So now that we have the ideal filters in place,

why don't we revisit the demodulation problem that you remember,

we left somewhat open a few lectures ago.

You remember the story, we applied sinusoidal modulation to an input signal

x(n) to obtain y(n), which is equal to x(n) times the cosine of omega 0n.

And we try to demodulate the modulated signal by multiplying

the modulated signal again by the carrier cosine omega 0n.

And we found out that the demodulated signal contains unwanted

high-frequency components that, at the time, we did not know how to remove.

Well, now we do, so let's look again at what happens.

So we have the spectrum of the signal.

We modulate it, and we obtain two copies half the amplitude,

at omega zero and minus omega zero.

When we multiply this signal by cosine of omega zero again, and we can see this

more in detail, if we explicitly show the periodicity of the spectrums.

So here you have the main minus pi to pi interval, and

here you have the repetition.

So we multiply that by cosine of omega 0 and

we get one copy of the repeated spectrum moved to the right by omega 0.

So you see here in green whereas with the dash line, you see the original modulated

spectrum, and the second component would be another copy of the modulated spectrum,

this time shifted to the left by omega zero.

And so, you have the two copies that now you have to sum together and what you get

is, of course, the reconstruction of the original signal centered around zero.

And these spurious components at high frequency.

So if we go back to the standard visualization of the spectrum over

a single period from minus pi to pi, this is the situation we have to deal with.

Well now we know what we need to do at this point is to eliminate this

the spurious components, and we can do that easily by using a low pass filter.

So, we put a low pass filter here, choosing a cutoff frequency omega c

that is large enough to include the main signal, the base band signal,

how it's called, and small enough in order to kill the high frequency components.

And when we do that,

we obtain the original signal back without any extraneous material.

4.5.d Demodulation revisited