Let us look at some applications of what we have just learned about
signal modulation.
So for example, voice and music are typically lowpass signals.
They don't have infinitely high frequencies because anyways,
it wouldn't be heard by the human hearing system.
Radio channels, on the other hand,
are bandpass signals because we need to modulate them high up,
otherwise there is too much interference or too much loss in transmission.
Modulation is a process of bringing a baseband signal, for example,
a voice signal into the transmission band for radio transmission.
And demodulation is the inverse or the dual of modulation,
and it will bring back the signal from a bandpass down to the basement.
So, let us look at this demodulation process.
It is simply done by multiplying the receive signal by the same carrier again.
So, we have y(n) is x(n) times cosine of omega c n.
Its spectrum, we have seen before, Y(e to the j omega) is
a combination of the two spectras shifted to omega c and -omega c.
The DTFT of y(n) multiplied by 2 cosine of (omega c n),
while it's going to be a combination of Y shifted to omega c and
to -omega c, then we replace the formula we just had before, so we have four terms.
One shifted by 2 omega c, another one by -2 omega c, and
two terms that are actually at the origin, and
so we have indeed X(e to the j omega) + 1/2 and
two modulated versions at 2 omega c and -2 omega c.