Lecture 5: Properties of Fourier Transforms
We saw earlier a variety of properties associated with the Laplace transform: linearity, time shift, convolution, differentiation, and integration. These opened up a variety of applications. There are similar the properties of F.T. which are also very useful.
 Linearity

The F.T. is linear:
This follows from the linearity of the integral.  Time/Frequency Duality

The duality property is one that is not
shared by the Laplace transform. While slightly confusing perhaps
at first, it essentially doubles the size of our F.T. table. The
duality property follows from the similarity of the forward and
inverse F.T. It states that if
then
where the function on the left is the function of time and the function on the right is the function of frequency.
In terms of the alternative notation (in Hz), we have that if
then
 Scaling Property

We have seen this for Laplace transforms: If
then
Discuss in terms of time and bandwidth: when a signal is expanded in time, it is compressed in frequency, and vice versa. We cannot be simultaneously short in time and short in frequency. This is the basis for the famed Heisenberg uncertainty principle .  TimeShift Property

If
then
In other words, a shift in time corresponds to a change in phase in the F.T.
What is interesting is that higher frequencies experience a greater phase shift than lower frequencies. A plot of the phase is linear in frequency.
 Frequencyshift Property

This innocuouslooking property forms
a basis for every radio and TV transmitter in the world! It simply
states that if
then
Note that this is a dual of the timeshift property.
Show a block diagram of what happens. Since is a complex signal, this is really only a mathematical description. But we can multiply by . Using linearity we get
What we get out is two images in the frequency domain, at positive and negative frequencies.As an application, suppose that is some informationbearing signal, say, the signal from a microphone. Plot. Now multiply by and plot the result, and show the effect in frequency. This is what KSL does! This kind of modulation is called amplitude modulation .
Observe that we can modulate signals onto a variety of different frequencies. This makes it possible to have several radio (or TV) stations. The higher frequencies also propagate further through the air than the baseband frequencies.
 Convolution Property

If
then
(where is convolution) and
 Time Differentiation

If
then
 Time Integration

We saw above that
If is zero mean (i.e. ) then