1 Chapter 3 Fourier Series Representations of Periodic Signals Chapter 3 Fourier Series...

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Chapter 3

Fourier Series Representations

of Periodic Signals

Chapter 3

Fourier Series Representations

of Periodic Signals

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Jean Baptiste Joseph Fourier, born in 1768, in France.

1807,periodic signal could be represented by sinusoidal series.

1829,Dirichlet provided precise conditions.

1960s,Cooley and Tukey discovered fast Fourier transform.

Chapter 3 Fourier Series

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§3.2 The Response of LTI Systems to Complex Exponentials

LTI 系统对复指数信号的响应 tyste th1. Continuous-time system

dhtxthtxty

)()()(*)()(

dhe ts )()(

dhee sst )(

)()()( sHtxsHest

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LTI 系统对复指数信号的响应

nh nynz

2. Discrete-time system

k

khknxnhnxny ][][][*][][

k

kn nhz ][

k

kn nhzz ][

)(][)( zHnxzHzn

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tyste th1. Continuous-time system

stH s h t e dt

Eigenfunction特征函数

——Eigenvalue (特征值）

nh nynz

2. Discrete-time system

Eigenfunction特征函数

——Eigenvalue (特征值） n

n

znhzH

ste stesH

nz nzzH

If set z=ejΩ, we can get the same conclusion: ejΩn is eigenfunction of DT LTI systems.

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(3) Input as a combination of Complex Exponentials

Continuous time LTI system:

N

k

tskk

N

k

tsk

k

k

esHaty

eatx

1

1

)()(

)(

Discrete time LTI system:

N

k

nkkk

N

k

nkk

zzHany

zanx

1

1

)(][

][


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Example 3.1

Consider an LTI system : 3 tth

3y t x t

2 1 j tx t e

2 3 3j ty t e x t

2 cos 4 cos 7x t t t

cos 4 3 cos7 3 3y t t t x t

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Example :

Consider an LTI system for which the input

and the impulse response determine the output

ttx 2cos2

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tueth t ty

2 2

1 14 41

1 2 1 2j t j ty t e e

j j

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3.3 Fourier Series Representation of Continuous-time

Periodic Signals

(1) General Form

2,1,0,)( )/2(0 keet tTjktjkk

3.3.1 Linear Combinations of Harmonically Related

Complex Exponentials

The set of harmonically related complex

exponentials:

Fundamental period: T ( common period )


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So, arbitrary periodic signal can be represented as

tjtj ee 00 , : Fundamental components

tjtj ee 00 22 , : Second harmonic components

tjNtjN ee 00 , : Nth harmonic components

k

tjkkeatx 0)(

( Fourier series )


ka ——Fourier Series Coefficients Spectral Coefficients （频谱系数）

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Example 3.2 tjkk

k

eatx 23

3

3/1 , 2/1

4/1 , 1

32

10

aa

aa

Consider a real periodic signal txtx

real periodic kk aa tx

0 01

a 2 cosk kk

x t a A k t

0 0 01

b 2 cos sink kk

x t a B k t C k t

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(2) Representation for Real Signal

Real periodic signal: x(t)=x*(t)

So a*k=a-k

k

tjkkeatx 0)(

10

10

]Re[2

][)(

0

00

k

tjkk

tjkk

k

tjkk

eaa

eaeaatx

Let (A) )( 00, kk tkjk

tjkk

jkk eAeaeAa

1

00 )cos(2)(k

kk tkAatx


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Let (A) )( 00, kk tkjk

tjkk

jkk eAeaeAa

1

00 )cos(2)(k

kk tkAatx

(B) kkk jCBa

]sincos[2)( 01

00 tkCtkBatxk

kk


kkk

kkk

aC

aB

sin

cos

14

2,1,0,)( )/2( ket tTjkk

3.3.2 Determination of the Fourier Series

Representation of a Continuous-time Periodic

Signal

k

tjkkeatx 0)(

( Orthogonal function set )

Determining the coefficient by orthogonality:

( Multiply two sides by )

k

tnkjk

tjn eaetx 00 )()(

tjne 0


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Fourier Series Representation:

nk

nkTdte

T

tnkj

,0

,0)(

Tadteadtetx kk

T

tnkjkT

tjn

00 )()(

T

tjnn dtetxT

a 0)(1

T

tjkk

k

tjkk

equationAnalysisdtetxT

a

equationSynthesiseatx

)()(1

)()(

0

0


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§3.3.2 Determination of Fourier Series Representation

tjkk

k

eatx 0

0

00

1 jk tk T

a x t e dtT

Synthesis equation综合公式

Analysis equation分析公式

ka ——Fourier Series Coefficients

Spectral Coefficients

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Example 3.5 Periodic square wave defined over one period as

2/ t T 0

t 1

1

1

T

Ttx

1 tx

-T -T/2 –T1 0 T1 T/2 T t

dttxT

aT

T

2/

/2- 0

1

Defining x

xxc

sinsin

101 sin

2Tkc

T

Tak

0 1sin 0k

k Ta k

k

T

T12

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Example 1: Periodic Square Wave (P135)

T

Edttx

Ta

T

1

)(1

10 )(tf

2

t1T1T

E

2

O

2

2

j

11

2

2

j

1

11 ej

1de

1 =

tntn

nT

EtE

T

2

j2

j

11

11

eej

nn

Tn

E

2sin

21

11

nTn

E

2

2sin

1

1

1

n

n

T

E

2

2

j

1

1

1

1 de)(1 T

Ttk

k ttfT

a

tk 1j

Defining x

xxc

sinsin

101 sin

2Tkc

T

Tak

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1

1sin

TTc

21

1sin

TTc

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1sin

TTc

T1 固定，的包络固定kTa 11 sin2 TcT

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TT /20 谱线变密

Figure 4.2

14 a TT

18 b TT

116 c TT

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Example Periodic Impulse Trains (周期冲激串 )

tx

tT 0 T T2T2

1

21 jk tT

k

x t eT

ka

T

1

-ω0 0 ω0 2ω0

2 2/ T/T,2,1,0 1

kT

ak

21 2 jk tT

k

y t H jk eT T

k

x t t kT

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Readlist

• Signals and Systems:– 3.4~3.5

• Question: – Proof properties of Fourier series.

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Problem Set

• 3.1 P250

• 3.3 P251

• 3.22(a.a) P255

1 Chapter 3 Fourier Series Representations of Periodic Signals Chapter 3 Fourier Series...

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