1 Signals and Systems Lecture 9 Fourier Series and LTI Systems Frequency Response and Filtering

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Signals and SystemsLecture 9

•Fourier Series and LTI Systems•Frequency Response and Filtering

2

dtethsH st

)()(

th stesHste

][nh nzzHnz

n

nznhzH ][)(

Chapter 3 Fourier Series

§3.8 Fourier Series and LTI Systems

Eigenfunctions of LTI System

System Function

System Function

3

dtethjH tj

)()(

n

njj enheH ][)(

)( jez

th tjejH tje

][nh njj eeH nje



Frequency Response of an LTI System

Frequency Response of CT LTI Systems

)( js

Frequency Response of DT LTI Systems

4



Linear Combinations of Eigenfunctions

Frequency Responseof LTI System

dtethjH tj

Periodic Signal

Continuous-time LTI System

k

gain

k ajkHa ""

0 )( )(00

0)()( jkHjejkHjkH including both amplitude & phase

th tjkk

k

ejkHaty 00

tjkk

k

eatx 0

)( jH must be well defined and finite.

5

Example:3.16

kk ajkHb )( 0 20

3

3

2)(k

tjkkeatx

Input signal:

jdeejH j

1

1)(

0

Frequency response:

)()( tueth tImpulse response:

3

3

2)(k

tjkkebty

Find out bk of output signal


6


Linear Combinations of Eigenfunctions

Frequency Responseof LTI System

nj

n

j enheH

Discrete-time LTI System

Periodic Signal

k

gain

Njk

k aeHa ""

2

)(

)(22 2

)()(N

jkeHjN

jkN

jkeeHeH

including both amplitude & phase

][nh

Nk

nN

jkN

jk

k eeHany 22

][

Nk

nN

jk

keanx2

][

)( jeH must be well defined and finite.

7

Example:3.17

NjN

e

jHjH

220

1

1)()(

N

nnx

2cos][Input

signal:

jn

njnj

eeeH

1

1)(

0

Frequency response:

][][ nunh nImpulse response:

nN

rny2

cos][Find out the output signal

))(tan(

20

1

1

1)(

jejHIf N=4, we get

211 rthen )(tan 1 and


8


Example

Consider an LTI system with input

the unit impulse response , determine the

Fourier Series Representation of output

nttx n

n

1

ty tueth t4

jk t jk tk k

k k

y t b e a H jk e

kb

0 k is even

4

1

jk k is odd

k

tjkkeatx 2

2

)1(1)(

12

2k

T

tjkk dtetxT

a

4

1)()(

jdtetxjH tj

9


§3.9 Filtering （滤波）

th tjkk

k

ejkHaty 00

tjkk

k

eatx 0

dtethjH tj

Filter

Frequency-Shaping Filter频率成形滤波器

Frequency-Selective Filter频率选择性滤波器

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1. Equalizer （均衡器）


§3.9.1 Frequency-Shaping Filter Change the shape of the spectrum

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2. Differentiator （微分器）

D tx dt

tdx

tuth 1

H j j

0

jH sgn2

jH

2

2

12


Figure 3.24

13


3. Discrete-time system

12

1 nxnxny

/ 2 cos / 2j jH e e

2/ jeH j

2

2

2/cos jeH

1

0

14

Filter out signals outside of the frequency range of interest

Lowpass Filters:

Only show amplitude here.

Note for DT:

)()( )2( jj eHeH


§3.9.2 Frequency-Selective Filter

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Highpass Filters

Remember: — = highest frequency in DT

njn e )1(



16Bandpass Filters



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1. Ideal Lowpass Filter (LPF)

c

cjH

0

1 0

1

c

jH

cPassband Stopband Stopband

Continuous-time system

Discrete-time system

k

keH

c

cj

2 0

2 1

0

1

c

jeH

c 22

Passband Stopband Stopband

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2. Ideal Highpass Filter (HPF)

c

cjH

0

1



1

0

cj

c

H e

0

1

jeH

22

0

1

c

jH

cPassband Stopband Passband

2j jH e H e

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3. Ideal Bandpass Filter (BPF)

others 0

1 21 jH


0

1

jH

1 212


others 0

1 21 jeH

2100

1

jeH

22

20


Example Consider an LTI system with input

the frequency response of this system is

as shown in Figure 1 , determine the output of system

kttxk

2

ty jH

0

1

2/3

jH

Figure 1 (a)

2/3 0

1

jH

Figure 1 (b)

2/52/3 2/3 2/5

1a cos

2y t t

b cos 2 y t t

k

jkkeatx 0)(

dtetxT

aT

tjkk

2

)(1

2

1

21


§3.10 Examples of continuous-time filters described

by differential equations

§3.10.1 A simple RC Lowpass Filter tvs

R

ti

C tvc

21

1

RCjH

RCarctgjH

2RC

1RC

5.0RC2RC

1RC

5.0RC

22


§3.10.2 A simple RC Highpass Filter

RCj

RCjjG

1

tvsR ti

C

rv t

jG tvs tvr

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][]1[][ nxnayny

njnjjnjj eeeaHeeH )1()()(

jj

aeeH

1

1)(

][][ nuanh n ][1

]1][

1

nua

ans

n


§3.11 Examples of discrete-time filters described

by difference equations

§3.11.1 First-Order Recursive DT Filter

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]}1[][]1[{3

1][ nxnxnxny

]}1[][]1[{3

1][ nnnnh

cos3

2

3

1]1[

3

1

][)(

jj

n

njj

ee

enheH


§3.11.2 Nonrecursive DT Filters Example : DT Averager/Smoother

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M

Nk

M

Nk

knMN

nhknxMN

ny ][1

1][][

1

1][

)2sin(

]2)1(sin[

1

1

1

1)(

2)(

NM

eMN

eMN

eH

MNj

M

Nk

jkj

Rolls off at nω lower as M+N+1 increases


§3.11.2 Nonrecursive DT Filters Example : DT Averager/Smoother

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Example : Simple DT “Edge” Detector

DT 2-points “differentiator”

Amplifies high-frequency components

]]1[][[2

1][ nxnxny

]]1[][[2

1][ nnnh

)2sin(

)1(2

1)(

2

j

jj

je

eeH

)2sin()( jeH


27

Summary

• Fourier Series and LTI Systems

• Frequency Response and Filtering

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Readlist

• Signals and Systems:– Chapter1~Chapter3

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Homework: 3.15 3.35

30


Homework:

3.1 3.13 3.15 3.34 3.35

31


dtethsH st

ste stesH Eigenfunction

Eigenvalue

k ks t s tk k k

k k

x t a e y t a H s e

nz nzzH Eigenfunction

Eigenvalue n

n

znhzH

n nk k k k k

k k

x n a z y n a H z z

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tjkk

k

eatx 0

dtetxT

a tjk

Tk0

00

1

Synthesis equation综合公式

Analysis equation分析公式


0 1 10 1

sin 2sin 0k

k T Ta c k T k

k T

1. Periodic square wave

2. Periodic Impulse Trains

,2,1,0 1

kT

ak

33


kk btyatxFSFS

kkk BbAactBytAxtz FS

00

FS

0tjk

keattx

kk aa txtx kk aa

or

txtx k ka a k ka a

34

221k

kT

adttxT

katx FS

knn ajktx 0

FS

kajktx 0FS


real odd tx kaPurely imaginary

odd

real even tx ka real even

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1 Signals and Systems Lecture 9 Fourier Series and LTI Systems Frequency Response and Filtering