Compressed Sensing Radar Detectors
based on Weighted LASSO

For the $N$ detection problems described in (5), we here define the following performance metrics. Suppose we perform the tests for $T$ trials. Denote by

for $i=1,\ldots,N$ and $t=1,\ldots,T$. Define the probability of false alarm $P_{fa,i}$ for the $i$-th problem or entry of ${\bm{x}}_{0}$ as

and the probability of detection $P_{d,i}$ as

where $S_{t}=\left\{i:x_{0,i}^{(t)}\neq 0\right\}$ is the support set of ${\bm{x}}_{0}^{(t)}$ with $S_{t}^{c}=\{1,\ldots,N\}\backslash S_{t}$.

The prior information ${\bm{p}}=[p_{1},p_{2},\ldots,p_{N}]^{T}$ of ${\bm{x}}_{0}$ is defined as the probability that each entry of ${\bm{x}}_{0}$ is non-zero, such that

where $Q_{i}(x_{0,i})$ is the distribution of non-zero entry of ${\bm{x}}_{0}$.

Since $N$ detection problems are processed simultaneously, we define the total false alarm rate $\bar{P}_{fa}$ as follow,

and similarly the total detection rate $\bar{P}_{d}$

With the definition of prior information $\bm{p}$ above, one can say that

Therefore,

Note that if we take the limit $N\to\infty$, these definition will be the same as that in [17], with (12) and (13) considering prior information.

Similarly to [17], we demonstrate and compare DWLD with NWLD. Their frameworks are shown in Fig. 1 (a) and (b). The main difference between the two detectors is in the test statistics and thresholds. The latter performs a “debiased” procedure, which is a linear transform of the weighted LASSO estimator, and applies calculable thresholds with given false alarm rate for each entry. The details of the superiority of DWLD and their proofs will be presented in Sec. II-C.

Due to the random nature of the row-orthogonal matrix ${\bm{A}}$, ${\hat{\bm{x}}}^{\rm{d}}$ is expected to follow Gaussian distribution with mean ${\bm{x}}_{0}$. Actually, this is the case as long as ${\bm{A}}$ is “right rotation invariant”, which means that the right singular basis of ${\bm{A}}$ is regarded as a random sample from the Haar measure of $N\times N$ orthogonal (or unitary) matrices. Further, in such cases, methods of statistical mechanics offer formulas to appropriately construct ${\hat{\bm{x}}}^{\rm{d}}$ from the (weighted) LASSO estimator and to analytically evaluate the variance of ${\hat{\bm{x}}}^{\rm{d}}$ depending on the asymptotic eigenvalue distribution $\rho_{\bm{J}}(s)$ of ${\bm{J}}={\bm{A}}^{T}{\bm{A}}$ (or ${\bm{J}}={\bm{A}}^{H}{\bm{A}}$) [16, 17]. Utilizing the formulas, we here propose DWLD for complex-valued row-orthogonal observation matrix, whose procedure is shown in Algorithm 1. Due to space limitation, we leave its derivation to [17].

One can deduce results similar to (17) for random Gaussian measurement matrices from some conclusions in [24]. More specifically, with Theorem 10 in [24], the debiased coefficient for random Gaussian design is $\Lambda_{G}=\gamma-\rho_{a}$, where $\rho_{a}=\#\{i|{\hat{x}_{i}^{\rm{WL}}}\neq 0\}/N$ denotes the active component density for real-valued situation. We emphasize that the methodologies in [16, 17] not only reproduce the same result for the Gaussian matrices but also can offer the debiased coefficients for general right rotation invariant matrices.

In general, the advantage of DWLD is that its threshold $\kappa_{d,i}$ can be calculated analytically from the false alarm rate $P_{fa,i}$. On the other hand, due to the fact that there is no analytic solution for weighted LASSO, one cannot establish the relationship between the threshold $\kappa_{WL,i}$ of NWLD and the false alarm rate $P_{fa,i}$. We will next introduce the two benefits brought by DWLD in Section II-C1 and II-C2, respectively.

One can easily conclude from the previous analysis that the smaller $\sigma_{w}^{2}$ is, the better the detection performance of DWLD can achieve. Given other parameters (e.g. distribution and size of measurement matrix ${\bm{A}}$, variance of the noise $\sigma^{2}$, distribution of ${\bm{x}}_{0}$, etc.), $\sigma_{w}^{2}$ can be expressed as a function of weights $\bm{\lambda}$ and prior information $\bm{p}$, given by

However, $f_{1}$ is unavailable because we do not know ${\bm{x}}_{0}$. Instead, we can obtain its estimation $\hat{\sigma}_{w}^{2}$ by the method developed in [16] and [17], which can be represented by

where the specific implementation of $f_{2}$ can be obtained from Algorithm 1. Therefore, we can find the optimal weights $\bm{\lambda}$ for the prior information ${\bm{p}}$ by optimizing $f_{2}$, given by

Unfortunately, $f_{2}$ dose not have an analytic expression. Employing heuristic algorithms to solve an $N$-dimensional parameter optimization problem is not very practical. Therefore, we reduced the degrees of freedom by parameterizing the weight vector $\bm{\lambda}$ with a few hyper-parameters, such as a linear model $\lambda^{*}_{i}\simeq\lambda_{0}-\alpha p_{i}$, or an exponential model $\lambda^{*}_{i}\simeq\lambda_{0}/p_{i}^{\alpha}$, and obtained sub-optimal results. Consequently, considering the linear model for example, the optimization problem in (26) becomes

The simulation results in Section III will verify the effectiveness of proposed weights optimization approach.

In all the numerical experiments, we artificially generate the original signal ${\bm{x}}_{0}$, observation matrix ${\bm{A}}$ and the noise ${\bm{\xi}}$. The original signal ${\bm{x}}_{0}$ is generated from the Bernoulli-Gaussian distribution and the entries are independent with prior information ${\bm{p}}$, thus

The measurement matrix ${\bm{A}}$ is decided by the radar system. The entries of the noise ${\bm{\xi}}$ are i.i.d. complex Gaussian variables: $\xi_{i}\sim{\cal{CN}}(0,\sigma^{2})$. We adopt the matched filtering (MF) definition of the signal-to-noise ratio (SNR) [5], such that

We apply DWLD and NWLD for solving the sub-Nyquist radar detection problem, in which the observation matrix $\bm{A}$ is partial Fourier, and examine their false alarm rate. In the following experiments, we set the length of ${\bm{x}}_{0}$ to $N=512$ and the variance of noise $\sigma^{2}$ to $0.01$. The probability of false alarm is set to $0.01$. All the results are obtained by $10^{4}$ Monte-Carol trials.

The result of the first experiment is shown in Fig. 2, where the prior information $\bm{p}$ is set as

The weight vectors ${\bm{\lambda}}_{1}$ and ${\bm{\lambda}}_{2}$ of the two detectors are set to $\lambda_{1,i}=0.1-0.1p_{i}$ and $\lambda_{2,i}=0.15-0.15p_{i}$, respectively. The threshold of NWLD is set to $0$, which corresponds to the naive decision based on the weighted LASSO estimator. From the result we can see that DWLD has a good ability to control the false alarm rate when different parameters vary while NWLD dose not.

The second one sets the prior information $\bm{p}$ as

The setting of weights and thresholds are the same as the former one. The result is shown in Fig. 3, in which DWLD also controls the false alarm rate completely.

We then compare the detection performance of DWLD and DLD, and the parameter settings of $\bm{A}$, ${\bm{x}}_{0}$ and ${\bm{\xi}}$ are the same as Sec. III-B. The result of the first experiment is shown in Fig. 4, where the prior information $\bm{p}$ is setting as (30). We compare the detection performance of DWLD with DLD for two regularization parameter settings. The regularization parameters of DLD $\lambda_{L}$ are set to be $0.1$ and $0.2$. Meanwhile, the weight vectors $\bm{\lambda}$ were optimized to $\lambda_{i}=0.1000-0.1002p_{i}$ for the linear model and $\lambda_{i}=0.0212/p_{i}^{0.3470}$ for the exponential model for $\bm{p}$ of (30). We demonstrate how the detection performance of these detectors varies with SNR. The results suggest that both detectors can maintain the false alarm rate with different SNR, and the total detection rate of DWLD is higher than DLD.

The second one sets the prior information $\bm{p}$ as (31). The regularization parameters of DLD $\lambda_{L}$ are set to be $0.1$ and $0.2$, and the optimal weight vectors $\bm{\lambda}$ were $\lambda_{i}=0.0963-0.1038p_{i}$ for the linear model and $\lambda_{i}=0.0281/p_{i}^{0.3594}$ for the exponential model. The result is shown in Fig. 5, which indicates that DWLD also presents better detection performance than DLD.