Chapter 9 computation of the dft

Biomedical Signal processing
Chapter 9 Computation of the Discrete
Fourier Transform
Zhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong
University

02/19/13 1
1 Zhongguo Liu_Biomedical Engineering_Shandong Univ

Chapter 9 Computation of the
Discrete Fourier Transform
9.0 Introduction
9.1 Efficient Computation of Discrete Fourier Transform
9.2 The Goertzel Algorithm
9.3 decimation-in-time FFT Algorithms
9.4 decimation-in-frequency FFT Algorithms
9.5 practical considerations （ software realization)

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] the inverse
as
DFT of X 3 [ k ]
Why not convolve the two sequences directly?

There are efficient algorithms called Fast
Fourier Transform (FFT) that can be orders of
3 magnitude more efficient than others.

9.1 Efficient Computation of Discrete
Fourier Transform
The DFT pair was given as
N −1
− j ( 2π / N ) kn 1 N −1
j ( 2π / N ) kn
X [ k ] = ∑ x[n]e x[n] = ∑ X [ k] e
n =0
N k =0

Baseline for computational 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 4

9.1 Efficient Computation of Discrete
Fourier Transform
N −1
− j ( 2π / N ) kn
X [ k ] = ∑ x[n]e
n =0

Complexity in terms of real operations
4N2 real multiplications
2N(N-1) real additions (approximate 2N2)

5 5

9.1 Efficient Computation of
Discrete Fourier Transform
Most fast methods are based on Periodicity
properties
( Periodicity in n−and /k;) Conjugate )symmetry( 2π / N ) kn
− j 2π / N ) k ( N − n ) j ( 2π N kN − j ( 2π / N k ( − n ) j
e =e e =e
− j ( 2π / N ) kn − j ( 2π / N ) k ( n + N ) j ( 2π / N ) ( k + N ) n
e =e =e
Re { } ]

6 6

Makes use of the periodicity j ( 2π / N ) Nk
e = e j 2π k = 1
Multiply DFT equation with this factor
j ( 2π / N ) kN
N −1
− j ( 2π / N ) rk N −1
j ( 2π / N ) k ( N −r )
X [ k] = e ∑ x[r ]e = ∑ x[r ]e
r =0 r =0
∞
j ( 2π / N ) k ( n −r )
Define yk [ n ] = ∑ x[r ]e u[ n − r]
r =−∞
using x[n]=0 for n<0 and n>N-1
X [ k ] = yk [ n ] n = N
X[k] can be viewed as the output of a filter to the input x[n]
Impulse response of filter: j ( 2π / N ) kn
h[n] = e u [ n]
X[k] is the output of the filter at time n=N
7 7

Goertzel j ( 2π / N ) kn
h[n] = e u[n] = W − knu[n]
Filter: N

1
Hk ( z ) =
1 − WN k z −1
−

−
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
N −1
X [ k ] = ∑ x[n]WN
kn

n =0
Computational complexity
4N real multiplications; 4N real additions
Slightly less efficient than the direct method
But it avoids computation and storage of kn
WN
8 8

Second Order Goertzel Filter
Goertzel Filter
1
Hk ( z ) = 2π
j k −1
1− e N z

Multiply both numerator and denominator
− j 2π k −j
2π
k
1− e N
z −1 1− e N
z −1
Hk ( z ) = =
 2π
−1 
− j k −1 
2π 2π k −1 −2
1 − e N z ÷ 1 − e N z ÷ 1 − 2 cos N z + z
j k

  
2π 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
9 9

2π 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
Complexity for one DFT coefficient ( x(n) is complex
sequence).
Poles: 2N real multiplications and 4N real additions
Zeros: Need to be implement only once:
4 real multiplications and 4 real additions
Complexity for all DFT coefficients
Each pole is used for two DFT coefficients
Approximately N2 real multiplications and 2N2 real
additions
10 10


2π 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 do not need to evaluate all N DFT coefficients
Goertzel Algorithm is more efficient than FFT
if
less than M DFT coefficients are needed,M <
log2N
11 11


Makes use of both periodicity and symmetry
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
N −1
− j ( 2π / N ) kn
X [ k ] = ∑ x[n]e
n =0

− j ( 2π / N ) kn − j ( 2π / N ) kn
= ∑ x[n]e
n even
+ ∑ x[n]e
n odd
12 12

− j ( 2π / N ) kn − j ( 2π / N ) kn
X [ k] = ∑ x[n]e + ∑ x[n]e
n even n odd
Substitute variables n=2r for n even and n=2r+1 for odd

N / 2 −1 N / 2 −1
X [ k] = ∑ x[2r ]W 2 rk
N + ∑ x[2r + 1]W ( 2 r +1) k
N
r =0 r =0
N /2 −1 N /2 −1
= ∑
r =0
x[2r ]WN /2 + WN
rk k
∑
r =0
x[2r + 1]WN / 2
rk

= G[ k] +W H [ k] k − j 2π 2 − j 2π
N W 2
N =e N = e N /2 = WN /2
G[k] and H[k] are the N/2-point DFT’s of each subsequence

13 13

N /2 −1 N /2 −1
X [ k] = ∑ x[2r ]W rk
N /2 +W k
N ∑ x[2r + 1]W rk
N /2
r =0 r =0

= G[ k] +W H [ k]k
− j 2π 2 rk − j 2π rk
N
e N = e N /2 = WNrk/2
N −1
k = 0,1,..., k = 0,1,..., N
2
 N  N
G k +  = G [ k ] H k +  = H [ k ]
 2  2

14 14

8-point DFT using decimation-in-time

15 Figure 9.3

computational complexity
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
16 2
 16


Repeat same process ,
Divide N/2-point DFTs
into
Two N/4-point DFTs
Combine outputs

N=8

17 17

After two steps of decimation in
time

Repeat until we’re left with two-point DFT’s
18 18

flow graph for 8-point decimation in time

Complexity:
19 Nlog2N complex multiplications and additions 19

Butterfly Computation

Flow graph constitutes of butterflies

We can implement each butterfly with one multiplication

Final complexity for decimation-in-time FFT
(N/2)log2N complex multiplications and additions
20 20

Final flow graph for 8-point decimation in
time

Complexity:
(Nlog2N)/2 complex multiplications and Nlog2N additions
21 21

9.3.1 In-Place Computation 同址运
算
Decimation-in-time flow graphs require two sets of
registers
Input and output for each stage

X 0 [ 0] = x [ 0] x [ 0] X 2 [ 0] X [ 0]
X 0 [ 1] = x [ 4] x [ 4] X 2 [ 1] X [ 1]
X 0 [ 2] = x [ 2] x [ 2] X 2 [ 2] X [ 2]
X 0 [ 3] = x [ 6] x [ 6] X 2 [ 3] X [ 3]
X 0 [ 4] = x [ 1] x [ 1] X 2 [ 4] X [ 4]
X 0 [ 5] = x [ 5 ] x [ 5] X 2 [ 5] X [ 5]
X 0 [ 6] = x [ 3] x [ 3] X 2 [ 6] X [ 6]
22X 0 [ 7] = x [ 7] x [ 7] X 2 [ 7] X [ 7] 22

9.3.1 In-Place Computation 同址运算

Note the arrangement of the input indices
Bit reversed indexing （码位倒置）

X 0 [ 0] = x [ 0] ↔ X 0 [ 000] = x [ 000] x [ 0] X [ 0]
X 0 [ 1] = x [ 4] ↔ X 0 [ 001] = x [ 100] x [ 4] X [ 1]
X 0 [ 2] = x [ 2] ↔ X 0 [ 010] = x [ 010] x [ 2] X [ 2]
X 0 [ 3] = x [ 6] ↔ X 0 [ 011] = x [ 110] x [ 6] X [ 3]
X 0 [ 4] = x [ 1] ↔ X 0 [ 100] = x [ 001] x [ 1] X [ 4]
X 0 [ 5] = x [ 5] ↔ X 0 [ 101] = x [ 101] x [ 5] X [ 5]
X 0 [ 6] = x [ 3] ↔ X 0 [ 110] = x [ 011] x [ 3] X [ 6]
X 0 [ 7 ] = x [ 7 ] ↔ X 0 [ 111] = x [ 111] x [ 7] X [ 7]
23 23

cause of bit-reversed order

binary coding for
position ：
000
001

010
011

100
101

110
111
must padding 0 to Figure 9.13
24 N = 2M

9.3.2 Alternative forms
Note the arrangement of the input indices

X 0 [ 0] = x [ 0] ↔ X 0 [ 000] = x [ 000] x [ 0] X [ 0]
X 0 [ 1] = x [ 4] ↔ X 0 [ 001] = x [ 100] x [ 4] X [ 1]
X 0 [ 2] = x [ 2] ↔ X 0 [ 010] = x [ 010] x [ 2] X [ 2]
X 0 [ 3] = x [ 6] ↔ X 0 [ 011] = x [ 110] x [ 6] X [ 3]
X 0 [ 4] = x [ 1] ↔ X 0 [ 100] = x [ 001] x [ 1] X [ 4]
X 0 [ 5] = x [ 5] ↔ X 0 [ 101] = x [ 101] x [ 5] X [ 5]
X 0 [ 6] = x [ 3] ↔ X 0 [ 110] = x [ 011] x [ 3] X [ 6]
X 0 [ 7 ] = x [ 7 ] ↔ X 0 [ 111] = x [ 111] x [ 7] X [ 7]
25 25


strongpoint ： in-place computations
shortcoming ： non-sequential access of data
Figure 9.14
26

Figure 9.15

shortcoming ： not in-place computation
non-sequential access of data
27

Figure 9.16

shortcoming ： not in-place computation
strongpoint: sequential access of data
28

− j ( 2π / N ) kn − j ( 2π / N ) kn
X [ k] = ∑ x[n]e + ∑ x[n]e
n even n odd
Substitute variables n=2r for n even and n=2r+1 for odd

N / 2 −1 N / 2 −1
X [ k] = ∑ x[2r ]W 2 rk
N + ∑ x[2r + 1]W ( 2 r +1) k
N
r =0 r =0
Review
N /2 −1 N /2 −1
= ∑
r =0
x[2r ]WN /2 + WN
rk k
∑
r =0
x[2r + 1]WN / 2
rk

= G[ k] +W H [ k] k − j 2π 2 − j 2π
N W 2
N =e N = e N /2 = WN /2

29 29

X 0 [ 000] = x [ 000] x [ 0] X [ 0]
X 0 [ 001] = x [ 100] x [ 4] X [ 1]
X 0 [ 010] = x [ 010] x [ 2] X [ 2]
X 0 [ 011] = x [ 110] x [ 6] X [ 3]
X 0 [ 100] = x [ 001] x [ 1] X [ 4]
X 0 [ 101] = x [ 101] x [ 5] X [ 5]
X 0 [ 110] = x [ 011] x [ 3] X [ 6]
X 0 [ 111] = x [ 111] x [ 7] X [ 7]

30 30


strongpoint ： in-place computations
shortcoming ： non-sequential access of data
Figure 9.14
31

9.4 Decimation-In-Frequency FFT Algorithm
N −1
The DFT equation X [ k ] = ∑ x[n]WN
nk

n =0
Split the DFT equation into even and odd frequency indexes

N −1 N / 2 −1 N −1
X [ 2r ] = ∑ x[n]WN 2 r =
n
∑ x[n]WN 2 r +
n
∑ x[n]WN 2 r
n

n =0 n =0 n= N / 2
N /2 −1 N / 2 −1
Substitute
variables
= ∑ x[n]W
n =0
n2r
N + ∑ x[n + N / 2]W
n =0
( n + N /2 ) 2 r
N

N / 2 −1
= ∑ ( x[n] + x[n + N / 2]) W
n =0
nr
N /2

N /2 −1
= ∑ rn
g (n)WN / 2
32 n =0
32

9.4 Decimation-In-Frequency FFT Algorithm
N −1
The DFT equation X [ k ] = ∑ x[n]WN
nk

n =0
N −1 N /2 −1 N −1
X [ 2r + 1] = ∑ x[n]W n (2 r +1)
N = ∑ x[n]W n (2 r +1)
N + ∑ x[n]W n (2 r +1)
N
n=0 n=0 n = N /2
N /2 −1 N /2 −1
= ∑
n =0
x[n]W n (2 r +1)
N + ∑ x[n + N / 2]W
n =0
N
( n + N / 2 ) (2 r +1)

N /2 −1
= ∑ ( x[n] − x[n + N / 2]) W
n =0
n (2 r +1)
N

N / 2 −1 N /2 −1
= ∑ ( x[n] − x[n + N / 2]) W W n
N
rn
N /2
= ∑n =0
h(n)WN WNn2
n r
/
n =0
N
n ( 2 r +1) (2 r +1)
W N =W W =W W
2 rn
N
n
N
rn
N /2
n
N W 2
= WNNrWNN / 2 = −1
33 N
33

decimation-in-frequency decomposition of an N-
point DFT to N/2-point DFT

N /2 −1 N /2 −1
X [ 2r ] = ∑ ( x[n] + x[n + N / 2]) WN /2=
nr
∑ rn
g (n)WN /2
n = 0 /2 −1
N n =0 N /2 −1
X [ 2r + 1] =
34 ∑
n =0
( x[n] − x[n + N / 2]) WN W
n rn
N /2
= ∑
n =0
h(n)WN WNn2
n r

34
/

decimation-in-frequency decomposition of an 8-
point DFT to four 2-point DFT

N / 4 −1 N / 4 −1
X [ 2* 2 s ] = ∑ [ g (n) + g (n + N / 4)]WNsn =
/4 ∑ p(n)WNsn
/4
n =0 n =0
N / 4 −1 N /4 −1
X [ 2*(2 s + 1) ] = ∑ [ g (n) − g (n + N / 4)]W W
2n sn
= ∑ q ( n)WN nWNn
2 s
35 n =0
N N /4
n =0 35
/4

2-point DFT

X v ( p ) = X v−1 ( p ) + X v −1 (q )

X v (q ) =  X v −1 ( p ) − X v−1 (q )  W80
  when N = 8

36 36

N /2 −1 N /2 −1
X [ 2r ] = ∑ ( x[n] + x[n + N / 2]) nr
WN /2 = ∑ rn
g (n)WN /2
n =0 n =0

N /4 −1 N /2 −1
X [ 2* 2 s ] = ∑ g (n)WN /2 +
2 sn
∑ 2 sn
g (n)WN /2
n =0 n = N /4
N /4 −1 N /4 −1
= ∑
n =0
g (n)WN /2 +
2 sn
∑
n =0
g (n + N / 4)WN /2( n + N /4)
2s

N /4 −1 N /4 −1
= ∑ g (n)W
n =0
sn
N /4 + ∑ g (n + N / 4)W
n =0
sn
N /4

N /4 −1 N /4 −1
= ∑ [ g (n) + g (n + N / 4)]W sn
N /4
= ∑
n =0
p(n)WNsn
/4
n =0

37

N /2 −1 N /2 −1
X [ 2r ] = ∑ ( x[n] + x[n + N / 2]) nr
WN /2 = ∑ rn
g (n)WN /2
n =0 n =0
N /2 −1
X [ 2*(2 s + 1) ] = ∑ g (n)WN /2 +1) n
(2 s

n =0
N /4 −1 N /2 −1
= ∑
n =0
g (n)WN /2+1) n +
(2 s
∑
n= N / 4
g (n)WN /2 +1) n
(2 s

N /4 −1 N /4 −1
= ∑
n =0
g (n)WNsn WN /2 +
/4
n
∑
n =0
g (n + N / 4)WN /2+1)( n + N /4)
(2 s

N /4 −1 N /4 −1
= ∑
n =0
g (n)WNsn WN n +
/4
2
∑
n =0
g (n + N / 4)WNsn WN nWN / 2+1) N /4
/4
2 (2 s

N /4 −1 N /4 −1

= ∑ [ g (n) − g (n + N / 4)]W 2n
N W sn
N /4
= ∑
n =0
q (n)WN nWNsn
2
/4
n =0
38 WN /2 +1) N /4 = WNsN /2WNN/2 = −1
(2 s
/2
/4

N /4 −1
X [ 2* 2 s ] = ∑ p (n)WNsn
/4
n =0

N /4 −1
X [ 2* 2* 2t ] = ∑ p (n)W 2 tn
N /4
n =0
N /8 −1 N /4 −1
= ∑
n =0
p (n)W 2 tn
N /4 + ∑
n = N /8
p (n)W 2 tn
N /4

N /8 −1 N /8 −1
= ∑n =0
p (n)W 2 tn
N /4 + ∑
n =0
p(n + N / 8)W 2 t ( n + N /8)
N /4

N /8 −1
= ∑ n =0
[ p(n) + p (n + N / 8)]WN /8
tn

= p(n) + p (n + 1) when N = 8
39

N /4 −1
X [ 2* 2 s ] = ∑ p (n)WNsn
/4
n =0
N /4 −1
X [ 2* 2*(2t + 1) ] = ∑ p(n)WN /4+1) n
(2 t

n =0
N /8 −1 N /4 −1
= ∑n =0
p (n)WN /4+1) n +
(2 t
∑
n = N /8
p (n)WN / 4+1) n
(2 t

N /8 −1 N /8 −1
= ∑n =0
p (n)WN /4+1) n +
(2 t
∑
n =0
p (n + N / 8)WN /4+1)( n + N /8)
(2 t

N /8 −1 N /8−1
= ∑
n =0
p (n)WN /4WN /4 +
2 tn n
∑
n =0
p (n + N / 8)WN /4WN /4WN /4+1) N /8
2 tn n (2 t

N /8 −1
= ∑ [ p(n) − p(n + N / 8)]WN /8WN n
tn 4

n =0 WN(2/4+1) N /8 = WNtN /4WNN/4 = − 1
t
/4
/8

= [ p (n) − p (n + 1)]W80 when N = 8
40

Final flow graph for 8-point DFT decimation
in frequency

41 41

9.4.1 In-Place Computation 同址运
算
DIF
FFT

DIT FFT

42 42


DIF
FFT

DIT FFT

43 43

decimation-in-frequecy Butterfly Computation

decimation-in-time Butterfly Computation

44 44

The DIF FFT is the transpose of the DIT FFT

DIF
FFT

DIT FFT

45 45


DIF
FFT

DIT FFT

46


DIF
FFT

DIT FFT

47

Figure 9.24 erratum
x [ 0]
x [ 4]
x [ 2]
x [ 6]
x [ 1]
x [ 5]
x [ 3]
x [ 7]
48


DIF
FFT

DIT FFT

49

Chapter 9 HW
9.1, 9.2, 9.3,

50
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Chapter 9 computation of the dft

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