In this lesson, we will cover additional properties of the Fourier
Transform. The most useful one is the Convolution Property. It tells us
that convolution in time corresponds to multiplication in the frequency
domain. Therefore, we can avoid doing convolution by taking Fourier
Transforms! In many cases, this will be much more convenient than
directly performing the convolution.

The Convolution Property

The convolution property states that:
Let us show that. To start, let
Then we can take the Fourier Transform of y(t) and
plug in the convolution integral for y(t) (notice how
we've marked the integrals with dt and dτ to keep
track of them):
Now, let's switch the order of the two integrals (this is valid
in all but the most pathological cases):

Therefore,

We've just shown that the Fourier Transform
of the convolution of two functions is simply the product of the Fourier
Transforms of the functions. This means that for linear, time-invariant
systems, where the input/output relationship is described by a
convolution, you can avoid convolution by using Fourier Transforms.
This is a very powerful result.

Multiplication of Signals

Our next property is the Multiplication Property. It states that the
Fourier Transform of the product of two signals in time is the
convolution of the two Fourier Transforms.

Now, write x_{1} (t) as an inverse Fourier Transform.

Therefore,

Example 1 Find the inverse Fourier Transform of

Here is a plot of this function:

Example 2 Find the Fourier Transform of
x(t) = sinc^{2}(t) (Hint: use the
Multiplication Property).

Example 3 Find the Fourier Transform of
y(t) = sinc^{2}(t) * sinc(t).
Use
the Convolution Property (and the results of Examples 1 and 2) to solve
this Example.

Frequency Shifting or Modulation

This tells us that modulation (such as
multiplication in time by a complex exponential, cosine wave, or sine
wave) corresponds to a frequency shift in the frequency domain.