This document contains lecture slides from a Digital Signal Processing course taught by Prof. Hamed Nassar at Beirut Arab University. The slides discuss topics related to fast Fourier transforms (FFTs) including window functions, circular convolution, and implementations of the discrete Fourier transform (DFT) and FFT using the divide and conquer algorithm. MATLAB code examples are provided to demonstrate concepts like windowing functions, circular shifts, and calculating the DFT and FFT.

1. Beirut Arab University
Computer Engineering Program
Spring 2015
Digital Signal Processing (COME 384)
Instructor: Prof. Dr. Hamed Nassar
Fast Fourier Transforms FFT
 Textbook:
◦ V. Ingle and J. Proakis, Digital Signal Processing
Using MATLAB 3rd Ed, Cengage Learning, 2012
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Dr. Hamed Nassar, Beirut Arab Univ

2. DSP Windows
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Prof. H. Nassar, BAU

3. DSP Windows
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Prof. H. Nassar, BAU

4. DSP Windows MATLAB code
 n=0:9 %stem ten samples
 RectWin=ones(1,10)
 sub1=subplot(4,1,1); stem(n, RectWin);grid
 title('Rectangular Window');
 TriWin=1-abs(length(n)-1-2*n)/(length(n)-1)
 sub2=subplot(4,1,2); stem(n, TriWin);grid
 title('Triangular Window');
 HannWin=0.5-0.5*cos(2*pi*n/(length(n)-1)))
 sub3=subplot(4,1,3); stem(n, HannWin);grid
 title('Hanning Window');
 HammWin=0.54-0.46*cos(2*pi*n/(length(n)-1))
 sub4=subplot(4,1,4); stem(n, HammWin);grid
 title('Hamming Window');
 linkaxes([sub1,sub2,sub3,sub4], 'y')%Use same y scale for
all subs Prof. H. Nassar, BAU

5. Constructing an infinite, periodic sequence from a finite
one, and vv
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Prof. H. Nassar, BAU

6. Circular Shift Example: 4 left
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Prof. H. Nassar, BAU

7. Circular convolution
 A linear convolution between two N-point sequences
results in a longer sequence (length=N+N-1, with indices:
0, 1, …, 2N-2).
 But we have to restrict our interval to 0 ≤ n ≤ N − 1,
therefore instead of linear shift, we should consider the
circular shift.
 A convolution operation that contains a circular shift, as
shown below, is called the circular convolution and is
given by
Prof. H. Nassar, BAU

8. Circular Convolution Example: Time domain approach (watch for circular
folding)
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Prof. H. Nassar, BAU

9. Circular Convolution Example: Frequency domain approach
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Prof. H. Nassar, BAU

10. FFT
 The DFT introduced earlier is the only transform that is
discrete in both the time and the frequency domains, and
is defined for finite-duration sequences.
 Although it is a computable transform, its straightforward
computation is very inefficient, especially when the
sequence length N is large.
 In 1965 Cooley and Tukey showed a procedure to
substantially reduce the amount of computations involved
in the DFT.
 This led to the explosion of other efficient algorithms
collectively known as fast Fourier transform (FFT)
algorithms.
 We will study one such algorithms in detail.
 One concept used that the book does not speak about
explicitly, but it is there (seen the previous example on
circular convolution) is circular folding.
Prof. H. Nassar, BAU

11. FFT
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12. Example
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13. )
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Prof. H. Nassar, BAU

14. Divide and Conquer Algorithm
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Prof. H. Nassar, BAU

15. Implementation
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16. Implementation
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Prof. H. Nassar, BAU

17. EXAMPLE 5.21 N = 15, L = 3, M =5
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Prof. H. Nassar, BAU

18. EXAMPLE 5.21 N = 15, L = 3, M =5
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Prof. H. Nassar, BAU

19. EXAMPLE 5.21 N = 15, L = 3, M =5
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Prof. H. Nassar, BAU

20. Reminder: Classical DFT MATLAB
script
 % This is a DFT program without loops
 clear all; %Write input sequence below
 nx=0:5; x=[0 1 7 3 1 2] %index vector nx, then input signal
x
 N=length(x);
 wN=exp(-2j*pi/N) %Base
 E=[0:N-1]' *[0:N-1] %Prepare structure of matrix W
 W=wN.^E %Matrix W=Base^Matrix (note the dot .)
 Y=x*W %DFT = input sequence * Matrix
 %To test, we'll obtain x(n=2)=1/N sum_k=0^N-1 Y
e^(j*2*pi*n*k)
 n=2
 k=nx;
 g=exp(2j*pi*n*k/N)
 G=1/N*(Y*g.') Prof. H. Nassar, BAU

21. FFT Example: N = 6, M=4, L=2
 x = {3, 1, 0, 2, 4, 1,3, 2} => x = {3, 1, 0, 2, 4, 1, 3, 2}
 This has DFT: 10.0, 0.5-2.6i, 2.5+2.6i, 2.0, 2.5-2.6i,
0.5+2.6i
 Always remember that:
◦ the 1D input vector x is first converted (reshaped) into a
row-major matrix xm of size LxM
◦ the corresponding output DFT vector X is obtained as a
column-major matrix X of size LxM, and thus has to be
converted (reshaped) back into a 1D vector.
◦ The best summary of the Divide and Conquer FFT algo:
Prof. H. Nassar, BAU

22.  clear all; % FFT using Divide & Conquer without loops
 x =[3 1 0 2 4 1 3 2];L=2;M=4; %%Write input sequence and your choice for L and M
 N=length(x)%N can be obtained auto, but M,L u should spec
 x=reshape(x, [L,M]) %Create an LxM matrix (col maj) from x
 wM=exp(-2j*pi/M) %Base M
 EM=[0:M-1]' *[0:M-1]%Prepare structure of matrix W: MxM
 WM=wM.^EM %Raise elements (note .) of Matrix to Base
 WLMF=x*WM %Obtain F matrix
 wN=exp(-2j*pi/N) %Base N
 ELM=[0:L-1]'*[0:M-1] %Prepare structure of twiddle matrix:LxM
 WLMT=wN.^ELM %Twiddle: Raise elements (.) of Matrix to Base
 WLMG=WLMF.*WLMT %Modify elements (note .) of F by Twiddle
 wL=exp(-2j*pi/L) %Base L
 EL=[0:L-1]'*[0:L-1] %Prepare structure of matrix W:LxL
 WL=wL.^EL %Raise elements (note .) of Matrix to Base
 X=WL*WLMG %Final DFT
 %To test, obtain x(n)=1/N sum_k=0^N-1 Y e^(j*2*pi*n*k) for some n
 n=6; k=0:N-1; %MATLAB: u involve vec in an op, u get vec
 XX=[X(1,:) X(2,:)] %Unfold matrix into a vector
 g=exp(2j*pi*n*k/N) %Note in IDFT sign of e is +ve
 G=1/N*(XX*g.') %IDFT for the given n. We could obtain whole x
Prof. H. Nassar, BAU

23. 5
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Prof. H. Nassar, BAU