Spectral Domain Noise Modeling in Compressive Sensing-Based Tonal Signal Detection

Abstract

Tonal signals are shown as spectral peaks in the frequency domain. When the number of spectral peaks is small and the spectral signal is sparse, Compressive Sensing (CS) can be adopted to locate the peaks with a low-cost sensing system. In the CS scheme, a time domain signal is modelled as $\boldsymbol{y}=\bPhi F^{-1}\boldsymbol{s}$, where y and s are signal vectors in the time and frequency domains. In addition, F^-1 and $\bPhi$ are an inverse DFT matrix and a random-sampling matrix, respectively. For a given y and $\bPhi$, the CS method attempts to estimate s with l₀ or l₁ optimization. To generate the peak candidates, we adopt the frequency-domain information of $\resmile{\boldsymbol{s}}$ = $\boldsymbol{F}\resmile{\boldsymbol{y}}$, where $\resmile{y}$ is the extended version of y and $\resmile{\boldsymbol{y}}\left(\boldsymbol{n}\right)$ is zero when n is not elements of CS time instances. In this paper, we develop Gaussian statistics of $\resmile{\boldsymbol{s}}$. That is, the variance and the mean values of $\resmile{\boldsymbol{s}}\left(\boldsymbol{k}\right)$ are examined.