Next, we'll derive the FT of some basic continuoustime signals. Table 1 summarizes these transform pairs.
Rectangular pulse
Let's begin with the rectangular pulse
The pulse function,
A plot of
Note that when
Impulse
The unit impulse function was described in a previous section. From the sifting property of the impulse function we find that
or
Complex Exponential
The complex exponential function,
This result follows from the sifting property of the impulse function. By linearity, we can then write
Cosine
The cosine signal can be expressed in terms of complex exponentials using Euler's Identity
Applying linearity and the Fourier transform of complex exponentials to the right side of Equation 8, we quickly get:
Real Exponential
The real exponential function is given by
therefore,
The Unit Step Function
In a previous section, we looked at the unit step function,
A closely related signal is the signum function, defined by
from which it follows that
The signum function can be described as follows:
Since we already have the Fourier transform of the exponential signal,
Using Equation 14 and linearity then leads to





1 








When working problems involving finding the Fourier transform, it is often preferable to use a table of transform pairs rather than to recalculate the Fourier transform from scratch. Often, transform pairs in can be combined with known Fourier transform properties to find new Fourier transforms.
Example 3.1 Find the Fourier transform of:
with the time reversal property:
to get the answer: