Given that an LTI system has the relationship
y(t) = x(t) * h(t) where x(t) is the input,
y(t) is the output, and h(t) is the impulse response of the system, we
know that Y(ω) = X(ω) H(ω).
H(ω) is a filter to eliminate unwanted
parts of the frequency spectrum. There are a number of different types of filters including ideal
low pass, high pass, band pass, and band stop, which
are plotted here. These
ideal filters have Fourier Transforms of the form
rect(ω/2ω_{c}), where ω_{c} is the cutoff frequency.
The corresponding time domain functions are sincs.

Example 1 Given two functions
x_{1}(t) = cos(500πt) and x_{2}(t) =
cos(1000πt), form a new signal x_{3}(t) =
x_{1}(t) x_{2}(t).

Draw the frequency response of a low pass filter that passes the low
frequency component of the signal and blocks the high frequency component of the signal.

Sinusoidal Amplitude Modulation

Modulation is multiplying a signal in time by
complex exponentials (such as sine waves and cosine waves) to shift the
signal to a desired frequency band. This is done, for example, by radio
(or television) to assign
different parts of the frequency spectrum to different radio (or television) stations. Remember that the Modulation Theorem told us that:

We will use the Modulation Theorem in an example of AM radio.

Example 2 AM radio - Double-sideband, suppressed carrier,
amplitude modulation

Let x(t) be a music signal with Fourier Transform
X(ω).

To modulate this signal, we'll form and transmit

y(t) = [x(t) + B] cos(ω_{c}t + φ)

where ω_{c} is the
carrier frequency of the radio station (e.g. 770 kHz), and
B is the carrier. Let B=0 which means that we
have a suppressed carrier
and let φ = 0 for simplification.
In this case, y(t) = x(t)
cos(ω_{c}t).

Then

Now, let's look at the Fourier Transform
of the modulated signal
y(t).
Y(ω) = ?

Your radio demodulates the signal
by multiplying the received signal y(t) by another
cosine wave to form a third signal z(t):

To recover
the original music signal
x(t)
from the
demodulated signal z(t),
you filter z(t) with a low-pass
filter
by multiplying by
a rect function in frequency.
This corresponds to convolving with a sinc
function in time.