The point of this lesson is that knowledge of the properties of the Fourier Transform can save you a lot of work. We will cover
some of the important Fourier Transform properties here.

Linearity

Because the Fourier Transform is linear, we
can write:

F[a x_{1}(t) + bx_{2}(t)] =
aX_{1}(ω)
+ bX_{2}(ω)

where
X_{1}(ω)
is the Fourier Transform of x_{1}(t) and
X_{2}(ω)
is the Fourier Transform of x_{2}(t).

Time Scaling

if a>0. If a< 0, then

(since u=at). Therefore

Time Shifting

Duality

Note the DUALITY when you compare
Examples 1 and 6 from Lesson 15.

Example 1 of Lesson 15 showed that the
Fourier Transform of a block (or rect)
function in time is a sinc in frequency. Example 6 of Lesson 15
showed that the Fourier Transform of a
sinc function in time is a block (or rect) function in frequency.

In general, the Duality property is very useful
because it can enable to solve Fourier Transforms that would be difficult
to compute directly (such as taking the Fourier Transform of a sinc
function). The Duality Property
tells us that if x(t) has a Fourier Transform
X(ω),
then if we form a new function of time that has the functional form of
the transform,
X(t),
it will have a Fourier Transform
x(ω)
that has the
functional form of the original time function (but is a function of
frequency).
Mathematically, we can write:

Notice that the second term in the last line is
simply the Fourier Transform integral of the function
X(t), i.e.

Therefore we get the Duality Property:

Example 1
Using the Fourier Transform integral equation, directly find
the Fourier Transform of

x(t) = e^{-at}u(t), a> 0

Example 2 Using the results of
Example 1 and the Duality Property, find the Fourier Transform of