## Doctoral Dissertations

#### Abstract

"A signal theory characterization of a time function or signal is a representation of the function throughout a sample interval by an orthogonal basis function expansion. The characterization described here obtains the coefficients in the expansion by processing the input waveform, in real time, through a system of three terminal passive RC filters. The outputs of the filters are sampled periodically and the coefficients of the basis function expansion in that interval are related to these values. The basis functions result from an exponential transformation applied to the Legendre polynomials and are orthogonal in time over the sample interval. The basis functions and the resulting reconstruction appear as summations of positive exponential terms. Different signals may require different sampling intervals and/or different number of terms in their orthogonal expansions. As a result, the input waveforms have been classified in the time domain by two methods. One is a graphical method. The worst case input to the system is bounded by simple test functions. The shortest duration of the resulting test functions gives the sample interval. The rate of convergence of the mean-square error of the approximation is also given for the various forms of the test function. The graphical technique is easy to use, but gives conservative error estimates. The input waveform is also classified in terms of the pole locations of the Laplace-transformed input function. Using conventional time domain synthesis techniques, the pole locations of the transformed input function can be located in the complex-S plane. Bounding these poles by circles, with center at the origin of the S plane, will give the maximum signal reconstruction error for a given number of filters. The sampling period is based on a normalized rate and the frequency scaling required to move the poles into desirable maximum error regions of the S plane determines the actual sampling rate. These regions are very broad and thus a considerable change in the pole positions can be tolerated without affecting the parameters of the system. This method is more analytical than the first method, although more work is required to find the pole locations"--Abstract, page ii-iii.

#### Advisor(s)

Bertnolli, Edward C.

#### Committee Member(s)

Antle, Charles E.

Dillman, Norman G., 1938-2010

Rivers, Jack L.

Betten, J. Robert

#### Department(s)

Electrical and Computer Engineering

#### Degree Name

Ph. D. in Electrical Engineering

#### Publisher

University of Missouri--Rolla

#### Publication Date

1968

#### Pagination

121 pages

#### Note about bibliography

Includes bibliographical references (pages 118-120).

#### Rights

© 1968 James Julius Baremore, All rights reserved.

#### Document Type

Dissertation - Open Access

#### File Type

text

#### Language

English

#### Subject Headings

Signal processing -- Mathematical models

Signal theory (Telecommunication)

#### Thesis Number

T 2149

#### Print OCLC #

5999043

#### Electronic OCLC #

794451797

#### Link to Catalog Record

#### Recommended Citation

Baremore, James Julius, "Signal theoretic characterization of a function using positive exponential basis functions" (1968). *Doctoral Dissertations*. 1917.

https://scholarsmine.mst.edu/doctoral_dissertations/1917