chapter 9 computation of the dft

123/4/10 1Zhongguo Liu_Biomedical Engineering_Shandong U

niv.

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

Baseline for computational 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

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

Re ]

7

X[k] can be viewed as the output of a filter to the 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:

8

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 [ 1] 0,1,...,[ ],


N

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

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,...,[ ],


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


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

16

computational complexityTwo 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

N2/2+N complex additions

More 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

20

Butterfly Computation

Flow graph constitutes of butterflies

20

We can implement each butterfly with one multiplication

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

21


Final flow graph for 8-point decimation in time

21

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

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

2

2

2

2

2

2

2

2

0

1

2

3

4

5

6

7

X

X

X

X

X

X

X

X

23


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 0 000 000

1 4 001 100

2 2 010 010

3 6 011 110

4 1 100 001

5 5 101 101

6 3 110 011

7 7 111 111

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

cause of bit-reversed order

binary coding for position ：000

001

010

011

100

101

110

111

MN 2

must padding 0 to

25

9.3.2 Alternative forms

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

25

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0 000 000

1 4 001 100

2 2 010 010

3 6 011 110

4 1 100 001

5 5 101 101

6 3 110 011

7 7 111 111

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

26

Figure 9.14


strongpoint ： in-place computations

shortcoming ： non-sequential access of data

27

Figure 9.15

shortcoming ： not in-place computation

non-sequential access of data

28

Figure 9.16

shortcoming ： not in-place computation

strongpoint: sequential access of data

29


29

/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


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

Review

30

9.3.1 In-Place Computation同址运算Bit reversed indexing（码位倒置）

30

0

0

0

0

0

0

0

0

000 000

001 100

010 010

011 110

100 001

101 101

110 011

111 111

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

31

Figure 9.14


strongpoint ： in-place computations

shortcoming ： non-sequential access of data

32

9.4 Decimation-In-Frequency FFT Algorithm

The DFT equation

32

1

0

[ ]N

nkN

n

X k x n W

1 / 2 1 1

2 2 2

0 0 / 2

2 [ ] [ ] [ ]N N N

n r n r n rN N N

n n n N

X r x n W x n W x n W

/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 /2 1

/2 22

0 0

[ ] [ / 2]N N

n N rn rN N

n n

x n W x n N W

/2 1

0/2( )

Nrn

nNg n W

33

9.4 Decimation-In-Frequency FFT Algorithm

The DFT equation

33

1

0

[ ]N

nkN

n

X k x n W

1 /2 1 1

(2 1) (2 1) (2 1)

0 0 /2

2 1 [ ] [ ] [ ]N N N

n r n r n rN N N

n n n N

X r x n W x n W x n W

/2 1

(2 1)

0

[ ] [ / 2]N

n rN

n

x n x n N W

/2 1 /2 1

/2 (2 1)(2 1)

0 0

[ ] [ / 2]N N

n N rn rN N

n n

x n W x n N W

(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

2 1

0/[ ] [ / 2]

Nn rnN

nNx n x n N W W

/2

/2 1

0

( )N

n rN N

n

n

h n W W

34

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

34

/2 1

0/22 [ ] [ / 2]

N

n

nrNX r x n x n N W

/2 1

0/22 1 [ ] [ / 2]

Nn rnN

nNX r x n x n N W W

/2

/2 1

0

( )N

n rN N

n

n

h n W W

/2 1

0/2( )

Nrn

nNg n W

35

/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

35

/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

36

2-point DFT

36

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

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

37

/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

38

/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

39

/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

40

/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

41

Final flow graph for 8-point DFT decimation in frequency

41

42


42

DIF FFT

DIT FFT

43


43

DIF FFT

DIT FFT

44


44

decimation-in-time Butterfly Computation

decimation-in-frequecy Butterfly Computation

45

The DIF FFT is the transpose of the DIT FFT

45

DIF FFT

DIT FFT

46


DIF FFT

DIT FFT

47


DIF FFT

DIT FFT

48

Figure 9.24 erratum

0

4

2

6

1

5

3

7

x

x

x

x

x

x

x

x

49


DIF FFT

DIT FFT

50

23/4/1050Zhongguo Liu_Biomedical Engineering_Shandong U

niv.

Chapter 9 HW9.1, 9.2, 9.3,

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