3.2 Continuous time convolution

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Defines convolution and derives the Convolution Integral.

Introduction

Convolution, one of the most important concepts in electrical engineering, can be used to determine the output a system produces for a given input signal. It can be shown that a linear time invariant system is completely characterized by its impulse response. The sifting property of the continuous time impulse function tells us that the input signal to a system can be represented as an integral of scaled and shifted impulses and, therefore, as the limit of a sum of scaled and shifted approximate unit impulses. Thus, by linearity, it would seem reasonable to compute of the output signal as the limit of a sum of scaled and shifted unit impulse responses and, therefore, as the integral of a scaled and shifted impulse response. That is exactly what the operation of convolution accomplishes. Hence, convolution can be used to determine a linear time invariant system's output from knowledge of the input and the impulse response.

Operation definition

Continuous time convolution is an operation on two continuous time signals defined by the integral

$\left(f*g\right)\left(t\right)={\int }_{-\infty }^{\infty }f\left(\tau \right)g\left(t-\tau \right)d\tau$

for all signals $f,g$ defined on $\mathbb{R}$ . It is important to note that the operation of convolution is commutative, meaning that

$f*g=g*f$

for all signals $f,g$ defined on $\mathbb{R}$ . Thus, the convolution operation could have been just as easily stated using the equivalent definition

$\left(f*g\right)\left(t\right)={\int }_{-\infty }^{\infty }f\left(t-\tau \right)g\left(\tau \right)d\tau$

for all signals $f,g$ defined on $\mathbb{R}$ . Convolution has several other important properties not listed here but explained and derived in a later module.

Definition motivation

The above operation definition has been chosen to be particularly useful in the study of linear time invariant systems. In order to see this, consider a linear time invariant system $H$ with unit impulse response $h$ . Given a system input signal $x$ we would like to compute the system output signal $H\left(x\right)$ . First, we note that the input can be expressed as the convolution

$x\left(t\right)={\int }_{-\infty }^{\infty }x\left(\tau \right)\delta \left(t-\tau \right)d\tau$

by the sifting property of the unit impulse function. Writing this integral as the limit of a summation,

$x\left(t\right)=\underset{\Delta \to 0}{lim}\sum _{n}x\left(n\Delta \right){\delta }_{\Delta }\left(t-n\Delta \right)\Delta$

where

${\delta }_{\Delta }\left(t\right)=\left\{\begin{array}{cc}1/\Delta & 0\le t<\Delta \\ 0& \mathrm{otherwise}\end{array}\right)$

approximates the properties of $\delta \left(t\right)$ . By linearity

$Hx\left(t\right)=\underset{\Delta \to 0}{lim}\sum _{n}x\left(n\Delta \right)H{\delta }_{\Delta }\left(t-n\Delta \right)\Delta$

which evaluated as an integral gives

$Hx\left(t\right)={\int }_{-\infty }^{\infty }x\left(\tau \right)H\delta \left(t-\tau \right)d\tau .$

Since $H\delta \left(t-\tau \right)$ is the shifted unit impulse response $h\left(t-\tau \right)$ , this gives the result

$Hx\left(t\right)={\int }_{-\infty }^{\infty }x\left(\tau \right)h\left(t-\tau \right)d\tau =\left(x*h\right)\left(t\right).$

Hence, convolution has been defined such that the output of a linear time invariant system is given by the convolution of the system input with the system unit impulse response.

Graphical intuition

It is often helpful to be able to visualize the computation of a convolution in terms of graphical processes. Consider the convolution of two functions $f,g$ given by

$\left(f*g\right)\left(t\right)={\int }_{-\infty }^{\infty }f\left(\tau \right)g\left(t-\tau \right)d\tau ={\int }_{-\infty }^{\infty }f\left(t-\tau \right)g\left(\tau \right)d\tau .$

The first step in graphically understanding the operation of convolution is to plot each of the functions. Next, one of the functions must be selected, and its plot reflected across the $\tau =0$ axis. For each real $t$ , that same function must be shifted left by $t$ . The product of the two resulting plots is then constructed. Finally, the area under the resulting curve is computed.

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