EE511 Day 5 Class Notes
Fourier Transform Continued
Laurence Hassebrook
Updated 9-8-03
Monday 9-8-03
4. Convolution in the time domain
Prove:

w1 t   w2 t    w1  w2 t   d  W1  f W2  f 

In order to solve this we need to introduce the Dirac Delta Function (t). The definition is:

  xdx  1

where
 x  0 for x  0
The Dirac Delta Function is used to evaluate integrals at specific time instances. For example: the
“unit impulse response” is obtained by

ht    h    t d

The above integral is evaluated when =t or the argument of the delta function is zero. At that time
all the other function in the integral that are a function of the variable integration, are sampled and
the integral is eliminated.
Now to prove that convolution in time is multiplication in frequency, we will start with the FT side
of the relationship. We will write the following equality:

w1 t  w2 t    W1  f W2  f exp  j 2ft df

By representing the FT of the integrand, we have
w1 t   w2 t   



 

w1  exp  j 2f d  w2  exp  j 2f d exp  j 2ft df

The integral over f is consolidated the integrals over  and  are moved to the outside s.t.
 w  w  
1
2


exp  j 2f    exp  j 2ft df d d
1
The inside integral is the inverse FT of a linear phase term which is a Dirac delta function.
2