Digital Signal Processing Using by ldd0229

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```									Chapter 5. The Discrete Fourier
Transform

Gao Xinbo
School of E.E., Xidian Univ.
xbgao@ieee.org
http://see.xidian.edu.cn/teach/matlabdsp/
Review 1
   The DTFT provides the frequency-domain (w)
representation for absolutely summable sequences.
   The z-transform provides a generalized frequency-
domain (z) representation for arbitrary sequences.
   Two features in common:
   Defined for infinite-length sequences
   Functions of continuous variable (w or z)
   From the numerical computation viewpoint, these two
features are troublesome because one has to evaluate
infinite sums at uncountably infinite frequencies.
Review 2
   To use Matlab, we have to truncate sequences and
then evaluate the expression at finitely many points.
   The evaluation were obviously approximations to the
exact calculations.
   In other words, the DTFT and the z-transform are not
numerically computable transform.
Introduction 1
   Therefore we turn our attention to a numerically computable
transform.
   It is obtained by sampling the DTFT transform in the
frequency domain (or the z-transform on the unit circle).
   We develop this transform by analyzing periodic sequences.
   From FT analysis we know that a periodic function can always
be represented by a linear combination of harmonically related
complex exponentials (which is form of sampling).
   This give us the Discrete Fourier Series representation.
   We extend the DFS to finite-duration sequences, which leads
to a new transform, called the Discrete Fourier Transform.
Introduction 2
   The DFT avoids the two problems mentioned above
and is a numerically computable transform that is
suitable for computer implementation.
   The numerical computation of the DFT for long
sequences is prohibitively time consuming.
   Therefore several algorithms have been developed to
efficiently compute the DFT.
   These are collectively called fast Fourier transform
(or FFT) algorithms.
The Discrete Fourier Series
   Definition: Periodic sequence
~(n)  ~(n  kN), n, k
x      x
   N: the fundamental period of the sequences
   From FT analysis we know that the periodic functions can be
synthesized as a linear combination of complex exponentials
whose frequencies are multiples (or harmonics) of the
fundamental frequency (2pi/N).
   From the frequency-domain periodicity of the DTFT, we
conclude that there are a finite number of harmonics; the
frequencies are {2pi/N*k,k=0,1,…,N-1}.
The Discrete Fourier Series
   A periodic sequence can be expressed as
N 1
~ ( n)  1     ~         

 X (k )e , n  0,1,
j 2N kn
x
N   k 0

~
{ X ( K ), k  0,1, } are called the discrete Fourier series
coefficients, which are given by
N 1
~             ~(n)e  j 2N kn , k  0,1,
X (k )   x
n 0

The discrete Fourier series representation of periodic sequences
The Discrete Fourier Series
   X(k) is itself a (complex-valued) periodic
sequence with fundamental period equal to N.

 j 2N
Let WN  e
N 1
~
X (k )  DFS[ ~ (n)]   ~ (n)WN
x             x  nk

n 0
N 1
~ (n)  IDFS[ X (k )]  1
~                ~
x
N
 X (k )WN
 nk

k 0

Example 5.1
Matlab Implementation
Matrix-vector multiplication:
Let x and X denote column vectors corresponding to the
primary periods of sequences x(n) and X(k), respectively.
Then X  W x x  1 W * X
N            N
N
1   1            1 
1 W 1       WN N 1) 
                   
(

WN  WN 0  k , n  N  1  
kn                        N                    
                
   ( N 1)      ( N 1) 2 
1 WN        WN           
DFS Matrix
dfs1 function(row vect.for dfs)
   Function [Xk] = dfs (xn, N)
   % Xk and xn are column(not row) vectors
   if nargin < 2 N = length(xn)
   n = 0: N-1; k = 0: N-1;
   WN = exp(-j*2*pi/N);
   kn = k’*n;
   WNkn = WN.^kn;
   Xk = WNkn * xn
Ex5.2 DFS of square wave seq.
 L, k  0, N ,2 N ,
~        
X (k )    j ( L 1) / N sin(kL / N )
e
                   sin(k / N )
Note
1.   Envelope like the sinc function;
2.   Zeros occur at N/L (reciprocal of duty cycle);
3.   Relation of N to density of freq. Samples;
Relation to the z-transform
Nonzero, 0  n  N  1           N 1
x(n)                         X ( z )   x(n) z n
 0,       elsewhere              n 0

~(n), 0  n  N  1 ~
x(n)  
x                           N 1
X (k )   x(n) e   

j 2N k    n

 0,    elsewhere             n 0

~
X (k )  X ( z ) |       j 2 k
z e N

The DFS X(k) represents N evenly spaced samples of the z-
transform X(z) around the unit circle.
Relation to the DTFT
N 1                   N 1
X (e )   x ( n )e
jw                   jwn
 x
~ (n)e  jwn
n 0                   n 0
~
X (k )  X (e jw ) |w 2 k
N

2                  2
Let w1          , and wk         k  kw1
N                   N
X (k )  X (e jwk )  X (e jkw1 )

The DFS is obtained by evenly sampling the DTFT at w1 intervals.
The interval w1 is the sampling interval in the frequency domain. It is called
frequency resolution because it tells us how close are the frequency samples.
Sampling and construction in the
z-domain
~
X (k )  X ( z ) |       j 2 k
, k  0,1,2, 
DFS & z-                               z e N
                         
transform                               

 x ( m)e                      x(m)WNkm
 j 2N km
                            
m                          m  

~ ( n)  1     ~     N 1
1 N 1             km 
IDFS x             X (k )WN     x(m)WN WN kn
 kn

N k 0 m  


Nk 0                                             

1 N 1  k ( n  m )                
  x ( m )  WN                   x(m)   (n  m  rN )
m       N k 0                 m         r  
                                               
      x(m) (n  m  rN )   x(n  rN )
m   r                                        r  
Comments
   When we sample X(z) on the unit circle, we
obtain a periodic sequence in the time domain.
   This sequence is a linear combination of the
original x(n) and its infinite replicas, each
shifted by multiples of N or –N.
   If x(n)=0 for n<0 and n>=N, then there will be
no overlap or aliasing in the time domain.
Comments
x(n)  ~(n) for 0  n  N  1
x
~(n) R (n)  ~(n)1 0  n  N  1
x ( n)  x            x 
N
0     else

RN(n) is called a rectangular window of length N.
THEOREM1: Frequency Sampling
If x(n) is time-limited (finite duration) to [0,N-1], then N
samples of X(z) on the unit circle determine X(z) for all z.
Reconstruction Formula
Let x(n) be time-limited to [0,N-1]. Then from Theorem 1 we should be able to
recover the z-transform X(z) using its samples X~(k).

~ ( n) z  n   1                  
N 1         N 1            N 1    N 1
~
X ( z )   x ( n) z   x
n
N          X (k )WN kn  z  n
n 0         n 0               n 0   k 0              
1 N 1 ~  N 1  kn  n  1 N 1 ~  N 1  k 1 n 
  X (k )  WN z    X (k )  WN z                         
N k 0         n 0         N k 0            n 0            
1   N 1
~ 1  WN kN z  N 

      X (k ) 1  W k z 1                 WN-kN=1
N   k 0           N       
 N N 1  ~
1 z            X (k )
X ( z) 
N
 1  W k z 1
k 0    N
The DTFT Interpolation Formula
~
1  e  jwN   N 1
X (k )             N 1
~            1  e  jwN
X (e ) 
jw

N
 1  e j 2k / N e  jw     X (k )

N 1  e j 2k / N e  jw   
k 0                         k 0

sin  wN   jw  N21             An interpolation polynomial
 ( w)         2
e
N sin  2 
w

N 1                            This is the DTFT interpolation
X (e )   X (k )w  2Nk 
jw    ~            
formula to reconstruct X(ejw) from
k 0                            its samples X~(k)

Since (0)  1 , we have that X(ej2pik/N)=X~(k), which
means that the interpolation is exact at sampling points.
The Discrete Fourier Transform
   The discrete Fourier series provided us a mechanism
for numerically computing the discrete-time Fourier
transform.
   It also alert us to a potential problem of aliasing in the
time domain.
   Mathematics dictates that the sampling of the discrete-
time Fourier transform result in a periodic sequences
x~(n).
   But most of the signals in practice are not periodic.
They are likely to be of finite duration.
The Discrete Fourier Transform
   Theoretically, we can take care of this problem by
defining a periodic signal whose primary shape is that
of the finite duration signal and then using the DFS on
this periodic signal.
   Practically, we define a new transform called the
Discrete Fourier Transform (DFT), which is the
primary period of the DFS.
   This DFT is the ultimate numerically computable
Fourier transform for arbitrary finite duration
sequences.
The Discrete Fourier Transform
   First we define a finite-duration sequence x(n) that has N
samples over 0<=n<=N as an N-point sequence
                             
~ ( n) 
x           x(n  rN ) ~(n)  x(n mod N )  x((n))N
x
r  

The compact relationships between x(n) and x~(n) are
~(n)  x((n)) ( Periodic extension)
x            N

x(n)  ~(n) R (n) (Window operation)
x        N

The function rem(n,N) can be used to implement our
modulo-N operation.
The Discrete Fourier Transform
   The Discrete Fourier Transform of an N-point sequence is
given by                    ~
 X (k ) 0  k  N  1 ~
X (k )  DFT[ x(n)]                        X ( k ) RN ( n )
 0           else
N 1
X (k )   x(n)WN , 0  k  N  1
nk

n 0

Note that the DFT X(k) is also an N-point sequence, that is, it is not
defined outside of 0<=n<=N-1.
DFT X(k) is the primary interval of X~(k).
N 1
~ ( n) R ( n)  1
x(n)  IDFT[ X (k )]  x       N
N
        
X (k )WN kn ,0  n  N  1
k 0
Matlab Implementation
X  WN x
1 *
x  WN X
N
1   1               1    
1 W 1         WN N 1) 
                   
(

WN  WN 0  k , n  N  1  
kn                          N                    
                
                ( N 1) 2 
1 WN N 1)
(
 WN         
Examples 5.6 5.7 Comments
   Zero-padding is an operation in which more zeros are appended
to the original sequence. The resulting longer DFT provides
closely spaced samples of the discrete-times Fourier transform
of the original sequence.
   The zero-padding gives us a high-density spectrum and
provides a better displayed version for plotting. But it does not
give us a high-resolution spectrum because no new information
is added to the signal; only additional zeros are added in the
data.
   To get high-resolution spectrum, one has to obtain more data
from the experiment or observations.
Properties of the DFT

1. Linearity: DFT[ax1(n)+bx2(n)]=aDFT[x1(n)]+bDFT[x2(n)]
N3=max(N1,N2): N3-point DFT
2. Circular folding:
 x(0)       k 0
x((n))N  
 x( N  k ) 1  k  N  1
 X (0)       k 0
DFT[ x((n))N ]  X ((k ))N  
 X ( N  k ) 1  k  N 1

Matlab: x=x(mod(-n,N)+1)
Properties of the DFT
3. Conjugation: DFT [ x * ( n)]  X * ((  k )) N

4. Symmetry properties for real sequences:
Let x(n) be a real-valued N-point sequence
X (k )  X * ((  k )) N
Re[ X (k )]  Re[ X ((k )) N ] : circular  even sequence
Im[X (k )]   Im[X ((N  k )) N ] : circular  odd sequence
| X (k ) || X ((k )) N |
X (k )  X ((k )) N
Comments:
    Circular symmetry
    Periodic conjugate symmetry
    About 50% savings in computation as well as in storage.
    X(0) is a real number: the DC frequency
    X(N/2)(N is even) is also real-valued: Nyquist component
    Circular-even and circular-odd components:
1
xec (n)  [ x(n)  x((n)) N ]   DFT[ xec (n)]  Re[ X (k )]  Re[ X ((k ))N ]
2
1
xoc (n)  [ x(n)  x((n)) N ]   DFT[ xoc (n)]  Im[X (k )]  Im[X ((k ))N ]
2

The real-valued signals                  Function, p143
Properties
5. Circular shift of a sequence
DFT [ x(( n  m)) N RN (n)]  WN X (k )
km

6. Circular shift in the frequency domain

DFT [WN ln x(n)]  X (( k  l )) N RN (k )
7. Circular convolution**
N 1
x1 (n)  x2 (n)   x1 (m) x2 ((n  m))N , 0  n N  1
m 0

DFT [ x1 (n)  x2 (n)]  X 1 (k ) X 2 (k )
Properties
1
8. Multiplication:      DFT[ x1 (n)  x2 (n)]  X 1 (k )  X 2 (k )
N
9. Parseval’s relation:
N 1                   N 1
1
E x   | x ( n) | 2
 | X ( k ) |2
n 0           N         k 0

Energy spectrum           | X ( k ) |2
N
Power spectrum             X (k )
2

N
Linear convolution using the DFT
In general, the circular convolution is an aliased version of the
linear convolution.
If we make both x1(n) and x2(n) N=N1+N2-1 point sequences by
padding an appropriate number of zeros, then the circular
convolution is identical to the linear convolution.
 N 1                     
x4 (n)  x1 (n)  x2 (n)    x1 (k ) x2 ((n  k ))N  RN (n)
 m 0                     
 N 1         

   x1 (k )  x2 (n  k  rN ) RN (n)
m 0        r               
   N1 1                                              
    x1 (k ) x2 (n  k  rN ) RN (n)    x3 (n  rN ) RN (n)
r   m 0                                r           
Error Analysis
   When N=max(N1,N2) is chosen for circular convolution, then
the first (M-1) samples are in error, where M=min(N1,N2).
             
e(n)  x4  x3    x3 (n  rN ) RN (n)  x3 (n)
r           
                
  x3 (n  rN ) RN (n) N  max(N1 , N 2 )
r 0            
e(n)  [ x3 (n  N )  x3 (n  N )]RN (n)
 x3 (n  N ) 0  n  N  1
n=0,1,…(N1+N2-1)-N
X3(n) is also causal
Block Convolution
   Segment the infinite-length input sequence into smaller sections (or blocks),
process each section using the DFT, and finally assemble the output sequence
from the outputs of each section. This procedure is called a block convolution
operation.
   Let us assume that the sequence x(n) is sectioned into N-point sequence and
that the impulse response of the filter is an M-point sequence, where M<N.
   We partition x(n) into sections, each overlapping with the previous one by
exactly (M-1) samples, save at last (N-M+1) output samples, and finally
concatenate these outputs into sequence.
   To correct for the first (M-1) samples in the first output block, we set the first
(M-1) samples in the first input blocks to zero.
Matlab Implementation
Function [y]=ovrlpsav(x,h,N)
Lenx = length(x); M=length(h); M1=M-1; L=N-M1;
H = [h zeros(1,N-M)];
X = [zeros(1,M1), x, zeros(1,N-1);
K = floor((Lenx+M1-1)/L);
For k=0:K, xk = x(k*L+1:k*L+N);
Y(k+1,: ) = circonvt(xk,h,N); end
Y=Y(:,M:N)’;
Y=(Y(:))’;
The Fast Fourier Transform
   Although the DFT is computable transform, the
straightforward implementation is very inefficient,
especially when the sequence length N is large.
   In 1965, Cooley and Tukey showed the a procedure to
substantially reduce the amount of computations
involved in the DFT.
   This led to the explosion of applications of the DFT.
   All these efficient algorithms are collectively known
as fast Fourier transform (FFT) algorithms.
The FFT
   Using the Matrix-vector multiplication to
implement DFT:
   X=WNx (WN: N*N, x: 1*N, X: 1*N)
    takes N×N multiplications and (N-1)×N
additions of complex number.
Number of complex mult. CN=O(N2)
   A complex multiplication requires 4 real
multiplications and 2 real additions.
Goal of an Efficient computation
   The total number of computations should be linear
rather than quadratic with respect to N.
   Most of the computations can be eliminated using the
symmetry and periodicity properties
k ( n N )    (k  N )n
W  kn
N    W N            W
N

WN  N / 2  WN
kn            kn

CN=N×log2N
If N=2^10, CN=will reduce to 1/100 times.
Decimation-in-time: DIT-FFT, decimation-in-frequency: DIF-FFT
4-point DFT→FFT example
3
X (k )   x(n)W4nk , 0  k  3; W4  e  j 2 / 4   j
n 0

Efficient         W40  W44  1; W41  W49   j
Approach          W42  W46  1; W43  j
   X=Wx
   X(0) = x(0)+x(2) + x(1)+x(3) = g1 + g2
   X(1) = x(0)-x(2) – j(x(1)-x(3)) = h1 - jh2
   X(2) = x(0)+x(2) - x(1)+x(3) = g1 - g2
   X(3) = x(0)-x(2) + j(x(1)-x(3)) = h1 + jh2
   It requires only 2 complex multiplications.
   Signal flowgraph
A 4-point DFT→FFT example
 X (0) X (1)   g1 g 2             1 1  g1 g 2 
 X (2) X (3)  h  jh  *W2  1  j  . * h h  *W2
              1          2            1 2
1 1             x(0) x(1) 
        . *W2 *  x(2) x(3) *W2
1  j                     
W2 0*0 W2 0*1  1 1 
where W2                
W2 W2  
1*0    1*1     1  1

              
so W2 * A or A *W2 no multiplic ation needed
Divide-and-combine approach
   To reduce the DFT computation’s quadratic dependence on N,
one must choose a composite number N=LM since L2+M2<<N2
for large N.
   Now divide the sequence into M smaller sequences of length L,
take M smaller L-point DFTs, and combine these into a large
DFT using L smaller M-point DFTs. This is the essence of the
divide-and-combine approach.

n  Ml  m, 0  l  L  1, 0  m  M  1
k  p  Lq , 0  p  L  1, 0  q  M  1
Divide-and-combine approach
M 1 L 1
X ( p, q )     x(l , m)WN Ml  m )( p  Lq )
(

m 0 l 0
M 1
 mp  L 1       
  WN  x(l , m)WNMlp  WNLmq
m 0      l 0       
                     
M 1                       mq
 mp  L 1      lp  
Twiddle factor     WN  x(l , m)WL  WM
m 0      l    
0         

 L int DFT   
   
po

M  po int DFT

Three-step procedure: P155
Divide-and-combine approach
   The total number of complex multiplications
for this approach can now be given by
   CN=ML2+N+LM2<o(N2)
   This procedure can be further repeat if M or L
are composite numbers.
   When N=Rv, then such algorithms are called
radix-R FFT algorithms.
A 8-point DFT→FFT example
   1. two 4-point DFT for m=1,2
1    pm  3         mq
           
X ( p, q)  W8  W4 x(l , m) W2
 l 0
pl


m 0 
             
W4 00 W4 01 W4 02 W4 03  x(0,0) x(0,1)
                                           
 3  W410 W411 W412 W413   x(1,0) x(1,1) 

F ( p, m)                                   *              
 l 0  W4 20 W4 21 W4 22 W4 23   x(2,0) x(2,1)
 
 30 31 32 33   x(3,0) x(3,1) 
W4 W4 W4 W4  
                            

2. G ( p, m)  W8         pm
F ( p, m)
is a 4  2 matrix dot mult. of 8 cplx multns
3. 2  poi nt DFTX with

 W8                   
1
( p, q )            pm
F ( p, m) W2     mq

m 0

G (0,0) G (0,1) 
G (1,0) G (1,1)  W 00 W 01 
                * 2          
2
G (2,0) G (2,1) W 10 W 11 
                  2     2    
G (3,0) G (3,1) 
Number of multiplications
   A 4-point DFT is divided into two 2-point
DFTs, with one intermedium matrix mult.
 number of multiplications=
4×4cplx→ 2 ×1+ 1 ×4 cplx       16 →6
   A 8-point DFT is divided into two 4-point
DFTs, with one intermedium matrix mult.
   8×8→2 ×6 + 2×4        64 →20
   For 16-point DFT:
   16×16→2 ×20 + 2×8          256 →56
In general the reduction of mult.

   If N=M*L
   N-pt DFT divided into
   M times L-pt DFT +
   Intermediate matrix transform +
   L times M-pt DFT
   CN=ML2+ML+LM2=N(M+L)<N2
Radix-2 FFT Algorithms
   Let N=2v; then we choose M=2 and L=N/2 and divide x(n) into
two N/2-point sequence.
   This procedure can be repeated again and again. At each stage
the sequences are decimated and the smaller DFTs combined.
This decimation ands after v stages when we have N one-point
sequences, which are also one-point DFTs.
   The resulting procedure is called the decimation-in-time FFT
(DIF-FFT) algorithm, for which the total number of complex
multiplications is: CN=Nv= N*log2N;
   using additional symmetries: CN=Nv= N/2*log2N
   Signal flowgraph in Figure 5.19
   function y=mditfft(x)
   % 本程序对输入序列 x 实现时域抽取快速傅立叶变换DIT-FFT基2算法
   m=nextpow2(x); n=2^m;             % 求x的长度对应的2的最低幂次m
   if length(x)<n
       x=[x,zeros(1,n-length(x))]; % 若x的长度不为2的幂次，补零到2的整数幂
   end
   nxd=bin2dec(fliplr(dec2bin([1:n]-1,m)))+1; % 求1:2^m数列的倒序
   y=x(nxd);                    % 将x倒序排列得y的初始值
   for mm=1:m % 将DFT作m次基2分解,从左到右，对每次分解作DFT运算
   le=2^mm;u=1;           % 旋转因子u初始化为w^0=1
   w=exp(-i*2*pi/le); % 设定本次分解的基本DFT因子w=exp(-i*2*pi/le)
     for j=1:le/2     % 本次跨越间隔内的各次蝶形运算
       for k=j:le:n   % 本次蝶形运算的跨越间隔为le=2^mm
        kp=k+le/2;     % 确定蝶形运算的对应单元下标
        t=y(kp)*u;     % 蝶形运算的乘积项
        y(kp)=y(k)-t; % 蝶形运算
        y(k)=y(k)+t; % 蝶形运算
       end
     u=u*w;           % 修改旋转因子,多乘一个基本DFT因子w
     end
   end
Decimation-in-frequency FFT
   In an alternate approach we choose L=2, M=N/2 and
follow the steps in (5.49).
   We can get the decimation-frequency FFT (DIF-FFT)
algorithm.
   Its signal flowgraph is a transposed structure of the
DIT-FFT structure.
   Its computational complexity is also equal to CN=Nv=
N/2*log2N
Matlab Implementation
   Function: X = fft(x,N)
   If length(x)<N, x is padded with zeros.
   If the argument N is omitted, N=length(x)
   If x is matrix, fft computes the N-point DFT of each
column of x
   It is written in machine languag and not use the
Matlab command. Therefore, it executes very fast.
   It is written as a mixed-radix algorithm.
   N=2v; N=prime number, it is reduced to the raw DFT.
Fast Convolutions
   Use the circular convolution to implement the linear
convolution and the FFT to implement the circular
convolution.
   The resulting algorithm is called a fast convolution algorithm.
   If we choose N=2v and implement the radix-2 FFT, then the
algorithm is called a high-speed convolution.
   If x1(n) is N1-point, x2(n) is N2-point, then
log 2 ( N1  N 2 1) 
N 2
   Compare the linear convolution and the high-speed conv.
High-speed Block Convolution
   Overlap-and-save method
   We can now replace the DFT by the radix-2
FFT algorithm to obtain a high-speed overlap-
and-save algorithm.
Examples
   Ex5.21 MATLAB verification of DFT time
versus data length N;
   Ex5.22 Convolution-time comparison of
using FFT versus DFT;
   Fast block-convolution using FFT;
参考书阅读和作业
   Textbook: pp.116~172
   Chinese reference book: pp.35~37,
pp.68~82 ,pp.97~109
   Exercises:
   1:   p5.1b,d; 选ch.1.3a
   2:   p5.3; p5.6; 选p5.9
   3:   p5.15;5.18a,d;
   4:   p5.23;p5.29;

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