Chapter 9 Computation of the Discrete Fourier Transform

9.1 decimation-in-time FFT Algorithms9.2 decimation-in-frequency FFT Algorithms9.3 IFFT Algorithm9.4 FFT Algorithm of real sequence9.5 practical considerations（ software realization)

Chapter 9 Computation of the Discrete Fourier Transform

10][1][

10][][

1

0

1

0

NnNkn

WkXN

nx

NkNkn

WnxkX

N

k

N

n

Direct computation:

Complex multiplication: 2N

)1( NNComplex addition:

Real multiplication:

Real addition:

24N

24N

9.1 decimation-in-time FFT Algorithms

-1

-1

-1

-1

WN0

WN1

WN2

WN3 x[7]

x[5]

x[3]

x[1]

x[6]

x[4]

x[2]

x[0]

H[2]

H[3]

H[1]

H[0]

G[3]

G[2]

G[1]

G[0] X[0]

X[1]

X[2]

X[3]

X[4]

X[5]

X[6]

X[7]

N/2 POINT DFT

N/2 POINT

DFT

Figure 9.3

Figure 9.9

)1(,22

22

:22

NNNN

tionsmultiplicacomplex

G1[0] G[0]

WN2

WN0

WN2

WN0

H1[0]

H1[0]

H2[0]

H2[0]

G1[1]

G2[1]

G2[0]

WN0 H[0]

-1

WN1

WN2

WN3 x[7]

x[3]

x[5]

x[1]

x[6]

x[2]

x[4]

x[0]

H[2]

H[3]

H[1]

G[3]

G[2]

G[1]

N/4point

DFT DFT

X[0]

X[1]

X[2]

X[3]

X[4]

X[5]

X[6]

X[7]

N/4point

DFT DFT N/4point

DFT DFT N/4point

DFT DFT

-1

-1

-1

-1

-1

-1

-1

Figure 9.5

Figure 9.10

stagelog*stage/nscomputatiobutterfly 2/*n computatiobutterfly / times2

addition complex of timesstagelog*stage/nscomputatiobutterfly 2/*ncomputatiobutterfly / time1

tionmultiplicacomplex of times

2

2

N

N

N

N：

：

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

compare the operation quantity

Figure 9.14alternative forms：

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

Figure 9.15

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

Figure 9.16

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

9.2 decimation-in-frequency FFT Algorithms

Figure 9.17

Figure 9.19

Figure 9.18

Figure 9.20

the two kinds of butterfly computation are transpose of each other.

Figure 9.22alternative forms：

Figure 9.23

Figure 9.24

10][1][

10][][

1

0

1

0

NnNkn

WkXN

nx

NkNkn

WnxkX

N

k

N

n

9.3 IFFT Algorithm

NWW

Xxmethodkn

NknN

/1

:1

attention： the difference to transpose

1。

X[4] 1/2WN-0

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2 1/2

1/2WN-2

1/2WN-0

1/2WN-0

1/2WN-0

1/2WN-0

1/2WN-0

1/2WN-0

-1

1/2WN-2

1/2WN-1

1/2WN-2

1/2WN-3 X[7]

X[3]

X[5]

X[1]

X[6]

X[2]

X[0] x[0]

x[1]

x[2]

x[3]

x[4]

x[5]

x[6]

x[7]

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

DIT-FFTDIF-IFFT

-1 1/2WN

1

1/2WN3 1/2WN

2

1/2WN2

1/2WN0

1/2WN0

1/2WN0

1/2WN0

1/2

-1

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

-1

-1

-1 1/2WN

0

1/2WN0

1/2WN0

1/2WN2

-1

-1

-1

-1 -1

-1

-1 X[7]

X[6]

X[5]

X[4]

X[3]

X[2]

X[1]

X[0] x[0]

x[4]

x[2]

x[6]

x[1]

x[5]

x[3]

x[7]

DIF-FFTDIT-IFFT

***1

0

*

1

0

][1][1

10][1][

:2

kXFFTNN

knWkX

N

NnNkn

WkXN

nx

method

N

k

N

k

step:（ 1） X*[k]（ 2） FFT{ }（ 3）（） */Ncan transfer FFT subprogram directly

2，

3。Method 3( exercise 9.1)

9.4 FFT Algorithm of real sequence

2])[][(

1,...0,2

][][][

*

*

2

kNYkYj

Nkj

kNYkYkX

2][][][)3(

*

1kNYkY

kX

1,...0]},[{][)2( NknyFFTkY

1,...0],[][][)1( 21 Nnnjxnxny

FFTcomplexpoNonebyFFTrealpoNtwocalculate

intint

12/,....0],[][]2/[

12/,....0],[][][)4(

21

21

NkkXWkXNkX

NkkXWkXkXkN

kN

12/,...0,2

]2/[][][*

2

Nkj

kNYkYkX

2]2/[][][)3(

*

1kNYkYkX

12/,...0]},[{][)2( NknyFFTkY

12/,...0],12[]2[][][][)1( 21 Nnnjxnxnjxnxny

FFTcomplexpoNonebyFFTrealpoNonecalculate

int2/int

9.5 practical considerations

Figure 9.11

in-place computati

on

Figure 9.13

cause of bit-reversed order

binary coding for position：000001

010011

100101

110111

MN 2

must padding 0 to

Coefficient: WN

r r=0,1,…N/2-1

summary:

9.1 decimation-in-time FFT Algorithms9.2 decimation-in-frequency FFT Algorithms9.3 IFFT Algorithm9.4 FFT Algorithm of real sequence9.5 practical considerations（ software realization)

requirements：1. derivation of decimation-in-time and decimation-in-frequency

FFT algorithms, and draw the graphs ;2. draw flow graph of IFFT;3. special arithmetic of real-sequence FFT;4. concepts of in-place computation and bit-reversed order.