Digital Signal Processing, Fall 2005

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Digital Signal Processing, X, Zheng-Hua Tan, 2005 1

Digital Signal Processing, Fall 2005

- E-Study -

Zheng-Hua Tan

Department of Communication Technology Aalborg University, Denmark

[email protected]

Lecture 10: Fast Fourier Transform

Course at a glance

Discrete-time signals and systems

Fourier-domain representation

DFT/FFT

System analysis

Filter design z-transform

MM1

MM2

MM9,MM10 MM3

MM6 MM4

MM7, MM8 Sampling and

reconstruction MM5

System structure System

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Digital computation of the DFT

The DFT of a finite-length sequence of length N

The inverse DFT

Due to the duality, focus on the DFT only.

Use the number of arithmetic multiplications and additions as a measure of computational complexity.

Fast Fourier transform (FFT) is a set of algorithms for the efficient and digital computation of the N- point DFT, rather than a new transform.

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

]

[ ¹

0

−

=

∑

⁻

=

− n N

W k N X

n

x ^N

k

kn N

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

[ ¹

0

−

=

∑

⁻

=

N k

W n x k

X ^N

n

kn N

Part I: Direct computation of the DFT

Direct computation of the DFT

Decimation-in-time FFT algorithms

Decimation-in-frequency FFT algorithms

Fourier analysis of signals using the DFT

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The DFT of a finite-length sequence of length N

Direct computation: N²complex multiplications and N(N-1) complex additions

Compute and store (only over one period)

Compute the DFT using stored and input 1 ,..., 1 , 0 ), / 2 sin(

) / 2 cos(

) / 2 (

−

= +

=

= ⁻

N k

N k j

N k e

W_N^k ^j ^N ^k

π π

π

Direct computation of the DFT

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

[ ¹

0

−

=

∑

⁻

=

N k

W n x k

X ^N

n

kn N

] [n x

k

WN

complex be

may ] [ and x n W_N^k

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

[ ¹

0

−

=

∑

⁻

=

N k

W n x k

X ^N

n

kn N

] [n

x X[k]

For each k

Therefore, for each value of k, the direct computation of X[k] requires 4N real multiplications and (4N-2) real additions.

The direct computation of the DFT requires real multiplications and real additions.

The efficiency can be improved by exploiting the symmetry and periodicity properties of

) 2 4 ( N− N

Direct computation of the DFT

4N2

kn

WN

1 ,..., 1 , 0 }), Re{

]}

[ Im{

} Im{

]}

[ (Re{

}) Im{

]}

[ (Im{

} Re{

]}

[ [(Re{

]

[ ¹

0

−

= +

+

−

=

∑

⁻

=

N k

W n

x W

n x j

W n

x W

n x k

X

kn N kn

N N

n

kn N kn

N

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Symmetry and periodicity of complex exponential

Complex conjugate symmetry

Periodicity in n and k

For example

The number of multiplications is reduced by a factor of 2.

} Im{

} Re{

)

( ^*

]

[ kn

N kn

N n N k

N W W W j W

W ⁻ = ⁻ = = −

n N k N N

n k N kn

N W W

W = ⁽ ⁺ ⁾ = ⁽ ⁺ ⁾

} Re{

]}) [

Re{

]}

[ (Re{

} Re{

]}

[ Re{

} Re{

]}

[

Re{ ^[ ^]

kn N

n N k N kn

N

W n

N x n

x

W n

N x W

n x

− +

=

−

+ ⁻

Part II: Decimation-in-time FFT algorithms

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FFT

Cooley and Tukey (1965) published an algorithm for the computation of the DFT that is applicable when N is a composite number, i.e., the product of two or more integers. Later, it resulted in a number of highly efficient computational algorithms.

The entire set of such algorithms are called the fast Fourier transform, FFT.

FFT decomposes the computation of the DFT of a sequence of length N into successively smaller DFTs.

Decimation-in-time FFT algorithms

Where

decomposition is done by decomposing the sequence into successively smaller subsequences,

and both the symmetry and periodicity of complex exponential are exploited.

Consider and separate x[n] into two (N/2)- point sequences

N =2v

kn N j kn

N e

W = ⁻ ⁽²^π^/ ⁾

∑

⁺

=

odd even

] [ ]

[ ]

[

n

kn N n

kn

N x nW

W n x k

X

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

[ ¹

0

−

=

∑

⁻

=

N k

W n x k

X ^N

n

kn N

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Decimation-in-time FFT algorithms

y periodicit the

to due ) 1 2 / ,..., 1 , 0 for compute only

(

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

[

] 1 2 [ ]

2 [

) ](

1 2 [ )

](

2 [

] 1 2 [ ]

2 [

] [ ]

[ ]

[

1 ) 2 / (

0

2 / 1

) 2 / (

0

2 /

1 ) 2 / (

0 1 2

) 2 / (

0

2

1 ) 2 / (

0

) 1 2 1 (

) 2 / (

0

2

odd even

−

=

−

= +

=

+ +

=

+ +

=

+ +

=

+

=

∑

−

=

−

=

−

=

−

=

−

=

− +

=

N k

k H W k G

W r x W

W r x

W r

x W

W r x

W r x W

r x

W n x W

n x k

X

k N

N r

rk N k

N N

r

rk N

N r

rk N k

N N

r

rk N

N r

k r N N

r

rk N

n

kn N n

kn N

Flow graph of the decimation-in-time

Periodicity is applied, e.g. G[7]=G[3]

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Decimation-in-time FFT

Further break down

∑

−

=

−

=

−

=

−

=

−

=

− +

=

−

=

+ +

=

+ +

=

+ +

=

1 ) 4 / (

0

4 / 2

/ 1

) 4 / (

0

4 /

1 ) 4 / (

0

4 / 2

/ 1

) 4 / (

0

4 /

1 ) 4 / (

0

) 1 2 (

2 / 1

) 4 / (

0

2 2 / 1

) 2 / (

0

2 /

] 1 2 [ ]

2 [ ]

[

] 1 2 [ ]

2 [

] 1 2 [ ]

2 [ ]

[ ]

[

N l

lk N k

N N

l

lk N

N l

lk N k

N N

l

lk N

N l

k l N N

l

lk N N

r

rk N

W l h W

W l h k

H

W l g W

W l g

W l g W

l g W

r g k

G

Combination of Fig. 9.3 and 9.4

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2-point DFT

Flow graph

2N

log stages and each stage has N complex multiplications and N complex

additions . N Nlog2

In total, complex multiplications and additions.

240 , 10 log

576 , 048 , 1

1024 2

e.g.

2 2

10

=

= N N N N

A reduction of 2 orders!

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Flow graph of butterfly computation

Flow graph

The number of complex multiplications are reduced by a factor of 2 over the number in Fig. 9.7.

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Part III: Decimation-in-frequency FFT algorithms

Direct computation of the DFT

Fig. 9.20

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Part IV: Fourier analysis of signals using the DFT

Direct computation of the DFT

Finite-duration requirement of DFT Æwindowing

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Fourier analysis of signals using the DFT

Fig. 10.2 Tapers off but is

not band-limited.

Not ideal.

Low-pass filtered and modified.

∑^∞

−∞

=

+

=

r c

j

T j r jT T X

e

X 2 )

1 ( )

( ω ω π

∫−

= ^π − π

θ ω θ

ω θ

π ^X ^e ^V^e ^d e

V ^j ( ^j ) ( ^j )

2 ) 1

( ⁽ ⁾

Windowing.

N n

kn N

j k N

e n v k

V ¹

0

) / 2

( , 0,1,..., 1 ]

[ )

( ⁻ ^π

=

− = −

=∑

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Digital Signal Processing, X, Zheng-Hua Tan, 2005 25 Effect of Windowing on Fourier analysis

Fig. 10.3

Effect of Windowing on Fourier analysis

A rectangular window of length 64.

) 2 (

) (

) cos(

] [ ) cos(

] [ ] [

) cos(

] [

) cos(

) (

) 1 (

) 0 (

1 1 1

0 0 0

1 1 1 0 0 0

1 1

0 0

ω ω θ ω

ω θ

ω ω θ ω

ω θ ω

θ ω θ

ω

θ ω θ

ω

θ θ

+

−

+

−

+ +

+

=

+ +

+

=

+ +

+

=

+ Ω +

+ Ω

=

j j j

j

j j j

j j

c

e W A e e

W A e

e W A e e

W A e e V

n n w A n

n w A n v

n A n

A n x

t A t

A t s

Effect of Windowing on Fourier analysis

Fig. 10.3

The DTFT of a sinusoidal signal is a pair of impulses.

Windowing broadens the impulses and reduces the distinction of signals that are close in frequency

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Summary

Course at a glance

Discrete-time signals and systems

Fourier-domain representation

DFT/FFT

System analysis

Filter design z-transform

MM1

MM2

MM6 MM4

MM7, MM8 Sampling and

reconstruction MM5

System structure System

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Digital Signal Processing, Fall 2005