Fourier Theory transform

8/10/2019 Fourier Theory transform

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Understanding of Fourier Theory

Yu Zhuliang

College of Automation Science and

Engineering

South China University of Technology


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Lecturer

Yu Zhuliang (

) Professor

Ph.D, NTU, M. Eng. and B. Eng. NUAA

Experiences in teaching,

,

Email: [email protected]

Phone: 18688899719, 020-22236326

QQ: 1049745720
mailto:[email protected]:[email protected]


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Fourier Transform

(FT)

Fourier Series (FS)

Discrete Time Fourier

Transform (DTFT)

Discrete Fourier Transform

(DFT)

Fast Fourier Transform (FFT)


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Signal Decomposition


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5

Unit impulse

Extremely important signal, unit impulse

0, 0[ ]

1, 0

nn

n


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Unit impulse and Sampling

Unit impulse sequence is always used to sample thevalue of signals at n=0 or n=n0

0 0 0

[ ] [ ] [0] [ ]

[ ] [ ] [ ] [ ]

x n n x n

x n n n x n n n


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Discrete-time system: Convolution sum

Express signal in terms of impulse

It is a weighted sum of shifted unit

impulses.

The response of a LTI system to the

input signal is just the weighted sum of

the shifted system response functions.

[ ] [ ] [ ]k

x n x k n k


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Convolution Sum

[ ] [ ] [ ]kk

y n x k h n

[ ] [ ] [ ] [ ] [ ]k

y n x k h n k x n h n

Linearity: Superposition

[ ] [ ]kn k h n

Impulse Response

0[ ] [ ] [ ] [ ]kn k h n h n k h n k Time Invariant

[ ] [ ] [ ]k

x n x k n k



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Convolution Integral

( ) ( ) ( )y t x h t d

( ) ( ) ( ) ( ) ( )y t x h t d x t h t

Linearity: Superposition

( ) ( )t h t

Impulse Response

( ) ( ) ( )t h t h t Time Invariant

( ) ( ) ( )x t x t d



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Convolution integral and Sum

The characteristics of an LTI system are

completely determined by its impulse response.

It is important to emphasize that this propertyholds in general only for LTI system.

( ) ( ) ( ) ( ) ( )

[ ] [ ] [ ] [ ] [ ]

k

y t x h t d x t h t

y n x k h n k x n h n


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Why need decomposition?

Easy in analysis

Provide deep insights of signals and

systems

Impulse is discrete in time domain:

convolution

How if we consider signal components

discrete in frequency domain?


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Historical Perspective

Joseph Fourier


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http://pic.baike.soso.com/p/20101111/20101111134200-1711877194.jpghttp://pic.baike.soso.com/p/20120224/20120224140031-134224729.jpg


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Thread of Fourier Theory

Fourier Transform

(FT)

Fourier Series (FS)


Transform (DTFT)


(DFT)



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Key in understanding Fourier Theory

Discrete vs. Periodic

Accept it first!

I promise to give proof later!


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18

Origin of idea

The set of basic signals can be used to construct a broad and

useful class of signals

The response for an LTI system to each signal should be

simple enough in structure to provide us with a convenient

representation for the response of the system to any signal

consctured as linear combination of the basic signals The importance for complex exponentials in the study of LTI

systems stems from the fact that the response of an LTI

system to a complex exponential input is the same complex

exponential with only a change in the amplitude,

Continuous time : ( )j t j te H j e


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Response of LTI Systems to Complex

Exponentials

What does these mean?

( ) ,

( ) ( )

k

k

s t

kk

s t

k k

k

x t a e

y t a H s e

0ks jk


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Response of LTI Systems to Complex

Exponentials

Differential equation Algebra equation

( ) kk

s tk

k

dx t a et

sd

0ks jk


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Fourier Transform

(FT)

Fourier Series (FS)


Transform (DTFT)


(DFT)



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Continuous-time periodic signal


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Fourier Series Representation

Periodicsignal

Representation of signals

02( ) ( ),x t x t TT

0

0

( )

1 ( )

jk t

k

k

jk t

kT

x t a e

a x t e dt T


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T=4T1

T=16T1

T=8T1



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Discrete-time periodic signal


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A discrete-time signal is periodic with period N if

Fundamental frequency Harmonic signals that are periodic with period N

is given by

There are only N distinct signals in the set

[ ] [ ]x n x n N

0 2 /N

0[ ] , 0, 1,jk nk

n e k

[ ] [ ]k k rN n n 0( ) , 0, 1,jk tk t e k


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(2 / )[ ] [ ] jk N nk k kk k

x n a n a e

(2 / )

[ ] [ ] jk N n

k k kk N k N

x n a n a e


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0[ ] jk nkk N

x n a e

01

[ ] j

k

k n

N

ka x n e

N


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Important property

So we select k from 0 to N-1

It is sometimes convenient to think of ak as a sequence

define for all values of k, but where only N successive

elements in the sequence will be used in the Fourier

Series representation.

k k Na a


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ExampleDiscrete vs. Periodic


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ComparisonDiscrete vs. Periodic


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Exercise you must do!

( ) ( )k

x t t kT

1k T

a


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Fourier Transform

(FT)

Fourier Series (FS)


Transform (DTFT)


(DFT)



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Motivations for Fourier Transform

For periodic signals, the complex exponential

building blocks are harmonically related. (Fourier

Series Theory)

Harmonically related complex exponentials mean

that the spectrum of periodic signal is discrete. How are aperiodic signals?

For aperiodic signals they are infinitesimally close in

frequency, and the representation in terms of a

linear combination takes the form of a integral ratherthan a sum.


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The Continuous-time Fourier Transform

For a signal

1

1

1, | |( )

0, | | / 2

t Tx t

T t T


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Its Fourier Coefficients are

0 10

0

2sin( ) 2,k k Tak T T


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Let me check

0

0 1

0

1

2sin( )

2sin( )

k

k

k T

a k

T

T

T=4T1

T=8T1

T=16T1


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What happens when ?

We think of an aperiodic signal as the limit of a

periodic signal as the period becomes arbitrary large

T


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Rewrite

0 01

( ) ( )jk t j t

k kk T

kx t a e a x t e dt

T


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Outside the range, x(t)=0

0

0

/ 2

/ 2

( )

)

1

(

1

1

( )

jkT

T

t

k

jk t

a e dt T

e dt

x

T

j

x t

XT

t


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If we define

Then

( ) ( ) j tX j x t e dt

00

1 1

( ) ( )k ka X jk X jT T


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Lets look at the extended periodic signal

0

0

0

0

0 0

1(

( )

1 ( )2

)

k

jk t

k

k

jk t

k

j t

k

X j

x t a e

e

X j e

kT

k


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Since

0

00

1( ) ( )

2 k

kj tx t kX j e


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Since

When T approaches infinity

0

00

1( ) ( )

2 k

kj tx t kX j e

1( ) ( ) ( )2

j t

x t X j e d x t


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1( ) ( )

2

( ) ( )

j t

j t

x t X j e d

X j x t e dt

Fourier Transform Pair


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1

1

1

( )

2sin

j t

Tj t

T

x t e dt

e dt

T



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1, | |( )

0, | |

WX j

W

sin( )

Wtx t

t


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Duality


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Does periodic signal have FT?

0( ) 2 ( )kk

X j a k

0( ) ( )kk

x t a k


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Exercise you must do again!

( ) ( )

k

x t t kT

1 2 2

, ( ) ( )k T k

a X j k T T


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Fourier Transform

(FT)

Fourier Series (FS)


Transform (DTFT)


(DFT)



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What is the Fourier transform of discrete-time

signal?


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Representation of Aperiodic Signals

N approaches infinity, [ ] [ ]x n x n


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(2 / )[ ] j

k

k N n

k

N

x n a e

(2 / )1 [ ] jk Nk

N

n

k

a x n eN


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2

1

( 2 / )

( 2 / )

( 2 / )

1 [ ]

1[ ]

1[ ]

jk N n

k

jk N n

k N

N

N

k N

k

jk n

a x n eN

x n eN

x n eN

?

?


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Lets define

We have

( ) [ ]j j n

k

X e x n e

0

1( )

j

k

ka X eN

0

1( ) |k ka X j

T

f A d l


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0

0

(2 / )

(2 / )

(

0

2 / )

[ ]

1 ( )

1

2 ( )

jk N n

k

jk N n

jk jk N

k N

N

N

n

k

k

k

jX e

x n a e

e

X e

N

e

i f A i di Si l


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0 0

2

0

1

2]

(1

2

[ ( )

)

k

jk jk n

j j n

N

x n X e e

X e e d

0 0

2, , 0N

N

R i f A i di Si l


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2

( ) [ ]

1[ ] ( )

2

n

j j n

j j n

X e x n e

x n X e e d

Discrete-time Fourier Transform Pair

R i f A i di Si l


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2

( ) [ ]

1[ ] ( )2

k

j j n

j j n

X e x n e

x n X e e d

( ) ( )

1( ) ( )

2

j t

j t

X j x t e dt

x t X j e d

Finite integral interval

Periodicity

P i di i


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PeriodicityDiscrete vs. Periodic


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Sampling Theorem

P i f h C i i F i


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Properties of the Continous-time Fourier

Transform

Multiplication Property

( ) ( ) ( )

1( ) ( ) ( ( ))

2

r t s t p t

R j S j P j d

C i b S li I l


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Connection between Sampling Interval

and Peoroid in Corresponding domain

( ) ( )

k

x t t kT

1 2 2

, ( ) ( )k Tk

a X j k T T

R t ti f ti ti i l


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Representation of a continuous-time signal

by its samples (Impulse-train Sampling)

1( ) ( ( ))

p s

k

X j X j kT



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I promise to give proof!

H i di i l?


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How to express an periodic signal?

E f


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Easy proof

Periodic vs Discrete

I t t l i !


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Important conclusions!

I t t l i !


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Important conclusions!

Sampling interval in one domain T

Period in corresponding domain2

T


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Fourier Transform

(FT)

Fourier Series (FS)


Transform (DTFT)


(DFT)


Dis r t F ri r Tr nsf rm


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If we sample the spectrum in frequency

domain, what happens?

( )X j

( ) [ ] j n

n

X j x n e

Discrete Fo rier Transform


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Uniquely sampling the spectrum infrequency domain with K samples

( )X j

2 21 1

0

21

0

22

( ) [ ]

[ ] [ ]

( )

n

Kj kn j knK K

n K n

K j knK

j kn

K

n l

X j k x nK

x n e x n e

x n lK e

e

[0,2 ]



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21

0

2( ) [ ]

[ ] ( )

K j knK

p

n

p

l

X j k x n eK

x n x n lK

[ ]px n Is periodic signal with period K andFourier series coefficients 1 2

( )X j kK K



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Sampling in

frequencydomain, the

time domain

signal

becomes

periodic.




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If the length of time domain signal is N, toguarantee the recoverability of time domain

signal, what is the minimum number of K?



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and Period in Corresponding domain

2

K

Connection between Sampling Interval and


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Connection between Sampling Interval and

Period in Corresponding domain

Sampling interval T, period

DTFT, sampling interval 1, period

DFT, sampling interval , period

2

2

K

22

K

K

2

T



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If the length of time domain signal is N, toguarantee the recoverability of time domain

signal, what is the minimum number of K?

K N



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That is why the sampled points in spectrumof a N point time-domain signal is N.

Both time domain and frequency domain are

discrete, it is suitable for computer

processing.

Questions?


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Questions?

Why we need to control the sampling rate in

time domain ? Answer: Able to recover original signal from

sampled signal

Explain.


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Fourier Transform(FT)

Fourier Series (FS)


Transform (DTFT)


(DFT)


Fast Fourier Transform


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Fast Fourier Transform

It is just a fast computation algorithm of DFT!

Last Question!


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Last Question!

( ) ( ) ( )j j jY e X e H e [ ] [ ] [ ]y n x n h n

DTFT

DTFTX

Inverse

DTFT

Linear convolution!

Last Question!


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Last Question!

DFT

DFTX

Inverse

DFT

Circular convolution!


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Fourier Transform

(FT)

Fourier Series (FS)


Transform (DTFT)


(DFT)


Fourier Theory transform

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