## Guided Tour Of Generalized Functions In Signal Processing

Conference

2005 Annual Conference

Location

Portland, Oregon

Publication Date

June 12, 2005

Start Date

June 12, 2005

End Date

June 15, 2005

ISSN

2153-5965

Conference Session

Mathematics Potpourri

Page Count

9

Page Numbers

10.679.1 - 10.679.9

DOI

10.18260/1-2--15478

Permanent URL

https://peer.asee.org/15478

71

#### Abstract NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

Session ####

Guided Tour of Generalized Functions in Signal Processing 1

John Schmeelk

Department of Mathematical Sciences Virginia Commonwealth University Doha, Qatar

Abstract

The paper considers generalized functions as a necessary ingredient in various signal- processing techniques. Oftentimes generalized functions are implemented in a casual way and not reflecting the need to establish results on a mathematical platform. A few of the important connections between the applications and mathematical foundations are included together with several illustrations.

I. Introduction

The rapid descent test functions with the inclusion of a few very needed principals are given in section 2. It also gives a brief introduction of tempered distributions including some very important theorems. The paper then moves into section 3 giving the fundamentals of a discrete Fourier transform pair. This is then embedded into a tempered distribution setting. Section 4 gives an introduction to windowing signal data and again embeds it into the tempered distribution setting. The paper concludes with a very brief overview on filtering frequency techniques.

II. The Test Space S

We adopt the following notation conventions. For positive integers, qi, (1 ≤ i ≤ n ) , the n length of q=(q1, q2,…, qn) is defined as q = ∑ q i . The absolute values in the paper also i =1 use the same notation. The context of the notation will indicate the appropriate meaning. n The Euclidean distance for x ∈ R will be denoted as a norm, x = n ∑x i =1 2 i , and for

differentiable functions, φ(x), the differential operator will be denoted as ∂ ( q1 ,..., qn ) D qφ ( x) = q1 φ ( x1 ,..., x n ) . ∂x1 ...∂x n n q

The test space S of rapid descent test functions are all infinitely differentiable and together with all of their partial derivatives decrease to zero faster than every power of

1 Funded by the Qatar Foundation, Doha, Qatar

“Proceedings of the 2005 American Society for engineering Education Annual Conferences & Exposition Copyright © 2005, American Society for Engineering Education”

Schmeelk, J. (2005, June), Guided Tour Of Generalized Functions In Signal Processing Paper presented at 2005 Annual Conference, Portland, Oregon. 10.18260/1-2--15478

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