Compressive detection of sparse signals in additive white Gaussian noise without signal reconstruction

doi:10.1016/j.sigpro.2016.08.020

Signal Processing

Volume 131, February 2017, Pages 376-385

https://doi.org/10.1016/j.sigpro.2016.08.020 Get rights and content

Highlights

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Some new descriptions for the detector and estimator have been presented.
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The equations with respect to σ₂ have been now written with respect to p in the whole paper.
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The Cramér-Rao lower bound (CRLB) for the proposed estimator, which is biased, has been computed.

Abstract

The main motivation behind compressive sensing is to reduce the sampling rate at the input of a digital signal processing system. However, if for processing the sensed signal one requires to reconstruct the corresponding Nyquist samples, then the data rate will be again high in the processing stages of the overall system. Therefore, it is preferred that the desired processing task is done directly on the compressive measurements, without the need for the reconstruction of the Nyquist samples. This paper addresses the case in which the processing task is “detection” (the existence) of a sparse signal in additive white Gaussian noise, with applications e.g. in radar systems. Moreover, we will propose two estimators for estimating the degree of sparsity of the detected signal. We will show that one of the estimators attains the Cramér–Rao lower bound of the problem.

Introduction

Detection of a sparse signal in additive white Gaussian noise (AWGN) from a small number of compressive measurements without signal reconstruction is an interesting subject that has received recent consideration [3], [11], [16], [2], [17]. This interest arises from a desire to avoid the difficulties caused by reconstructing the original signal for detection. Note that by using compressed sensing, the data rate is greatly reduced, that is, the rate of measurements is much lower than the Nyquist rate. However, if for processing the measurements, one needs to reconstruct the original signal (i.e. reconstruct the Nyquist samples), then the data rate is again very high. In other words, compressed sensing will not be very useful if the Nyquist samples are needed again to be reconstructed in some part of the system, and it is desirable to do the required process (sparse signal detection and parameter estimation) directly based on compressed measurements [4].

To the best of our knowledge, this problem has not yet been modeled using the sparse signal as a random process with an appropriate a priori probability density function (PDF). In [3], [2], [17], the signal is not random and there is no specific property in the signal modeling to express its sparsity. Authors of [16] exploit sparsity by performing the detection for all possible support patterns with fixed size on average. Therefore, the degree of sparsity (or equivalently, the support size) must be known exactly and this is difficult in practice.

In some studies (e.g. [11], [16]), it is assumed that the random signal has a Gaussian PDF, which is not a suitable distribution for modeling sparsity. Here, in contrast to these works, a sparsity-enforcing a priori PDF is assumed for the signal, and a new detection algorithm along with a sparsity degree estimation is proposed.

The paper is organized as follows. In Section 2, the problem of detecting a sparse signal in AWGN is modeled mathematically and a solution is provided for this model. In fact, by using a random model for the sparse signal and assuming that the number of Nyquist samples is large, the distribution of the measurement vector when the signal exits, can be approximated by a Gaussian distribution. This is achieved by using the central limit theorem (CLT). Moreover, since the sparsity degree of the signal is unknown, it is needed to rely on a generalized likelihood ratio test (GLRT) approach. In Section 3, the relationship between the signal to noise ratio (SNR) at the input of the detector and the SNR of the presented detector statistic is studied for high input SNRs. In Section 4, the performance of the proposed estimator for the sparsity degree is investigated and it is shown that the estimator attains the Cramér–Rao lower bound (CRLB), that is, it is an efficient estimator. However, there is no guarantee that the estimated probability of activity is less than 1. Therefore, another estimator is proposed that is guaranteed to estimate this probability less than 1 but it does not attain the CRLB. Nevertheless, the variance of the estimator is close to the CRLB. Finally, Section 5 presents simulation results for detection of the sparse signal and estimation of the degree of sparsity.

Notations: For any vector $y$ , its transpose is $y^{T}$ . For any square matrix $A$ , its determinant is denoted by $| A |$ . The probability of an event A is represented by $P {A}$ . The expectation and covariance matrix of a random vector $y$ are denoted by $E {y}$ and $cov {y}$ , respectively. Moreover, $I_{M}$ stands for the M×M identity matrix.

Section snippets

Our problem modeling and solution

Detecting a sparse signal in AWGN is mathematically expressed as the hypothesis testing problem ${\begin{matrix} H_{0} : y = Φ n \\ H_{1} : y = Φ (s + n), \end{matrix}$ where $y_{M \times 1}$ is the measurements vector, $Φ_{M \times N}$ is the compressed sensing measurement matrix where elements are drawn from a random Gaussian matrix with independent identically distributed (i.i.d.) elements having zero mean and unit variance (it should be noted that at the detector side, the matrix $Φ$ is randomly chosen and then kept fixed), and $M ⪡ N$ . Moreover, $n_{N \times 1}$ is the AWGN samples

The SNR of the presented detector

It seems that obtaining a closed form formula for the receiver operating characteristic (ROC) of the presented detector would be too tricky. This is mainly due to the presence of the natural logarithm term in (6). Therefore, as a measure of the detector performance, the relationship between the SNR at the input of the detector and the SNR at the threshold comparison is computed. The signal at the input of the detector is $y_{in} = s + n$ , in which $s$ is the signal term and $n$ is the noise term. Therefore

Performance evaluation of the sparsity degree estimator

From (3), the proposed estimator for the sparsity degree is $\hat{p} = \frac{1}{σ_{on}^{2} - σ_{off}^{2}} (\frac{1}{M} y^{T} A^{- 1} y - σ_{n}^{2} - σ_{off}^{2}),$ where $y = Φ (s + n)$ . Note that, if the following two parameters are defined as $σ_{Δ}^{2} ≜ σ_{on}^{2} - σ_{off}^{2}$ , and $σ_{η}^{2} ≜ σ_{off}^{2} + σ_{n}^{2}$ , it is clear that $σ^{2} + σ_{n}^{2} = σ_{η}^{2} + p σ_{Δ}^{2}$ and hence, $\hat{p} = \frac{\frac{1}{M} y^{T} A^{- 1} y - σ_{η}^{2}}{σ_{Δ}^{2}}$ .

Theorem 3

The proposed estimator (17) is unbiased and it attains the CRLB of the problem which equals $\frac{2 {(σ^{2} (p) + σ_{n}^{2})}^{2}}{M {(σ_{on}^{2} - σ_{off}^{2})}^{2}} = \frac{2}{M} {(\frac{σ_{η}^{2}}{σ_{Δ}^{2}} + p)}^{2}$ .

Proof

See Appendix D.

From the above theorem, it is obvious that a small CRLB is obtained for large number

Results and simulations

In this section, seven simulations are conducted to experimentally evaluate the performance of the proposed algorithm for sparse signal detection and sparsity degree estimation. In all of these simulations, the following parameters are fixed $p = 0.1, σ_{off} = 0.1, σ_{on} = 1, p_{0} = 0.5, N = 1320 .$ In what follows, for each ROC figure, the area under the curve (AUC) is given in a table (Tables 1–4). Moreover, by the compression ratio (CR) we mean the fraction $\frac{N}{M}$ .

Conclusion

In this paper, it has been shown that the detection of a sparse signal in AWGN and the estimation of its degree of sparsity can be accomplished using compressive measurements without signal reconstruction. In fact, performing the detection and estimation directly in the compressive space allows us to keep the rate of measurements low through all the parts of the digital processing system. Here, the proposed approach for deriving the detector and the estimators is related to the signal sparsity

Acknowledgments

The authors would like to thank anonymous reviewers for their comments and suggestions.

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View full text

Compressive detection of sparse signals in additive white Gaussian noise without signal reconstruction

Highlights

Abstract

Introduction

Section snippets

Our problem modeling and solution

The SNR of the presented detector

Performance evaluation of the sparsity degree estimator

Results and simulations

Conclusion

Acknowledgments

Inf. Sci.

Signal Process.

Signal Process.

Phys. Commun.

Convergence of Probability Measures

Signal processing with compressive measurements

IEEE J. Sel. Top. Signal Process.

Joint compressive single target detection and parameter estimation in radar without signal reconstruction

IET Radar Sonar Navig.

Fundamentals of Statistical Signal Processing: Estimation Theory