Digital Signal Processing, Fall 2006

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Digital Signal Processing, IX, Zheng-Hua Tan, 2006 1

Digital Signal Processing, Fall 2006

Zheng-Hua Tan

Department of Electronic Systems Aalborg University, Denmark

[email protected]

Lecture 9: The Discrete 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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The discrete-time Fourier transform (DTFT)

The DTFT is useful for the theoretical analysis of signals and systems.

But, according to its definition

computation of DTFT by computer has several problems:

The summation over nis infinite

The independent variable wis continuous

n j n

j x ne

e

X ^ω ⁻ ^ω

∞

−∞

∑

=

= [ ] )

(

The discrete Fourier transform (DFT)

In many cases, only finite duration is of concern

The signal itself is finite duration

Only a segment is of interest at a time

Signal is periodic and thus only finite unique values

For finite duration sequences, an alternative Fourier representation is DFT

The summation over nis finite

DFT itself is a sequence, rather than a function of a continuous variable

Therefore, DFT is computable and important for the implementation of DSP systems

DFT corresponds to samples of the Fourier transform

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Part I: The discrete Fourier series

The discrete Fourier series

The Fourier transform of periodic signals

Sampling the Fourier transform

The discrete Fourier transform

Properties of the DFT

Linear convolution using the DFT

A periodic sequence with period N

Periodic sequence can be represented by a Fourier series, i.e. a sum of complex exponential sequences with frequencies being integer multiples of the

fundamental frequency associated with the

Only Nunique harmonically related complex exponentials since

so

∑

=

k

kn N

ej

k N X

n

x 1 ~[ ] ⁽² ^/ ⁾ ]

~[ ^π

]

~[ ]

~x[n = x n+rN

]

~x[n )

/ 2 ( π N

kn N j mn j kn N j n mN k N

j e e e

e ⁽²^π^/ ⁾⁽ ⁺ ⁾ = ⁽²^π^/ ⁾ ²^π = ⁽²^π^/ ⁾

∑

⁻

= 1 ¹~[ ] ⁽² ^/ ⁾ ]

~[ ^N X k e^j ^N ^kn

n N

x ^π

The frequency of the periodic sequence.

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The Fourier series coefficients

The coefficients

The sequence is periodic with period N

For convenience, define

]

~[ ]

~[ ¹

0

) )(

/ 2

( X k

e n x N

k X

N

n

n N k N

j =

= +

∑

⁻

=

+

− π

∑

−

=

−

=

1

0

) / 2 ( 1

0

) / 2 (

]

~[ ]

~[

]

~[ ] 1

~[

N

n

kn N j N

k

kn N j

e n x k

X

e k N X

n x

π π

) / 2

( N

j

N e

W = ⁻ ^π

∑

−

=

−

=

−

=

1

0 1

0

]

~[ ]

~[ equation Analysis

]

~[ ] 1

~[ equation Synthesis

N

n

kn N N

k

kn N

W n x k

X

W k N X

n x

Very similar equations Æduality

DFS of a periodic impulse train

Periodic impulse train

The discrete Fourier series coefficients

By using synthesis equation, an alternative representation of is

∑

^∞

−∞

=

−

=

r

rN n n

x[ ] [ ]

~ δ

∑

⁻

=

−

=

− −

=

− = =

= ¹

0 ) / 2 ( 1

0 1

0

1 ] 1

~[ ] 1

~[ ^N

k

kn N j N

k kn N N

k

kn

N e

W N W N

k N X

n

x ^π

]

~[n x

1 ]

[ ]

~[ ¹

0

=

∑

⁻

= N

n

kn

WN

n k

X δ

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Part II: The Fourier transform of periodic signals

The discrete Fourier series

Fourier transform of complex exponentials

Fourier transform of

has the required periodicity with period

∑

∞

−∞

=

−

=

−

=

k j

N

k

kn N j

N k k

N X e

X

e k N X

n x

2 ) ( ]

~[ ) 2

~(

]

~[ ] 1

~[ ¹

0

) / 2 (

ω π π δ

ω

π

]

~x[n

∑ ∑

∑

∞

−∞

=

+

−

=

∞

<

∞

−

=

r k

k k

j k

n j k

r a

e X

n e

a n

x ^k

) 2 (

2 )

(

, ]

[

π ω ω δ

ω π

ω

)

~( _j_ω e

X 2π

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Fourier transform of a periodic impulse train

Periodic impulse train

The discrete Fourier series coefficients

Fourier transform

Finite duration signal ( )

Construct

Its Fourier transform

∑

^∞

−∞

=

−

=

r

rN n n

p[ ] [ ]

~ δ

∑

^∞

−∞

=

−

=

k j

N k e N

P 2 )

2 ( )

~( _ω πδ ω π

] [n x

1 ] [ ]

~[ ¹

0

=

∑

⁻

= N

n

kn

WN

n k

P δ

∑

^∞

−∞

=

−

=

k

k N j j

j j

N e k

N X e

P e X e

X 2 )

( ) 2 (

)

~( ) ( )

~( ω ω ω π (2π/ ) δ ω π

] 1 , 0 [ of outside 0

]

[n = N−

x

∑

^∞

−∞

=

∞

−∞

=

−

=

−

=

r r

rN n x rN

n n

x n p n x n

x[ ] [ ]*~[ ] [ ]* ( ) ( )

~ δ

]

~[n x

Compare

Conclude that

i.e. the DFS coefficients of are samples of the Fourier transform of the one period of

∑

^∞

−∞

=

−

=

k

k N j j

j j

N e k

N X e

P e X e

X 2 )

( ) 2 (

)

~( ) ( )

~( ω ω ω π ₍₂π_/ ₎ δ ω π

]

~[n x ]

~[n x

∑

^∞

−∞

=

−

=

k j

N k k

N X e

X 2 )

( ]

~[ ) 2

~( ω π δ ω π

k N j

k N

j X e

e X k

X~[ ] ( ⁽² ^/ ⁾ ) ( )| ₍₂ _/ ₎

π ω ω

π = =

=

⎩⎨

⎧ ≤ ≤ −

= 0, otherwise 1 0

],

~[ ]

[ x n n N

n x

ÆFirst represent it as Fourier series and then calculate Fourier transform

ÅÆ

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Part III: Sampling the Fourier transform

An aperiodic sequence and its Fourier transform

generates a periodic sequence in kwith period Nsince the Fourier transform is periodic in with period

∑

^∞ ⁻

∫

₋

−∞

=

↔

= ^π

π

ω ω ω

ω ω

π ^X ^e ^e ^d n

x e

n x e

X ^j ⁿ ^j ^j ⁿ

n

j ( )

2 ] 1 [ ]

[ )

(

) (

| ) ( ]

~[ ₍₂ _/ ₎

) / 2 (

k N j k

N

j X e

e X k

X = ^ω _ω₌ _π = ^π

ω 2π

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Sampling the Fourier transform

Now we want to see if the sampling sequence is the sequence of DFS coefficients of a sequence this can be done by using the synthesis equation

]

~[ k X

]

~[

] [

]

~[ ] 1 [

] [

] ]

[ 1 [

]

~[ 1

1

0

) ( 1

0

) / 2 ( 1

0

n x

rN n x

m n p m x N W

m x

W e

m N x

W k N X

r

m m

N

k

m n k N N

k

kn N m

km N j N

k

kn N

=

−

=

−

⎥=

⎦

⎢ ⎤

⎣

= ⎡

=

∑

∑ ∑

∑

∞

−∞

=

∞

−∞

=

∞

−∞

=

−

=

−

=

∞ −

−∞

=

−

=

−

π

]

~[n x

A periodic sequence resulting from aperiodic convolution

Examples

Case 1 Fig 8.8

In this case, the Fourier series coefficients for a periodic sequence are samples of the Fourier transform of one period

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Examples

Case 2 Fig 8.9

In this case, still the Fourier series coefficients for are samples of the Fourier transform of . But, one period of is no longer identical to

This is just sampling in the frequency domain as compared in the time domain discussed before.

] [n x

]

~x[n

] [n x ]

~[n x

Sampling in the frequency domain

The relationship between and one period of in the undersampled case is considered a form of time domain aliasing.

Time domain aliasing can be avoided only if has finite length, just as frequency domain aliasing can be avoided only for signals being bandlimited.

If has finite length and we take a sufficient number of equally spaced samples of its Fourier transform (specifically, a number greater than or equal to the length of ), then the Fourier transform is recoverable from these samples, equivalently is recoverable from .

] [n

x ~x[n]

] [n x

] [n x ]

[n

x ~x[n]

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Sampling in the frequency domain

Recovering

i.e. recovering does not require to know its Fourier transform at all frequencies

Application: represent finite length sequence by using Fourier series (coefficients) ÆDFT

] [n x

⎩⎨

⎧ ≤ ≤ −

= 0, otherwise 1 0

],

~[ ]

[ x n n N

n x

] [n x

] [ ]

~[ ]

~[ , ]

~[ ]

[n x n DFS X k x n xn

x → → → →

Fourier transform

Discrete-time Fourier transform

Discrete Fourier transform

∫

∑

−

∞ −

−∞

=

π π

ω ω

π ^X ^e ^e ^dω n

x

e n x e

X

n j j

n j n

j

) 2 (

] 1 [

] [ )

(

∫

∞

−

Ω

∞

−

Ω

−

Ω Ω

=

= Ω

d e j X t

x

dt e t x j

X

t j t j

) 2 (

) 1 (

) ( )

( π

kn N N j

n

e k N X

n x

e n x k

X

) / 2 1 (

0

] 1 [

] [

] [ ]

[

π π

∑

−

− −

=

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Part IV: The DFT

Consider a finite length sequence of length N samples (if smaller than N, appending zeros)

Construct a periodic sequence

Assuming no overlap btw

Recover the finite length sequence

To maintain a duality btw the time and frequency domains, choose one period of as the DFT

] [n x

∑

^∞

−∞

=

−

=

r

rN n x n

x[ ] [ ]

~

] [n rN x −

⎩⎨

⎧ ≤ ≤ −

= 0, otherwise 1 0

],

~[ ]

[ x n n N

n x

] )) [((

)]

modulo [(

]

~[

n N

x N n

x n

x = =

]

~[ k X

⎪⎩

⎪⎨

⎧ ≤ ≤ −

= ~[ ], 0 1 ]

[ X k k N

k X

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The DFT

Periodic sequence and DFS coefficients

Since summations are calculated btw 0 and (N-1)

∑

−

=

−

=

1

0 1

0

]

~[ ] 1

~[

]

~[ ]

~[

N

k

kn N N

n

kn N

W k N X

n x

W n x k

X

⎪⎩

⎪⎨

⎧ ≤ ≤ −

=

∑

⁻

=

−

otherwise

, 0

1 0

, ] 1 [

] [

1

0

N n W

k N X

n x

N

k

kn N

⎪⎩

⎪⎨

⎧ ≤ ≤ −

=

∑

⁻

=

otherwise

, 0

1 0

, ] ] [

[

1

0

N k W

n k x

X

N

n

kn N

∑

⁻

=

= ¹ − 0

] 1 [

] [

N

k

kn

WN

k N X

n x

∑

⁻

=

= ¹

0

] [ ] [

N

n

kn

WN

n x k

X

Generally

The DFT

A finite or periodic sequence has only Nunique values, x[n]for 0<=n<N

Spectrum is completely defined by N distinct frequency samples

DFT: uniform sampling of DTFT spectrum

(13)

The DFT of a rectangular pulse

Example 8.7 pp.561

The DFT of a rectangular pulse

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Part V: Properties of the DFT

Properties of the DFT – linearity

Linearity

The lengths of sequences and their DFTs are all equal to the maximum of the lengths of and

] [ ] [ ]

[ ]

[ ₂ ₁ ₂

1 n bx n aX k bX k

ax

DFT↔ +

+

]

1[n

x x₂[n]

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Circular shift of a sequence

Given

Then

] [ ]

[ ] [

] [ ] [

) / 2 ( 1

1 n X k e X k

x

k X n x

m N k DFT j

DFT

π

= −

↔

⎩⎨

⎧ = − = − ≤ ≤ −

= 0, otherwise 1 0

], )) [((

]

~[ ]

[ ¹

1

N n m

n x m n x n n x

x ^N

Circular shift of a sequence – an example

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Duality

1 0

], )) [((

] [

] [ ] [

−

≤

−

↔

N k k

Nx n

X

k X n x

N DFT

DFT

Circular convolution

In linear convolution, one sequence is multiplied by a time –reversed and linearly shifted version of the other. For convolution here, the second sequence is circularly time reversed and circularly shifted. So it is called an N-point circular convolution

1 0

, ] )) [((

] [

1 0

, ] )) [((

] )) [((

1 0

, ]

~ [ ]

~[ ] [

1

0

2 1 1

0

2 1

1

0

2 1 3

−

≤

−

=

−

≤

−

=

−

≤

−

=

∑

−

=

−

=

−

=

N n m

n x m x

N n m

n x m x

N n m

n x m x n

x

N

m

N N

m

N N

N

m

] [ N ] [ ]

[ ₁ ₂

3 n x n x n

x =

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Circular convolution with a delayed impulse

The delayed impulse sequence x1^[n^]=δ^[n−n0^]

] [ ]

[ ] [

2 3

1

0 0

k X W k X

W k X

kn N kn N

=

Summary of properties of the DFT

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Part VI: Linear convolution of the DFT

The discrete Fourier transform

Procedure

Compute the N-point DFTs and of two sequences and , respectively

Compute the product of

Compute the sequence as the inverse DFT of

As we know, the multiplication of DFTs corresponds to a circular convolution of the sequences. To obtain a linear convolution, we must ensure that circular convolution has the effect of linear convolution.

]

1[k

X X₂[k] ]

1[n x

1 0

for ] [ ] [ ]

[ ₁ ₂

3 k = X k X k ≤k ≤N−

X ]

3[k X

]

2[n x

] [ N ] [ ]

[ ₁ ₂

3 n x n x n

x =

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Linear convolution of two finite-length sequences

∑

^∞

−∞

=

−

=

m

m n x m x n

x₃[ ] ₁[ ] ₂[ ]

The circular convolution corresponding to is identical to the linear convolution corresponding to if the length of DFTs satisfies

Circular convolution as linear convolution with alaising

⎪⎩

⎪⎨

⎧ − ≤ ≤ −

=

−

≤

=

−

≤

=

∑

^∞

−∞

=

otherwise

, 0

1 0

, ] ] [

[

: ] [ of DFT inverse the

] [ ] [ ] [ So,

1 0

), (

) (

] [ Also

1 0

), (

] [ : DFT a Define

) ( ) ( ) ( : ] [ of ansform Fourier tr

3 3

3 2 1 3

) / 2 ( 2 ) / 2 ( 1 3

) / 2 ( 3 3

2 1

3 3

N n rN

n n x

x

k X

k X k X k X

N k e

X e

X k X

N k e

X k X

e X e X e

X n x

p r

N k j N

k j

N k j

j j

j

π π

π

ω ω

ω

] [ ]

[ ₂

1k X k X

] [ N ] [ ]

[ ₁ ₂

3 n x n x n

x _p =

) ( )

( ₂

1

ω

ω j

j X e

e X

−1 +

≥L P N

(20)

Circular convolution as linear convolution with alaising

Summary

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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