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