Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We...

Chapter 3Discrete Fourier Transform

Review

Features in common

We need a numerically computable transform, that is

Discrete Fourier Transform (DFT)

The DTFT provides the frequency-domain ( ) representation for absolutely summable sequences.

The z-transform provides a generalized frequency-domain ( ) representation for arbitrary sequences.

Defined for infinite-length sequences. Functions of continuous variable ( or ). They are not numerically computable transform.

Chapter 3Discrete Fourier Transform

Content

The Family of Fourier Transform

The Discrete Fourier Series (DFS)

The Discrete Fourier Transform (DFT)

The Properties of DFT

The Sampling Theorem in Frequency Domain

Approximating to FT (FS) with DFT (DFS)

Summary

Introduction

Fourier analysis is named after Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist.

A signal can be either continuous or discrete, and it can be either periodic or aperiodic. The combination of these two features generates the four categories of Fourier Transform.

Aperiodic-Continuous － Fourier Transform

dejXtx

dtetxjX

Periodic-Continuous － Fourier Series

ejkXtx

dtetxT

Aperiodic-Discrete － DTFT

deeXnx

Periodic-Discrete － DFS (DFT)

Summary

Time function Frequency function

Continuous and Aperiodic Aperiodic and Continuous

Continuous and Periodic( ) Aperiodic and Discrete( )

Discrete ( ) and Aperiodic Periodic( ) and Continuous

Discrete ( ) and Periodic ( )

Periodic( )

and Discrete( )

return

DefinitionPeriodic time functions can be synthesized as a linear combination of complex exponentials whose frequencies are multiples (or harmonics) of the fundamental frequency

Periodic continuous-time function )()( rTtxtx

Periodic discrete-time function )()( rNnxnx

ktTjekXtx

knNjekX

fundamental frequency

2fundamental frequency

elsewhere ,0

nrkNjN

enxmNkX

nmNkNj

Because:

The is a periodic sequence with fundamental period equal to N

[IDFS)(~

)(~)](~[DFS)(~

WnxnxkX

Relation to the z-transform

elsewhere,0

10),(~)(

Nnnxnx

zXkX 2|)()(~

The DFS represents N evenly spaced samples of the z-transform around the unit circle.

))(()(~

, )()(2

n NenxkXznxzX

Relation to the DTFT

enxkXenxeX

2|)()(~

elsewhere,0

10),(~)(

Nnnxnx

The DFS is obtained by evenly sampling the DTFT at

intervals. It is called frequency resolution and represents the

sampling interval in the frequency domain.

jIm[z]

2|)()(~

N=8 frequency resolution

The properties of DFS

Linearity

)](~)(~[DFS 2121 kXbkXanxbnxa

Shift of a sequence

)](~[DFS2

kXekXWmnxmk

Modulation

)](~[DFS lkXnxW lnN

Periodic convolution

)(~)(~

)(~)(~)](~

[IDFS)(~ then

mnxmxkYny

kXkXkY

)(~)(~)(~)(~

[IDFS)(~

mnxmxmnxmx

WkXWmxN

WkXkXN

kXkXny

return

Introduction

The DFS provided us a mechanism for numerically computing the discrete-time Fourier transform. But most of the signals in practice are not periodic. They are likely to be of finite length.

Theoretically, we can take care of this problem by defining a periodic signal whose primary shape is that of the finite length signal and then using the DFS on this periodic signal.

Practically, we define a new transform called the Discrete Fourier Transform, which is the primary period of the DFS.

This DFT is the ultimate numerically computable Fourier transform for arbitrary finite length sequences.

Finite-length sequence & periodic sequence

)(nx Finite-length sequence that has N samples

)(~ nx periodic sequence with the period of N

)()(~)(

10 ),(~)(

nRnxnx

elsewhere

Nnnxnx

))(()(~

rNnxnx

Window operation

Periodic extension

The definition of DFT

10 ,)(1

)]([IDFT)(

10 ,)()]([DFT)(

NnWkXN

NkWnxnxkX

)()(~)()(1

)()()(

nRnxnRWkXN

kRkXkRWnxkX

return

Linearity

)()()]()([DFT 2121 kbXkaXnbxnax N3-point DFT, N3=max(N1,N2)

Circular shift of a sequence

)()]())(([DFT kXWnRmnx kmNNN

)())(()]([DFT kRlkXnxW NNnl

Circular shift in the frequency domain

The sum of a sequence

)()()(N

nxWnxkX

The first sample of sequence

)())(()]([

)()]([

kRkNNxnXDFT

kXnxDFT

Circular convolution

)( )()())(()(

)())(()()( )(

nxnxnRmnxmx

nRmnxmxnxnx

)()()]( )([DFT 2121 kXkXnxnx N

)( )(1

)]()([DFT 2121 kXkXN

nxnx N

Multiplication

Circular correlation

xy nymnxmnynxmr )(*)()(*)()(

Linear correlation

Circular correlation

)()(*))((

)())((*)()(

mRnymnx

mRmnynxmr

)()(*))((

)())((*)(

)]([IDFT)(

)()()(

mRnymnx

mRmnynx

kRmrthen

kYkXkRif

Parseval’s theorem

* )()(1

)()( then

)()( let

Conjugate symmetry properties of DFT

and)(nxep )(nxop

Let be a N-point sequence)(nx Nnxnx ))(()(~

]))(())(([2

1)](~)(~[

]))(())(([2

1)](~)(~[

nNxnxnxnxnx

It can be proved that

)(~)(~)(~)(~

)())(())((2

)()(~)(

)())(())((2

)()(~)(

nRnNxnx

nRnxnx

nRnNxnx

nRnxnx

Circular conjugate symmetric

component

Circular conjugate

antisymmetriccomponent

)()()( nxnxnx opep

)())(()(

)())(()(*

nRnNxnx

NNopop

NNepep

)()()( kXkXkX opep

)())(()(

)())(()(*

kRkNXkX

NNopop

NNepep

and)(kX ep )(kX op

)]())((Im[)](Im[

)]())((Re[)](Re[

kRkNXkX

NNepep

)]())((Im[)](Im[

)]())((Re[)](Re[

kRkNXkX

NNopop

)())(()( then

)())(()( if

kRkNXkX

nRnNxnx

Circular even sequences

Circular odd sequences

)())(()( then

)())(()( if

kRkNXkX

nRnNxnx

)()())((

)())(()]([DFT**

kNXkRkNX

kRkXnx

Conjugate sequences

)()]())(([DFT

)]())(([DFT**

kXnRnNx

)())(())((2

)()](Re[DFT

* kRkNXkX

Complex-value sequences

)())(())((2

)()](Im[DFT

* kRkNXkX

)(]))(())(([2

)](Re[)]([DFT

* nRnNxnx

)(]))(())(([2

)](Im[)]([DFT

* nRnNxnx

)())(()( then

sequence value-real is )( if* kRkNXkX

Real-value sequences

Imaginary-value sequences

)())(()( then

part imaginary has only )( if* kRkNXkX

Summary

)( )( )(

)](Im[ )](Re[)(

kXkXkX

nxjnxnx

)](Im[ )](Re[)(

)( )( )(

kXjkXkX

nxnxnx opep

example

Linear convolution & circular convolution

)()()()(

)()()(N

mnxmxmnxmx

nxnxny

Linear convolution

)(1 nx

)(2 nx

N1 point sequence, 0≤n≤ N1-1

N2 point sequence, 0≤n≤ N2-1

)(nyl L point sequence, L= N1+N2-1

Circular convolution

10 ),()(

111 LnN

Nnnxnx

We have to make both and L-point

sequences by padding an appropriate number of zeros

in order to make L point circular convolution.

)(1 nx )(2 nx

10 ),()(

222 LnN

Nnnxnx

)()()(

)())(()()( )()(

nRrLny

nRmrLnxmx

nRmnxmxnxnxny

)()()( nRrLnyny Lr

)()()( )( isthat

)()( then

nxnxnxnx

return

Sampling in frequency domain

WmxzXkX kN

)(|)()(~

rNnxWN

[IDFS)(~

N rNnxnx )()(~

Frequency Sampling TheoremFor M point finite duration sequence, if the frequency sampling number N satisfy:

MN then

)()()(~)( nxnRnxnx NNN

Interpolation formula of )(zX

Interpolation function

()()()()(N

kXekXeX

Interpolation function

Interpolation formula of )( jeX

return

Approximating to FT of continuous-time aperiodic signal with DFT

dejXtx

dtetxjX

TdtTdtnTt ,,

Sampling in time domain

nTjtj enTxTdtetxjX )()()(

dejXntx

dejXtx

Truncation in time domain

)1~0(:,,)~0(: 00 NnNTTTt

nTjenTxTjX

dejXnTx nTj

Sampling in frequency domain

)]([DFT

)()()(1

enxTenTxTjkXN

)]([IDFT1

)]([IDFT

ejkXnTx

Approximating to FS of continuous-time periodic signal with DFS

ejkXtx

dtetxT

TTdtNTTTdtnTt

Sampling in time domain

)]([DFS1

Truncating in frequency domain

)1,0(: let, , 00 NkNFfNTT s

)]([IDFS)(1

)()()(

jkXNejkXN

ejkXejkXnTx

ejkXtx

Some problems Aliasing

Otherwise, the aliasing will occur in frequency domainhs

hs ffTff

11,2 Sampling in time domain:

Sampling in frequency domain:0

Period in time domain0T Frequency resolution 0F

andis contradictory

Spectrum leakage

sequence length-finite ),()()(

sequence length-infinite ),(

nRnxnx

)()()( 12 j

Rjj eWeXeX

Spectrum extension (leakage)

Spectrum aliasing

Fence effect

Frequency resolution

Comments

return

Zero-padding is an operation in which more zeros are appended to the original sequence. It can provides closely spaced samples of the DFT of the original sequence.

The zero-padding gives us a high-density spectrum and provides a better displayed version for plotting. But it does not give us a high-resolution spectrum because no new information is added.

To get a high-resolution spectrum, one has to obtain more data from the experiment or observation.

example

Summary

return

The frequency representations of x(n)

)( jeX

Time sequence

z-transform of x(n)

Complex frequency

domain

DTFT of x(n)Frequency

domainDFT of x(n)

Discrete frequency domain

jez kNjez

interpolation

Illustration of the four Fourier transforms

Discrete Fourier SeriesSignals that are discrete and periodic

DTFTSignals that are discrete and aperiodic

Fourier SeriesSignals that are continuous and periodic

Fourier TransformSignals that are continuous and aperiodic

)(~ ny

n0 1 2 3 4 5 6

)(~1 mx

)(~2 mnx

1n2n3n4n5n6n

return

)())(()(2

1)( nRnNxnxnx NNep

NnNx ))((

)(nxep

)())(()( * nRnNxnx NNepep

Nep nNx ))((

0 1 2 3 4 5 6 7 8 9 10

Original sequence

0 1 2 3 4 5 6 7 8 9 10

Circular conjugate symmetric component

0 1 2 3 4 5 6 7 8 9 10-4

4Circular conjugate antisymmetric component

return

)()8.0(10 11 nRn

0 1 2 3 4 5 6 7 8 9 10

Circular even sequence x(n)

0 1 2 3 4 5 6 7 8 9 100

The DFT of x(n)

0 1 2 3 4 5 6 7 8 9 100

return

)())(( nRkNX NN

0 1 2 3 4 5 6 7 8 9 10-4

4Circular odd sequence x(n)

0 1 2 3 4 5 6 7 8 9 10

The imaginary part of DFT[x(n)]

0 1 2 3 4 5 6 7 8 9 10

return

)())(( nRkNX NN

)())(()( * kRkNXkX NN

number real a is )0(

)0()())(()0( *

XkRkNXXkNN

number real a is )2

()())(()2

even is if

NXkRkNX

)())(()( * kRkNXkX NN

number imaginary an is )0(

)0()())(()0( *

XkRkNXXkNN

number imginary an is )2

()())(()2

even is if

NXkRkNX

)( , )( 21 nxnx N-point real-value sequences

)]([DFT)( )],([DFT)( 2211 nxkXnxkX

)()()]([DFT)]([DFT

)]()([DFT)]([DFT)(

)()()(

kjXkXnxjnx

njxnxnykY

njxnxny

)())(()(2

1)()](Re[DFT)(1 kRkNYkYkYnykX NNep

)())(()(2

1)](Im[DFT)(2 kRkNYkY

jnykX NNop

0 1 2 3 4 5 6 7 8 9

Linear convolution

n 0 1 2 3 4 5 6 7 8 9

Circular convolution N = 6

0 1 2 3 4 5 6 7 8 9

n 0 1 2 3 4 5 6 7 8 9

return

],2,3,2,1[)( ],2,2,1[)( 21 nxnx

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

Magnitude Response, N = 8

frequency in pi units

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

1Phase Response

return

0 1 2 3 4 5 6 70

X(k),N = 8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

return

)()0( X )2

0 5 10 15 20 250

t-1 -0.5 0 0.5 10

0 5 10 15 20 250

n-2 -1 0 1 2

ta tx )8.0(10)( )( jX a

)(nx )( jeX

-10 -5 0 5 100

n-2 -1 0 1 2

-10 0 100

n-10 0 10

)()( 11 nRnx )()( jj eReX

)(~ nxN )(~kX N

-10 0 100

n-10 0 10

return

)(nxN )(kX N

)( jeR

)(2jeX

)(2 nx

)(1 nx

)(1jeX

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

DTFT DFT

return

]1,1,1,1[)( nx

]0,0,0,0,1,1,1,1[)( nx

]0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1[)( nx

0 2 4 6 8 10 12 14 16 18 20-2

2signal x(n), 0<=n<=19

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

)(20 kX

)52.0cos()48.0cos()( nnnx

0 10 20 30 40 50 60 70 80 90 100-2

2signal x(n), 0<=n<=19+80 zeros

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

)(100 kX

0 10 20 30 40 50 60 70 80 90 100-2

2signal x(n), 0<=n<=99

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

)(100 kX

0 50 100 150 200 250 300 350 400-2

2signal x(n), 0<=n<=99+300 zeros

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

return

)(400 kX

Suppose

kHz 4 Hz, 100 hfF

Determine

N , ,0 TT

Solution sF

T 1.010

125.01042

102422

80010125.0

Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We...

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