Department of Computer Eng. Sharif University of Technology Discrete-time signal processing Chapter...
-
Upload
thomasina-dixon -
Category
Documents
-
view
215 -
download
2
Transcript of Department of Computer Eng. Sharif University of Technology Discrete-time signal processing Chapter...
Department of Computer Eng.
Sharif University of Technology
Discrete-time signal processing
Chapter 8:Chapter 8:THE DESCRETE FOURIER THE DESCRETE FOURIER
TRANSFORMTRANSFORM
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer and Buck, ©1999-2000 Prentice Hall Inc.
8.0 Introduction
• 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).
2
DSP
Chapter 8: The Discrete Fourier Transform
8.1 Discrete Fourier Series
• 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
3
DSP
[ ] [ ]x n x n rN
2 / 2 / 2 2 /j N k mN n j N kn j mn j N kne e e e
1
2 / 2 /
0
1 1[ ]
Nj N kn j N kn
k N k
x n X k e X k eN N
[ ]x n
Chapter 8: The Discrete Fourier Transform
Discrete Fourier Series Pair
• 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
4
DSP
1
2 /
0
[ ]N
j N kn
n
X k x n e
1
2 /
0
1[ ]
Nj N kn
k
x n X k eN
2 /j NNW e
1
0
1[ ]
Nkn
Nk
x n X k WN
1
0
[ ]N
knN
n
X k x n W
Chapter 8: The Discrete Fourier Transform
Example 1 1
• DFS of a periodic impulse train
• Since the period of the signal is N
• We can represent the signal with the DFS coefficients as
5
DSP
1[ ]
0r
n rNx n n rN
else
1 1
2 / 2 / 2 / 0
0 0
[ ] [ ] 1N N
j N kn j N kn j N k
n n
X k x n e n e e
1
2 /
0
1[ ]
Nj N kn
r k
x n n rN eN
Chapter 8: The Discrete Fourier Transform
Example 2: Duality in the DFT
• 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].
6
DSP
Chapter 8: The Discrete Fourier Transform
[ ] [ ]r
Y k N k rN
10
0
1[ ] [ ] 1
Nkn
N Nk
y n N k W WN
Example 3
• DFS of an periodic rectangular pulse train
• The DFS coefficients
7
DSP
2 /10 54
2 /10 4 /10
2 /100
sin / 21
sin /101
j kj kn j k
j kn
keX k e e
ke
Chapter 8: The Discrete Fourier Transform
8.2 Properties of DFS
• Linearity (all signals have the same period)
• Shift of a Sequence
• Duality
8
DSP
1 1
2 2
1 2 1 2
DFS
DFS
DFS
x n X k
x n X k
ax n bx n aX k bX k
2 /
2 /
DFS
DFS j km N
DFSj nm N
x n X k
x n m e X k
e x n X k m
DFS
DFS
x n X k
X n Nx k
Chapter 8: The Discrete Fourier Transform
0 1m N
Periodic Convolution
• Take two periodic sequences
• Let’s form the product
• The periodic sequence with given DFS can be written as
• Periodic convolution is commutative
9
DSP
1 1
2 2
DFS
DFS
x n X k
x n X k
3 1 2X k X k X k
1
3 1 20
N
m
x n x m x n m
1
3 2 10
N
m
x n x m x n m
Chapter 8: The Discrete Fourier Transform
Periodic Convolution Cont’d
• Substitute periodic convolution into the DFS equation
• Interchange summations
• The inner sum is the DFS of shifted sequence
• Substituting
10
DSP
1
3 1 20
N
m
x n x m x n m
1 1
3 1 20 0
[ ] [ ]N N
knN
n m
X k x m x n m W
1 1
3 1 20 0
[ ] [ ]N N
knN
m n
X k x m x n mW
1
2 20
[ ]N
kn kmN N
n
x n mW W X k
1 1 1
3 1 2 1 2 1 20 0 0
[ ] [ ] [ ]N N N
kn kmN N
m n m
X k x m x n mW x m W X k X k X k
Chapter 8: The Discrete Fourier Transform
8.3 The Fourier Transform of Periodic Signals
• 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
14
DSP
2 2j
k
kX e X k
N N
2 2
0 0
212
00
1 1 2 2
2 2
1 2 1
j j n j n
k
kN j nj n N
k k
kX e e d X k e d
N N
kX k e d X k e
N N N
Chapter 8: The Discrete Fourier Transform
Example
• Consider the periodic impulse train
• The DFS was calculated previously to be
• Therefore the Fourier transform is
15
DSP
[ ]r
p n n rN
1 for all kP k
2 2j
k
kP e
N N
Chapter 8: The Discrete Fourier Transform
Relation between Finite-length and Periodic Signals
• 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
16
DSP
[ ] [ ] [ ] [ ]r r
x n x n p n x n n rN x n rN
2
2 2
2 2
j j j j
k
kjj N
k
kX e X e P e X e
N N
kX e X e
N N
2
2
kj jN
k
N
X k X e X e
Example
• Consider the following sequence
• The Fourier transform
• The DFS coefficients
17
DSP
2 /j NNW e
2 sin 5 / 2
sin / 2j jX e e
4 /10 sin / 2
sin /10j k k
X k ek
Chapter 8: The Discrete Fourier Transform
8.4 Sampling the Fourier Transform
• 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
18
DSP
2 /
2 /
j N kj
N kX k X e X e
[ ] DTFT jx n X e
2 /
2 /N k
j N k
z eX k X z X e
X k
1
2 /
0
1[ ]
Nj N kn
k
x n X k eN
Chapter 8: The Discrete Fourier Transform
Sampling the Fourier Transform Cont’d
• The only assumption made on the sequence is that DTFT exist
• Combine equation to get
• Term in the parenthesis is
• So we get
19
DSP
j j m
m
X e x m e
1
2 /
0
1[ ]
Nj N kn
k
x n X k eN
2 /j N kX k X e
12 / 2 /
0
12 /
0
1[ ]
1
Nj N km j N kn
k m
Nj N k n m
m k m
x n x m e eN
x m e x m p n mN
1
2 /
0
1 Nj N k n m
k r
p n m e n m rNN
[ ]r r
x n x n n rN x n rN
Chapter 8: The Discrete Fourier Transform
Sampling the Fourier Transform Cont’d
• 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
21
DSP
0 1
0
x n n Nx n
else
Chapter 8: The Discrete Fourier Transform
8.5 The Discrete Fourier Transform
• 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]
22
DSP
0 outside of 0 1x n n N
r
x n x n rN
mod NN
X k X k X k
X k
0 1
0
X k k NX k
else
mod NN
x n x n x n
Chapter 8: The Discrete Fourier Transform
The Discrete Fourier Transform Cont’d
• The DFS pair
• The equations involve only on period so we can write
23
DSP
1
2 /
0
1[ ]
Nj N kn
k
x n X k eN
1
2 /
0
[ ]N
j N kn
n
X k x n e
12 /
0
[ ] 0 1
0
Nj N kn
n
x n e k NX k
else
1
2 /
0
10 1
[ ]
0
Nj N kn
k
X k e n Nx n N
else
Chapter 8: The Discrete Fourier Transform
The Discrete Fourier Transform Cont’d
• The DFT pair can also be written as
• The Discrete Fourier Transform
24
DSP
Chapter 8: The Discrete Fourier Transform
[ ]DFTX k x n
12 /
0
[ ] 0 1
0
Nj N kn
n
x n e k NX k
else
1
2 /
0
10 1
[ ]
0
Nj N kn
k
X k e n Nx n N
else
Example
• 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]
25
DSP
42 /5
0
2
2 /5
1
15 0, 5, 10,...
0
j k n
n
j k
j k
X k e
e
ek
else
Chapter 8: The Discrete Fourier Transform
Example Cont’d
• 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
26
DSP
Chapter 8: The Discrete Fourier Transform
8.6 Properties of DFT
• Linearity
• Duality
• Circular Shift of a Sequence
27
DSP
1 1
2 2
1 2 1 2
DFT
DFT
DFT
x n X k
x n X k
ax n bx n aX k bX k
2 / 0 n N-1
DFT
j k N mDFT
N
x n X k
x n m X k e
DFT
DFT
N
x n X k
X n Nx k
Chapter 8: The Discrete Fourier Transform
Circular Convolution
• Circular convolution of two finite length sequences
30
DSP
1
3 1 20
N
Nm
x n x m x n m
1
3 2 10
N
Nm
x n x m x n m
Chapter 8: The Discrete Fourier Transform
Example
• Circular convolution of two rectangular pulses L=N=6
• DFT of each sequence
• Multiplication of DFTs
• And the inverse DFT
31
DSP
1 2
1 0 1
0
n Lx n x n
else
21
1 20
0
0
N j knN
n
N kX k X k e
else
2
3 1 2
0
0
N kX k X k X k
else
3
0 1
0
N n Nx n
else
Chapter 8: The Discrete Fourier Transform
Example
• We can augment zeros to each sequence L=2N=12
• The DFT of each sequence
• Multiplication of DFTs
32
DSP
2
1 2 2
1
1
LkjN
kjN
eX k X k
e
22
3 2
1
1
LkjN
kjN
eX k
e
Chapter 8: The Discrete Fourier Transform
8.7 Linier convolution using the DFT
• 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.
33
DSP
Chapter 8: The Discrete Fourier Transform
Linear convolution of two finite-length sequences
34
DSP
Chapter 8: The Discrete Fourier Transform
3 1 2m
x k x m x n m
Circular convolution as linear convolution
35
DSP
Chapter 8: The Discrete Fourier Transform
With aliasing
Without aliasing
Implementing LTI systems using the DFT
• 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.
39
DSP
Chapter 8: The Discrete Fourier Transform
Implementing LTI systems using the DFT cont’d
• 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.
40
DSP
Chapter 8: The Discrete Fourier Transform
Overlap-add method
42
DSP
Chapter 8: The Discrete Fourier Transform
0
[ ], 0 1[ ]
0,
rr
r
x n x n rL
x n rL n Lx n
otherwise
0
[ ] [ ]
[ ] [ ]* [ ]
rr
r r
y n x n h n y n rL
y n x n h n
Overlap-add method
• 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.
43
DSP
Chapter 8: The Discrete Fourier Transform
Overlap-save method
• 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.
45
DSP
Chapter 8: The Discrete Fourier Transform