1. CE 40763 Digital Signal Processing Fall 1992 Discrete Fourier Transform (DFT)
Hossein Sameti
Department of Computer Engineering
Sharif University of Technology

2. Motivation
The DTFT is defined using an infinite sum over a discrete time signal and yields a continuous function X(ω)
not very useful because the outcome cannot be stored on a PC.
Now introduce the Discrete Fourier Transform (DFT), which is discrete and can be stored on a PC.
We will show that the DFT yields a sampled version of the DTFT.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

3. Review of Transforms C. C. D. C. D. Complex Inf. or Finite Int. D.
periodic
periodic D.
finite
finite
Int.

4. Review of Transforms

5. Discrete Fourier Series (DFS)
Decompose in terms of complex exponentials that are periodic with period N.
How many exponentials? N
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

6. Discrete Fourier Series (DFS)
Exponentials that are periodic with period N.
arbitrary integer 1
* Proof:

7. Discrete Fourier Series (DFS)
How to find X(k)?
Answer:
Proof : substitute X(k) in the first equation.
It can also easily be shown that X(k) is periodic with period N:
arbitrary integer
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

8. DFS Pairs Analysis: Synthesis: Periodic N pt. seq. in time domain
seq. in freq. domain
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

9. Example … …
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

10. Example (cont.) (eq.1) (eq.2) (eq.1) & (eq.2)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

11. Properties of DFS Shift property: Periodic convolution: Period N
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

12. Periodic convolution - Example
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

13. Properties of DFS In the list of properties: 𝑊 𝑁 = 𝑒 −𝑗2𝜋 𝑁
𝑊 𝑁 = 𝑒 −𝑗2𝜋 𝑁
𝑊 𝑁 𝑘𝑛 = 𝑒 𝑘 (𝑛)
Where:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

14. Properties of DFS
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

15. Properties of DFS
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

16. Discrete Fourier Transform
N pt.
N pt.
DFT
N pt.
N pt.
DTFT
DFT
DFS
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

17. Deriving DFT from DFS
1) Start with a finite-length seq. x(n) with N points (n=0,1,…, N-1).
2) Make x(n) periodic with period N to get
Extracts one period of
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

18. Deriving DFT from DFS (cont.)
3) Take DFS of
4) Take one period of to get DFT of x(n):
N pt.
N pt.
periodic
N pt.
periodic
N pt.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

19. Example

20. Discrete Fourier Transform
Definition of DFT:
N pt. DFT of x(n)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

21. Example

22. Example, cont’d
Mehrdad Fatourechi, Electrical and Computer Engineering, University of British Columbia, Summer 2011

23. Relationship between DFT and DTFT
DFT thus consists of equally-spaced samples of DTFT.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

24. Relationship between DFT and DTFT
8 pt. sequence
8 pt. DFT
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

25. M pt. DFT of N pt. Signal
So far we calculated the N pt. DFT of a seq. x(n) with N non-zero values:
Suppose we pad this N pt. seq. with (M-N) zeros to get a sequence with length M.
We can now take an M-pt. DFT of the signal x(n)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

26. M pt. DFT of N pt. Signal DFT N pt. M pt.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

27. Example 4 pt. DFT: 6 pt. DFT: How are these related to each other?

28. M pt. DFT of N pt. Signal Going from N pt. to 2N pt. DFT
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

29. M pt. DFT of N pt. Signal N pt. DFT 2N pt. DFT N pt. seq. N pt. 2N pt.
padded with N zeros
What is the minimum number of N needed to recover x(n)?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

30. Problem Statement
Assume y(n) is a signal of finite or infinite extent.
Sample at N equally-spaced points.
N pt. sequence.
N pt. sequence.
What is the relationship between x(n) and y(n)?
What happens if N is larger , equal or less than the length of y(n)?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

31. Solution to the Problem Statement
We start with x(n) and find its relationship with y(n):
Change the order of summation:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

32. Solution to the Problem Statement
However, we have shown that:
Convolution with train of delta functions
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

33. Solution to the Problem Statement
One period of the replicated version of y(n)
Examples
If we sample at a sampling rate that is higher than the number of points in y(n), we should be able to recover y(n).
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

34. Properties of DFT Shift property: N pt. seq.
The above relationship is not correct, because of the definition of DFT.
The signal should only be non-zero for the first N points.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

35. Properties of DFT In the list of properties: 𝑊 𝑁 = 𝑒 𝑗2𝜋 𝑁 ,
𝑊 𝑁 = 𝑒 𝑗2𝜋 𝑁 ,
𝑊 𝑁 𝑘𝑛 = 𝑒 𝑘 (𝑛)
where:
and
where:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

36. Summery of Properties of DFT
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

37. Summery of Properties of DFT
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

38. Using DFT to calculate linear convolution
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

39. Convolution We are familiar with, “linear convolution”.
Question: Can DFT be used for calculating the linear convolution?
The answer is: NO! (at least not in its current format)
We now examine how DFT can be applied in order to calculate linear convolution.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

40. Definitions of convolution
Linear convolution:
Application in the analysis of LTI systems
Periodic convolution:
A seq. with period N
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

41. Definitions of convolution(cont.)
Circular convolution:
N pt. seq.
Circular convolution is closely related to periodic convolution.
N pt. DFT of x3
N pt. DFT of x2
N pt. DFT of x1
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

42. Example: Circular Convolution
Make an N pt. seq.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

43. Example: Circular Convolution

44. Circular Convolution & DFT
We know from DFS properties:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

45. Circular Convolution & DFT
If we multiply the DFTs of two N pt. sequences, we get the DFT of their circular convolution and not the DFT of their linear convolution.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

46. Example : Circular Convolution
Calculate N pt. circular convolution of x1 and x2 for the following two cases of N:
N=L
N=2L
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

47. Case 1: N=L IDFT N pt. DFT of x1
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

48. Case 1: N=L
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

49. Case 2: N=2L Pad each signal with L extra zeros to get an 2L pt. seq.:
N=2L pt. DFT of x1
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

50. Case 2: N=2L
Same as linear convolution!!

51. Using DFT to Calculate linear Convolution
Our hypothesis is that if we pad two DT signals with enough zeros so that its length becomes N, we can use DFT to calculate linear convolution.
L pt. seq.
P pt. seq.
Goal: calculate
Using DFT
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

52. Using DFT to Calculate linear Convolution
To get DFT, we have to sample the above DTFT at N equally-spaced points:
Solution to the problem statement
(Eq.1)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

53. Using DFT to Calculate linear Convolution
On the other hand, we know that:
N pt. DFT of x1
N pt. DFT of x2
Circular convolution
(Eq.2)
(Eq.1)
(Eq.2)
Replicated version of the linear convolution
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

54. Using DFT to Calculate linear Convolution
In other words, the N pt. circular convolution of two DT signals is the same as their linear convolution, if we make the result of linear convolution periodic with period N and extract one period.
To avoid aliasing:
We can thus use DFT in order to calculate the linear convolution of two sequences!
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

55. Algorithm for calculating linear convolution using DFT
1) Start with
L pt. seq.
P pt. seq.
2) Choose
3) Pad with N-L zeros to get N points.
4) Pad with N-P zeros to get N points.
5) Calculate the N pt. DFTs of the above two sequences and multiply them together.
6) Calculate IDFT of the resulting N pt. sequence.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

56. Using DFT for linear filtering when one sequence is very long
h(n)
A very long sequence
An FIR filter with a limited number of taps (P)
Examples of this situation?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

57. Using DFT for linear filtering when one sequence is very long
Solution to the problem
Overlap - add
Main idea: using the following property:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

58. Overlap- add method
Segment the long sequence into non-overlapping chunks of data with the length of L.
Convolve each chunk with h(n) to get (L+P-1) new points.
Add the results of the convolution of all chunks to get the final answer.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

59. Overlap- add method- Example
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

60. Overlap- add method- Example
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

61. Overlap- add method- Example
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

62. Summary
Discussed DFS and DFT and examined the relationship between DFT and DTFT
Showed how DFT can be used for calculating convolution sum.
Next: Fast Fourier Transform
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology