0. Chapter 8: THE DESCRETE FOURIER TRANSFORM
Department of Computer Eng.
Sharif University of Technology
Discrete-time signal processing
Chapter 8: THE DESCRETE FOURIER TRANSFORM
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer and Buck, © Prentice Hall Inc.

1. Chapter 8: The Discrete Fourier Transform
8.0 Introduction
DSP
For finite-duration sequences, it is possible to develop an alternative Fourier representation referred to as the discrete Fourier transform(DFT).
It is a sequence rather than a continuous variable.
It corresponds to samples, equally spaced in frequency, of the Fourier transform of the signal.
The importance of DFT is because efficient algorithms exists for the computation of the DFT (FFT).
Chapter 8: The Discrete Fourier Transform 1

2. 8.1 Discrete Fourier Series
DSP
Given a periodic sequence with period N so that
The Fourier series representation of continuous-time periodic signals require infinite complex exponentials.
But for discrete-time periodic signals we have
Due to the periodicity of the complex exponential we only need N exponentials for discrete time Fourier series
Chapter 8: The Discrete Fourier Transform 2

3. Discrete Fourier Series Pair
DSP
A periodic sequence in terms of Fourier series coefficients
The Fourier series coefficients can be obtained via
For convenience we sometimes use
Analysis equation
Synthesis equation
Chapter 8: The Discrete Fourier Transform 3

4. Chapter 8: The Discrete Fourier Transform
Example 1 1
DSP
DFS of a periodic impulse train
Since the period of the signal is N
We can represent the signal with the DFS coefficients as
Chapter 8: The Discrete Fourier Transform 4

5. Example 2: Duality in the DFT
DSP
Here let the Fourier series coefficients be the periodic impulse train Y[k] , and given by this equation:
Substituting Y[k] in to DFS equation gives
Comparing this result with the results for example1 we see that Y[k]=Nx[k] and y[n]=X[n].
Chapter 8: The Discrete Fourier Transform 5

6. Chapter 8: The Discrete Fourier Transform
Example 3
DSP
DFS of an periodic rectangular pulse train
The DFS coefficients
Chapter 8: The Discrete Fourier Transform 6

7. Chapter 8: The Discrete Fourier Transform
8.2 Properties of DFS
DSP
Linearity (all signals have the same period)
Shift of a Sequence
Duality
Chapter 8: The Discrete Fourier Transform 7

8. Chapter 8: The Discrete Fourier Transform
Periodic Convolution
DSP
Take two periodic sequences
Let’s form the product
The periodic sequence with given DFS can be written as
Periodic convolution is commutative
Chapter 8: The Discrete Fourier Transform 8

9. Periodic Convolution Cont’d
DSP
Substitute periodic convolution into the DFS equation
Interchange summations
The inner sum is the DFS of shifted sequence
Substituting
Chapter 8: The Discrete Fourier Transform 9

10. Graphical Periodic Convolution
DSP
Chapter 8: The Discrete Fourier Transform 10

11. Chapter 8: The Discrete Fourier Transform
Symmetry Properties
DSP
Chapter 8: The Discrete Fourier Transform 11

12. Symmetry Properties Cont’d
DSP
Chapter 8: The Discrete Fourier Transform 12

13. 8.3 The Fourier Transform of Periodic Signals
DSP
Periodic sequences are not absolute or square summable
Hence they don’t have a Fourier Transform
We can represent them as sums of complex exponentials: DFS
We can combine DFS and Fourier transform
Fourier transform of periodic sequences
Periodic impulse train with values proportional to DFS coefficients
This is periodic with 2 since DFS is periodic
The inverse transform can be written as
Chapter 8: The Discrete Fourier Transform 13

14. Chapter 8: The Discrete Fourier Transform
Example
DSP
Consider the periodic impulse train
The DFS was calculated previously to be
Therefore the Fourier transform is
Chapter 8: The Discrete Fourier Transform 14

15. Relation between Finite-length and Periodic Signals
DSP
Consider finite length signal x[n] spanning from 0 to N-1
Convolve with periodic impulse train
The Fourier transform of the periodic sequence is
This implies that
DFS coefficients of a periodic signal can be thought as equally spaced samples of the Fourier transform of one period 15

16. Chapter 8: The Discrete Fourier Transform
Example
DSP
Consider the following sequence
The Fourier transform
The DFS coefficients
Chapter 8: The Discrete Fourier Transform 16

17. 8.4 Sampling the Fourier Transform
DSP
Consider an aperiodic sequence with a Fourier transform
Assume that a sequence is obtained by sampling the DTFT
Since the DTFT is periodic resulting sequence is also periodic
We can also write it in terms of the z-transform
The sampling points are shown in figure
could be the DFS of a sequence
Write the corresponding sequence
Chapter 8: The Discrete Fourier Transform 17

18. Sampling the Fourier Transform Cont’d
DSP
The only assumption made on the sequence is that DTFT exist
Combine equation to get
Term in the parenthesis is
So we get
Chapter 8: The Discrete Fourier Transform 18

19. Sampling the Fourier Transform Cont’d
DSP
Chapter 8: The Discrete Fourier Transform 19

20. Sampling the Fourier Transform Cont’d
DSP
Samples of the DTFT of an aperiodic sequence can be thought of as DFS coefficients of a periodic sequence obtained through summing periodic replicas of original sequence.
If the original sequence is of finite length and we take sufficient number of samples of its DTFT the original sequence can be recovered by
It is not necessary to know the DTFT at all frequencies To recover the discrete-time sequence in time domain
Discrete Fourier Transform representing a finite length sequence by samples of DTFT
Chapter 8: The Discrete Fourier Transform 20

21. 8.5 The Discrete Fourier Transform
DSP
Consider a finite length sequence x[n] of length N
For given length-N sequence associate a periodic sequence
The DFS coefficients of the periodic sequence are samples of the DTFT of x[n]
Since x[n] is of length N there is no overlap between terms of x[n-rN] and we can write the periodic sequence as
To maintain duality between time and frequency
We choose one period of as the Fourier transform of x[n]
Chapter 8: The Discrete Fourier Transform 21

22. The Discrete Fourier Transform Cont’d
DSP
The DFS pair
The equations involve only on period so we can write
Chapter 8: The Discrete Fourier Transform 22

23. The Discrete Fourier Transform Cont’d
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The DFT pair can also be written as
The Discrete Fourier Transform
Chapter 8: The Discrete Fourier Transform 23

24. Chapter 8: The Discrete Fourier Transform
Example
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The DFT of a rectangular pulse
x[n] is of length 5
We can consider x[n] of any length greater than 5
Let’s pick N=5
Calculate the DFS of the periodic form of x[n]
Chapter 8: The Discrete Fourier Transform 24

25. Chapter 8: The Discrete Fourier Transform
Example Cont’d
DSP
If we consider x[n] of length 10
We get a different set of DFT coefficients
Still samples of the DTFT but in different places
Chapter 8: The Discrete Fourier Transform 25

26. Chapter 8: The Discrete Fourier Transform
8.6 Properties of DFT
DSP
Linearity
Duality
Circular Shift of a Sequence
Chapter 8: The Discrete Fourier Transform 26

27. Chapter 8: The Discrete Fourier Transform
Example: Duality
DSP
Chapter 8: The Discrete Fourier Transform 27

28. Chapter 8: The Discrete Fourier Transform
Symmetry Properties
DSP
Chapter 8: The Discrete Fourier Transform 28

29. Chapter 8: The Discrete Fourier Transform
Circular Convolution
DSP
Circular convolution of two finite length sequences
Chapter 8: The Discrete Fourier Transform 29

30. Chapter 8: The Discrete Fourier Transform
Example
DSP
Circular convolution of two rectangular pulses L=N=6
DFT of each sequence
Multiplication of DFTs
And the inverse DFT
Chapter 8: The Discrete Fourier Transform 30

31. Chapter 8: The Discrete Fourier Transform
Example
DSP
We can augment zeros to each sequence L=2N=12
The DFT of each sequence
Multiplication of DFTs
Chapter 8: The Discrete Fourier Transform 31

32. 8.7 Linier convolution using the DFT
DSP
Efficient algorithms are available for computing the DFT of finite-duration sequence, therefore it is computationally efficient to implement a convolution of two sequences by the following procedure:
Compute the N point discrete Fourier transforms X1[k] and X2[k] for the two sequences given.
Compute the product X3[k]=X1[k]X2[k].
Compute the inverse DFT of X3[k].
The multiplication of discrete Fourier transforms corresponds to a circular convolution. To obtain a linear convolution, we must ensure that circular convolution has the effect of linear convolution.
Chapter 8: The Discrete Fourier Transform 32

33. Linear convolution of two finite-length sequences
DSP
Chapter 8: The Discrete Fourier Transform 33

34. Circular convolution as linear convolution
DSP
With aliasing
Without aliasing
Chapter 8: The Discrete Fourier Transform 34

35. DSP

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37. DSP

38. Implementing LTI systems using the DFT
DSP
Let us consider an L point input sequence x[n] and a p point impulse response h[n].
The linear convolution has finite-duration with length L+P-1.
consequently for linear convolution and circular convolution to be identical, the circular convolution must have the length of at least L+p-1 points.
i.e. both x[n] and h[n] must be augmented with sequence amplitude with zero amplitude.
This process is often referred to as zero-padding.
Chapter 8: The Discrete Fourier Transform 38

39. Implementing LTI systems using the DFT cont’d
DSP
In many applications the input signal is an indefinite duration.
We have to store all the input data.
No filtered samples calculated until all the input samples have been collected.
Implementing FFT for such large number of points is impractical.
The solution is to use block convolution.
In this method signal is segmented into sections and each section can then be convolved with the finite length impulse response and the filtered sections fitted together in an appropriate way.
Chapter 8: The Discrete Fourier Transform 39

40. Implementing LTI systems using the DFT cont’d
DSP
Chapter 8: The Discrete Fourier Transform 40

41. Chapter 8: The Discrete Fourier Transform
Overlap-add method
DSP
Chapter 8: The Discrete Fourier Transform 41

42. Chapter 8: The Discrete Fourier Transform
Overlap-add method
DSP
xr[n] has L nonzero points and h[n] is of length P, each of yr[n] has length L+P-1.
A method to implement linear convolution is the nonzero points in the filtered sections will overlap by P-1 points and these overlap samples must be added to compute linear convolution.
This method is called overlap-add method.
Chapter 8: The Discrete Fourier Transform 42

43. Overlap-Save Method
DSP

44. Chapter 8: The Discrete Fourier Transform
Overlap-save method
DSP
It corresponds to implementing an L-point circular convolution of a P- point impulse response h[n] with an L-point segment xr[n] and identifying the part of singular convolution that corresponds to linear convolution.
We showed that if an L-point sequence is circularly convolved with a P-point sequence (P<L) then the first P-1 point of result are incorrect.
This method is called overlap-save method.
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