Next: Discrete-Time Fourier Transform Up: No Title Previous: Properties of Fourier Transform

# Examples

• 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 f0=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|

 X(f) = = =

Based on the scaling property, we have

• Gaussian function

The Fourier transform of a Gaussian or bell-shaped function is

 X(f) = = =

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 bell-shaped function x(t) is also a bell-shaped 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 .

Next: Discrete-Time Fourier Transform Up: No Title Previous: Properties of Fourier Transform
Ruye Wang
2000-09-06