Compressively sensing nonadjacent block-sparse spectra via a block discrete chirp matrix

Abstract

The block-sparse structure is shared by many types of signals, including audio, image, and radar-emitted signals. This structure can considerably improve compressive sensing (CS) performance and has attracted much attention in recent years. However, when fitting this model in practical applications, the nonzero blocks are always separated by one or more zero blocks to avoid interference between active emitters. (Generally, a block is occupied by an emitter.) In this paper, we coin a new phrase, ‘nonadjacent block sparse,’ or NBS, to describe this new structure. Our contributions are threefold. First, from a statistical probability perspective, the mean value and variance of block sparsity are evaluated and used to describe an NBS signal. Second, by employing the block discrete chirp matrix (BDCM), we propose and prove a condition that ensures the successful recovery of NBS signals from their linear measurements with high probability. Specifically, as long as a condition involved in mean value and variance of block sparsity is satisfied, an NBS signal can be successfully recovered with a high probability. Third, extensive experiments are simulated, and deep theoretical implications are discussed. The analyzed results demonstrate the progress we have made toward block-sparse CS.

Appendix

To prove Proposition 1, we leverage the following theorem.

Theorem 1

[22] If a sensing matrix${\varvec{\Phi}}$satisfies the block RIP with$\delta_{2k\left| b \right.} \le \sqrt 2 - 1$, then there is a unique block-k-sparse vector${\hat{\mathbf{x}}}$equal to the solution of Eq. (2), where${\varvec{\Phi}}$satisfies the block RIP with$\delta_{k\left| b \right.}$. Therefore, for any k-block-sparse signalx, $\left\| {\left( {1 - \delta_{k\left| b \right.} } \right){\mathbf{x}}} \right\|_{2}^{2} \le \left\| {\varvec{\Phi}}{\mathbf{x}} \right\|_{2}^{2} \le \left\| {\left( {1 + \delta_{k\left| b \right.} } \right){\mathbf{x}}} \right\|_{2}^{2}$holds.

We use $\delta_{k}$ for $\delta_{k\left| b \right.}$ to denote a parameter that is in context with the block-sparse case only. Considering Theorem 1, Proposition 1 gives a condition under which the block RIP holds with high probability.

Next, we provide a proof of proposition 1. Our proof strategy is to first evaluate the expectation of the spectral norm of the submatrix drawn from ${\varvec{\Phi}}$. Then, we derive the probability of block RIP being satisfied. Before providing the substance of the proof, we need to define some notations.

Let $\mathcal{K} = \{ {\varvec{\Theta}}|\left\| {\varvec{\Theta}} \right\|_{1} = k\}$ be the set that contains all the block index vectors for extracting k blocks from ${\varvec{\Phi}}$, where ${\varvec{\Theta}} = \left[ {{\varvec{\uptheta}}_{1} , \ldots ,{\varvec{\uptheta}}_{r} } \right]^{\text{T}} = \left[ {\theta_{(1 - 1)n + 1} , \ldots ,\theta_{(1 - 1)n + n} , \ldots ,\theta_{(r - 1)n + 1} , \ldots ,\theta_{(r - 1)n + n} } \right]^{\text{T}}$$\in {\mathbb{R}}^{L \times 1}$ is an arbitrary realization of V. For any matrix M with dL columns and a block index vector ${\varvec{\Theta}} \in {\mathbb{R}}^{L \times 1}$, we define the extracting operation as ${\mathbf{M}}|_{{\varvec{\Theta}}} = \mathcal{E}\left( { \, {\mathbf{M}} \cdot \left( {diag({\varvec{\Theta}}) \otimes {\mathbf{I}}_{d} } \right) \, } \right)$, where $\mathcal{E}\left( \cdot \right)$ is a function to remove zero blocks. If a set $\varOmega$ contains several block index vectors, ${\mathbf{M}}|_{\varOmega }$ stands for a set of submatrices drawn from M according to elements of $\varOmega$. Furthermore, we define $\mathcal{C}\left( {\varvec{\Theta}} \right)$ as the function that calculates the cardinality of set $\varGamma { = }\left\{ {\left\| {{\varvec{\uptheta}}_{i} } \right\|_{1} |\left\| {{\varvec{\uptheta}}_{i} } \right\|_{1} > 0} \right\}$. According to the definition of the cell Eq. (12), $\mathcal{C}\left( {\varvec{\Theta}} \right)$ is the number of active cells.

Based on Eq. (14), we have ${\varvec{\Phi}}_{i}^{\text{T}} {\varvec{\Phi}}_{j} = \left( {{\mathbf{q}}_{i} \otimes {\mathbf{U}}} \right)^{\text{T}} \left( {{\mathbf{q}}_{j} \otimes {\mathbf{U}}} \right) = {\mathbf{q}}_{i}^{\text{T}} {\mathbf{q}}_{j} {\mathbf{I}}$ for any ${\mathbf{q}}_{i}$ and ${\mathbf{q}}_{j}$. Therefore, we study the properties of the submatrix extracted from Q instead of directly studying the submatrix drawn from ${\varvec{\Phi}}$. We start by evaluating a surrogate expectation of a spectral norm of ${\mathbf{Q}}_{\mathcal{K}}$ as follows:

$$ \begin{aligned} {\mathbb{E}}_{s} & = \sum\limits_{{{\varvec{\Theta}} \in \mathcal{K}}} {\left\| { \, \left( {{\mathbf{Q}}|_{{\varvec{\Theta}}} } \right)^{\text{T}} {\mathbf{Q}}|_{{\varvec{\Theta}}} - {\mathbf{I}}_{k} \, } \right\|_{2} P\left( {\varvec{\Theta}} \right)} \\ & = \sum\limits_{{{\varvec{\Theta}} \in \mathcal{K}}} {\left\| {\left[ {{\mathbf{Q}}_{1} |_{{{\varvec{\uptheta}}_{1} }} ,{\mathbf{Q}}_{2} |_{{{\varvec{\uptheta}}_{2} }} , \ldots ,{\mathbf{Q}}_{r} |_{{{\varvec{\uptheta}}_{r} }} } \right]^{*} \left[ {{\mathbf{Q}}_{1} |_{{{\varvec{\uptheta}}_{1} }} ,{\mathbf{Q}}_{2} |_{{{\varvec{\uptheta}}_{2} }} , \ldots ,{\mathbf{Q}}_{r} |_{{{\varvec{\uptheta}}_{r} }} } \right] - {\mathbf{I}}_{k} } \right\|_{2} P\left( {\varvec{\Theta}} \right)} \\ & = \sum\limits_{{{\varvec{\Theta}} \in \mathcal{K}}} {\frac{1}{\sqrt r }\left\| {{\mathbf{1}}_{k} - blkdiag\left( {\underbrace {{{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{1} } \right\|_{1} }} ,{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{2} } \right\|_{1} }} , \ldots ,{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{r} } \right\|_{1} }} }}_{k}} \right)} \right\|_{2} P\left( {\varvec{\Theta}} \right)} \\ \end{aligned} $$

(15)

where the last step uses the properties of the DCM that ${\mathbf{Q}}_{i}^{\text{T}} {\mathbf{Q}}_{j}$ equals $\sqrt {{1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-0pt} r}} \, {\mathbf{1}}$ and 0 for $i \ne j$ and $i = j$, respectively.

Since $\mathcal{C}\left( {\varvec{\Theta}} \right)$ is fixed, the maximum of $\rho = \left\| {{\mathbf{1}}_{k} - blkdiag\left( {{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{1} } \right\|_{1} }} ,{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{2} } \right\|_{1} }} , \ldots ,{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{r} } \right\|_{1} }} } \right)} \right\|_{2}$ is achieved when the variance of $\varGamma$ is its minimum. Furthermore, the maximum of $\rho = \left\| {{\mathbf{1}}_{k} - blkdiag\left( {{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{1} } \right\|_{1} }} ,{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{2} } \right\|_{1} }} , \ldots ,{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{r} } \right\|_{1} }} } \right)} \right\|_{2}$ can be bounded by $k - \left\lfloor {k/\mathcal{C}\left( {\varvec{\Theta}} \right)} \right\rfloor$, as shown in Figs. 9 and 10. Let $\mathcal{K} = \mathcal{H}_{1} \cup \mathcal{H}_{2} \cdots \cup \mathcal{H}_{k}$, where $\mathcal{H}_{i} = \left\{ {{\varvec{\Theta}}\left| {\mathcal{C}\left( {\varvec{\Theta}} \right) = i} \right.} \right\}$. Then, we have

Fig. 9

$\rho = \left\| {{\mathbf{1}}_{k} - blkdiag\left( {{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{1} } \right\|_{1} }} ,{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{2} } \right\|_{1} }} , \ldots ,{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{r} } \right\|_{1} }} } \right)} \right\|_{2}$ versus variances of different $\varGamma$, it is easy to observe that the maximum of $\rho$ is achieved when the variance is minimum

Fig. 10

The maximum of $\rho = \left\| {{\mathbf{1}}_{k} - blkdiag\left( {{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{1} } \right\|_{1} }} ,{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{2} } \right\|_{1} }} , \ldots ,{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{r} } \right\|_{1} }} } \right)} \right\|_{2}$ under different block sparsities (denoted by k). The dashed lines are $k - \left\lfloor {k/\mathcal{C}\left( {\varvec{\Theta}} \right)} \right\rfloor$. Obviously, $k - \left\lfloor {k/\mathcal{C}\left( {\varvec{\Theta}} \right)} \right\rfloor$ can well approximate and upper bound the maximum of $\rho$

$$ \begin{aligned} {\mathbb{E}}_{s} & = \sum\limits_{i = 1}^{k} {\sum\limits_{{{\varvec{\Theta}} \in \mathcal{H}_{i} }} {\frac{1}{\sqrt r }\left\| {{\mathbf{1}}_{k} - blkdiag\left( {{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{1} } \right\|_{1} }} ,{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{2} } \right\|_{1} }} , \ldots ,{\mathbf{1}}_{{\left\| {{\varvec{\uptheta}}_{r} } \right\|_{1} }} } \right)} \right\|_{2} P\left( {\varvec{\Theta}} \right)} } \\ & \le \sum\limits_{i = 1}^{k} {\sum\limits_{{{\varvec{\Theta}} \in \mathcal{H}_{i} }} {\frac{1}{\sqrt r }\left( { \, k - \left\lfloor {k/i} \right\rfloor \, } \right)P\left( {\varvec{\Theta}} \right)} } \\ & \le \sum\limits_{i = 1}^{k} {\frac{k}{\sqrt r }\left( { \, 1 - 2^{1 - i} \, } \right)\sum\limits_{{{\varvec{\Theta}} \in \mathcal{H}_{i} }} {P\left( {\varvec{\Theta}} \right)} } \\ & \le \sum\limits_{i = 1}^{k} {\frac{k}{\sqrt r }\left( { \, 1 - 2^{1 - k} \, } \right)\sum\limits_{{{\varvec{\Theta}} \in \mathcal{H}_{i} }} {P\left( {\varvec{\Theta}} \right)} } \, = \, \frac{k}{\sqrt r }\left( { \, 1 - 2^{1 - k} \, } \right)P\left( {\left\| {\mathbf{V}} \right\|_{1} = k} \right) \\ \end{aligned} . $$

(16)

In Eq. (16) line 3, we use the inequation $k - \left\lfloor {k/i} \right\rfloor \le k\left( { \, 1 - 2^{1 - i} \, } \right)$; see Fig. 11. Since $P\left( {\left\| {\mathbf{V}} \right\|_{1} = k} \right)$ can be well approximated by a Gaussian function, the maximum of $k \cdot P\left( {\left\| {\mathbf{V}} \right\|_{1} = k} \right)$ is obtained when k equals the mean value. Therefore,

Fig. 11

Comparison of $k - \left\lfloor {k/i} \right\rfloor$ and $k\left( { \, 1 - 2^{1 - i} \, } \right)$. Solid lines represent the values of $k - \left\lfloor {k/i} \right\rfloor$, while dashed lines denote the value of $k\left( { \, 1 - 2^{1 - i} \, } \right)$. Obviously, the dashed lines outperform the solid lines, that is, $k - \left\lfloor {k/i} \right\rfloor \le k\left( { \, 1 - 2^{1 - i} \, } \right)$

$$ \begin{aligned}{\mathbb{E}}_{s} &\le \, \frac{k}{\sqrt r }\left( { \, 1 - 2^{1 - k} \, } \right)P\left( {\left\| {\mathbf{V}} \right\|_{1} = k} \right)\\ & \le \frac{\mu }{\sqrt r }\left( { \, 1 - 2^{1 - \mu } \, } \right)P\left( {\left\| {\mathbf{V}} \right\|_{1} = \mu } \right) = \frac{{\mu \left( {1 - 2^{ 1- \mu } } \right)}}{{\sqrt {2\pi r\sigma^{2} } }} .\end{aligned} $$

(17)

With the application of Markov’s inequality, the result is

$$ \begin{aligned} & P\left( {\left\| { \, \left( {{\mathbf{Q}}|_{{\varvec{\Theta}}} } \right)^{\text{T}} {\mathbf{Q}}|_{{\varvec{\Theta}}} - {\mathbf{I}} \, } \right\|_{2} \ge \alpha } \right) \le {{\mathcal{E}_{s} } \mathord{\left/ {\vphantom {{\mathcal{E}_{s} } \alpha }} \right. \kern-0pt} \alpha }\\ & \Rightarrow P\left( {\left\| { \, \left( {{\mathbf{Q}}|_{{\varvec{\Theta}}} } \right)^{\text{T}} {\mathbf{Q}}|_{{\varvec{\Theta}}} - {\mathbf{I}} \, } \right\|_{2} \le \alpha } \right) \ge 1 - {{\mathcal{E}_{s} } \mathord{\left/ {\vphantom {{\mathcal{E}_{s} } \alpha }} \right. \kern-0pt} \alpha } \\ & \Rightarrow P(1 - \alpha \le \left\| { \, {\mathbf{Q}}|_{{\varvec{\Theta}}} \, } \right\|_{2}^{2} \le 1 + \alpha ) \ge 1 - {{\mathcal{E}_{s} } \mathord{\left/ {\vphantom {{\mathcal{E}_{s} } \alpha }} \right. \kern-0pt} \alpha } \\ \end{aligned} . $$

(18)

By the assumption $\mu \left( {1 - 2^{1 - \mu } } \right)/\sqrt {2\pi r\sigma^{2} } \le \varepsilon \left( {\sqrt 2 - 1} \right)$, it is obvious that

$$ \begin{aligned}& P\left( { \, 1 - \alpha \le \left\| { \, {\mathbf{Q}}|_{{\varvec{\Theta}}} } \right\|_{2}^{2} \le 1 + \alpha \, } \right) \\ & \ge 1 - {{{\mathbb{E}}_{s} } \mathord{\left/ {\vphantom {{{\mathbb{E}}_{s} } \alpha }} \right. \kern-0pt} \alpha } = 1 - \frac{{\mu \left( {1 - 2^{1 - \mu } } \right)}}{{\alpha \sqrt {2\pi r\sigma^{2} } }} \, \ge \, 1 - \frac{{\varepsilon \left( {\sqrt 2 - 1} \right)}}{\alpha } .\end{aligned} $$

(19)

Finally, recalling Theorem 1 and substituting $\alpha$ for $\delta_{2k} \le \sqrt 2 - 1$, we complete the proof. □