Suboptimum Low Complexity Joint Multi-target Detection and Localization for Noncoherent MIMO Radar with Widely Separated Antennas

# Suboptimum Low Complexity Joint Multi-target Detection and Localization for Noncoherent MIMO Radar with Widely Separated Antennas

Wei Yi,  Tao Zhou, Mingchi Xie,  Yue Ai and Rick S. Blum,  The work of R. S. Blum was supported by the National Science Foundation under Grant No. ECCS-1405579.W. Yi, T. Zhou, M. Xie and Y. Ai are with the University of Electronic Science and Technology of China, Chengdu 611731, China. W. Yi is the corresponding author (e-mail: kussoyi@gmail.com).R. S. Blum is with the Electrical and Computer Engineering Department, Lehigh University, Bethlehem, PA 18015 USA (e-mail: rblum@eecs.lehigh.edu)
###### Abstract

In this paper, the problems of simultaneously detecting and localizing multiple targets are considered for noncoherent multiple-input multiple-output (MIMO) radar with widely separated antennas. By assuming a prior knowledge of target number, an optimal solution to this problem is presented first. It is essentially a maximum-likelihood (ML) estimator searching parameters of interest in a high-dimensional space. However, the complexity of this method increases exponentially with the number of targets. Besides, without the prior information of the number of targets, a multi-hypothesis testing strategy to determine the number of targets is required, which further complicates this method. Therefore, we split the joint maximization into disjoint optimization problems by clearing the interference from previously declared targets. In this way, we derive two fast and robust suboptimal solutions which allow trading performance for a much lower implementation complexity which is almost independent of the number of targets. In addition, the multi-hypothesis testing is no longer required when target number is unknown. Simulation results show the proposed algorithms can correctly detect and accurately localize multiple targets even when targets share common range bins in some paths.

EDICS Category: RAS-DMMP, RAS-LCLZ.

## I Introduction

Recently, multiple-input multiple-output (MIMO) radar, inspired by wireless communications, has drawn more and more attention from researchers [1, 2, 3, 4, 5]. Generally, MIMO radar can be classified into two categories, namely, co-located MIMO radar [1] and MIMO radar with widely separated antennas [3]. The former one, similar to conventional phase array radar [2], employs multiple independent signals transmitted by the closely spaced antennas to obtain waveform diversity [6]. The latter one, observes a target at different angles to achieve spatial diversity [7]. Among these studies, both coherent and non-coherent processing has been considered. Non-coherent processing requires time synchronization between the nodes. Besides time synchronization, coherent processing requires additional phase synchronization [8]. Both categories have been shown to offer considerable advantages over conventional radar system in various aspects, such as target detection [9], target tracking [10, 23] and target localization [11, 12, 13, 14]. In particular, position information supports an increasing number of location-based applications and services [15, 16, 17], therefore target localization is of critical importance for MIMO system.

In general, there are basically two kinds of target localization methods. One is based on the time of arrival (TOA) or angle of arrival (AOA) information from the received signals, which are used to calculate the position via triangulation [18, 12, 11]. Such an algorithm is categorized as an indirect localization approach. The other one, called a direct localization approach, jointly processes the raw signal echos to acquire the maximum-likelihood estimation (MLE) [8, 10, 13, 19, 20, 21]. The latter method takes full advantage of received echo information, and thus leads to a higher localization accuracy, especially for weak targets. To obtain the solutions of this method, one of the basic ideas is to employ an iteration algorithm [22], but it requires a proper initial solution from the prior position information, which can restrict the application of this approach in real applications. The other approaches, known as grid-searching methods [10], obtain the target location estimates by searching for the coordinate position that maximizes the likelihood ratio. If only a single target is present, it can be effectively localized using the MLE. However, in many practical situations, there are multiple targets in the coverage area of the system, and multi-target localization is a very challenging problem, for simply expanding the searching dimension to match the number of targets is computationally prohibited.

So far, several problems have been addressed regarding the multi-target localization in radar networks [23, 24, 25, 26]. In [23], the multiple-hypothesis (MH)-based algorithm is applied to estimate the number of targets and further achieve the localization for these targets. In [24] a sparse modeling is proposed for distributed MIMO radar to achieve joint position and velocity estimation of multiple targets. Moreover, motivated by [24], [25] uses a block sparse Bayesian learning method to estimate the multi-target positions. While in [26], the multi-target localization problem is researched using only Doppler frequencies in MIMO radar networks.

Inspired by those works, in this paper, we study the problem of multi-target joint detection and localization for MIMO radar with widely separated antennas. This work is an extension of our previous work [27]. Firstly, we present an optimal high dimension localization method based on joint MLE, whose complexity increases exponentially with the number of targets. Besides, without the prior information of the number of targets, a multi-hypothesis testing strategy is required [28], which further complicates this method. To tackle this problem, we then derive two reduced-complexity strategies, specifically, the successive space removal (SSR) algorithm and the successive interference cancellation (SIC) algorithm. The main idea is to split the dimensional joint maximization into disjoint optimization problems. It allows the information of each target to be extracted one by one from the original received signal. It is worth mentioning that our proposed algorithms are based on the threshold decision in detection theory [29], hence the target detection information can be simultaneously obtained. In other words, our algorithms belong to a joint multi-target detection and localization procedure, which trades off the algorithm performance for implementation complexity. Numerical examples are provided to assess the detection and localization performances of the our proposed multi-target localization algorithms.

The rest of the paper is organized as follows. The system model is introduced in Section II. In Section III, and the definitions of partially separable and isolated targets are clarified and the high dimensional optimal joint multi-target detection and localization method is derived. In Section IV, two suboptimal algorithms are proposed under the condition that targets are isolated or arbitrarily located, and then the performance of these algorithms is assessed by simulation results in Section V. Finally, Section VI concludes this paper.

## Ii Models and Notation

We assume a typical MIMO radar scenario with transmitters located at , and receivers located at , respectively, in a two-dimensional Cartesian coordinate system. The antennas of both transmitters and receivers are widely separated. A set of mutually orthogonal signals are transmitted, with the lowpass equivalents , .

The focus in this paper is on simultaneously detecting and localizing multiple targets, therefore only static targets are considered here. Suppose that (, is a variable and usually unknown before joint detection and localization) static targets appear in the radar surveillance region, with the th target located at . For convenience, we define a two-dimensional vector of the unknown location of the th target as

 θgΔ=[xg,yg]′, (1)

where “ ” denotes the matrix transpose. It should be pointed out that although a 2-dimensional model is adopted here, the extension to a higher dimensional case is direct.

For noncoherent MIMO radar, the received signal reflected from all targets at the th receiver due to the signal transmitted from the th transmitter (defined as the th transmit-receive path) is, for , given by 111Due to the assumed orthogonality of the signals, it is possible to separate the signal traveling over the th path.:

 rlk(t)=G∑g=1αlkgsk(t−τlkg)+nlk(t)+clk(t), (2)

where is the observation time interval. The reflection coefficient of the th path for the th target is assumed to be a deterministic unknown complex constant with amplitude and phase during the observation time . In practice, is related to the Radar Cross Section (RCS) of the th target, and is time varying and unknown before localization in most cases. The term denotes the time delay of the received signal from the th target at the th receiver due to the th transmitter, and can be expressed as

 τlkg=√(xg−xtk)2+(yg−ytk)2+√(xg−xrl)2+(yg−yrl)2c, (3)

with the speed of light. The terms and in (2) represents the thermal noise and clutter of the th path. Note that, to accommodate the more general case of moving targets, the signal model with target velocity taken into account can be found in [8].

After sampling, the continuous signal of (2) can be written in a vector form

 rlk=G∑g=1αlkg~slkg+nlk+clk, (4)

where

 rlkΔ=[rlk[0],rlk[1],...,rlk[NT−1]]′, (5)
 ~slkgΔ=[~slkg[0],~slkg[1],...,~slkg[NT−1]]′, (6)

with a sampling interval , thus the sampled signal is , , . Note that is a function of the unknown target location. The sampled version of the noise and the clutter in (2), i.e., and in (4), are defined similarly as in (5) as

 nlkΔ=[nlk[0],nlk[1],...,nlk[NT−1]]′,clkΔ=[clk[0],clk[1],...,clk[NT−1]]′. (7)

The thermal noise and clutter at the th receive antenna are assumed to be zero-mean complex white Gaussian noise with the correlation matrixes and respectively, where denotes the identity matrix and the superscript “ ” denotes conjugate transpose. The temporal correlation matrix of the thermal noise and clutter return is then

 Rlk=σ2lkINT+Clk (8)

For simplicity, we assume that for a given transmitter-receiver pair, the clutter temporal correlation matrix is known or estimated a priori. Thus can be diagonalized by a whitening process. With a slight abuse of notation, we assume such a whitening has been applied prior to (4), but we employ the same notation employed in (4).

Both the thermal noise and clutter echo are assumed to be independent between different transmit-receive paths, thus, for any or

 (9)

This assumption is justified for widely spread antennas.

## Iii Joint Multi-target Detection and Localization

As discussed in [28], the MLE of the unknown parameter vector can be found by examining the likelihood ratio for the hypothesis pair, with corresponding to the target presence hypothesis and corresponding to the noise only hypothesis. As for multi-target estimation, the observation vector is related to the parameters of all targets . Thus for the joint estimation of all targets, we introduce a high dimensional parameter vector , which is the concatenation of the individual target parameters, defined as,

 Θ=[θ′1,θ′2,...,θ′G]′∈R2G. (10)

Before proceeding, it is necessary to introduce the following Definition, which is instrumental to the development of the subsequent algorithms.

Definition 1: Consider a scenario with targets and an MIMO radar. The th and th targets ( , and ) are said to be separable over the th path, if the time difference of arrival between these two targets is larger than the radar effective pulse width . That means

 ∣∣τlkg−τlkj∣∣>τc, (11)

where is the effective duration of the time-correlation of the transmitted waveform , [28] (for example, if a rectangular pulse with pulse width is employed, then ). Conversely, the th and th targets are called inseparable over the th path if (11) is not satisfied, indicating that the th target shares one range bin in the th path with the th target. If the th target is separable with any other targets in the data plane over all the transmit-receive paths, the th target is referred to as an isolated target. Otherwise, the th target is partially separable. Furthermore, if any pairs of targets is mutually separable over all paths, then all the targets are completely isolated.

Take an MIMO radar as an example, where each antenna receives the signals transmitted from other antennas. A scenario with two partially separable targets is plotted in Fig. 1 in which only two of the total four paths are plotted. It shows that the two targets are separable in the th propagation path but inseparable in the th path.

### Iii-a Optimal High-dimensional Method

In order to simplify the problem, we first assume that the number of targets is known before localization. Let represent the target presence hypothesis as modeled in (4) and represents target absence hypothesis, and we can write the likelihood functions of the received vectors of the th path, i.e., , conditioned on the hypotheses and parameters as

 (12)

and

 p(rlk|H0)=κ0exp{−12rHlkR−1lkrlk}, (13)

where is composed of the unknown complex reflection coefficients of all targets and denotes a constant independent of . Since is not a function of , for the estimation of , the likelihood function is equivalent to the likelihood ratio [31]

 ℓ(rlk|Θ,αlk)∝p(rlk|Θ,αlk,H1)p(rlk|H0)=exp{12rHlkR−1lk(G∑g=1αlkg~slkg)+12(G∑g=1αlkg~slkg)HR−1lkrlk−12(G∑g=1αlkg~slkg)HR−1lk(G∑g=1αlkg~slkg)}. (14)

For any parameter , the likelihood ratio (14) is maximized using [32], where is calculated as the solution to

 ∂∂αlklnℓ(rl,k|Θ,αlk)|αlk=^αlk=0. (15)

Note that (15) can be written as a group of equations, with the th () equation expressed as

 ~sHlkgR−1lkrHlk−αlkg~sHlkgR−1lk~slkg−G∑g1=1,g1≠g~sHlkgR−1lkαlkg1~slkg1=0 (16)

and the detailed derivation is shown in Appendix A. It can be seen that (16) is a linear equation in . Therefore, for compactness, we rewrite the equations of (16) in the following matrix form (also see in Appendix A),

 ~SHlkR−1lk~Slk^αlk=~SHlkR−1lkrlk, (17)

with an matrix, and the term expressed as follows

 ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝~sHlk1R−1lk~slk1~sHlk1R−1lk~slk2⋯~sHlk1R−1lk~slkG~sHlk2R−1lk~slk1~sHlk2R−1lk~slk2⋯~sHlk2R−1lk~slkG⋮⋮⋯⋮~sHlkGR−1lk~slk1~sHlkGR−1lk~slk2⋯~sHlkGR−1lk~slkG⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (18)

If (18) is invertible (the invertibility of matrix (18) will be discussed later in this section), using (17), we have the ML estimation of as

 ^αlk=(~SHlkR−1lk~Slk)−1~SHlkR−1lkrlk. (19)

In order to obtain the likelihood of the th transmit-receive path without parameter , we rewrite the logarithmic form of (14) as

 lnℓ(rlk|Θ,αlk)=12{rHlkR−1lk~Slkαlk+αHlk~SHlkR−1lkrlk−(~Slkαlk)HR−1lk(~Slkαlk)}. (20)

Substitution of (19) into the third term on the right-hand side of (20), we have

 (~Slkαlk)HR−1lk(~Slkαlk)=αlkH~SlkHR−1lk~Slk(~SlkHR−1lk~Slk)−1~SlkHR−1lkrlk=αlkH~SlkHR−1lkrlk. (21)

Therefore the summation of the second and third terms on the right-hand side of (20) is zero and only the first term remains. Then inserting (19) into (20), we have

 lnℓ(rlk|Θ,αlk)=12rHlkR−1lk~Slk(~SlkHR−1lk~Slk)−1~SlkHR−1lkrlk. (22)

Due to the independence of observations over different paths, the ML joint detection and estimation of locations of the targets over all transmit-receive paths can be formulated as

 ^ΘML=argmaxΘ∈R2GN∑k=1M∑l=1lnℓ(rlk|Θ,^αlk) (23) subject to N∑k=1M∑l=1lnℓ(rlk|^ΘML,^αlk)≥λ, (24)

where is a detection threshold determined by the detection or false alarm probabilities. If the summation of the log-likelihood functions exceeds , a detection of targets is made, otherwise no target is declared.

Recall that in the beginning of the Section III-A, the number of targets was assumed to be known before the development of the high-dimensional localization method. The dimension of the multi-target location parameter has to be predefined before carrying out the maximization search. If is unknown, which is the usual case for practical applications, all possible hypotheses of the number of targets have to be evaluated (i.e., a multiple hypotheses testing problem). Owing to the limits of computational complexity, usually an upper bound to the number of prospective targets has to be preset. The number should be set large enough to cover the possibility of the largest number of targets. However a big causes unnecessary computational expense 222Since one has to evaluate all the hypothesis before making a decision, even if no target is present, searches over the discretized data plane must be performed. and performance loss due to the increased number of admissible hypotheses.

### Iii-B Discussion

#### Iii-B1 The invertibility of matrix (18)

There are cases where (18) is not invertible. Assume there are targets, then (18) becomes

 (~sHlk1R−1lk~slk1~sHlk1R−1lk~slk2~sHlk2R−1lk~slk1~sHlk2R−1lk~slk2). (25)

If the time delays of the reflected signals from the two targets over the th path are the same, i.e., , then and the four elements of (25) are exactly the same. This means that the rank of the matrix (25) is one, i.e., (25) is not invertible, and one can not compute the ML estimation of using (19). Actually, when , the matrix version of (17) is composed of two identical equations from (16), thus only one ML estimation of the reflection coefficient can be obtained. This can be explained from a physical point of view, since it is impossible to distinguish and estimate the reflection coefficients for targets with the same time delays over this path. Those cases might be avoided by not looking for targets at these locations, meaning that the search points and which satisfy are eliminated in the optimization process.

On the contrary, when the time delays of the two targets satisfy , the th and th elements of (25) are approximately equal to zero. The th and th elements can be viewed as the two reflected signals with different time delays. Thus (18) is invertible, and two ML estimates of reflection coefficients for each target can be obtained using (19). Based on the foregoing discussion, we can see that the invertibility of matrix (18) relates to the geometric layout of the antennas and targets .

#### Iii-B2 The curse of dimensionality

since no analytic solution exists for the MLE of (23), numerical methods are required. For the grid-search method, in the area of interest (-dimensional), assume that along the and dimensions there are and grid points respectively, implying a total of grid points. The unit size of each dimension is chosen based on the characteristics of radar system (e.g., range resolution), the geographical setting of the radar antennas with respect to the area of interest and the computational resources. After the grid search, standard optimization methods can also be employed to refine the estimation [10]. Although the grid-search implementation of (23) is straightforward in principle, it involves a high-dimensional joint maximization. Since the discretized data plane contains grid cells, the total complexity increases exponentially with the number of targets . Therefore this high-dimensional multi-target localization method is computationally prohibitive if there are more than a few targets.

The above problems and the multi-hypothesis testing problem mentioned before heavily restrict the applications of the high-dimensional method. Hence, suboptimal algorithms are also investigated in the subsequent sections to trade off algorithm performance for implementation complexity.

## Iv Suboptimum Strategies

### Iv-a Successive-Space-Removal Algorithm

The aim of this subsection is to derive reduced-complexity strategies for implementing the MLE (23), at the price of estimation performance tradeoff. The main idea is to split the -dimensional joint maximization into disjoint optimization problems, which allows information about each target to be extracted one-by-one from the original received signal.

The design of this suboptimal algorithm is based upon the assumption that the targets present in the radar surveillance region are completely isolated. Normally a MIMO radar receiver incorporates thousands of resolution range bins, so completely isolated targets are not rare. In this case, from Definition 1 and the fact that is a diagonal matrix, for any and , as discussed in Section III-B, and corresponding to the th and th target respectively must effectively meet the condition

 ~sHlkgR−1lk~slkj=0. (26)

Thus the matrix is diagonal, and then the closed-form ML estimation of is obtained, by using (16), as

 ^αlkg=~sHlkgR−1lkrlk~sHlkgR−1lk~slkg. (27)

By substituting (27) back into (20), we have

 lnℓ(rlk|Θ,^αlk)=12rHlkR−1lkG∑g=1^αlkg~slkg=12G∑g=1rHlkR−1lk~slkg~sHlkgR−1lkrlk~sHlkgR−1lk~slkg=12G∑g=11~sHlkgR−1lk~slkg|~sHlkgR−1lkrlk|2. (28)

Substitution of (28) into (23) gives

 ^ΘML=argmax(θ1,⋯,θG)∈R2GN∑k=1M∑l=1G∑g=1ℓlk(θg)=argmax(θ1,⋯,θG)∈R2GG∑g=1F(θg),subject to^θ1,⋯,^θG are comletely \emph{% islated}. (29)

where

 ℓlk(θg)=121~sHlkgR−1lk~slkg∣∣~sHlkgR−1lkrlk∣∣2 (30)

is the log-likelihood function for a single target location for the th transmit-receive path, and

 F(θg)=N∑k=1M∑l=1ℓlk(θg) (31)

is defined as the objective function of the th single target location . The right-hand side of (30) implies that will be large only when can be well matched with .

For the scenario with completely isolated targets, the maximum of the summation of objective functions in (29) is equal to the summation of maximums of the objective functions, because this special scenario excludes the case where two targets are are in a common range bin for any path. According to this fact, we can reasonably simplify the high-dimensional optimization problem in (29) by reducing the dimension of the search space. So (29) can be approximately expressed as

 ^ΘML=[^θ′1,^θ′2,⋯,^θ′G]′ (32)

with its th element estimated as

 ^θg=argmaxθg∈SgF(θg), (33)

where the initial parameter space , and for ,

 Sg=Sg−1∖B(θ,^θg−1).\lx@notefootnoteThesymbol$A∖B$denotesthesetdifferenceofset$A$and$B$. (34)

, written succinctly as , is defined as that subset of the search area, which includes the range bins for those paths which are occupied (see Fig. 1) by the target located at , which is written as

 B(θg)=M⋃l=1N⋃k=1Blk(θ,θg), (35)

where , similarly succinctly written as , denotes the part of corresponding to the th path.

In order to solve the optimization problem, we need to accurately find the maximums of each objective function constrained by different conditions. The estimator (33) can provide a computational efficient and practical method to find the maximums at the scenario with completely isolated targets. From the previous analysis, we can know two targets are not present in the area defined in (34), indicating no target shares a common range bin. In fact, this is actually implied by the constraint corresponding to the optimization problem in (32) and (33). It also means that it is reasonable to find each maximum one by one by eliminating the areas corresponding to every determined target.

Because the true location corresponding to the th target is unknown, we need to substitute the estimation result for . Taking the potential estimation error between and the true position into consideration, the correctness of the decision of range bins in each path is not guaranteed. Thus, in implementation, is defined as follows (set the error margin as a range bin)

 Blk(^θg)={(x,y)|⌊τlkg(x,y)/τc⌋−⌊^τlkg/τc⌋≤1}, (36)

where

 ^τlkg=1c(√(^x−xtk)2+(^y−ytk)2+√(^x−xrl)2+(^y−yrl)2) (37)

is the time delay of the estimated target located at in the th path, is the effective duration of the time-correlation function of the transmitted waveform and is the maximum integer not greater than . Therefore, in (34) represents the search space after removing the area affected by the first declared targets.

A variant of (22) with much lower complexity can be expressed as follows

 ^ΘML=[^θ′1,^θ′2,⋯,^θ′G]′with^θg=argmaxθg∈SgF(θg)subject toG∑g=1F(θg)≥λ. (38)

As is mentioned before, the structure of (38) indicates that the 2-dimensional maximization of (29) can be replaced by sequentially implementing 2-dimensional maximizations, i.e., finding the , , which maximize , then removing the search area affected by to form the search space for the next maximization until . By doing this, the complexity is reduced significantly, and we refer to this algorithm as the successive-space-removal (SSR) multi-target localization method.

However, SSR would also face the cumbersome multi-hypothesis testing problem when target number is unknown. To deal with this, we propose a step-by-step detection procedure for SSR. Since the existence of a certain target is irrelevant to other targets under the assumption that the targets are completely isolated, we can approximately replace the detection process in (38) with single target detection problems as

 F(^θg)H1≷H0λg,    g=1,2,…,G, (39)

where is the threshold of the th single target detection process. Usually threshold is chosen to achieve a certain false alarm probability. If the background is homogeneous, one can use the same threshold for all detection processes. In cases where the number of targets is not available, the localization process can be terminated automatically if the th estimated location is determined as not target, i.e., . This simply relies on the fact that when the background is homogeneous, meaning that every estimate in the subsequent search will be decided as . Since the threshold remains the same on the whole data plane in each detection process, the decision of the threshold for all detection processes is made only once to narrow down the possible locations of the targets. A summary of the proposed SSR algorithm under homogeneous background is given in Algorithm 1. It should be noted that for the non-homogeneous environment, in order to achieve the desired constant false alarm rate, the value of detection threshold in (40) needs to be adapted along with the variety of the noise/clutter, i.e., false alarm rate approach [33, 34, 29]. Besides, in Algorithm 1, we set an upper bound for the maximum number of the potential targets. Thus when estimated locations have been obtained, the iteration ends automatically to avoid the overload of the system.

When the assumption that all targets are completely isolated holds, SSR can sequentially localize multiple targets efficiently with no need for a multi-hypotheses testing algorithm. However, for more general cases, targets located arbitrarily may share range bins with each other in one or more transmit-receive paths, i.e., partially separable. In this case, the direct removal of the search space of detected targets using (34) and (35) will result in the miss-detection of subsequent targets which are inseparable over certain pathes with the previously detected targets. Fig. 2(a) shows a scenario with three targets wherein the two targets on the left-hand side are inseparable. It can be seen in Fig. 2(b) that the elimination of the area corresponding to the target on the lower left-hand corner (stronger one) will hinder the subsequent detection and localization of the target on the upper left-hand side. In order to deal with this problem, a carefully designed suboptimal strategy is given in the next subsection.

### Iv-B Successive-Interference-Cancellation Algorithm

The new algorithm differs from SSR in that it does not directly clear search space affected by the targets detected as in (34) and instead only eliminates the interference of the extracted targets from the objective function. As a consequence, the objective function changes every time after a target is detected. In this way, another variant of (23) for the ML joint detection and localization of multiple targets can be formulated as

 (42)

where is the objective function for the th maximization (i.e., extraction of the th target) and is defined as follows

 Fg+1(θ)=Fg(θ)−N∑k=1M∑l=1Mlkg(θ)=F(θ)−g∑i=1N∑k=1M∑l=1Mlki(θ), (43)

with the term in (43) referred to as the modified term related to the th detected target over the th path. In order to eliminate the interference to the likelihood from the previously detected targets, the modified term of the th detected target over the th path is defined as

 Mlkg(θ)={ℓlk(θ),θ∈Blk(^θg)∖Clk(^θg)0,otherwise (44)

with

 Clk(^θg)=Blk(^θg)∩{Blk(^θ1)∪⋯∪Blk(^θg−1)} (45)

where the terms and are the defined by (30) and (36) respectively. In essence, the modified term is equal to the log-likelihood over the th transmit-receive path for the parameter space that is affected by the estimated target , i.e., , otherwise it is zero. However, for a certain parameter , its log-likelihood over the th path may have already been subtracted in the previous modifications of the objective function, namely, . Hence, is equal to only for the parameter space , otherwise zero.

Then can be viewed as a modified form of the original objective function , wherein the likelihood interference from the previously detected targets has been eliminated. Hence we refer to this algorithm as a successive-interference-cancellation (SIC) algorithm. The idea of SIC is similar in spirit to the well known CLEAN algorithm [30]. To further reduce complexity, it should be noted that the log-likelihood values of all the paths have already been calculated when we compute the original objective function . So there is no need to recalculate the log-likelihood values to generate the modified term.

It can be seen that (38) and (42) have exactly the same structure. Therefore, similar to the implementation of SSR, SIC can also be performed sequentially to break down the high-dimensional joint maximization and avoid the multiple hypotheses testing problem. Nevertheless, there are two differences between SSR and SIC. Firstly, for each iteration, SIC only modifies the objective function to clear the interference of detected targets and keeps the search space intact, rather than deleting the search area as in SSR. This greatly facilitates the detection and localization of inseparable targets. We still consider the same scenario shown in Fig. 2(a) wherein the two targets in the left-hand side are inseparable. The modified objective function after eliminating the interference of the target on the lower left-hand corner (stronger one) using SIC is shown in Fig. 3. It can be seen that compared to the objective function in Fig. 2(b), SIC is able to reserves more information of the target on the upper left-hand side (inseparable with the detected and located one), making the subsequent detection and localization of this target possible.

Secondly, the setting of the detection threshold for SIC is more complicated. The reason is that for different parts of the parameter space, the modified objective function defined in (43) is composed of the likelihood summation of different number of paths. Thus even for the homogeneous background, the value of the detection threshold may change for different parts of the parameter space to prevent missing targets. For this reason, we define the detection threshold of the parameter for the th iteration as

 λg(θ)=N∑k=1M∑l=1ωkl−g∑i=1N∑k=1M∑l=1χBlk(^θi)(θ)ωklN∑k=1M∑l=1ωklλ′, (46)

where denotes the indicator function on the set , is the threshold for the original objective function which contains all the paths for and is a coefficient which accounts for the impact of the th path on the calculation of the threshold. For instance, could be the intensity of noise power of the th path. If we approximately assume the coefficients are the same for all paths, then (46) becomes

 λg(θ)=MN−g∑i=1N∑k=1M∑