But now we should go back to the reason why we started this detour into

the modulation theorem, and

try to understand what the Gibbs phenomenon is all about.

So remember, in the beginning of our detour, we were at a point

where we had expressed the approximate filter, ĥ of n, as the product

between the ideal impulse response, h of n times an indicator function.

That serves the purpose of isolating

the points of the ideal impulse response that we want to keep.

So if this is the ideal impulse, h of n, and this is w of n, you can see that this

window basically kills all these guys here and kills all these guys here and

leaves us with an FIR approximation to the impulse response.

In the frequency domain,

this corresponds to the convolution of the Fourier Transforms of the two actors.

The Fourier Transform of the ideal impulse response is direct function

that we know very well.

The Fourier Transform of the indicator function, we have seen many times before

and it is actually the Fourier Transform of a zero-centered moving average filter.

So it's formula is here and it's sin(ω(2N + 1) /2) / sin(ω/2).

So here's we're going to try and

compute the convolution between the Fortier Transforms in a graphical way.

Here are the actors involved in the play.

We have H(e^jω), which is the Fourier Transform of the idea of filter.

We have W(e^jω), which is a Fourier Transform of the indicator function.

And here, we have the result which is the integral of the product of these two guys.

As ω moves along the axis here,

we will recenter the Fourier Transform of the indicator function,

take the product of the two, and compute the integral.

In the beginning, we're just integrating the wiggles of the function

over the non-zero part of the ideal impulse response.

So what we have is oscillatory behavior, of small amplitude.

Things start to become interesting when the main part of the Fourier Transform of

the indicator function starts to overlap with the support of the erect filter.

As we approach the transition band of the filter,

we see that the value of the convolution starts to grow.

And as a matter of fact, it grows even more with the entire main lobe

of the Fourier Transform of the window is under direct function.

As we move along, the ripples that trail the main lobe starts to get integrated and

so the behavior in the past band is again oscillatory as those

little ripples enter and exit the main integration interval.