Wiener's Generalized Harmonic Analysis and Waveform Design

Abstract

Bounded codes or waveforms are constructed whose autocorrelation is the inverse Fourier transform of certain positive functions. For the positive function F=1 the corresponding unimodular waveform of infinite length, whose autocorrelation is the inverse Fourier transform of F, is constructed using real Hadamard matrices. This waveform has a autocorrelation function that vanishes everywhere on the integers except at zero where it is one. In this case error estimates have been calculated which suggest that for a pre-assigned error the number (finite) of terms from this infinite sequence that are needed so that the autocorrelation at some non-zero k is within this given error range is `almost' independent of k. In addition, such unimodular codes (both real and complex) whose autocorrelation is the inverse Fourier transform of F=1 has also been constructed by extending Wiener's work on Generalized Harmonic Analysis (GHA) and a certain class of exponential functions. The analogue in higher dimensions is also presented. Further, for any given positive and even function f defined on the integers that is convex and decreasing to zero on the positive integers, waveforms have been constructed whose autocorrelation is f. The waveforms constructed are real and bounded with a bound that depends on the value of f at zero.