Lecture Digital signal processing: Lecture 9 - Zheng-Hua Tan

Lecture Digital signal processing - Lecture 4 introduce the discrete fourier transform. This lesson presents the following content: 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.

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

Fourier-domain

representation

DFT/FFT

System analysis

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3

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 n is infinite

The independent variable w is continuous

n j n

j

e n x e

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 n is finite

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

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5

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

The discrete Fourier series

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 N unique harmonically related complex

e k X N n

x[ ] 1 ~[ ] (2 / )

][

~][

~x n = x n+rN

] [

~ n x

)/2( π N

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

e e

~ 1 ] [

e k X n

The frequency of the periodic sequence.

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7

The Fourier series coefficients

The coefficients

The sequence is periodic with period N

For convenience, define

] [

~ ]

[

~ ] [

0

) )(

/ 2 (

k X e

n x N

= + ∑−

1

0

) / 2 (

] [

~ ]

[

~

] [

~ 1 ]

N

k

kn N j

e n x k

X

e k X N n

x

π π

) / 2

~ ] [

~ equation Analysis

] [

~ 1 ] [

~ equation Synthesis

N

n

kn N

N

k

kn N

W n x k

X

W k X N n x

Very similar equations

Æ duality

DFS of a periodic impulse train

Periodic impulse train

The discrete Fourier series coefficients

By using synthesis equation, an alternative

0 1

0

1 1

] [

~ 1 ]

[

k

kn N j N

k

kn N N

k X N n

] [

~ n

x

1 ]

[ ]

W n k

X δ

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9

Part II: The Fourier transform of periodic signals

The Fourier transform of periodic signals

Fourier transform of complex exponentials

N

k

kn N j

N

k k

X N e

X

e k X N n

x

)

2 ( ] [

~ 2 )

(

~

] [

~ 1 ]

[

0

) / 2 (

π ω δ

π

ω

π

] [

j k

n j k

r a

e

X

n e

a n

)2(

2)

(

,]

[

πωωδπ

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11

Fourier transform of a periodic impulse train

Periodic impulse train

The discrete Fourier series coefficients

e

P~( ω) 2πδ(ω 2π ) ]

[n

x

1 ] [ ]

W n k

j j

N

k e

X N e

P e X e

X~( ω ) ( ω )~( ω ) 2π ( (2π/ ) )δ(ω 2π )

] 1 , 0 [ of outside 0

rN n x rN

n n

x n p n x

n

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

] [

i.e the DFS coefficients of are samples of the

Fourier transform of the one period of

j j

N

k e

X N e

P e X e

X~( ω ) ( ω )~( ω ) 2π ( (2π/ ) )δ(ω 2π )

] [

~ n

x

] [

X N e

X~( ω ) 2π ~[ ]δ(ω 2π )

k N j

e X e

, 0

1 0

], [

~ ]

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13

Part III: Sampling the Fourier transform

Sampling the Fourier transform

An aperiodic sequence and its Fourier transform

generates a periodic sequence in k with period N since

the Fourier transform is periodic in with period

π X e e d

n x e

n

j

)(2

1][]

[)

(

)(

|)(][

) / 2 (

k N j k

N j

e X e

X k

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15

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 (

m n p m x W

N m x

W e

m x N

W k X N

r

m m

N

k

m n k N

N

k

kn N m

km N j

N

k

kn N

In this case, the Fourier series coefficients for a

periodic sequence are samples of the Fourier

transform of one period

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17

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

] [

~ 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

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19

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) Æ DFT

, 0

1 0

], [

~ ]

~ ] [

~ , ]

[

~ ]

[n x n DFS X k x n x n

Fourier transform

Discrete-time Fourier transform

Discrete Fourier transform

ω ω

ω

π X e e d

n x

e n x e

X

n j j

n j n

j

)(2

1][

][)

=

= Ω

d e j X t

x

dt e t x j

X

t j

) ( 2

1 ) (

) ( )

(

π

kn N j N

n

e k X N n x

e n x k

X

) / 2 ( 1

0

] [

1 ] [

] [ ]

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21

Part IV: The DFT

The discrete Fourier transform

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

x[ ] [ ]

~

] [n rN

, 0

1 0

], [

~ ]

x

] )) [((

)]

modulo [(

] [

~

N

n x N n

x n

] [

~

k X

X

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The DFT

Periodic sequence and DFS coefficients

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

~ 1 ]

[

~

] [

~ ]

N

n

kn N

W k X N n

x

W n x k

, 0

1 0

, ] [ 1

k X N

,

0

1 0

, ] [

N

k

kn N

W k X N n x

N

n

kn N

W n x k

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27

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

] [ ] [ ]

[ ]

] [

1 n

x x2[n]

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Circular shift of a sequence

Given

Then

] [ ]

[ ] [

] [ ] [

) / 2 ( 1

x

k X n

x

m N k j DFT

DFT

π

−

=

↔

⎩

⎨

=

otherwise

, 0 1 0 ], )) [((

] [

~ ] [

~

]

1

N n m

n x m n x n x

n

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Duality

1 0

], )) [((

]

[

] [ ]

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 3

n x m x

N n m

n x m x

N n m

n x m x

N

m

] [ 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

=

Summary of properties of the DFT

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Part VI: Linear convolution of the DFT

Linear convolution using the DFT

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 ] [ ] [ ]

X

] [

3 k X

] [

2 n x

] [ N ] [ ]

x =

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x3[ ] 1[ ] 2[ ]

The circular convolution corresponding to is identical

to the linear convolution corresponding to if the length

Circular convolution as linear convolution with alaising

,0

10

,][

]

[

:][ofDFTinverse

the

][][]

[

So,

10

),(

)(

]

[

Also

10

),(

][ :DFT

a

Define

)()()( :][ofansform

Fourier tr

3 3

3

2 1 3

) / 2 ( 2 ) / 2 ( 1 3

) / 2 ( 3 3

2 1

3 3

N n rN

n x n

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

r

p

N k j N

k j

N k j

j j

j

π π

π

ω ω

ω

] [ ]

X

][N][]

x p =

) ( ) ( jω jω

e X e X

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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

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