# Lecture 5: Signal Distortion During Transmission

Fourier transforms form a natural way of looking at the frequency response of filters, since they examine the response due to a a trigonometric exponential. We have already seen how to characterize a filter by its frequency response, using the Laplace transform. We can do the same thing (somewhat more naturally) using Fourier transforms.

The output of a signal with transfer function
is

Let us express this in polar form:

The magnitude response of the output is therefore

so the magnitude response is the input magnitude response ``shaped'' by the system magnitude response. In a word, the output is

*filtered.*We can also consider the phase response:

The phase angle of the output is the phase angle of the input plus whatever phase the system contributes.

There are applications in which we would like the system to not change
the input signal as it passes through the system, other than to delay
the signal and possibly uniformly attenuating the whole signal. For
example, the signals that pass through the air to our radios: we would
hope that they would arrive at the radio in the same form as they left
the transmitter, except delayed by the propagation through the air and
attenuated. A system which does not distort a signal (other than
delaying it and possibly changing the overall gain) is said to be a
*
distortionless system
*
. The output of a distortionless system may
be written as

where is the overall gain and is the delay introduced by the system. In terms of Fourier transforms, we have

So the transfer function of a distortionless system is

Observe that the magnitude does not change with frequency, and the phase is a linear function of :

In practice, distortionless transmission is never exactly obtained
over all frequencies. Over a localized region of frequency, however,
the magnitude response may be flat enough, and the phase response may
be linear enough, so the distortion is acceptable.
For any system, including those with nonlinear phase,
we can compute the time delay of a signal as it passes through a
system by