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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:

 

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

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 Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE)  for which Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE)  is one period. As we chose period N to be larger than the duration of  Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE) ,  is identical to Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE) . As the period  Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE)  for any finite value of n.
The Fourier series representation of Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE)  is :

 

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

Since Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE)   over a period that includes the interval Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE)  , it is convenient to choose the interval of summation to be this period, so that  Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE) can be replaced by Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE) in the summation. Therefore,

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

 

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

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  Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE)  is an impulse at  Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE)  .However in discrete time ,for signal  Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE) the discrete time fourier transform is periodic in  Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE) with period 2π  . The DTFT of Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE) is a train of impulses at  Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE)  i.e Fourier Transform can be written as 

 

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

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

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

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

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

Properties of DTFT Periodicity:

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

Linearity:
The DTFT is linear.
If

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

 

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,

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

 

Time and Frequency Scaling: 

Time reversal

Let us find the DTFT of x[-n]

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

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

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

 

Convolution Property : 

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

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

 

Proof:

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

now put n-k =m, for fixed k, 
 

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

This is a very useful result.


Symmetry Property:


If

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

Furthermore if x[n] is real then,

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

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,

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

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]' .
The document Discrete Time Fourier Transform & Its Properties | Signals and Systems - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Signals and Systems.
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FAQs on Discrete Time Fourier Transform & Its Properties - Signals and Systems - Electrical Engineering (EE)

1. What is the Discrete Time Fourier Transform (DTFT)?
Ans. The Discrete Time Fourier Transform (DTFT) is a mathematical technique used to analyze the frequency content of discrete-time signals. It transforms a discrete-time signal from the time domain into the frequency domain, providing information about the amplitudes and phases of its constituent frequencies.
2. How is the Discrete Time Fourier Transform (DTFT) different from the Discrete Fourier Transform (DFT)?
Ans. The Discrete Time Fourier Transform (DTFT) is a continuous function of frequency, while the Discrete Fourier Transform (DFT) is a discrete function of frequency. The DTFT is defined for continuous frequencies, whereas the DFT is defined for discrete frequencies. Additionally, the DTFT assumes an infinite duration signal, while the DFT assumes a finite duration signal.
3. What are the properties of the Discrete Time Fourier Transform (DTFT)?
Ans. The Discrete Time Fourier Transform (DTFT) has several important properties, including linearity, time shifting, frequency shifting, convolution, and duality. Linearity means that the transform of a linear combination of signals is equal to the linear combination of their individual transforms. Time shifting involves shifting a signal in the time domain, which results in a phase shift in the frequency domain. Frequency shifting involves shifting a signal in the frequency domain, which results in a time shift in the time domain. Convolution in the time domain corresponds to multiplication in the frequency domain. Lastly, duality refers to the relationship between the DTFT of a signal and the DTFT of its inverse in the time domain.
4. How is the Discrete Time Fourier Transform (DTFT) computed?
Ans. The Discrete Time Fourier Transform (DTFT) can be computed using the formula: X(e^jω) = Σ[x[n] * e^(-jωn)], where X(e^jω) represents the DTFT of the discrete-time signal x[n], ω represents the angular frequency, and n represents the discrete time index. This formula calculates the sum of the signal multiplied by a complex exponential for each discrete time index.
5. What are the applications of the Discrete Time Fourier Transform (DTFT)?
Ans. The Discrete Time Fourier Transform (DTFT) has various applications in signal processing and communication systems. It is used for signal analysis, such as determining the frequency content of a signal, identifying harmonics, and detecting periodicity. The DTFT is also essential in system analysis and design, such as filter design and equalization. Additionally, it is used in image processing for tasks like image enhancement and compression.
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