Describe in detail how the ConvolutioninTime rule provides t

Describe in detail how the Convolution-in-Time rule provides the basis for Filtering. I encourage you to draw pictures to support your claim. x(t)*z(t)=X(omega)Z(omega)

Solution

We have two functions h(t) and X(t)

We made the change t=, just made the variable change and reflecting h(t)

Now, h() moves “t” units in time, getting h(t-) that is equal to h(-(-t)), replacing this expression in the function and in the intervals we get:

The next thing to do is intercept the red function with the blue function using x ( t ) · h ( t - ) for each different intervals of time

- Firts interval of intersection

-<t<-1

There is not intersection Y(t)=0

- Second interval of intersection

-1<t<-1

We have intersection resolving the integral of convolution we have Y(t)=((t+1)^2)/(2) for -1<t<1

If you observe the result signal Y(t) is as a portion of a filtering signal

- Third interval of intersection

1<t<2

We have as result Y(t)=2t   1<t<2

-Fourth interval of intersection

2<t<4

-Fifth, the last interval

4<t<

There is not intersection so Y(t)=0 for 4<t<

The result convulotion is