 Constant function
The forward Fourier transform of a constant function x(t)=1 is
This impulse spectrum
indicates that the signal contains only the
DC component at zero frequency f=0. The inverse transform is
 Impulse function
The forward Fourier transform of an impulse function is
This constant spectrum indicates that the signal is a superposition of
sinusoids of all frequencies, which cancel each other any where along the
time axis except at t=0 where they add up to form an impulse. The inverse
transform is
 Comb function
The comb function defined as
is a function with period T, whose Fourier series coefficient is
Written as a continuous function, the comb function's spectrum is
We see that the spectrum of an impulse train with time interval T is also an
impulse train with frequency interval f_{0}=1/T. Moreover, according to the
definition of the Fourier transform, we also have
COMB(f) 
= 



= 


Therefore we have this equation
which can be compared with the equation in continuous case:
 Step function
Unit step function is defined as
and its Fourier transform is
As this integral does not converge, we first consider the Fourier
transform of an exponentially decaying step function
():
Letting
,
we get
 Square impulse
A square impulse or rectangular function of width a and height 1/a can
be considered as the difference between two unit step functions
and due to linearity, its Fourier spectrum is the difference between
the two corresponding spectra:
 Triangle function
As x(t) is an even function, its Fourier transform is
Noticing that this triangle function is the convolution of 2 rectangular
impulses rect(t) (a=1), we can get the same result more conveniently based
on the convolution theorem:

x(t)=e^{t}
Based on the scaling property, we have
 Gaussian function
The Fourier transform of a Gaussian or bellshaped function
is
Taking derivative of X(f) with respect to f, we get:
Now X(f) can be obtained as the solution of a differential equation:
which can be solved as below:
or
When f=0, we have C=X(0), which can be found as
(Note
)
We see that the Fourier transform of a bellshaped function x(t) is also a
bellshaped function X(f):
Moreover, due to the property of time and frequency scaling, we have:
(Note that if
,
then
above is a normal
distribution with variance
and mean .) If we let
,
x(t) becomes narrower and taller and approaches
,
and X(f) becomes wider and approaches constant 1. On the
other hand, if we rewrite the above as
and let
,
x(t) approaches 1 and X(f) approaches
.