Discrete Time Convolution is Multiplication without Carry

Innocent E Okoloko

doi:10.24018/ejece.2021.5.5.358

Abstract

In this paper an analysis of discrete-time convolution is performed to prove that the convolution sum is polynomial multiplication without carry, whether the sequences are finite or not, by using several examples to compare the results computed using the existing approaches to the polynomial multiplication approach presented here. In the design and analysis of signals and systems the concept of convolution is very important. While software tools are available for calculating convolution, for proper understanding it is important to learn now to calculate it by hand. To this end, several popular methods are available. The idea that the convolution sum is indeed polynomial multiplication without carry is demonstrated in this paper. The concept is further extended to deconvolution, N-point circular convolution and the Z-transform approach.

Highlights

In the design and analysis of signals-whether discrete-time (DT) or continuous-time (CT), the concept of convolution is an indispensable and basic foundation
Linear convolution has been described in the major authoritative scholarly textbooks on signals and systems analysis
This paper attempts to prove and to generalize that it does not matter whether sequences are finite, or infinite or which method is used to calculate, the convolution sum obtained will correspond to the polynomial multiplication of both signals without carry

Summary

INTRODUCTION

In the design and analysis of signals-whether discrete-time (DT) or continuous-time (CT), the concept of convolution is an indispensable and basic foundation. It is acknowledged that it is indispensable to understand analytical convolution because is the only method that can obtain a closed form solution. This paper attempts to prove and to generalize that it does not matter whether sequences are finite, or infinite or which method is used to calculate, the convolution sum obtained will correspond to the polynomial multiplication of both signals without carry. [8] shows that for finite length signals, vector multiplication of both signals in the time domain can be directly performed without taking the Ztransforms and it will produce the same result as the Ztransform approach.

CONVOLUTION SUM IS MULTIPLICATION WITHOUT

Non-Finite Sequence Example

Finite Sequence Example

Z-transform Example

DECONVOLUTION

CONCLUSION