WITH: Weighted Truncated Hadamard-Matrix-Based Deterministic Compressive Sampling for Sparse Multiband Signals

Abstract

This paper considers the orthogonal observation matrix design of deterministic compressive sampling (CS). An observation matrix called the weighted truncated Hadamard-modulated wideband converter (WITH-MWC) is deterministically designed based on the truncated Hadamard matrix. This matrix can meet the restricted isometry property (RIP) condition with overwhelming probability by randomly or strategically extracting from a standard Hadamard matrix. Most compressed sampling systems are highly sensitive to noise. To reduce the adverse effects of noise interference, partial specific matrix elements are weighted according to the sparse characteristics of multiband signals, and the recovery probability is provably better than that of the original system and other deterministic observation matrices, especially in the low signal-to-noise ratio scenario. Compared to the random Gaussian and random Bernoulli matrices, WITH-MWC is much easier to implement in hardware. The simulations verify the above analysis.

Appendices

Appendix

Proof of lemma 1

According to the JL lemma, for all $n \times n$ matrices decimated from a Gaussian random variables set (J), the probability that the inner product of any two different vectors deviates from 0 decreases exponentially with the increase in n. Therefore, the similarity of a Gaussian distributed matrix to an orthogonal matrix can be expressed by the following probability:

$$\begin{aligned} P( \vert \left\langle u,v\right\rangle \vert \ge \epsilon ) \le 4e^{-\frac{n\epsilon ^{2}}{8}}, 0<\epsilon <1 \end{aligned}$$

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where $u,v \in R^{n}$ are independently and randomly sampled from J.

Because $u,v\sim N(0,\frac{1}{n})$, then $\dfrac{u\pm v}{\sqrt{2}}\sim N(0,\frac{1}{n})$.

From the JL lemma,

$$\begin{aligned} P(\vert \vert \dfrac{u + v}{\sqrt{2}} \vert \vert -1 \ge \epsilon ) \le e^{-\frac{n\epsilon ^{2}}{8}}, 0<\epsilon <1 \end{aligned}$$

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$$\begin{aligned} P( 1 - \vert \vert \dfrac{u - v}{\sqrt{2}} \vert \vert \ge \epsilon ) \le e^{-\frac{n\epsilon ^{2}}{8}}, 0<\epsilon <1 \end{aligned}$$

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Because

$$\begin{aligned} \vert \vert \dfrac{u + v}{\sqrt{2}} \vert \vert - 1 + 1 - \vert \vert \dfrac{u - v}{\sqrt{2}} \vert \vert= & {} \vert \vert \dfrac{u + v}{\sqrt{2}} \vert \vert - \vert \vert \dfrac{u - v}{\sqrt{2}} \vert \vert \nonumber \\= & {} \left\langle u,v\right\rangle \end{aligned}$$

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Therefore,

$$\begin{aligned} P( \left\langle u,v\right\rangle \ge \epsilon )\le & {} P( \vert \vert \dfrac{u + v}{\sqrt{2}} \vert \vert -1 \ge \epsilon ) + P( 1 - \vert \vert \dfrac{u - v}{\sqrt{2}} \vert \vert \ge \epsilon ) \nonumber \\\le & {} 2e^{-\frac{n\epsilon ^{2}}{8}}, 0<\epsilon <1 \end{aligned}$$

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Similarly,

$$\begin{aligned} P(-\left\langle u,v\right\rangle \ge \epsilon ) \le 2e^{-\frac{n\epsilon ^{2}}{8}}, 0<\epsilon <1 \end{aligned}$$

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Finally, Theorem 1 can be obtained:

$$\begin{aligned} P(\vert \left\langle u,v\right\rangle \vert \ge \epsilon ) \le 4e^{-\frac{n\epsilon ^{2}}{8}}, 0<\epsilon <1 \end{aligned}$$

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