The simplex projection method as a spectral extrapolation technique

Abstract

The resolving power of measuring instruments are limited by their instrumental bandwidth W_d. If such an instrument samples an analog signal s(x) at a rate less than the signal bandwidth W_s then its output {s_u(i)} will lack the finer details of s(x), where: i = –M, –M+1, ..., M; x = (2i – 1)Δ/2, T = sampling period = 2MΔ, and Δ = sampling interval = 1/W_d. Correct analytic reconstruction of s(x) directly from its sampled representation is possible only if W_d ≥ W_s. If before its sampling, s(x) with W_d < W_s, is properly conditioned by low-pass filtering then {s_u(i)} is (1) unaliased, (2) lacks details and (3) may contain spurious oscillations.
The recovery of the lost high-frequency components by post-detection processing has been the subject of attention for many years in instrument design and development. In general, the recovery problem is ill-posed because {s_u(i)} admits many possible solutions and regularization is often used to pick out the acceptable solution from the admissible ones. The basic feature is the introduction of a compromise between fidelity of the solution to {s_u(i)} and fidelity to some prior information about the analytic properties of s(x) such as finite support, and minimum negativity.
We demonstrate an approach for recovering the lost high-frequency components using only the available frequency components. No other prior information about s(x) is required to establish the cost function used to search for the acceptable solution. The lost frequency components {S(M+n)} are extrapolated from the known low-frequency Fourier components {S(m)} using the simplex projection method (SPM) where: {S(m)} is the Fourier transform of {s_u(i)}, f = m/T, m = –M, –M+1, ..., M; n = M+1, M+2, ..., N; and W_s = 2(M+N)/T. Spectral extrapolation implies the interpolation of new s(x) values within T.