Unified Vector Multiplication Approach for Calculating Convolution and Correlation

Innocent E Okoloko

doi:10.24018/ejers.2021.6.4.2488

Abstract

This paper is a theoretical analysis of discrete time convolution and correlation and to introduce a unified vector multiplication approach for calculating discrete convolution and correlation, both of which are important concepts in the design and analysis of signals and systems and are usually encountered in the first course in signals and systems analysis. There are software tools for calculating them, however, it is important to learn now to compute them by hand. Several methods have been proposed to compute them by hand, most of which can be very involving. However, a closer look at the concepts reveal that the convolution and correlation sums are actually vector multiplication with diagonalwise addition and for finite sequences, can be computed by hand the same way. The method is also extended to N-point circular convolution. The method also makes it clearer to see the similarities and differences between convolution and correlation.

Highlights

Convolution and correlation are basic foundations in the analysis and design of signals and systems
To show that the convolution sum is vector multiplication with left diagonal addition several examples will be presented from the reviewed books
It is shown that for finite sequences, one does not need to apply the Z-transform to a pair of sequences before multiplication in order to simplify the computation of the convolution sum

Summary

INTRODUCTION

Convolution and correlation are basic foundations in the analysis and design of signals and systems. An attempt is made to generalize and show that whichever way the convolution sum is calculated, and whether the sequences are finite or not, the convolution sum corresponds to the vector multiplication of both signals followed by diagonal addition. It will be shown here that for finite length signals, one can directly perform the vector multiplication of both signals in the time domain without taking the Z-transforms and obtain the same result as the Ztransform approach.

CONVOLUTION AS VECTOR MULTIPLICATION WITH LEFT

Finite Sequence Example

Non-finite Sequence Example

Z-transform Example

CORRELATION SUM AS VECTOR MULTIPLICATION WITH

CONCLUSION