work tracks might extend the proposed framework to consider spectral constraint [37] and MIMO radar beampattern design by optimizing integrated sidelobe level (ISL) with practical constraints.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61771109 and 61501083. The authors like to thank Dr. Augusto Aubry for his constructive comments.

Let us denote Sin block matrix form, i.e.,

S=Sn1n2K×K,E36

where the block matrix Sn1n2CNT×NTcan be computed as

Sn1n2=sn1sn2,n1n212K2.E37

Hence, exploiting the fact that vd0and θ0are statistically independent random variables, the block matrix Γm1m2of ΓSin (21) can be expressed as

Γm1m2=σ02Eej2πm1m2vd0EAθ0Sm1m2Aθ0,m1m212K2.E38

Since vd0is a uniformly distributed random variable, e.g., vd0Uv¯d0ε02v¯d0+ε02, the first expectation of (38) can be computed as

Eej2πm1m2vd0=1ε0v¯d0ε02v¯d0+ε02ej2πm1m2vd0dvd0=ej2πv¯d0m1m2sinπε0m1m2πε0m1m2,m1m212K2.E39

Let Φϑ0θ¯0denote the second expectation of (38) whose q1q2entry is given by

Φϑ0θ¯0q1q2=Ea˜q1Tθ0Sm1m2a˜q2θ0=trSm1m2Φ¯q1q2,q1q212NR2,E40

where

a˜qθ0=1NRej2πsinθ0λdrq1atθ0,q12NR,E41

and

Φ¯q1q2=Ea˜q2θ0a˜q1Tθ0.E42

Based on θ0as a uniformly distributed random variable, e.g., θ0Uθ¯0ϑ02θ¯0+ϑ02, the q1q2entry of expectation Φ¯q1q2can be computed as

Φ¯q1q2p1p2=1NTNRϑ0θ¯0ϑ02θ¯0+ϑ02ej2πsinθ0λdrq2q1+dtp1p2dθ0q1q212NR2,p1p212NT2.E43

As to the computation of (43), we can adopt numerical integration.

Next, we focus on the computation of ΘW. Similarly, let us write Win block matrix structure, given by

W=Wi1i2K×K,E44

where block matrix Wi1i2CNR×NRis given by

Wi1i2=wi1wi2,i1i212K2.E45

As a consequence, based on the statistical independence of vd0and θ0, the block matrix Θi1i2of ΘWin (22) is

Θi1i2=σ02Eej2πi1i2vd0EAθ0Wi1i2Aθ0,i1i212K2.E46

Following the same lines of reasoning in (39) and (43), both expectations in (46) can be evaluated.

The Θwwcan be rewritten as

Θww=a1a2aKNT,E47

where an=αn,1αn,2αn,KNTTCKNT, for n=1,2,,KNT. Hence, the sΘwwscan be expressed as

s¯Θwws¯=n=1niKNTs¯ansn+s¯ais¯i=n=1niKNTs¯iαn,isn+s¯ais¯i+k=1kiKNTl=1liKNTslαk,lsk.E48

Using the property αn,i=αi,nsince Θwwis a positive semidefinite matrix, (48) can be computed as

s¯Θwws¯=αi,is¯i2+n=1niKNT2s¯iαi,nsn+k=1kiKNTl=1liKNTslαk,lsk.E49

Hence, we obtain

a0,i=αi,i,a1,i=2n=1niKNTαi,nsn,a2,i=k=1kiKNTl=1liKNTslαk,lsk.E50

Following the same line of reasoning, the coefficients b0,b1,b2are given by,

b0,i=βi,i,b1,i=2n=1niKNTβi,nsn,b2,i=k=1kiKNTl=1liKNTslβk,lskE51

where βm,ndenotes the mnth entry of Σ¯dvww.

Notes

  • Notice that based on its definition, the shift matrix satisfies the condition J r = J − rT .
  • Notice that the similar convergence analysis can be obtained in [23].
  • Notice that LFM waveforms have good properties in the pulse compression and ambiguity feature.
  • Notice that we consider the exit condition A / 10 4 both for Algorithms 1 and 2, where A denotes the upper bound of the objective function neglecting the signal-dependent interference (for example, A = 10 is considered in this simulation).

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Guolong Cui, Xianxiang Yu and Lingjiang Kong (December 20th 2017). Space-Time Transmit-Receive Design for Colocated MIMO Radar, Topics in Radar Signal Processing, Graham Weinberg, IntechOpen, DOI: 10.5772/intechopen.71946. Available from:

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