Signal Reconstruction as a Wiener Filter Approximation

Abstract

The problem of reconstructing a non-negative signal from a finite number of spectral data is a problem of finding an optimal approximation to one function by another. For example, for velocity measurement by crossed beam laser Doppler anemometry, a limited number of channels can provide high quality data on the autocorrelation function of the intensity of the scattered light. However, extrapolation of these data is required in order to estimate velocity distributions narrower than the point spread function determined by the number of channels, e.g. in the case of laminar flow. We describe here methods based on the theory of best approximation in weighted Hilbert spaces, (1). These methods have been under development for some time for use in a variety of 1-D and 2-D estimation problems. A new interpretation of these methods is now possible based on the close analogy between the reconstruction of a non-negative function from finitely many values of its Fourier transform, and the design of approximate Wiener filters,(2).