Wiener's Generalized Harmonic Analysis and Waveform Design

Wiener's Generalized Harmonic Analysis and Waveform Design

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##### Date

2007-02-21

##### Authors

Datta, Somantika

##### Advisor

Benedetto, John J

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##### 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.