chapter 9 computation of the discrete fourier transform

104/19/2023 1Zhongguo Liu_Biomedical Engineering_Shandong

Univ.

Chapter 9 Computation of the Discrete

Fourier TransformZhongguo Liu

Biomedical Engineering

School of Control Science and Engineering, Shandong University

Biomedical Signal processing

2

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)

Chapter 9 Computation of the Discrete Fourier Transform

3

9.0 Introduction

1. Compute the N-point DFT and of the two sequence and

kX1 kX 2

nx1 nx2

2. Compute for kXkXkX 213 10 Nk

3. Compute as the inverse DFT of kX 3

nxNnxnx 213

Implement a convolution of two sequences by the following procedure:

Why not convolve the two sequences directly? There are efficient algorithms called Fast Fourier Transform (FFT) that can be orders of magnitude more efficient than others.

4


The DFT pair was given as

4

1

0

2 /1[ ]

N

k

j N knx n X k

Ne

1

0

2 /[ ]

N

n

j N knX k x n e

DFTcomputational complexity: Each DFT coefficient requires

N complex multiplications; N-1 complex additions

All N DFT coefficients requireN2 complex multiplications; N(N-1) complex additions

5


5

Complexity in terms of real operations4N2 real multiplications2N(N-1) real additions (approximate 2N2)

1

0

2 /[ ]

N

n

j N knX k x n e

2 / ( ), ( )* j N k N k kN N N

n nN

ne W W W W

6


Most fast methods are based on Periodicity propertiesPeriodicity in n and k; Conjugate symmetry

6

2 / 2 / 2 / 2 /j N k N n j N kN j N k n j N kne e e e 2 / 2 / 2 / j N kn j N k n N j N k N ne e e

( ) ( ) nk k N N kN N N

n nW W W

( )( )* ( )* n nk N k kN N N

nW W W

( )( )* ( )* n nk N k kN N N

nW W W

Re ]

7

X[k] can be viewed as output of a filter to input x[n]

Impulse response of filter:

X[k] is the output of the filter at time n=N

9.2 The Goertzel Algorithm

Makes use of the periodicityMultiply DFT equation with this factor

7

2 / 2 1j N Nk j ke e

1

0

1

0

2 / 2 / 2 /[ ] [ ]

N N

r

k

r

j N kN j N r j N k N rX k x r x re e e

k n NX k y n

2 /[ ]

j N knh n u ne

2 /[ ]k

r

j N k n ry n x r u n re

Define

using x[n]=0 for n<0 and n>N-1,

8

9.2 The Goertzel AlgorithmGoertzel

Filter:

Computational complexity4N real multiplications; 4N real additionsSlightly less efficient than the direct method

2 /[ ] [ ] [ ]nk

N

j N knh n u n W u ne

1

1

1k kN

H zW z

[ ] [ 0,1 1,...] [ , ,],kk k Ny n y n W x n n N [ 1] 0ky

0,1,., ..,k n NkX k Ny n

nkNWBut it avoids computation and storage of

1

0

[ ]N

knN

n

X k x n W

9

Second Order Goertzel Filter

Goertzel Filter

9

2 2

1 1

2 21 21 1

1 12

1 2cos1 1

j k j kN N

kj k j kN N

e z e zH z

kz ze z e z N

Multiply both numerator and denominator

1

2

1

1k

j kN

H z

ze

2[ ] [ 2] 2cos [ 1] 0,1,...,[ ],

ky n y n y n x n n N

N

[ ] [ ] [ 1]kk Ny y WN yN N 0,1,, ...,Nk kX

10


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 additionsComplexity for all DFT coefficients

Each pole is used for two DFT coefficients Approximately N2 real multiplications and 2N2 real

additions10

2[ ] [ 2] 2cos [ 0,1,...,1] [ ],

y n y n y n

N

kn nx N

[ ] [ ] [ 1] kk Ny y W yN N N 0,1,, ...,X kk N

2 ( )[ ] [ 2] 2 0,cos [ 1] [ ], 1,...,

y n y n y n x n

Nn

N kN

, X k X N k

11


If do not need to evaluate all N DFT coefficientsGoertzel Algorithm is more efficient than FFT if less than M DFT coefficients are needed,M < log2N

11

2[ ] [ 2] 2cos [ 1] 0,1,...,[ ],

ky n y n y n x n n N

N

[ ] [ ] [ 1]kk Ny y WN yN N 0,1,, ...,Nk kX

12


Makes use of both periodicity and symmetryConsider special case of N an integer power of 2Separate x[n] into two sequence of length N/2

Even indexed samples in the first sequenceOdd indexed samples in the other sequence

12

1

0

n even n odd

2 /

2 / 2 /

[ ]

[ ] [ ]

N

n

j N kn

j N kn j N kn

X k x n

x n x n

e

e e

13


13

/2 1 /2 1

2 12

r 0 r 0

[2 ] [2 1]N N

r krkN NX k x r W x r W

Substitute variables n=2r for n even and n=2r+1 for odd

G[k] and H[k] are the N/2-point DFT’s of each subsequence

n even n odd

2 / 2 /[ ] [ ]

j N kn j N knX k x n x ne e

, kNG k W H k

/2 1 /2 1

/2 /2r 0 r 0

[2 ] [2 1]N N

rk k rkN N Nx r W W x r W

2

/2

2/2

22

N NN

jjNW We e

14


14

G[k] and H[k] are the N/2-point DFT’s of each subsequence

kNG k W H k

/2 1 /2 1

/2 /2r 0 r 0

[2 ] [2 1]N N

rk k rkN N NX k x r W W x r W

/2

22 2/2 rk

N

rrN N

j kj kWe e

2

NG k G k

2

NH k H k

10,1,...,

2

Nk

0,1,...,k N

15

8-point DFT using decimation-in-time

Figure 9.3

/2 1 /2 1

/2 /2r 0 r 0

[2 ] [2 1]N N

rk k rkN N NX k x r W W x r W

k

NG k W H k

16

computational complexityTwo N/2-point DFTs

2(N/2)2 complex multiplications2(N/2)2 complex additions

Combining the DFT outputsN complex multiplicationsN complex additions

Total complexityN2/2+N complex multiplicationsN2/2+N complex additionsMore efficient than direct DFT

16

17


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

17

N=8

18


After two steps of decimation in time

18

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

19


flow graph for 8-point decimation in time

19

Complexity:Nlog2N complex multiplications and additions

N=2m

m=log2N

20

Butterfly Computation

Flow graph of basic butterfly computation

20

We can implement each butterfly with one multiplication

Final complexity for decimation-in-time FFT

(N/2)log2N complex multiplications and Nlog2N additions

21

Nlog2N complex multiplications and additions


Final flow graph for 8-point decimation in time

21

Complexity:

(Nlog2N)/2 complex multiplications and Nlog2N additions

N=2m

m=log2N

22

9.3.1 In-Place Computation同址运算Decimation-in-time flow graphs require two sets of registersInput and output for each stage

22

0

0

0

0

0

0

0

0

0 0

1 4

2 2

3 6

4 1

5 5

6 3

7 7

X x

X x

X x

X x

X x

X x

X x

X x

0

4

2

6

1

5

3

7

x

x

x

x

x

x

x

x

0

1

2

3

4

5

6

7

X

X

X

X

X

X

X

X

1

1

1

1

1

1

1

1

0

1

2

3

4

5

6

7

X

X

X

X

X

X

X

X

one set of registers

23

9.3.1 In-Place Computation同址运算

Note the arrangement of the input indicesBit reversed indexing （码位倒置）

23

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 000

1 001

2 010

3 0

0 000

4 100

2 010

6 110

1 001

5 101

3 011

7 111

11

4 100

5 101

6 110

7 111

X x X x

X x X x

X x X x

X x X x

X x X x

X x X x

X x X x

X x X x

0

4

2

6

1

5

3

7

x

x

x

x

x

x

x

x

0

1

2

3

4

5

6

7

X

X

X

X

X

X

X

X

24

Figure 9.13

normal order binary coding for position ：

000

001

010

011

100

101

110

111

0

4

2

6

1

5

3

7

x

x

x

x

x

x

x

x

2 1 0x n n n

000

001

010

011

100

101

110

111

x

x

x

x

x

x

x

x

2 1 0n n n2 1 0n n n

25

Figure 9.13

cause of bit-reversed order binary coding for position ：

000

001

010

011

100

101

110

111

0

4

2

6

1

5

3

7

x

x

x

x

x

x

x

x

2 1 0x n n n

000

100

010

110

001

101

011

111

x

x

x

x

x

x

x

x

2 1 0n n n 0 1 2n n n

26

9.3.2 Alternative forms

Note the arrangement of the input indicesBit reversed indexing （码位倒置）

26

0

4

2

6

1

5

3

7

x

x

x

x

x

x

x

x

0

1

2

3

4

5

6

7

X

X

X

X

X

X

X

X

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 000

1 001

2 010

3 0

0 000

4 100

2 010

6 110

1 001

5 101

3 011

7 111

11

4 100

5 101

6 110

7 111

X x X x

X x X x

X x X x

X x X x

X x X x

X x X x

X x X x

X x X x

27

Figure 9.14


strongpoint ： in-place computationsshortcoming ： non-sequential access of data

input in normal order and output in bit-reversed order

28

Figure 9.15

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

Input, output same normal order

29

Figure 9.16

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

Same structure, Input bit-reversed order , output normal order

30

9.4 Decimation-In-Frequency FFT Algorithm

The DFT equation

30

1

0

[ ]N

nkN

n

X k x n W

1

/2

1 /2 2 2

2 1

0 0

[ ] [ ] [ ]2

r rN N

n n nN N N

N

nn n N

rX x n W x n W x n Wr

/2 1

/20

[ ] [ / 2]N

nrN

n

x n x n N W

Split the DFT equation into even and odd frequency indexes

Substitute variables

/2 1

0

/2 12

0

2 2/[ ] [ / 2]

N

rn rN N

n

NN

n

nx n xW N Wn

/2 1

0/2( )

Nrn

nNg n W

/ 2 n Nn n

31

9.4 Decimation-In-Frequency FFT Algorithm

The DFT equation

31

1

0

[ ]N

nkN

n

X k x n W

2 1 21 /2 1 1

( ) ( )1 2( )

0 0

1

/2

[ ] [ ] [2 ]1 r rN N N

n n nN N N

n n

r

n N

X x n W x n W x n Wr

/2 1

(2 1)

0

[ ] [ / 2]N

n rN

n

x n x n N W

/2 1 /2 1

(2 1)(2 1) /2

0 0

2[ /[ ] ]N N

rn rN N

n n

n Nx n W x n WN

(2 1) /22 1N

r Nr NN N NW W W

2

/22 1n r rn n rn n

N NN N NW W W W W

/2 1

0/2[ ] [ / 2] n

Nrn

Nn

Nx n x n N W W

/2

/2 1

0

( )N

rnN N

n

n

Wh n W

/ 2 n Nn n

32

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

32

/2 1

0/22 [ ] [ / 2]

N

n

nrNr x n x n NX W

/2 1

0/22 1 [ ] [ / 2]

Nrn

nN

nNr x n x n N WX W

/2

/2 1

0

( )N

rnN N

n

n

Wh n W

/2 1

0/2( )

NrnN

n

Wg n

33

/4

/4 12

0

( )N

n sN N

n

n

q n W W

/

2

0/4

4 1

2*(2 1) [ ( ) ( / 4)] N N

Nn sn

n

X s g n g n N W W

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

33

/4 1

0/4( )

Nsn

nNp n W

/

0/4

4 1

2*2 [ ( ) ( / 4)]N

sn

nNX s g n g n N W

34

2-point DFT

34

081 1( ) ( ) ( )v v vX q X p X q W 8when N

1 1( ) ( ) ( )v v vX p X p X q

35

/2 1

0/22 [ ] [ / 2]

N

n

nrNX r x n x n N W

/2 1

0/2( )

Nrn

nNg n W

/4 1 /2 1

2 2

0 /4

/4 1 /4 12 2 ( /4)

0 0

/4 1 /4

/2 /2

/2 /2

/4 /4

/

1

0 0

04

/4 1

2*2 ( ) ( )

( ) ( / 4)

( ) ( / 4)

[ ( ) ( / 4)]

N Nsn sn

n n N

N Nsn s n N

n n

N Nsn sn

N N

N N

N N

N

n n

Nsn

n

X s g n W g n W

g n W g n N W

g n W g n N W

g n g n N W

/4 1

0/4( )

Nsn

nNp n W

36

/2 1

0/22 [ ] [ / 2]

N

n

nrNX r x n x n N W

/2 1

0/2( )

Nrn

nNg n W

/2 1

(2 1)

0

/4 1 /2 1(2 1)

/2

/2 /2

/4 /2

(2 1)

0 /

/2

/4 /4 /

4

/4 1 /4 1(2 1)( /4)

0 0

/4 12 2 (2 1)

02

2*(2 1) ( )

( ) ( )

( ) ( / 4)

( ) ( / 4)

Ns n

n

N Ns n s n

n n N

N

N

N N

N N N

N N N N

Nsn n s n N

n n

Nsn n s

Nn n s N

n

X s g n W

g n W g n W

g n W W g n N W

g n W W g n N W W W

/4 1

/4

0

/4 12

0/4[ ( ) ( / 4)] N N

N

n

Nn sn

n

g n g n N W W

/2 /2

(2 1) /4 /2 /4/2 1s N sN N

N N NW W W

/4

/4 12

0

( )N

n sN N

n

n

q n W W

37

/4

/4 1

0

2*2 ( )N

sn

nNX s p n W

/4

/4

/4 12

0

/8 1 /4 1

/2

/4

2

0 8

2*2*2 ( )

( ) ( )

Ntn

n

N Nt

N

n tn

n n NN N

X t p n W

p n W p n W

/8 1 /8 1

2 2 ( /8)

04

0/4 /( ) ( / 8)

N Ntn t n N

Nn n

Np n W p n N W

/8 1

0/8[ ( ) ( / 8)]

Ntn

nNp n p n N W

( ) ( 1)p n p n 8when N

38

/4

/4 1

0

2*2 ( )N

sn

nNX s p n W

/4

/4

/4 1(2 1)

0

/8 1 /4 1(2 1) (2 1)

04

//

8

2*2*(2 1) ( )

( ) ( )

Nt n

n

N Nt n t n

N

Nn n N

N

X t p n W

p n W p n W

/8 1 /8 1

(2 1) (2 1)( /8)

0 0/4 /4( ) ( / 8)

N Nt n t n N

nN

nNp n W p n N W

/8 1

0

4/8[ ( ) ( / 8)] N

Ntn n

Nn

p n p n N W W

0

8[ ( ) ( 1)]p n p n W 8when N

/8 1 /8 12 2 (2

/4 /4 /4 /4 /41) /8

0 0

( ) ( / 8)N N

tn nN N N

tn n t N

n nN Np n W W p n N W W W

/4 /4(2 1) /8 /4 /8

/4 1t N tN NN N NW W W

39

Final flow graph for 8-point DFT decimation in frequency

39

N=8

4-point DFT 2-point DFT

40


40

DIF FFT

DIT FFT

41


41

DIF FFT

DIT FFT

transpose

42


42

decimation-in-time Butterfly Computation

decimation-in-frequecy Butterfly Computation

transpose

43

The DIF FFT is the transpose of the DIT FFT

43

DIF FFT

DIT FFT

transpose

44


DIF FFT

DIT FFT

transpose

45


DIF FFT

DIT FFT

transpose

46

Figure 9.24 erratum

0

4

2

6

1

5

3

7

x

x

x

x

x

x

x

x

47


DIF FFT

DIT FFT

transpose

48 04/19/202348Zhongguo Liu_Biomedical

Engineering_Shandong Univ.

Chapter 8 HW9.1, 9.2, 9.3,

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