Convolution

Ther convolution of and , denoted by f*g( t ) , is given by

f*g( t ) =

Convolution is a useful concept and can be found in various places in applied mathematics since it plays an important role in for instance heat conduction, wave motion and time series analysis. Here we intend to give an interpretation of f*g by animation of the convolution product of
1) two rectangular windows where and and

2) a rectangular window with an exponential function .

> restart: alias(u=Heaviside):

Let us define the convolution product h= f*g by the integral

> h:=t->Int(f(tau)*g(t-tau),tau=0..t):

> 'h(t)'=h(t);

The convolution of and , where u(t) is the Heaviside unit step function, is

> f:=t->u(t)-u(t-1):g:=t->u(t)-u(t-2):

> 'h(t)'=h(t);

The graph of this convolution product is shown in Figure 11

> plot(value(h(t)),t=0..3,labels=[`t`, `h(t)`],color=blue);

Figure 11 The convolution product ,

,

If we move the green rectangle (window) from left to right towards the stationary red window on Figure 12,

Figure 12 Rectangular windows

the overlapping area will increase as the green window crosses into the red window. Up until t = 1, the area increases proportional to t, see Figure 11.

> assume(t >0, t<1);

> 'h(t)'=value(h(t));

When the right side of the green window is in the region , the two windows overlap completely and the value of the convolution product .

> assume(t>1,t<2):

> 'h(t)'=value(h(t));

For the overlapping area drop linearly from 1 to 0 . Hence on [2, 3] , .

> assume(t>2,t<3):

> 'h(t)'=value(h(t));

After t = 3 there is no longer overlap and

> assume(t>3):

> 'h(t)'=value(h(t));

• Convolution (f, g) animates the convolution product f*g

Figure 13 shows the animation of the convolution product of and

> t:='t':

> Convolution(f,g);

Figure 13 Animation of the convolution product f*g .

,

The counter displays the value of the overlapping area between the green and the red windows. The green disk moves on the blue convolution curve and indicate graphically the value of .

Figure 14 shows the animation of the convolution product of a rectangular window with an exponential function.

.

> g:=t->exp(-t):

> Convolution(f,g);

Figure 14 Animation of the convolution product f*g .

,

The convolution product reach a maximum when the right hand side of the red rectangle is at t = 1 and the green disk is at the top of the convolution curve h(t).