Sub-Nyquist Sampling and Measurement of FRI Signals With Additive Shot Noise

Abstract

To mitigate the impact of shot noise on the parameter measurement process of Finite Rate of Innovation (FRI) signals, this paper proposes a sub-Nyquist modulation sampling system with two measurement channels. The noisy signal is sampled in the main channel after being filtered by a sinc kernel to obtain the aliasing Fourier coefficients, whereas the FRI signal is shifted out of the baseband with a cosine modulation function in the accessory channel to expose the shot noise information to the low frequency band. The burst time parameters of shot noise can be estimated using the annihilating filter method from the samples in the accessory channel, while the parameters of the FRI signal will be measured from the observations in the main channel after the shot noise information has been removed. If the Rate of Innovation of the FRI signal is ρ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>L</i></sub> and is affected by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> impulses of shot noise, the entire measurement process only requires ρ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>L</i></sub> ∙ τ + 2 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> points to be sampled per unit time τ. To enhance the robustness of the measurement process in actual circuit systems, a corrected annihilating filter method is proposed, and the parameter measurement error when employing this method is analyzed. Numerical and hardware experiments demonstrate that, compared to classical anti-noise algorithms, the proposed method not only measures the parameters of FRI signals contaminated by shot noise more accurately, but also significantly improves the robustness of the measurement process in non-ideal environments.