Recovery of lost signal features without a priori knowledge

Abstract

Sampling a bandlimited signal s(x) at a rate f_s that is less than its bandwidth W, yields a data representation {s(i)} where index i = −M, −M+1, ..., M; with incomplete information about the high-frequency behavior of s(x). The relation between f_s and the sampling interval Δ is fs = 1/Δ. Undersampling occurs when: Δ > 1/W.
An undersampled signal representation exhibits ringings particularly near locations where s(x) show sudden amplitude changes. This unwanted behavior occurs because all frequency components of s(x), which are greater than f_c = f_s/2, are absent in {s(i)}.
In this paper, we present a new analytic continuation (AC) technique for recovering the lost frequency components. The AC technique does not explicitly require any a priori information regarding the behavior of s(x).
Standard AC techniques work only when s(x) either has a known finite support (finite-support constraint) or has a known floor for its minimum amplitude value (minimum negativity constraint).
We utilize the simplex projection method (SPM) to predict the values of the lost high-frequency components from a set of data consisting of the low-frequency components (f < f_s/2) of s(x). Previous applications of the SPM has been in the prediction of future values of a time-series data of simulated chaotic signals, annual occurrences of measles and chicken pox, and population behavior of marine plankton. To our knowledge, our work is the first prediction application of SPM to frequency-domain data.
We compare the performance of the SPM against an AC technique that uses the minimum-negativity constraint (MNC), to establish the limits of its reliability. Off hand, the SPM is non-iterative and therefore is very fast to implement unlike constraint-based techniques.