Paley–Wiener Characterization of Kernels for Finite-Rate-of-Innovation Sampling

Abstract

Exact reconstruction of finite-rate-of-innovation signals can be achieved by employing customized sampling kernels that satisfy certain frequency-domain properties. We impose compact support in time as an additional constraint. Considering frequency-domain reconstruction, we derive conditions for admissible sampling kernels and corresponding sampling rates. Our constructive kernel design methodology is based on the Paley-Wiener theorem for compactly supported functions. The new kernels satisfy generalized Strang-Fix conditions and have specific polynomial-modulated-exponential-reproducing properties. Unlike exponential splines, which have a support that is directly proportional to the number of exponentials they can generate, the proposed kernels have a support that is independent of that number. To analyze noise robustness, we consider a special member of the class that has a sum-of-modulated splines (SMS) form in the time domain and optimize its parameters to minimize the noise variance. The sum-of-sincs (SoS) kernel reported in the literature is an instance of this construction. In noise robustness analysis, SMS kernels show improvement in mean-squared error (MSE) compared with the state-of-the-art alternatives. In continuous-time noise, the improvement in MSE is about 2 dB for low signal-to-noise ratio (SNR) and 7 dB for high SNR. In the case of discrete white Gaussian noise, the MSE is lower by as much as 25 dB by using a higher-order SMS kernel compared with the SoS kernel for SNRs in the range of 10-15 dB.