Example Find the Fourier Transform of the constant 1. Use duality and the fact that
the transform of δ(t) is 1

Fourier Transform of Periodic Signals

We can also
define a Fourier Transform for periodic signals. If a
signal has both periodic and aperiodic components, then this will enable
us to use one transform to deal with both the periodic and aperiodic
components.

Can we take the Fourier Transform of a periodic
signal? For example, we can write:

but we can not directly calculate this integral because it does not
converge.
So we will
generalize the Fourier Transform to include impulses in the
frequency domain.

We can use either the Duality or Modulation
Properties to show that:

We can check
this by taking Inverse Fourier Transform (and using the
sifting property):

Now let's consider a
general periodic signal x(t) that we can represent
as a Fourier Series:

Since we know that:

by linearity of the Fourier Transform,
we get that:

Thus
we see that a periodic signal has a Fourier Transform that is
an infinite impulse train at discrete frequencies kω_{0} with weights of
2πC_{k} (many or most of the weights are usually 0).

Example Find the Fourier Transform of
cos(ω_{0}t).

Pulsed Cosine

Example Find the Fourier Transform of the pulsed cosine:

You can do it directly and also using the multiplication property.