Hossein Sameti Department of Computer Engineering Sharif University of Technology

2  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 C. D. C. D. Complex Inf. or Finite D. Int. periodic D. Int. finite

4

5 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 Exponentials that are periodic with period N. arbitrary integer * Proof: 1

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

9 … …

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

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

12 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

 In the list of properties: 13 Where: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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

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

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

 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 17 Extracts one period of Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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

19

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

21

Mehrdad Fatourechi, Electrical and Computer Engineering, University of British Columbia, Summer

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

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

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

27 4 pt. DFT: 6 pt. DFT: 8 pt. DFT: 100 pt. DFT: How are these related to each other?

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

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

 Assume y(n) is a signal of finite or infinite extent. 30 Sample at N equally-spaced points. 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

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

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

33 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

 Shift property: 34 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

 In the list of properties: 35 where: and where: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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

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

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

 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. 39 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

40 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 Circular convolution: N pt. seq. Circular convolution is closely related to periodic convolution. N pt. DFT of x 1 N pt. DFT of x 2 N pt. DFT of x 3 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

42 Circular convolution? Make an N pt. seq. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

43

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

45 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 Calculate N pt. circular convolution of x 1 and x 2 for the following two cases of N: 1)N=L 2)N=2L Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

47 N pt. DFT of x 1 IDFT Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

48 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

49 Pad each signal with L extra zeros to get an 2L pt. seq.: N=2L pt. DFT of x 1 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

50 Same as 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. 51 L pt. seq. P pt. seq. Using DFT Goal: calculate Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

52 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 N pt. DFT of x 1 N pt. DFT of x 2 (Eq.2) Circular convolution (Eq.1) (Eq.2) Replicated version of the linear convolution On the other hand, we know that: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

 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. 54 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

1) Start with 55 L pt. seq. P pt. seq. 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. 2) Choose Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

56 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 Solution to the problem Overlap - add Main idea: using the following property: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

 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. 58 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

59 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

60 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

61 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

 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 62 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology