Biomedical Signal processing
Chapter 9 Computation of the Discrete
Fourier Transform
Zhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong
University
2017/7/28
1
Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 9 Computation of the
Discrete Fourier Transform
9.0 Introduction
9.1 Direct Computation Of The Discrete Fourier
Transform
9.2 decimation-in-time FFT Algorithms
9.3 decimation-in-frequency FFT Algorithms
9.4 practical considerations
9.5 ......
2
9.0 Introduction
Implement a convolution of two sequences
by the following procedure:
1. Compute the N-point DFT X 1k  and X 2 k 
of the two sequence x1n and x2 n
2. Compute X 3 k   X 1k X 2 k  for 0  k  N 1
3. Compute x3 n  x1 n N x2 n as the inverse
DFT of X 3 k 
Why not convolve the two sequences directly?
DFT has efficient FFT(Fast Fourier Transform)
algorithms that can be orders of magnitude
more efficient than others.
3
9.1 Direct Computation of the
Discrete Fourier Transform
The DFT pair was given as
N 1
X k   
n 0
1
 j  2 / N  kn
x[n]e
, x[n]  N
N 1
 X k  e
k 0
DFTcomputational complexity:
Each DFT coefficient requires
N complex multiplications;
N-1 complex additions
All N DFT coefficients require
N2 complex multiplications;
N(N-1) complex additions
4
j  2 / N  kn
0k , n  N 1
4
9.1 Direct Computation of the
Discrete Fourier Transform
Most fast methods are based on Periodicity properties
Periodicity in n and k;
 j  2 / N  k  N n
 j  2 / N kn
 j  2 / N k  n N 
e
e
e
kn
N
W
k ( N n)
N
W
( N k )n
N
W
,
e
 j  2 / N 
 WN
Complex conjugate symmetry
WNk ( N  n )  WN kn  (WNkn )*,
e
 j  2 / N k  N n 
e
(WNk ( N  n ) )*  (WN kn )*  WNkn
 j  2 / N kN  j  2 / N k   n 
e
e
j  2 / N kn
5
5
9.1.1 Direct Evalutation of the
Defination of DFT
N 1
X  k    x[n]e
 j  2 / N  kn
n 0
N 1
N 1
  x  n WNkn
n 0
 
 
 j  Re x[n] ImW   Imx[n] Re W 

kn
kn

X  k     Re  x[n] Re WN  Im  x[n] Im WN
n 0
kn
N
kn
N
k  0,1, 2, , N  1
Complexity in terms of real operations
4N2 real multiplications
N(4N-2) real additions : N (2(N-1) +2N )
6
6
9.1.1 Direct Evalutation of the Defination of DFT
k ( N n)
 kn
kn
Conjugate symmetry:
WN
 WN  (WN ) *
N 1
 
 
Re  X  k     Re  x[n] Re WNkn  Im  x[n] Im WNkn 
n 0
N 1
2
group terms in the summation for n and (N-n):
 
N 1


kn
kn

Re
x
[
n
]
Re
W
+
Re
x
[
n
]
Re
W
 
 


N
N
n 0
n  N
2
n=N  n，n =N  n
N
2
N 1
2
1

k [ N n ]
Re
x
[
N

n
]
R
e
W

 
N
 
n 0



 
kn
k [ N n ] 

Re  x[n] Re WN  Re x[ N  n] Re WN

 N

n 0
kn
1
2
Re
W
N
kn 

   Re  x[n]  Re  x[ N  n] Re WN


乘法次数减少一半
n 0
 
7
9.1.1 Direct Evalutation of the Defination of DFT
Conjugate symmetry:
WNk ( N  n )  WN kn  (WNkn ) *
N 1
 
 
Re  X  k     Re  x[n] Re WNkn  Im  x[n] Im WNkn 
n 0
N 1
2
group terms in the summation for n and (N-n):
 


 
kn
k [ N n ] 

Re  x[n] Re WN  Re x[ N  n] Re WN



n 0
N 1
kn
2
Re
W
N
kn
   Re  x[n]  Re  x[ N  n] Re WN 


n 0
 
N 1
2
similarly,
 


 
kn
k [ N n ] 


Im
x
[
n
]
Im
W

Im
x
[
N

n
]
I
m
W
 



N
N

 kn
n 0 N
1
 Im WN
乘法次数减少一半
2
     Im  x[n]  Im  x[ N  n] Im WNkn 


n 0
 
8
9.1.2 The Goertzel Algorithm
Makes use of the periodicity W  kN  e j  2 / N  Nk  e j 2 k  1
N
Multiply DFT equation with this factor
X k   e
j  2 / N  kN
Define yk  n  
N 1
 x[r ]e
r 0

 j  2 / N  rk
 x[r ]e
N 1
  x[r ]e
r 0
j  2 / N  k  n r 
r 
j  2 / N  k  N  r 
h[n  r ]
u  n  r   x[n]* h[n]
n  N，0  r  N -1
0  r  N -1
using x[n]=0 for n<0 and n>N-1, X  k   yk  n 
n N
yk[n] can be viewed as output of a filter to input x[n]
Filter impulse response :
h[n]  e
j  2 / N kn
u  n
X[k] is the output of the filter at time n=N.
DFT of x[n]
Goertzel Filter
9
9
9.1.2 The Goertzel Algorithm
 kn
j  2 / N kn
Goertzel Filter: h[n]  e
u[n]  WN u[n]
1
Hk  z  
1  WN k z 1
k  0,1,..., N -1
for causal LTI system
x[n] is a complex
x[n]=0, n<0
yk [n]  yk [n  1]WN k  x[n], n  0,1,..., N , yk [1]  0
X  k   yk  n  n  N , k  0,1,..., N -1
N 1
X  k    x[n]WNkn
n 0
Computational complexity for one X[k].
4N real multiplications; 4N real additions;
Slightly less efficient than the direct method(4N, 4N-2);
But it avoids computation and storage of WNkn (N2 个)
k
WN (N 个)
10
Second Order Goertzel Filter
Goertzel Filter
1

Hk  z  
 k 1
1  WN z
y[n]
为便于分析
y[n]
yk [ n ]
1
1 e
j
2
k 1
N z
x[n]=0, n<0
Multiply both numerator and denominator for causal LTI system
 j 2 k 1
k 1
N
1  WN z
1 e
z

Hk  z  
2
2
2 k 1 2
j k

j k



1
1
1  2 cos
z z
N
N
1

e
z
1

e
z



N



y[1]  y[2]  0
2 k
y[n]   y[n  2]  2 cos
y[n  1]  x[n], n  0,1,..., N
N
k
实现极点因
yk [n]  y[n]  WN y[n  1]
实现零点因
式
k
yk [ N ]  y[ N ]  WN y[ N  1] 式
 X  k  , k  0,1,..., N -1
每个K只需计算 1次
11
Second Order Goertzel Filter 除n=0时y[0]=x[0], y[n]对每
个n有两个实数乘4个实数加
实现极点因式:
2 k
y[n]   y[n  2]  2 cos
y[n  1]  x[n], n  0,1,..., N
N
实现零点因式:
yk [ N ]  y[ N ]  WNk y[ N  1]  X  k  , k  0,1,..., N -1
same
Complexity for one X[k] ( x[n], y[n] is complex).
Poles: 2N real multiplications
2 ( N  k ) and 4N real additions
y[n]   y[n  2]  2 cos
y[n  1]  x[n], n  0,1,..., N
Zeros: computed onlyNonce: 4 real ×and 4 real + .
total: 2(N+2) real ×and 4(N+1) real + .
乘 N /2
Complexity for all N X[k]
N
Each pole is used for X[k], X[N-k] , k  0,1,..., 2 -1
Approximately N2 real multiplications and 2N2 real
additions. More efficient than the direct method.
12
12
Second Order Goertzel Filter
2 k
y[n]   y[n  2]  2 cos
y[n  1]  x[n], n  0,1,..., N
N
yk [ N ]  y[ N ]  WNk y[ N  1]  X  k  , k  0,1, ..., N
If we only need to evaluate M X[k] , the total
computation is proportional to NM,
Goertzel method is attractive when M is small.
 The computation for FFT algorithm is Nlog2N.
if M < log2N, either Goertzel Algorithm or
direct method is more efficient than FFT.
13
13
9.2 Decimation-in-Time FFT Algorithms
Makes use of periodicity and symmetry of WNkn
Consider special case of N=2ν : integer ν power of 2
Separate x[n] into two sequences of length N/2
Even-numbered sequence ;
Odd-numbered sequences .
N 1
X  k    x[n]e
 j  2 / N  kn
,
n 0

 x[n]e
n even
14
 j  2 / N kn

e
 j  2 / N 
 x[n]e
 WN
 j  2 / N  kn
n odd
14
9.2 decimation-in-time FFT Algorithms
X k  
 x[n]e
 j  2 / N kn
n even
 x[n]e

 j  2 / N kn
n odd
Substitute variables n=2r for n even and n=2r+1 for odd
X k  

N /2 1
 x[2r ]W
2 rk
N
r 0
N /2 1

r 0
rk
N /2
x[2r ]W
g[r]
W
k
N
 G  k   WNk H k  ,

 x[2r  1]W

2
N
W
N/2-point DFT
N /21
x[2r  1]W
h[r]
 2 r 1 k
N
r 0
N /2 1
r 0
N /2 1
rk
N /2
 j 2 2
e N
X 1k  x1[m]W
m0
 j 2
 e N /2
mk
N /2
 WN /2
G[k] and H[k] are the N/2-point DFTs of each subsequence:
N
g[n]=x[2n] and h[n] =x[2n+1] , n  0,1,..., -1.
2
15
15
9.2 decimation-in-time FFT Algorithms
X k  
N /2 1
N /2 1
 x[2r ]W  W  x[2r  1]W N
k  ,..., N -1




N
2
G
G kkW  H kW
k
,



, kH

 0,1,...,

1


k  0,1,..., N -1

rk
N /2
r 0
k
NN
2

k
N
k
N

 N
 N
X k    G k    WN
2
2


k
N
2
r 0
2
N
2

 N
H  k    G  k   WN
2

2 r ( k  N / 2)

j
r ( k  N /2 )
W
 e N /2
N /2
rk
N /2
k
 j 2 rk
 e N /2
N
2
H k 
 WNrk/2
N

 G k    G k  ,
2

N

H k    H k 
2

G[k] and H[k] are the N/2-point DFTs of each subsequence:
N
g[n]=x[2n] and h[n] =x[2n+1] , n  0,1,..., -1.
16
2
16
X k  
N /2 1

r 0
N /2 1
k
using
DIT
x[2r ]WNrk/28-point
 WNk  DFT
x[2r  1]
WNrk/2  G
k

W
  N H k 
r 0
N
k

N


X k    G  k   WN 2 H  k 
 2
 N
G k +   G k 
 2
 N
H k +   H k 
 2
W
0
N
W
1
N
WN2
WN3
WN4
W
W
17
Figure 9.4
5
N
6
N
0
1
2
3
N
k  0,..., -1
2
4
5
6
7
WN7 k  N ,..., N -1
2
computational complexity
X k   G k   WNk H k 
 
Two N/2-point DFTs
N N 1
exactly:
2
2 2
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
 
2
N
N
N
complex additions. exactly: 2
1 N 
2 2
2
More efficient than direct DFT when N>2
N2/2+N
18
18
9.2 decimation-in-time FFT Algorithms
Repeat same process , Divide
N/2-point DFTs into
Two N/4-point DFTs
Combine outputs
N=8
0
N /2
1
N /2
W
W
2
N /2
W
W
W
W
0
N
2
N
4
N
WN3/2  WN6
19
19
9.2 decimation-in-time FFT Algorithms
After two steps of decimation in time
W20  WN0
Repeat until we’re left with two-point DFT’s
20
W21  WNN /2
20
9.2 decimation-in-time FFT Algorithms
flow graph for 8-point decimation in time
binary code:
n2 n1n0
N=8=23
3 stages:
N=2m
m=log2N
m级: 每级乘法
和加法个数为
N
Complexity:
Nlog2N complex multiplications and additions
21
21
Butterfly Computation
Flow graph of basic butterfly computation
W
r
N
WNr
 r  N /2
WN
r
N
N /2
N
=W W
WNN /2  1
We can implement each butterfly with one multiplication
Final complexity for decimation-in-time FFT
(N/2)log2N complex multiplications and Nlog2N additions
22
22
9.2 decimation-in-time FFT Algorithms
DIT flow graph for 8-point decimation in time
N=8=23
generalized to
N=2m
m=log2N
m级: 每级乘法
和加法个数为
N
WN0
WN2
WN4
WN2
WN2WNN /2  WN6
WN6  WN2WNN /2
Complexity:
Nlog2N complex multiplications and Nlog2N additions
23
23
9.2.1 Generalization and Programming the FFT
Final flow graph for 8-point decimation in time
N=8=23
0
generalized to
N=2m
m=log2N
m级: 每级乘法
N/2个,加法N个
WN
W
W
0
N
2
N
WN2
WN4-1
WN6
WN2
-1
the figure
serves as
proof,
WN2
and for FFT
WN6
-1
programming.
Complexity:
(Nlog2N)/2 complex multiplications and Nlog2N additions
24
24
9.2.2 In-Place Computation同址运算
DIT flow graphs require two sets of registers: Input, output
 In-Place Computation needs one or memory units
set of registers.
X 0  0  x  0
X 0  1  x  4
X 0  2  x  2
X 0 3  x  6
X 0  4  x 1
X 0  5  x  5
X 0  6  x 3
X 0 7  x 7
25
x  0
x  4
x  2
x  6
x 1
x  5
x 3
x 7
X 1  0
X 1  1
X 1  2
X 1  3
X 1  4
X 0
X 1
X  2
X  3
X  4
X 1  5
X  5
X 1  6
X 6
X1 7
X 7
25
9.2.2 In-Place Computation同址运算
Note the arrangement of the input indices
Bit reversed indexing（码位倒置）
X 0  000 X 0  0 x  000 x  0
X 0  001 X 0  1 x 100 x  4
X 0  010 X 0  2 x  010 x  2
X 0  011 X 0 3 x 110 x  6
X 0 100 X 0  4 x  001 x 1
X 0 101 X 0 5 x 101 x 5
X 0 110 X 0  6 x  011 x 3
X 0 111 X 0  7  x 111 x  7 
X 0
X 1
X  2
X  3
X  4
X  5
X 6
X 7
26
26
normal order
x  n2n1n0 
binary coding for position：
对应x[n]
从上向下出现的顺
的序号
n2 n1n0 n2 n1序
n0
x  000
000
x  001
001
x  010
010
x  011
011
x 100
100
x 101
101
x 110
110
x 111
111
x  0
出
现 x  4
的 x  2
顺
序 x  6
不
是 x 1
输 x  5
入
序 x 3
号 x 7
Figure 9.13
27
cause of bit-reversed order
序号的
3次奇
偶分解
x  n2n1n0 
binary coding for position：
对应x[n]
从上向下出现的顺
的序号
n2 n1n0 n0 n1序
n2 是把二进制序号翻转了
x  000
000
x  0
x 100
001
x  4
x  010
010
x  2
x 110
011
x  001
100
x 101
101
x  011
110
x 111
111
x  6
x 1
x  5
x 3
x 7
Figure 9.13
28
9.2.3 Alternative forms
Note the arrangement of the input indices
Bit reversed indexing（码位倒置）
X 0  000 X 0  0 x  000
x  0
X 0
X 0  001 X 0  1 x 100
x  4
X 1
X 0  010 X 0  2 x  010
x  2
X 0  011 X 0 3
x 110
x  6
X  2
X 0 100 X 0  4 x  001
x 1
X 0 101
X 0  5
x 101
X 0 110 X 0  6 x  011
X 0 111
29
X 0  7  x 111
x  5
x 3
x 7
X  3
X  4
X  5
X 6
X 7
29
9.2.3 Alternative forms
Figure 9.14
 input in normal order and output in bit-reversed order
 strongpoint：in-place computations
 shortcoming：non-sequential access of elements
of intermediate arrays.
30
Figure 9.15
 strongpoint： Input, output same normal order
 shortcoming：not in-place computation; non-sequential
access of elements of intermediate arrays.
31
Figure 9.16
Same structure, Input bit-reversed order , output normal order
shortcoming：not in-place computation
strongpoint: sequential access of data
32
9.3 Decimation-In-Frequency FFT Algorithm
N 1
X  k    x[n]WNnk
The DFT equation
n 0
Split the DFT equation into even and odd frequency indexes
N 1
X  2r    x[n]WNn 2 r 
n 0


N /2 1
n 0
N / 2 1
n  N /2

n  n  N / 2  n WN
nr
 x[n]  x[n  N / 2]WN / 2
n 0
N / 2 1
n2r
N

N /2 1

g[n]WNnr/2
 x[n ' N / 2]W

x[n]WNn 2 r
x[n]W
n 0
33
W

x[n]WNn 2 r 
N 1

n 0

nr
N /2
N /2 1
nn' 0
 n 'NN//2222rr
N
WNn 2 r =WNnr/2
 N /22 r
Nr
N
=W =1
N/2-point DFT
33
9.3 Decimation-In-Frequency FFT Algorithm
N 1
X  k    x[n]WNnk
The DFT equation
N 1
X  2r  1   x[n]W
n (2 r 1)
N
n 0
N /2 1
n 0
N /2 1


n 0
N /2 1

n 0
 x[n]W


N /2 1
n (2 r 1)
N

n 0
x[n]WNn ( 2 r 1) 
NN/2
/211

 xx[[nn' N / 2]W
n
N
n 0
W
34
rn
N /2
n
N
rn
N /2
 n ' NN/2/2(2(2rr1)1)
n (2 r 1)
N
W
n
N
N/2-point DFT

N /2 1

n 0
N
(2 r 1)
2
N
W W W W , W
2 rn
N
n  N /2
n  n  N / 2  n
nn'00
n (2 r 1)
x
[
n
]

x
[
n

N
/
2]
W

 N
n 2 r 1
N

x[n]WNn ( 2 r 1)
N
  x[n]  x[n  N / 2]W W

N 1
h[n]WNnWNrn/2
N
2
N
 WNNrW
 1
34
DIF decomposition: N-point DFT to N/2-point DFT
X  2r 
X  2r  1
X  2r  
N /2 1
  x[n]  x[n  N / 2]
n N
0 /2 1
X  2r  1 
35

n 0
WNnr/2
N /2 1

g[n]WNrn/2
n  0 N /2 1
rn
N /2
n
x
[
n
]

x
[
n

N
/
2]
W

 NW


n 0
h[n]WNnWNrn/2
35
DIF decomposition: N-point DFT to N/2-point DFT
8=23
N=2m
g[0]
p[0]
g[1]
p[1]
X [4]
g[2]
WN0
q[0]
X [2]
g[3]
m=log2N
X [0]
WN2
q[1]
2点
DFT
X [6]
X  2r 
X  2* 2s  
N /4 1

n 0
X  2*(2s  1)  
36
sn
g
[
n
]

g
[
n

N
/
4]
W

 N /4 
N /4 1
  g[n]  g[n  N / 4]W
n 0
2n
N
sn
N /4
W
N /4 1

n 0

p[n]WNsn/4
N /4 1

n 0
q[n]WN2 nWNsn/4
36
2-point DFT in the last stage of
decimation-in-frequency decomposition
X v [ p]  X v1[ p]  X v1[q]
X v [q]   X v1[ p]  X v1[q] W80 ,
37
when N  8
37
Final flow graph of decimation-in-frequency
decomposition of 8-point DFT
N=8=23
generalized to
N=2m
m=log2N
m级: 每级
乘法N/2个,
加法N个
N/2=4-point DFT N/4=2-point DFT
Complexity:
(Nlog
2N)/2 complex multiplications and Nlog2N additions
42
42
9.3 .1 In-Place Computation同址运算
DIF FFT
结构相同,
乘数排列
区别大
DIT FFT
43
43
9.3 .1 In-Place Computation同址运算
DIF FFT
transpose
DIT FFT
44
44
9.3 .2 Alternative forms
transpose
decimation-in-frequecy Butterfly Computation
transpose
decimation-in-time Butterfly Computation
45
45
The DIF FFT is the transpose of the DIT FFT
DIF FFT
transpose
DIT FFT
46
46
9.3 .2 Alternative forms
DIF FFT
transpose
DIT FFT
49
Chapter 9 HW
9.1, 9.6, 9.7,
50
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Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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