Lecture 5: Representation of Nonperiodic Functions
We have seen how we can represent periodic functions in terms of sines and cosines. This is a natural thing to contemplate (at least retrospectively). What if we want to represent nonperiodic functions in terms of periodic functions? This seems kind of strange at first. Why would we want to? Part of the answer comes because we know how LTI systems respond to sines and cosines. It turns out to be closely related to sinusoidal steadystate analysis of circuits and provide some useful tools for design of signal processing and communication systems.
To represent a nonperiodic function in terms of periodic functions, one thing we will need to to use periodic functions that might have infinite period. It will also turn out to add up a whole lot of them: more than a countable infinite. Here is one approach to the problem.
Suppose we have a function
. We construct a periodic extension
of
called
, where
is repeated every
seconds. Since
is periodic, it has a Fourier series
where
and . The integral is the same (since is zero outside of ) as
Now let's define the function
Comparing these, we can see that
So we get the F.S. coefficients by sampling the function . In the limit as , we sample infinitely often.
Now the F.S. for
is
Let and write
Now in the limit as we get
So we have been able to reconstruct our nonperiodic function from . These are important enough they bear repeating:
This is called the Fourier Transform of . The inverse Fourier transform is
We also write
or
This is notation similar to what we used for Laplace transforms.
Some things to notice:
 Notice that to get we are doing the same kind of thing we did for F.S. except that we add up a continuum of frequencies.

Suppose that
is causal. In the Laplace transform, set
Then we get the Laplace transform back! But notice
that the integral goes from
to
. So we explicitly
don't assume that
is causal. We thus do not typically use
the F.T. to examine things like transient response: use the Laplace
transform for that.
As will be mentioned below, however, the F.T. cannot be used for unstable signals, since the integral does not exist in this case. The F.T. is not usually used to examine stability, as the L.T. can.
 Notice again that in the forward transform, a negative is used in the exponent.

The frequency variable in the F.T. is
, which is
frequency in radians/sec. If we make the change of variable
and find the transform in terms of
we get the
following transform pair:
As we did for F.S., we can plot the magnitude and phase of the F.T.
We can write
Notice that if is real, then (they are complex conjugates) so that
or, in other words, if is a real function, then the magnitude function is even and the phase function is odd.