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Course by: Davide Rocchesso. E-mail the author

# Convolution - Discrete time

Module by: Anders Gjendemsjø. E-mail the author

Summary: Time discrete convolution

## Introduction

The idea of discrete-time convolution is exactly the same as that of continuous-time convolution. For this reason, it may be useful to look at both versions to help your understanding of this extremely important concept. Convolution is a very powerful tool in determining a system's output from knowledge of an arbitrary input and the system's impulse response.

It also helpful to see convolution graphically, i.e. by using transparencies or Java Applets. Johns Hopkins University has an excellent Discrete time convolution applet. Using this resource will help understanding this crucial concept.

## Derivation of the convolution sum

We know that any discrete-time signal can be represented by a summation of scaled and shifted discrete-time impulses, see (Reference). Since we are assuming the system to be linear and time-invariant, it would seem to reason that an input signal comprised of the sum of scaled and shifted impulses would give rise to an output comprised of a sum of scaled and shifted impulse responses. This is exactly what occurs in convolution. Below we present a more rigorous and mathematical look at the derivation:

Letting be a discrete time LTI system, we start with the folowing equation and work our way down the the convoluation sum.

yn=xn= k =xkδnk= k =xkδnk= k =xkδnk=k=xkhnk y n x n k x k δ n k k x k δ n k k x k δ n k k x k h n k
(1)
Let us take a quick look at the steps taken in the above derivation. After our initial equation we rewrite the function xn x n as a sum of the function times the unit impulse. Next, we can move around the ℋ operator and the summation because ˙ ˙ is a linear, DT system. Because of this linearity and the fact that xk x k is a constant, we pull the constant out and simply multiply it by ˙ ˙ . Finally, we use the fact that ˙ ˙ is time invariant in order to reach our final state - the convolution sum!

Above the summation is taken over all integers. Howerer, in many practical cases either xn x n or hn h n or both are finite, for which case the summations will be limited. The convolution equations are simple tools which, in principle, can be used for all input signals. Following is an example to demonstrate convolution; how it is calculated and how it is interpreted.

### Graphical illustration of convolution properties

A quick graphical example may help in demonstrating why convolution works.

## Convolution Sum

As mentioned above, the convolution sum provides a concise, mathematical way to express the output of an LTI system based on an arbitrary discrete-time input signal and the system's response. The convolution sum is expressed as

yn= k =xkhnk y n k x k h n k
(2)
As with continuous-time, convolution is represented by the symbol *, and can be written as
yn=xn*hn y n x n h n
(3)
By making a simple change of variables into the convolution sum, k=nk k n k , we can easily show that convolution is commutative:
yn=xn*hn=hn*xn y n x n h n h n x n
(4)
From Equation 4 we get a convolution sum that is equivivalent to the sum in Equation 2:
yn= k =hkxnk y n k h k x n k
(5)
For more information on the characteristics of convolution, read about the Properties of Convolution.

## Convolution Through Time (A Graphical Approach)

In this section we will develop a second graphical interpretation of discrete-time convolution. We will begin this by writing the convolution sum allowing x x to be a causal, length-m signal and h h to be a causal, length-k, LTI system. This gives us the finite summation,

yn= l =0m1xlhnl y n l 0 m1 x l h n l
(6)
Notice that for any given n n we have a sum of the mm products of xl x l and a time-delayed hnl h n l . This is to say that we multiply the terms of x x by the terms of a time-reversed h h and add them up.

Going back to the previous example:

What we are doing in the above demonstration is reversing the impulse response in time and "walking it across" the input signal. Clearly, this yields the same result as scaling, shifting and summing impulse responses.

This approach of time-reversing, and sliding across is a common approach to presenting convolution, since it demonstrates how convolution builds up an output through time.

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