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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”

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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