signal processing first ch 8 iir filters 1. 8-1 the general iir difference equation 2 feedback term...

Signal Processing First

CH 8

IIR Filters

1

8-1 The General IIR Difference Equation

2

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feedback termrecursive filter

FIR part

No. of coeff. = N+M+1

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101

8-2 Time-Domain Response

3

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8-2.1 Linearity and Time Invariance of IIR Filters


5Invariant Time

],[][

...

]1[]0[]1[]0[]0[]1[][1][

]0[]0[][

until 0 ...

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8-2.1 Linearity and Time Invariance of IIR Filters


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8-2.2 Impulse Response of a First-Order IIR System


7

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8-2.2 Impulse Response of a First-Order IIR System


8

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]3[2]1[3][2][

]3[2]1[3][2][

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8-2.3 Response to Finite-Length Inputs

Characteristics of IIR filter• With the Initial Rest Cond., output is 0 until the input comes.• In case of the 1-st order IIR, impulse response is infinite & becomes geometric series as input disappears & time goes by.


9ealternativ 0 ,][lim 1; iv)

unstable ,][lim 1; iii)

stable ,1

][lim 1; ii)

unstable ,][lim 1; i)

11

1

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8-2.4 Step Responses of a First-Order Recursive System


10

1),1(

1,1

1

)1...(

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][][][*][][

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8-2.4 Step Responses of a First-Order Recursive System

8-3 System Function of an IIR Filter

11

partfeedback

part FIR

)(

)(

-1)(

)((z)

)()()()-(1

)()()()(

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110

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8-3.1 The General First-Order Case


12

][]1[][

)()()( ),(1

1)(

)(

1)(

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)()()(

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1)()(

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8-3.2 The System Function and Block-Diagram Structures


13

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)()()1( )()(

1)(

)()()()(

1)()( :II Case

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8-3.3 Relation to the Impulse Response

8-4 Poles and Zeros

15

.0at zero ,5.0at pole

5.0

2

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2)( )(2)(5.0)(

][2]1[5.0][ Ex)

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8-4.2 Pole Locations and Stability

by. goes timeas decreases

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stable isfilter IIRorder first general at Pole

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8-4 Poles and Zeros

16

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1

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8-5 Frequency Response of an IIR Filter

.)ˆ( meaningful for the demanded is 1condition thei.e.

ROC, in the be should circle)unit 1(

, is 1

)( stablefor ROC filter, IIRorder st -1 theof caseIn

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8-7 The Inverse z-Transform and Some Applications

17

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8-7.1 Revisiting the Step Response of a First-Order System


19

.][)( of form in the are ][ & ],[ ],[

expansion.fraction partialby 1

of form in the )( & ),( ),(Represent

1

11

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k

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8-7.2 A General Procedure for Inverse z-Transformation

?][ 4.03.01

1.21)( 8.10)Ex

21

1

nxzz

zzX


20

8-7.2 A General Procedure for Inverse z-Transformation

?][ 4.03.01

4.04.22)( 8.11)Ex

21

21

nyzz

zzzY

8-8 Steady-State Response and Stability

21

8-8 Steady-State Response and Stability

22

][

][

;2.0 i.e. ],[][When

],[5]1[8.0][ ;8.0 ,5 If 8.12)Ex

05

10

ny

ny

nuenx

nxnynyab

ss

t

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23

HW #5(Do not submit ! But prepare the final exam.)

HW #5 - Textbook Problems: p238-242

P-8.1, 8.4, 8.6, 8.8, 8.10, 8.11, 8.12, 8.15

8-9 Second Order Filters

24]2[]1[2][]2[64.0]1[8.0][64.08.01

21)( 8.14)Ex

3.05.01

231)(

]2[2]1[3][]2[3.0]1[5.0][ 8.13)Ex

1)(

)()(

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

]2[]1[][]2[]1[][

Eq. Diff. General The

21

21

21

21

22

11

22

110

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nxbnxbnxbnyanyany

8-9.1 z-Transform of Second-Order Filters


25

]2[]1[][][

][]2[]1[][

)()()( ),(1

1)(Let

)(1

1)()(

210

21

22

1102

21

1

22

11

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110

nwbnwbnwbny

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8-9.2 Structures for Second-Order IIR Systems


26

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j

reaa

ja

p

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p

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2

)4(

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roots conjugatecomplex 2

root real 1

roots real 2

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if

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1)(

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2

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221

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212

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1

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8-9.3 Poles and Zeros


27

1 ,0at zeros ,2

31at poles

1

)(2

1

22)( 8.14)Ex

3/

2

2

21

1

zej

z

zz

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zz

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j

8-9.3 Poles & Zeros


28

8-9.4 Impulse Response of a Second-Order IIR System

?][]2[6

1]1[

6

5][

][]2[6

1]1[

6

5][

0at zero & 31 ,2/1at poles

)31

)(21

()31

1)(21

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1(z) 8.15)Ex

poles; real , 1)

][][][][

expansion 11

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)(

2

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21

22112

2

12

21

1

1

2

22

21

1

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110

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29

][)3

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3 ,2

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found. are rules No

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30



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211

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)(cos2

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)(cos21)(

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34

8-10 Frequency Response of Second-Order IIR Filter

signal processing first ch 8 iir filters 1. 8-1 the general iir difference equation 2 feedback term...

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