Digital Signal Processing - 2018 Faculty of Engineering Helwan University

1. ‫ر‬َ‫ـد‬ْ‫ق‬‫ِـ‬‫ن‬،،،‫لما‬‫اننا‬ ‫نصدق‬ْْ‫ق‬ِ‫ن‬‫ر‬َ‫د‬
LECTURE (8)
The Discrete Fourier Transform
Assist. Prof. Amr E. Mohamed

2. Introduction
 The discrete-time Fourier transform (DTFT) provided the frequency-
domain (ω) representation for absolutely summable sequences.
 The z-transform provided a generalized frequency-domain (z)
representation for arbitrary sequences.
 These transforms have two features in common.
 First, the transforms are defined for infinite-length sequences.
 Second, and the most important, they are functions of continuous variables
(ω or z).
 In other words, the discrete-time Fourier transform and the z-transform
are not numerically computable transforms.
 The Discrete Fourier Transform (DFT) avoids the two problems
mentioned and is a numerically computable transform that is suitable
for computer implementation.
2

3. Discrete Fourier Transform (DFT)
 Given a finite-duration discrete-time signal, a corresponding periodic discrete-time signal
can be generated which has a discrete Fourier transform (DFT) that happens to be a
discrete-frequency spectrum.
 Thus given a signal that can be represented by a sequence of numbers a spectral
characterization of the signal can be obtained, which can also be represented by a
sequence of numbers.
 Consequently, the DFT is highly amenable to processing by computers and digital signal
processors. The DFT turns out to be a sampled version of the frequency spectrum of the
original finite-duration nonperiodic signal and, therefore, it is a very important tool for
digital signal processing (DSP).
 It was demonstrated that the direct evaluation of the DFT entails a considerable amount
of redundancy and through an ingenious method that has come to be known as the fast
Fourier transform (FFT).
 It was shown that a huge amount of computation can be eliminated without degrading the
precision of the DFT in any way.
 The FFT approach renders a highly amenable tool for computation also a very efficient
one and for these reasons the DFT has found widespread applications over the years.
3

4. Discrete Fourier Transform(DFT)
4

5. Recall of DTFT
 DTFT is not suitable for DSP applications because
 In DSP, we are able to compute the spectrum only at specific discrete values of ω
 Any signal in any DSP application can be measured only in a finite number of points.
 A finite signal measures at N-points:
 Where y(n) are the measurements taken at N-points
5









Nn
Nnny
n
nx
,0
)1(0),(
00
)(

6. Computation of DFT
 Recall of DTFT
 The DFT can be computed by:
 Truncate the summation so that it ranges over finite limits x[n] is a finite-length
sequence.
 Discretize ω to ωk  evaluate DTFT at a finite number of discrete frequencies.
 For an N-point sequence, only N values of frequency samples of X(ejw) at N
distinct frequency points, are sufficient to determine x[n]
and X(ejw) uniquely.
 So, by sampling the spectrum X(ω) in frequency domain
6
DFTenxX
N
X
N
n
N
kn
j





1
0
2
)(k)(
2
),X(kk)(



10,  Nkk

7. Computation of DFT
 The inverse DFT is given by:
 Proof
7
IDFTekX
N
x
N
k
N
kn
j
 


1
0
2
)(
1
n)(


 
 




























1
0
1
0
1
0
)(
2
1
0
21
0
2
)()()(n)(
1
)(n)(
)(
1
n)(
N
m
N
m
N
n
N
nmk
j
N
k
N
kn
jN
m
N
mk
j
nxnmmxx
e
N
mxx
eemx
N
x




8. The DFT Pair
8
N
j
N
N
k
kn
N
N
k
N
kn
j
N
n
kn
N
N
n
N
kn
j
eWwhere
NnWkX
N
x
ekX
N
xSynthesis
NkWnxX
enxXAnalysis



2
1
0
1
0
2
1
0
1
0
2
1,...,1,0)(
1
n)(
)(
1
n)(
1,...,1,0)(k)(
)(k)(





















9. Example
 Find the discrete Fourier Transform of the following N-points discrete
time signal
 Solution:
 On the board
9
   )1(,...),1(),0()(  Nxxxnx

10. Nth Root of Unity
102
1
2
1
10)
19)
2
1
2
1
8)
7)
2
1
2
1
6)
15)
2
1
2
1
4)
3)
2
1
2
1
2)
11)
9
8
2
9
8
8
8
2
8
8
7
8
2
7
8
6
8
2
6
8
5
8
2
5
8
4
8
2
4
8
3
8
2
3
8
2
8
2
2
8
1
8
2
1
8
0
8
2
0
8
jeW
eW
jeW
jeW
jeW
eW
jeW
jeW
jeW
eW
j
j
j
j
j
j
j
j
j
j






























1*
2/
2
)2/(







NN
k
N
k
N
k
N
Nk
N
k
N
Nk
N
WW
WW
WW
WW
N
j
N eW
2



11. DFT Examples
 Find the 4-Point DFT the discrete time periodic signal, x(n):
 If the sampling frequency, fs = 2 Hz.
a) Plot the signal x(n) as sampled and as a time signal.
b) Plot the resulting spectrum, X(k) as sampled and as a frequency signal.
11
 







2
1
0
2
1
1 ,,,nxp
The signal xP(n) as
sampled and as a
time signal.

12. DFT Examples
12
   





3
0
2
3
0
4
2
)(
n
knj
P
n
knj
P enxenxkX

kjkjkjkj
eeee


3
2
2
2
1
2
0
2
2
1
0
2
1
1

kj
kj
kj
eee



 2
3
2
2
1
0
2
1
11



  kj
kj
kj
eee





 2
34
2
2
1
0
2
1
11



SYMMETRY
 







2
1
0
2
1
1 ,,,nxp
3,2,1,0n

















keekX
kjkj
2
cos
2
2
1
2
1
1)( 22



13. DFT Examples
13
3210
2
cos21)( ,,,, kkkX 







21)0( X
1)1( X
21)2( X
1)3( X
HzNffHzf ss 5.04/2/2 
sec/4/22/2 radxNf sss  
}1,21,1,21{)( kX

14. Relation To The Z-transform
 Let 𝑥(𝑛) be a finite-duration sequence of duration 𝑁 such that
 Then we can compute its z-transform:
 Now we construct a periodic sequence 𝑥 𝑛 by periodically repeating
𝑥(𝑛) with period 𝑁, that is,
 The DFS of 𝑥 𝑛 is given by

14


 

Elsewhere
NnNonzero
nx
,0
)1(0,
)(





1
0
)(z)(
N
n
n
znxX


 

Elsewhere
Nnnx
nx
,0
)1(0),(~
)(











1
0
2
)(k)(
~ N
n
n
k
N
j
enxX

k
N
j
ez
zXX 2
)(k)(
~



15. Relation To The Z-transform
 which means that the DFS 𝑿(𝑘) represents 𝑁 evenly spaced samples of
the z-transform 𝑋(𝑧) around the unit circle.
15
Re(z)
Im(z)
z0
z1
z2
z3
z4
z5
z6
z7
1
N = 8

16. 

 
,)()( j
ez
j
zHeH
10,][
)()(
21
0
/2





 Nkenx
eXkX
N
nk
jN
n
Nk
j
k










N
kj
zk
2
exp
Re(z)
Im(z)
z0
z1
z2
z3
z4
z5
z6
z7
1
N = 8
16
Relation To The DTFT

17. DFT - Matrix Formulation
xWX
Nx
x
x
x
WWW
WWW
WWW
NX
X
X
X
NN
N
N
N
N
N
N
NNN
N
NNN


















































 


]1[
...
]2[
]1[
]0[
...1
...............
...1
...1
1...111
)1(
...
)2(
)1(
)0(
)1)(1()1(21
)1(242
121
17

18. Computation Complexity
 To calculate the DFT of N-Points discrete time signal, we need:
 (N-1)2 Complex Multiplications
 N(N-1) Complex Additions.
18

19. IDFT - Matrix Formulation
XW
N
x
XWx
NX
X
X
X
WWW
WWW
WWW
N
Nx
x
x
x
H
NN
N
N
N
N
N
N
NNN
N
NNN
1
]1[
...
]2[
]1[
]0[
...1
...............
...1
...1
1...111
1
)1(
...
)2(
)1(
)0(
1
)1)(1()1(2)1(
)1(242
)1(21
























































19

20. Matrix Formulation
 Result: Inverse DFT is given by
 which follows easily by checking WHW = WWH = NI, where I denotes the
identity matrix. Hermitian transpose:
 Also, “*” denotes complex conjugation.
20

21. DFT Interpretation
 DFT sample X(k) specifies the magnitude and phase angle of the kth spectral
component of x[n].
 The amount of power that x[n] contains at a normalized frequency, fk, can be
determined from the power density spectrum defined as
21
)(spectrumPhase
|)(|spectrumMagnitude
kX
kX


10,
|)(|
)(
2
 Nk
N
kX
kSN

22. Periodicity of DFT Spectrum
 The DFT spectrum is periodic with period N (which is expected, since the DTFT
spectrum is periodic as well, but with period 2π).
22
)()(N)k(
)(N)k(
)(N)k(
2
2
1
0
2
1
0
)(
2
kXekXX
eenxX
enxX
nj
nj
N
n
N
nk
j
N
n
N
nNk
j























23. Example
 Example: DFT of a rectangular pulse:
 the rectangular pulse is “interpreted” by the DFT as a spectral line at
frequency ω = 0. DFT and DTFT of a rectangular pulse (N=5)
23

24. Zero Padding
 What happens with the DFT of this rectangular pulse if we increase N by
zero padding:
 where x(0) = · · · = x(M − 1) = 1. Hence, DFT is
24
   0,...,0,0,0),1(,...),1(),0()(  Mxxxnx
(N-M) Positions

25. Zero Padding
 DFT and DTFT of a Rectangular Pulse with Zero Padding (N = 10, M = 5)
25

26. Zero Padding
 Zero padding of analyzed sequence results in “approximating” its DTFT
better,
 Zero padding cannot improve the resolution of spectral components,
because the resolution is “proportional” to 1/M rather than 1/N,
 Zero padding is very important for fast DFT implementation (FFT).
26

27. Frequency Interval/Resolution
 Frequency Interval/Resolution: DFT’s frequency resolution
 and covered frequency interval
 Frequency resolution is determined only by the length of the observation
interval, whereas the frequency interval is determined by the length of
sampling interval. Thus
 Increase sampling rate  Expand frequency interval,
 Increase observation time  Improve frequency resolution.
 Question: Does zero padding alter the frequency resolution?
 Answer: No, because resolution is determined by the length of observation
interval, and zero padding does not increase this length.
27

28. Example (DFT Resolution)
 Example (DFT Resolution): Two complex exponentials with two close frequencies
F1 = 10 Hz and F2 = 12 Hz sampled with the sampling interval T = 0.02 seconds.
Consider various data lengths N = 10, 15, 30, 100 with zero padding to 512 points.
 DFT with N=10 and zero padding to 512 points. Not resolved: F2−F1 = 2 Hz < 1/(NT) = 5 Hz.
28

29. Example (DFT Resolution)
 DFT with N = 30 and zero padding to 512 points. Resolved: F2 − F1 = 2 Hz > 1/(NT)=1.7 Hz.
29

30. Example (DFT Resolution)
 DFT with N=100 and zero padding to 512 points. Resolved: F2−F1 = 2 Hz > 1/(NT) = 0.5 Hz.
30

31. Linear & Circular Convolution
31

32. Linear Convolution
 By Discrete Time Fourier Transform (DTFT)
32









-k
-k
x(k)k)-h(ny(n)
k)-x(nh(k)y(n)
x(n)*h(n)y(n)
)()()(  jjj
eXeHeY 
 )()( j
eYIDTFTny 

33. Circular Convolution
 Circular convolution of length N is
 By DFT
33
     
     
     








1
0k
C
1
0k
C
NC
kxk-nhny
k-nxkhny
nxnhny
N
N
N
N
)()()( kXkHkY 
 )()( kYIDFTnyC 

34. Convolution of two periodic sequences
34

35. How to Compute Circular Convolutions
 Method #1:
35

36. How to Compute Circular Convolutions
 Method #2:
 Compute the linear convolution and then alias it:
36

37. How to Compute Circular Convolutions
 Method #3:
 Compute 4-point DFTs, multiply, compute 4-point inverse DFT:
37

38. Using Cyclic Convs and DFTs to Compute Linear Convs:
38

39. Fast Fourier Transform (FFT)
Algorithm
39

40. Fast Fourier Transform (FFT) Algorithm
 The direct evaluation of the DFT involves N complex multiplications and N −1 complex additions for
each value of X(k), and since there are N values to determine, N2 multiplications and N(N−1)
additions are necessary.
 Consequently, for large values of N, say in excess of 1000, direct evaluation involves a considerable
amount of computation.
 It turns out that the direct evaluation of the DFT on the basis of its definition entails a large
amount of redundancy and through some clever strategies, a huge reduction in the amount of
computation can be achieved.
 These strategies have come to be known collectively as fast Fourier transforms (FFTs) although, to
be precise, these methods have nothing to do with transforms in the true sense of the word.
 FFT is an algorithm which calculates the DFT and the IDFT efficiently.
 It has a very small complexity of calculations compared to the direct calculations of the DFT or
the IDFT.
 There are 2 possible algorithms:
- the decimation in time algorithm. - the decimation in frequency algorithm.
 The decimation in time algorithm needs the arrangement of the input time sequence {x(n)}.
 The decimation in frequency algorithm needs the arrangement of the output frequency sequence
{X(k) }. 40

41. Fast Fourier Transform (FFT) Algorithm
 The basic idea behind the fast Fourier transform (FFT)
 Decompose the N-point DFT computation into computations of smaller-size DFTs
 Take advantage of the periodicity and symmetry of the complex number .
 Assuming that N is an even number. Divide the data sample into two groups by
decimation
41
1
2
,...1,0'
)1'2()'(
)'2()'(





 N
nfor
nxnv
nxnu
)...7()3();5()2();3()1();1()0(
)...6()3();4()2();2()1();0()0(
xvxvxvxv
xuxuxuxu



42. Fast Fourier Transform (FFT) Algorithm
 Consider special case of N an integer power of 2
 Separate x[n] into two sequence of length N/2
 Even indexed samples in the first sequence
 Odd indexed samples in the other sequence
 Substitute variables n=2r for n even and n=2r+1 for odd
42
       









1
oddn
/2
1
evenn
/2
1
0
/2
][][][
N
knNj
N
knNj
N
n
knNj
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43. Computation Complexity
 To calculate the DFT of N-Points discrete time signal, we need:
 (N-1)2 Complex Multiplications
 N(N-1) Complex Additions.
 To calculate the FFT of N-Points discrete time signal, we need:
 Complex Multiplications
 Complex Additions.
43
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44. Decimation In Time
 8-point DFT example using decimation-in-time
 Two N/2-point DFTs
 2(N/2)2 complex multiplications
 2(N/2)2 complex additions
 Combining the DFT outputs
 N complex multiplications
 N complex additions
 Total complexity
 N2/2+N complex multiplications
 N2/2+N complex additions
 More efficient than direct DFT
 Repeat same process
 Divide N/2-point DFTs into
 Two N/4-point DFTs
 Combine outputs
44

45. 45
 After two steps of decimation in time
On
the whiteboard
 Repeat until we’re left with two-point DFT’s
On
the whiteboard
Decimation In Time

46. 46
Decimation-In-Time FFT Algorithm
 Final flow graph for 8-point decimation in time
On
the whiteboard

47. 47