fir and iir filter asdsa

8/12/2019 FIR and IIR Filter asdsa

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Feb.2008 DISP Lab 1

FIR and IIR Filter DesignTechniques

FIR IIR Speaker: Wen-Fu Wang

Advisor: Jian-Jiun Ding

E-mail: [email protected] Graduate Institute of Communication Engineering

National Taiwan University, Taipei, Taiwan, ROC


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Feb.2008 DISP Lab 2

Outline

Introduction

IIR Filter Design by Impulseinvariance method

IIR Filter Design by Bilineartransformation method

FIR Filter Design by Window functiontechnique


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Feb.2008 DISP Lab 3

Outline

FIR Filter Design by Frequency

sampling technique FIR Filter Design by MSE

Conclusions

References


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Feb.2008 DISP Lab 5

Introduction

IIR is the infinite impulse responseabbreviation.

Digital filters by the accumulator, themultiplier, and it constitutes IIR filterthe way, generally may divide intothree kinds, respectively is Direct

form, Cascade form, and Parallelform.


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Feb.2008 DISP Lab 6

Introduction

IIR filter design methods include theimpulse invariance, bilineartransformation, and step invariance.

We must emphasize at impulseinvariance and bilineartransformation.


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Feb.2008 DISP Lab 7

Introduction

IIR filter design methods

Continuous frequency

band transformation

ImpulseInvariance

method

Bilineartransformation

method

Step invariancemethod

IIR filter

Normalized analog

lowpass filter


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Feb.2008 DISP Lab 8

Introduction

The structures of IIR filter

Direct

form 1

Direct form2

b0

b1

b2 b2

b1

b0

-a1

-a2

-a1

-a2

x(n) x(n)Y(n) Y(n)

1z

1z

1z

1z

1z

1z


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Feb.2008 DISP Lab 9

Introduction

The structures of IIR filter

Cascade form

x(n) Y(n)b0

b1

b2

-a1

-a2

-c1

-c2

d1

d2

Parallel form

Y(n)x(n)

b1

b0

d1

d0

E

-c1

-c2

-a1

-a2

1z

1z

1z

1z

1z

1z

1z

1z


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Feb.2008 DISP Lab 10

Introduction

FIR is the finite impulse responseabbreviation, because its designconstruction has not returned to thepart which gives.

Its construction generally uses Directform and Cascade form.


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Introduction

FIR filter design methods include thewindow function, frequency sampling,minimize the maximal error, and MSE.

We must emphasize at window function,frequency sampling, and MSE.

Window function

technique

Frequency

sampling technique

Minimize the

maximal error

FIR filter

Mean square

error


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Introduction

The structures of FIR filter

x(n) x(n)

b1

b2

b3

b4

b0Y(n) Y(n)

Direct form Cascade form

b1

b2

d1

d2

b0

1z

1z

1z

1z

1z

1z

1z

1z


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The most straightforward of these isthe impulse invariance transformation

Let be the impulse responsecorresponding to , and define thecontinuous to discrete timetransformation by setting

We sample the continuous timeimpulse response to produce thediscrete time filter

( )ch t ( )cH s

( ) ( )c

h n h nT


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The system function is

It is the many-to-one transformationfrom the s plane to the z plane.

1 2( ) | )sT cz e

k

H z H s jk

T T


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The impulse invariancetransformation does map the -axisand the left-half s plane into the unitcircle and its interior, respectively

j

Re(Z)

Im(Z)

1

S domain Z domain

sTe

j


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The Butterworth and Chebyshev-Ilowpass designs are more appropriatefor impulse invariant transformationthan are the Chebyshev-II and ellipticdesigns.

This transformation cannot be applied

directly to highpass and bandstopdesigns.


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is expanded a partial fractionexpansion to produce

We have assumed that there are nomultiple poles

And thus

( )c

H s

1

( )N

kc

k k

AH s

s s

1

( ) ( )kN

s t

c kk

h t A e u t

1( ) ( )k

Ns nT

kkh n A e u n

11

( )1 k

Nk

s Tk

AH z

e z


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Example:

Expanding in a partial fractionexpansion, it produce

The impulse invariant transformationyields a discrete time design with the

system function

2 2( )

( )c

s aH s

s a b

1/ 2 1/ 2( )cH s

s a jb s a jb

( ) 1 ( ) 1

1/ 2 1/ 2( )

1 1a jb T a jb T H z

e z e z


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The most generally useful is the

bilinear transformation.

To avoid aliasing of the frequencyresponse as encountered with theimpulse invariance transformation.

We need a one-to-one mapping from

the splane to thezplane. The problem with the transformation

is many-to-one.sTz e


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The axis is compressed into theinterval for in a one-to-one method

The relationship between andis nonlinear, but it is approximatelylinear at small .

( , )T T

'

'

'

-

'/ T

/ T


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The desired transformation to isnow obtained by invertingto produce

And setting , which yields

12' tanh ( )2

sTs

T

2 '

tanh( )2

s Ts

T

s z

1' ( ) lns z

T

2 lntanh( )

2

zs

T

1

12 1( )

1z

T z

Re(Z)

Im(Z)

1

S domain Z domain

12

12

Ts

zT

s

j


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The discrete-time filter design isobtained from the continuous-timedesign by means of the bilinear

transformation

Unlike the impulse invarianttransformation, the bilineartransformation is one-to-one, andinvertible.

1 1(2/ )(1 )/(1 )( ) ( ) |c s T z z H z H s


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FIR Filter Design by Windowfunction technique

Simplest FIR the filter design iswindow function technique

A supposition ideal frequencyresponse may express

where

( ) [ ]j j nd dn

H e h n e

1[ ] ( )

2

j j n

d dh n H e e d


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To get this kind of systematic causalFIR to be approximate, the mostdirect method intercepts its idealimpulse response!

[ ] [ ] [ ]dh n w n h n

( ) ( ) ( )dH W H


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1.Rectangular window

2.Triangular window (Bartett window)

1, 0[ ]

0,

n Mw n

otherwise

2 , 02

2[ ] 2 ,2

0,

n MnM

n Mw n n M M

otherwise


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1.Rectangular window

2.Triangular window (Bartett window)

0 10 20 30 40 50 600

0.5

1

sequence (n)

T(n)

Rectangular window

0 10 20 30 40 50 600

0.5

1

sequence (n)

T(n)

Bartlett window

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-50

0

50

100

pi unitsFrequencyresponseT

(jw)(dB) Rectangular window

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-50

0

50

100

pi unitsFrequencyresponseT(jw)(dB) Bartlett window


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3.HANN window

4.Hamming window

1 21 cos , 0

[ ] 2

0,

nn M

w n M

otherwise

20.54 0.46cos , 0

[ ]0,

nn M

w n Motherwise


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-50

0

50

100


(jw)(dB) Hanning window

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-50

0

50

100

pi unitsFrequencyresponseT(jw)(dB) Hamming window

0 10 20 30 40 50 600

0.5

1

sequence (n)

T(n)

Hanning window

0 10 20 30 40 50 600

0.5

1

sequence (n)

T(n)

Hamming window


3.HANN window

4.Hamming window


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5.Kaisers window

6.Blackman window

2

0

0

2[ 1 (1 ) ]

[ ] , 0,1,...,

[ ]

nI

Mw n n M

I

2 40.42 0.5cos 0.08cos , 0

[ ]0,

n nn M

w n M Motherwise


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-50

0

50

100


(jw)(dB) Blackman window

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-150

-100

-50

0

50

100

pi unitsFrequencyresponseT(

jw)(dB) Kaiser window

5.Kaisers window

6.Blackman window

0 10 20 30 40 50 600

0.5

1

sequence (n)

T(n)

Blackman window

0 10 20 30 40 50 600

0.5

1

sequence (n)

T(n)

Kaiser window



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( / )s

M

Window Peak sidelobe level(dB)

Transitionbandwidth

Max. stopband

ripple(dB)

Rectangular -13 0.9 -21

Hann -31 3.1 -44

Hamming -41 3.3 -53

Blackman -57 5.5 -74


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FIR Filter Design by Frequencysampling technique

For arbitrary, non-classicalspecifications of , the calculation

of ,n=0,1,,M, via an appropriate

approximation can be a substantialcomputation task.

It may be preferable to employ adesign technique that utilizesspecified values of directly,without the necessity of determining

' ( )dH

( )dh n

' ( )dH

( )dh n


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We wish to derive a linear phase IIRfilter with real nonzero . Theimpulse response must be symmetric

where are real and denotesthe integer part

( )h n

[ /2]

0

1

2 ( 1/ 2)( ) 2 cos( )

1

M

k

k

k nh n A A

M

kA [ / 2]M

0,1,...,n M


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It can be rewritten as

where and

Therefore, it may write

where

1/ 2 /

0

/ 2

( )N

j k N j kn N

k

k

k N

h n A e e

0,1,..., 1n N

1N M k N k

A A

/ 2 /( ) j k N j kn N k kh n A e e

1

0/ 2

( ) ( )N

kkk N

h n h n

0,1,..., 1n N


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with corresponding transform

where

Hencewhich has a linear phase

1

0

/2

( ) ( )N

k

k

k N

H z H z

/

2 / 1

(1 )( )

1

j k N N

kk j k N

A e zH z

e z

' ( 1)/2 sin / 2

( ) sin[( / / 2)]

j T N

k k

TN

H A e k N T


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The magnitude response

which has a maximum value

at where

' sin / 2( )sin[( / / 2)]

k k

TNH A

k N T

kN A

/k sk N 2 /s T


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The only nonzero contribution toat is from , and hencethat

Therefore, by specifying the DFTsamples of the desired magnitude

response at the frequencies ,

and setting

'( )H

k ' ( )kH

'( )k kH N A

' ( )dH k

' ( ) /k d kA H N


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We produce a filter design fromequation (5.1) for which

The desired and actual magnituderesponses are equal at the Nfrequencies

'

'( ) ( )k d kH H

k


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In between these frequencies, isinterpolated as the sum of theresponses , and its magnitude

does not, equal that of

'( )H

' ( )kH

' ( )dH


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Example: For an ideal lowpass filter

from , we wouldchoose

The frequency samples areindeed equal to the desired

' 1, 0,1,...,

( )0, 1,...,[ / 2]

d k

k PH

k P M

' ( ) /k d kA H N

( 1) / ( 1), 0,1,...,

0, 1,...,[ / 2]

k

k

M k PA

k P M

' ( )k

H

' ( )d kH


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The response is very similar to theresult form using the rectangularwindow, and the stopband is similarly

disappointing.

We can try to search for the optimumvalue of the transition sample would

quickly lead us to a value ofapproximately , k p0.38( 1) /( 1)ppA M


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FIR Filter Design by MSE

: The spectrum of the filter weobtain

: The spectrum of the desiredfilter

MSE=

( )H f

( )dH f

2/

2/

21 s

s

f

f ds dffHfHf

0 0.1 0.2 0.3 0.4 0.5-0.5

0

0.5

1

1.5


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Larger MSE, but smaller maximalerror

Smaller MSE, but larger maximalerror

0 0.1 0.2 0.3 0.4-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4-0.5

0

0.5

H(F) H(F) - H (F)d

0 0.1 0.2 0.3 0.4-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4-0.5

0

0.5

H(F)H(F) - H (F)

d


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2. when n,

when n= , n0,

when n= , n= 0,

3. The formula can be repressed as:

02cos2cos2/1

2/1 dFFFn

2/12cos2cos2/1

2/1 dFFFn

12cos2cos2/1 2/1 dFFFn

dFFHdFFHFnnsnssMSE ddk

n

k

n 2/1

2/12

2/1

2/101

22 2cos][22/][]0[


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4. Doing the partial differentiation:

5. Minimize MSE: for all ns

2/1

2/12]0[2

]0[dFFHs

s

MSEd

2/12/1

2cos2][][

dFFHFnnsns

MSEd

0][

ns

MSE

2/1

2/1]0[ dFFHs d

2/1

2/12cos2][ dFFHFnns d

[ ] [0]

[ ] [ ]/ 2 for n=1,2,...,k

[ ] [ ]/ 2 for n=1,2,...,k

[ ] 0 for n


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Conclusions

FIR advantage:

1. Finite impulse response

2. It is easy to optimalize3. Linear phase

4. Stable

FIR disadvantage:1. It is hard to implementation than IIR


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Conclusions

IIR advantage:

1. It is easy to design

2. It is easy to implementation IIR disadvantage:

1. Infinite impulse response

2. It is hard to optimalize than FIR3. Non-stable


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References

[1]B. Jackson, Digital Filters and Signal

Processing, Kluwer Academic Publishers 1986 [2]Dr. DePiero, Filter Design by Frequency

Sampling, CalPoly State University

[3]W.James MacLean, FIR Filter DesignUsing Frequency Sampling

[4],

,

2005 [5]Maurice G.Bellanger, Adaptive Digital

Filters second edition, Marcel dekker 2001


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Feb 2008 DISP Lab 55

References

[6] Lawrence R. Rabiner, Linear ProgramDesign of Finite Impulse Response DigitalFilters, IEEE 1972

[7] Terrence J mc Creary, On FrequencySampling Digital Filters, IEEE 1972

fir and iir filter asdsa

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