Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE) PDF Download

epresentation of Discrete periodic signal.
A periodic discrete time signal x[n] with period N can be represented as a Fourier series:

Here the summation ranges over any consecutive N integers of x[n],
where N is the period of the discrete time signal x[n].
Here equation
(i) is called the Synthesis Equation and equation
(ii) is called the Analysis Equation.
Now since x[n] is periodic with period N; the Fourier series coefficients are related as;

a_k = a_k+N
 

Discrete Time Fourier Transform of an aperiodic discrete time signal 

Given a general aperiodic signal  X[n] of finite duration, that is; for some integer N, X[n]=0 if |n|>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,

X(ω) is the angular representation of the Discrete time Fourier Transform (DFTF) of the signal x[n].

 Another way of representing DTFT of a periodic discrete signal
In continuous time, the fourier transform of    is an impulse at    .However in discrete time ,for signal   the discrete time fourier transform is periodic in   with period 2π  . The DTFT of  is a train of impulses at    i.e Fourier Transform can be written as 

Consider a periodic sequence x[n] with period N and with fourier series representation

Then discrete time Fourier Transform of a periodic signal x[n] with period N can be written as :

Properties of DTFT Periodicity:

Linearity:
The DTFT is linear.
If

Stability:

The DTFT is an unstable system i.e the input x[n] gives an unbounded output.
Example :

 If x[n] = 1 for all n
then DTFT diverges i.e Unbounded output. 

Time Shifting and Frequency Shifting:
 If,

Time and Frequency Scaling: 

Time reversal

Let us find the DTFT of x[-n]

Time expansion:

It is very difficult for us to define x[an] when a is not an integer. However if a is an integer other than 1 or -1 then the original signal is not just speeded up. Since n can take only integer values, the resulting signal consists of samples of x[n] at an.

If k is a positive integer, and we define the signal

Convolution Property : 

Let h[n] be the impulse response of a discrete time LSI system. Then the frequency response of the LSI system is

Proof:

now put n-k =m, for fixed k, 
 

This is a very useful result.

Symmetry Property:

If

Furthermore if x[n] is real then,

The DTFT of Cross-Correlation Sequence between x[n] and h[n]

If the DTFT of the cross correlation sequence between x[n] and h[n] exists then,

Conclusion: 

In this lecture you have learnt:

• For a Discrete Time Periodic Signal the Fourier Coefficients are related as .
• DTFT is unstable which means that for a bounded 'x[n]' it gives an unbounded output.
• We saw its time shifting & frequency shifting properties & also time scaling & frequency scaling.
• Convolution Property for an LSI system is given as, if 'x[n]' is the input to a system with transfer function 'h[n] then the DTFT of the output 'y[n]' is the multiplication of the DTFTs of 'x[n]' and 'h[n]'.
• We saw symmetry properties and DTFT of cross-correlation between 'x[n]' and 'h[n]' .