Digital Signal Processing, Fall 2006 - sharif.irsharif.ir/~bahram/sp4cl/DigitalSignalProcessing-DTFT-DFT/Digital... · 1 1 Digital Signal Processing, IX, Zheng-Hua Tan, 2006 Digital

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Digital Signal Processing, IX, Zheng-Hua Tan, 20061

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

Systemanalysis

Filter design

z-transform

MM1

MM2

MM9, MM10MM3

MM6

MM4

MM7, MM8

Sampling andreconstruction MM5

Systemstructure

System

2


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 infiniteThe independent variable w is continuous

nj

n

j enxeX ωω −∞

−∞=∑= ][)(


The discrete Fourier transform (DFT)

In many cases, only finite duration is of concern The signal itself is finite durationOnly a segment is of interest at a timeSignal 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 continuous variableTherefore, DFT is computable and important for the implementation of DSP systemsDFT corresponds to samples of the Fourier transform

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Part I: The discrete Fourier series

The discrete Fourier seriesThe Fourier transform of periodic signalsSampling the Fourier transformThe discrete Fourier transformProperties of the DFTLinear 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 exponentials since

so

∑=k

knNjekXN

nx )/2(][~1][~ π

][~][~ rNnxnx +=

][~ nx)/2( Nπ

knNjmnjknNjnmNkNj eeee )/2(2)/2())(/2( ππππ ==+

∑−

=

=1

0

)/2(][~1][~ N

k

knNjekXN

nx π

The frequency of the periodic sequence.

4


The Fourier series coefficients

The coefficients

The sequence is periodic with period N

For convenience, define

][~][~][~ 1

0

))(/2( kXenxNkXN

n

nNkNj ==+ ∑−

=

+− π

∑

∑−

=

−

−

=

=

=

1

0

)/2(

1

0

)/2(

][~][~

][~1][~

N

n

knNj

N

k

knNj

enxkX

ekXN

nx

π

π

)/2( NjN eW π−=

∑

∑−

=

−

=

−

=

=

1

0

1

0

][~][~ equation Analysis

][~1][~ equation Synthesis

N

n

knN

N

k

knN

WnxkX

WkXN

nxVery similar equations

duality


DFS of a periodic impulse train

Periodic impulse train

The discrete Fourier series coefficients

By using synthesis equation, an alternative representation of is

∑∞

−∞=

−=r

rNnnx ][][~ δ

∑∑∑−

=

−

=

−−

=

− ===1

0

)/2(1

0

1

0

11][~1][~ N

k

knNjN

k

knN

N

k

knN e

NW

NWkX

Nnx π

][~ nx

1][][~ 1

0== ∑

−

=

N

n

knNWnkX δ

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Part II: The Fourier transform of periodic signals



The Fourier transform of periodic signals

Fourier transform of complex exponentials

Fourier transform of

has the required periodicity with period

∑

∑∞

−∞=

−

=

−=

=

k

j

N

k

knNj

NkkX

NeX

ekXN

nx

)2(][~2)(~

][~1][~ 1

0

)/2(

πωδπω

π

][~ nx

∑ ∑

∑∞

−∞=

+−=

∞<<∞−=

r kkk

j

k

njk

raeX

neanx k

)2(2)(

,][

πωωδπω

ω

)(~ ωjeX π2

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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

rNnnp ][][~ δ

∑∞

−∞=

−=k

j

Nk

NeP )2(2)(~ πωδπω

][nx

1][][~ 1

0== ∑

−

=

N

n

knNWnkP δ

∑∞

−∞=

−==k

kNjjjj

NkeX

NePeXeX )2()(2)(~)()(~ )/2( πωδπ πωωω

]1,0[ of outside 0][ −= Nnx

∑∑∞

−∞=

∞

−∞=

−=−==rr

rNnxrNnnxnpnxnx )()(*][][~*][][~ δ][~ nx


The Fourier transform of periodic signals

Compare

Conclude that

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

∑∞

−∞=

−==k

kNjjjj

NkeX

NePeXeX )2()(2)(~)()(~ )/2( πωδπ πωωω

][~ nx][~ nx

∑∞

−∞=

−=k

j

NkkX

NeX )2(][~2)(~ πωδπω

kNjkNj eXeXkX )/2(

)/2( |)()(][~πω

ωπ===

⎩⎨⎧ −≤≤

=otherwise ,0

10 ],[~][

Nnnxnx

First represent it as Fourier series and then calculate Fourier transform

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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

πωωωω ω

πdeeXnxenxeX njjnj

n

j )(21][][)(

)(|)(][~ )/2()/2(

kNjkN

j eXeXkX ππω

ω == =

ω π2

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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

][~ kX

][~

][

][~][1][

]][[1

][~1

1

0

)(

1

0

)/2(

1

0

nx

rNnx

mnpmxWN

mx

WemxN

WkXN

r

mm

N

k

mnkN

N

k

knN

m

kmNj

N

k

knN

=

−=

−=⎥⎦

⎤⎢⎣

⎡=

=

∑

∑∑ ∑

∑ ∑

∑

∞

−∞=

∞

−∞=

∞

−∞=

−

=

−−

−

=

−∞

−∞=

−

−

=

−

π

][~ nx

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

][nx][~ nx

][nx][~ nx


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 .

][nx ][~ nx

][nx

][nx

][nx

][nx ][~ nx

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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) DFT

][nx

⎩⎨⎧ −≤≤

= otherwise ,0

10 ],[~][

Nnnxnx

][nx

][][~][~,][~][ nxnxkXDFSnxnx →→→→



Fourier transform

Discrete-time Fourier transform

Discrete Fourier transform∫

∑

−

−∞

−∞=

=

=

π

πωω

ωω

ωπ

deeXnx

enxeX

njj

nj

n

j

)(21][

][)(

∫

∫∞

∞−

Ω

∞

∞−

Ω−

ΩΩ=

=Ω

dejXtx

dtetxjX

tj

tj

)(21)(

)()(

π

knNjN

k

knNjN

n

ekXN

nx

enxkX

)/2(1

0

)/2(1

0

][1][

][][

π

π

∑

∑−

=

−−

=

=

=

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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

][nx

∑∞

−∞=

−=r

rNnxnx ][][~

][ rNnx −

⎩⎨⎧ −≤≤

= otherwise ,0

10 ],[~][

Nnnxnx

]))[(()] modulo [(][~NnxNnxnx ==

][~ kX

⎪⎩

⎪⎨⎧ −≤≤

= otherwise ,0

10 ],[~][ NkkXkX

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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

knN

N

n

knN

WkXN

nx

WnxkX

⎪⎩

⎪⎨⎧

−≤≤= ∑−

=

−

otherwise ,0

10 ,][1][

1

0NnWkX

NnxN

k

knN

⎪⎩

⎪⎨⎧

−≤≤= ∑−

=

otherwise ,0

10 ,][][1

0NkWnxkX

N

n

knN

∑−

=

−=1

0][1][

N

k

knNWkX

Nnx

∑−

=

=1

0][][

N

n

knNWnxkX

Generally


The DFT

A finite or periodic sequence has only N unique values, x[n] for 0<=n<NSpectrum is completely defined by N distinct frequency samplesDFT: uniform sampling of DTFT spectrum

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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

][][][][ 2121 kbXkaXnbxnaxDFT

+↔+

][1 nx ][2 nx

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Circular shift of a sequence

Given

Then ][][][

][][

)/2(11 kXekXnx

kXnx

mNkjDFT

DFT

π−=↔

↔

⎩⎨⎧ −≤≤−=−=

= otherwise ,0

10 ],))[((][~][~][ 1

1

Nnmnxmnxnxnx N


Circular shift of a sequence – an example

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Duality

10 ],))[((][

][][

−≤≤−↔

↔

NkkNxnX

kXnx

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

10 ,]))[((][

10 ,]))[((]))[((

10 ,][~][~][

1

021

1

021

1

0213

−≤≤−=

−≤≤−=

−≤≤−=

∑

∑

∑

−

=

−

=

−

=

Nnmnxmx

Nnmnxmx

Nnmnxmxnx

N

mN

N

mNN

N

m

][ N ][][ 213 nxnxnx =

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Circular convolution with a delayed impulse

The delayed impulse sequence ][][ 01 nnnx −= δ

][][

][

23

1

0

0

kXWkX

WkXkn

N

knN

=

=


Summary of properties of the DFT

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Part VI: Linear convolution of the DFT



Linear convolution using the DFT

ProcedureCompute the N-point DFTs and of two sequences and , respectivelyCompute the product ofCompute 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 kX ][2 kX][1 nx

10for ][][][ 213 −≤≤= NkkXkXkX

][3 kX

][2 nx

][ N ][][ 213 nxnxnx =

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Linear convolution of two finite-length sequences

∑∞

−∞=

−=m

mnxmxnx ][][][ 213


The circular convolution corresponding to is identicalto the linear convolution corresponding to if the lengthof DFTs satisfies

Circular convolution as linear convolution with alaising

⎪⎩

⎪⎨⎧

−≤≤−=

=−≤≤=

−≤≤=

=

∑∞

−∞=

otherwise ,0

10 ,][][

:][ of DFT inverse the][][][ So,

10 ),()(][ Also

10 ),(][ :DFT a Define

)()()( :][ of ansformFourier tr

33

3

213

)/2(2

)/2(13

)/2(33

2133

NnrNnxnx

kXkXkXkX

NkeXeXkX

NkeXkX

eXeXeXnx

rp

NkjNkj

Nkj

jjj

ππ

π

ωωω

][][ 21 kXkX

][ N ][][ 213 nxnxnx p =

)()( 21ωω jj eXeX

1−+≥ PLN

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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

Systemanalysis

Filter design

z-transform

MM1

MM2

MM9, MM10MM3

MM6

MM4

MM7, MM8

Sampling andreconstruction MM5

Systemstructure

System

Digital Signal Processing, Fall 2006 - sharif.irsharif.ir/~bahram/sp4cl/DigitalSignalProcessing-DTFT-DFT/Digital... · 1 1 Digital Signal Processing, IX, Zheng-Hua Tan, 2006 Digital

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