Lecture 28 : Discrete time Fourier transform and its Properties
Discrete Time Fourier Transform of an aperiodic discrete time signal
Given a general aperiodic signal of finite duration, that is; for some integer N, . From this aperiodic signal we can construct a periodic signal for which is one period. As we chose period N to be larger
than the duration of , is identical to .
As the period , for any finite value of n.
The Fourier series representation of is :
Since over a period that includes the interval , it is convenient to choose the interval of summation to be this period, so that can be replaced by in the summation. Therefore,
of finite duration, that is; for some integer N, 
. From this aperiodic signal we can construct a periodic signal 
for which 
, 
for any finite value of n.
, it is convenient to choose the interval of summation to be this period, so that 
