The modulated E-spline with multiple subbands and its application to sampling wavelet-sparse signals

The theory of Finite Rate of Innovation (FRI) can be applied to sampling and reconstructing certain classes of parametric signals. The objective of this paper is to have a sub-Nyquist sampling scheme for continuous-time wavelet-sparse signals within the general framework of FRI theory. Though the signal has a parametric representation in the wavelet basis, it is not possible to recover the signal merely from its low-pass samples, which makes the problem different from the conventional FRI settings. The need for the Fourier coefficients at frequencies widely spread over the spectrum puts challenges on the design of the sampling kernel. This paper presents a new family of sampling kernels that are able to stably reproduce exponentials over a wide range of frequencies and gives numerical examples on applying the new kernel to sampling wavelet-sparse signals.