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;
ak = ak+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:
41 videos|52 docs|33 tests
|
1. What is the Discrete Time Fourier Transform (DTFT)? |
2. How is the Discrete Time Fourier Transform (DTFT) different from the Discrete Fourier Transform (DFT)? |
3. What are the properties of the Discrete Time Fourier Transform (DTFT)? |
4. How is the Discrete Time Fourier Transform (DTFT) computed? |
5. What are the applications of the Discrete Time Fourier Transform (DTFT)? |
|
Explore Courses for Electrical Engineering (EE) exam
|