basic iir digital filter structures - signal and image...

Copyright © 2001, S. K. Mitra1

Basic IIR Digital FilterBasic IIR Digital FilterStructuresStructures

• The causal IIR digital filters we areconcerned with in this course arecharacterized by a real rational transferfunction of or, equivalently by a constantcoefficient difference equation

• From the difference equation representation,it can be seen that the realization of thecausal IIR digital filters requires some formof feedback

1−z


Basic IIR Digital FilterBasic IIR Digital FilterStructuresStructures

• An N-th order IIR digital transfer function ischaracterized by 2N+1 unique coefficients,and in general, requires 2N+1 multipliersand 2N two-input adders for implementation

• Direct form IIR filters: Filter structures inwhich the multiplier coefficients areprecisely the coefficients of the transferfunction


Direct Form IIR Digital FilterDirect Form IIR Digital FilterStructuresStructures

• Consider for simplicity a 3rd-order IIR filterwith a transfer function

• We can implement H(z) as a cascade of twofilter sections as shown on the next slide

33

22

11

33

22

110

1 −−−

−−−

++++++==

zdzdzdzpzpzpp

zDzPzH)()(

)(



where

33

22

1101

−−− +++=== zpzpzppzPzXzWzH )()()(

)(

)(zH1 )(zH2)(zW

)(zX )(zY

33

22

11

2 111

−−− +++===

zdzdzdzDzWzYzH

)()()(

)(



• The filter section can be seen to bean FIR filter and can be realized as shownbelow

]3[]2[]1[][][ 3210 −+−+−+= nxpnxpnxpnxpnw

)(zH1



• The time-domain representation of isgiven by

Realization offollows from theabove equationand is shown onthe right

)(zH2

][][][][][ 321 321 −−−−−−= nydnydnydnwny)(zH2



• A cascade of the two structures realizingand leads to the realization ofshown below and is known as the directform I structure

)(zH2

)(zH1)(zH



• Note: The direct form I structure isnoncanonic as it employs 6 delays to realizea 3rd-order transfer function

• A transpose of the direct form I structure is shown on the rightand is called the directform I structuret



• Various other noncanonic direct formstructures can be derived by simple blockdiagram manipulations as shown below



• Observe in the direct form structure shownbelow, the signal variable at nodes andare the same, and hence the two top delayscan be shared

1 '1



• Likewise, the signal variables at nodesand are the same, permitting the sharingof the middle two delays

• Following the same argument, the bottomtwo delays can be shared

• Sharing of all delays reduces the totalnumber of delays to 3 resulting in a canonicrealization shown on the next slide alongwith its transpose structure

2'2



• Direct form realizations of an N-th order IIRtransfer function should be evident


Cascade Form IIR DigitalCascade Form IIR DigitalFilter StructuresFilter Structures

• By expressing the numerator and thedenominator polynomials of the transferfunction as a product of polynomials oflower degree, a digital filter can be realizedas a cascade of low-order filter sections

• Consider, for example, H(z) = P(z)/D(z)expressed as

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

)()()(

321

321zDzDzD

zPzPzPzDzPzH ==



• Examples of cascade realizations obtainedby different pole-zero pairings are shownbelow



• Examples of cascade realizations obtainedby different ordering of sections are shownbelow



• There are altogether a total of 36 differentcascade realizations of

based on pole-zero-pairings and ordering• Due to finite wordlength effects, each such

cascade realization behaves differently fromothers

)()()()()()()(zDzDzD

zPzPzPzH321

221=



• Usually, the polynomials are factored into aproduct of 1st-order and 2nd-orderpolynomials:

• In the above, for a first-order factor

∏

++++= −−

−−

k kk

kkzzzzpzH 2

21

1

22

11

0 11

ααββ

)(

022 == kk βα



• Consider the 3rd-order transfer function

• One possible realization is shown below

++

= −−

−−

−

− ++

+

+2

221

12

222

112

111

111

11

11

0 zzpzH zz

zz

ααββ

αβ)(



• Example - Direct form II and cascade formrealizations of

are shown on the next slide

321

321

201804010203620440

−−−

−−−

−++

++=zzzzzzzH

...

...)(

= −

−

−−

−−

−++

++1

1

21

21

401508010203620440

zz

zzzz

...

...



Direct form II Cascade form


Parallel Form IIR Digital FilterParallel Form IIR Digital FilterStructuresStructures

• A partial-fraction expansion of the transferfunction in leads to the parallel form Istructure

• Assuming simple poles, the transfer functionH(z) can be expressed as

• In the above for a real pole

∑

+= −−

−

++

+

k zzz

kk

kkzH 22

11

110

10 ααγγγ)(

012 == kk γα

1−z



• A direct partial-fraction expansion of thetransfer function in z leads to the parallelform II structure

• Assuming simple poles, the transfer functionH(z) can be expressed as

• In the above for a real pole

∑

+= −−

−−

++

+

k zzzz

kk

kkzH 22

11

22

10

10 ααδδδ)(

022 == kk δα



• The two basic parallel realizations of a 3rd-order IIR transfer function are shown below

Parallel form I Parallel form II



• Example - A partial-fraction expansion of

in yields

321

321

201804010203620440

−−−

−−−

−++

++=zzzzzzzH

...

...)(

21

1

1 508012050

4016010 −−

−

− ++

−−

−++−=

zzz

zzH

..

..

.

..)(

1−z



• The corresponding parallel form I realizationis shown below



• Likewise, a partial-fraction expansion ofH(z) in z yields

• The correspondingparallel form IIrealization is shownon the right

21

11

1

1

5080125020

401240

−−

−−

−

−

++

+

−+=

zzzz

zzzH

..

..

.

.)(


Realization Using MATLABRealization Using MATLAB• The cascade form requires the factorization

of the transfer function which can bedeveloped using the M-file zp2sos

• The statement sos = zp2sos(z,p,k)generates a matrix sos containing thecoefficients of each 2nd-order section of theequivalent transfer function H(z) determinedfrom its pole-zero form


Realization Using MATLABRealization Using MATLAB• sos is an matrix of the form

whose i-th row contains the coefficientsand , of the the numerator anddenominator polynomials of the i-th 2nd-order section

6×L

=

2L1L0L2L1L0L

221202221202211101211101

dddppp

dddpppdddppp

sos

}{ lip}{ lid


Realization Using MATLABRealization Using MATLAB

• L denotes the number of sections• The form of the overall transfer function is

given by

• Program 6_1 can be used to factorize anFIR and an IIR transfer function

∏∏=

−−

−−

= ++++==

L

i iii

iiiL

ii zdzdd

zpzppzHzH1

22

110

22

110

1)()(



• Note: An FIR transfer function can betreated as an IIR transfer function with aconstant numerator of unity and adenominator which is the polynomialdescribing the FIR transfer function



• Parallel forms I and II can be developedusing the functions residuez andresidue, respectively

• Program 6_2 uses these two functions


Realization of Realization of Allpass Allpass FiltersFilters• An M-th order real-coefficient allpass

transfer function is characterized byM unique coefficients as here the numeratoris the mirror-image polynomial of thedenominator

• A direct form realization of requires2M multipliers

• Objective - Develop realizations ofrequiring only M multipliers

)(zAM

)(zAM

)(zAM


Realization Using MultiplierRealization Using MultiplierExtraction ApproachExtraction Approach

• Now, an arbitrary allpass transfer functioncan be expressed as a product of 2nd-orderand/or 1st-order allpass transfer functions

• We consider first the minimum multiplierrealization of a 1st-order and a 2nd-orderallpass transfer functions


First-OrderFirst-Order Allpass Allpass Structures Structures• Consider first the 1st-order allpass transfer

function given by

• We shall realize the above transfer functionin the form a structure containing a singlemultiplier as shown below1d

1d1X

2X1Y

2Y

11

11

1 1)( −

−

++=

zdzdzA


First-OrderFirst-Order Allpass Allpass Structures Structures

• We express the transfer functionin terms of the transfer parameters of thetwo-pair as

• A comparison of the above with

yields1

1

11

11 −

−

++=

zdzdzA )(

221

21122211111

221

1211211 111 td

ttttdttddtt

tzA−

−−=

−+= )()(

111 XYzA /)( =

1211222111

221

11 −=−−== −− ttttztzt ,,



• Substituting and in we get

• There are 4 possible solutions to the aboveequation:Type 1A:Type 1B:

121122211 −=− tttt1

11−= zt 1

22−−= zt

22112 1 −−= ztt

11 212

121

221

11 =−=−== −−− tztztzt ,,,

121

112

122

111 11 −−−− −=+=−== ztztztzt ,,,



• Type 1A :• Type 1B :

• We now develop the two-pair structure forthe Type 1A allpass transfer function

22112

122

111 11 −−− −==−== zttztzt ,,,

121

112

122

111 11 −−−− +=−=−== ztztztzt ,,,

t

t



• From the transfer parameters of this allpasswe arrive at the input-output relations:

• A realization of the above two-pair issketched below

21

12 XzXY −−=

221

22

11

1 1 XYzXzXzY +=−+= −−− )(



• By constraining the , terminal-pairwith the multiplier , we arrive at theType 1A allpass filter structure shownbelow

2X 2Y1d

Type 1A


First-Order First-Order Allpass Allpass StructuresStructures• In a similar fashion, the other three single-

multiplier first-order allpass filter structurescan be developed as shown below

Type 1B Type 1A t

Type 1B t


Second-OrderSecond-Order Allpass AllpassStructuresStructures

• A 2nd-order allpass transfer function ischaracterized by 2 unique coefficients

• Hence, it can be realized using only 2multipliers

• Type 2 allpass transfer function:

221

11

21121

2 1 −−

−−

++

++=

zddzdzzdddzA )(


Type 2 Type 2 AllpassAllpass Structures Structures



• Type 3 allpass transfer function:

22

11

2112

3 1 −−

−−

++

++=

zdzdzzddzA )(


Realization Using MultiplierRealization Using MultiplierExtraction ApproachExtraction Approach

• Example - Realize

• A 3-multiplier cascade realization of theabove allpass transfer function is shownbelow

321

321

201804014018020

3 −−−

−−−

−++

+++−=zzzzzzzA

...

...)(

)..)(.(

)..)(.(211

211

50801401805040

−−−

−−−

++−

+++−=zzzzzz


Realization Using Two-PairRealization Using Two-PairExtraction ApproachExtraction Approach

• The stability test algorithm described earlierin the course also leads to an elegantrealization of an Mth-order allpass transferfunction

• The algorithm is based on the developmentof a series of th-order allpass transferfunctions from an mth-order allpasstransfer function for

)( 1−m)(zAm 1−

)(zAm 11 ,...,, −= MMm



• Let

• We use the recursion

where• It has been shown earlier that is

stable if and only if

mm

mm

mmmmm

zdzdzdzdzzdzdzdd

m zA −−−−

−−

−−−−−

−−

+++++

+++++= )(

)(

...

...)( 1

12

21

1

11

22

11

1

11 ,...,, −= MMm],[)()(

)(zAk

kzAm

mm

mmzzA−

−− =

11

mmm dAk =∞= )(

)(zAM

12 <mk for 11 ,...,, −= MMm



• If the allpass transfer function isexpressed in the form

then the coefficients of are simplyrelated to the coefficients of through

)()(

)()(

''...'

'...'')( 1

12

21

1

121

121

11 −−−

−−−

−

−−−−−−−

++++++++

− = mm

mm

mmmm

zdzdzdzzdzdd

m zA

)(zAm 1−)(zAm

111 2 −≤≤−

−= − midddddm

immii ,'

)(zAm 1−



• To develop the realization method weexpress in terms of :

• We realize in the form shown below

)(zAm 1−)(zAm

)(zAm

)(

)()(zAzkzAzk

mmm

mmzA1

11

1

1 −−

−−

+

+=

)(zAm

)(zAm 1−1X

1Y2X

2Y

22211211tttt



• The transfer function of theconstrained two-pair can be expressed as

• Comparing the above with

we arrive at the two-pair transfer parameters

)(

)()()(

zAtzAttttt

mm

mzA122

121122211111 −

−

−

−−=

11 XYzAm /)( =

)(

)()(zAzkzAzk

mmm

mmzA1

11

1

1 −−

−−

+

+=



• Substituting and in theequation above we get

• There are a number of solutions for and

12211

−−== zktkt mm,1

21122211−−=− ztttt

122

−−= zkt mmkt =11

122112 1 −−= zktt m )(

21t12t



• Some possible solutions are given below:

11 2112

121

2211 =−=−== −− tzktzktkt mmm ,)(,,

mmmm ktzktzktkt +=−=−== −− 11 211

121

2211 ,)(,,2

2112

121

2211 11 mmmm ktzktzktkt −=−=−== −− ,,,

221

112

12211 1 mmm ktztzktkt −==−== −− ,,,



• Consider the solution

• Corresponding input-output relations are

• A direct realization of the above equationsleads to the 3-multiplier two-pair shown onthe next slide

11 2112

121

2211 =−=−== −− tzktzktkt mmm ,)(,,

212

11 1 XzkXkY mm−−−= )(

21

12 XzkXY m−−=



• The transfer parameters

lead to the 4-multiplier two-pair structureshown below


121

2211 ,)(,,



• Likewise, the transfer parameters

lead to the 4-multiplier two-pair structureshown below

221

1212

12211 11 mmmm ktzktzktkt −=−=−== −− ,,,



• A 2-multiplier realization can be derived bymanipulating the input-output relations:

• Making use of the second equation, we canrewrite the first equation as

212

11 1 XzkXkY mm−−−= )(

21

12 XzkXY m−−=

21

21 XzYkY m−+=



• A direct realization of

lead to the 2-multiplier two-pair structure,known as the lattice structure, shown below

21

21 XzYkY m−+=

21

12 XzkXY m−−=



• Consider the two-pair described by

• Its input-output relations are given by

• Define


121

2211 ,)(,,

21

11 1 XzkXkY mm−−+= )(

21

12 1 XzkXkY mm−−+= )(

21

11 XzXkV m )( −−=



• We can then rewrite the input-outputrelations as and

• The corresponding 1-multiplier realizationis shown below

21

11 XzVY −+= 112 VXY +=

1V



• An mth-order allpass transfer functionis then realized by constraining any one ofthe two-pairs developed earlier by the

th-order allpass transfer function)( 1−m

)(zAm

)(zAm 1−



• The process is repeated until theconstraining transfer function is

• The complete realization of based onthe extraction of the two-pair lattice isshown below

)(zAM

10 =)(zA



• It follows from our earlier discussion that is stable if the magnitudes of all

multiplier coefficients in the realization areless than 1, i.e., for

• The cascaded lattice allpass filter structurerequires 2M multipliers

• A realization with M multipliers is obtained ifinstead the single multiplier two-pair is used

1<|| mk 11 ,...,, −= MMm

)(zAM



• Example - Realize

33

22

11

321

121

1 −−−

−−−

++++++=

zdzdzdzzdzdd

321

3213 2.018.04.01

4.018.02.0)( −−−

−−−

−+++++−=

zzzzzzzA



• We first realize in the form of alattice two-pair characterized by themultiplier coefficientand constrained by a 2nd-order allpassas indicated below

)(3 zA

)(2 zA2.033 −== dk

2.03 −=k



• The allpass transfer function is of theform

• Its coefficients are given by

)(2 zA

22

11

2112

''1

''2 )(

−−

−−

++

++=zdzdzzddzA

4541667.0223

231)2.0(1

)18.0)(2.0(4.01

'1 ===

−−−−

−−

ddddd

2708333.0223

132)2.0(1

)4.0)(2.0(18.01

'2 ===

−−−−

−−

ddddd



• Next, the allpass is realized as alattice two-pair characterized by themultiplier coefficientand constrained by an allpass asindicated below

)(2 zA

)(1 zA2708333.0'

22 == dk

,2.03 −=k 2708333.02=k



• The allpass transfer function is of theform

• It coefficient is given by

)(1 zA

11

11

"1

"1 )(

−

−

+= +

zd

zdzA

3573771.02708333.14541667.0

1)(1"1 '

2

'1

2'2

'1

'2

'1 ====

+−−

dd

ddddd



• Finally, the allpass is realized as alattice two-pair characterized by themultiplier coefficientand constrained by an allpass asindicated below

)(1 zA

1)(0 =zA3573771.0"

11 == dk

,2.03 −=k,2708333.02 =k 3573771.01 =k

)(2 zA )(1 zA

)(3 zA


Cascaded Lattice RealizationCascaded Lattice RealizationUsing MATLABUsing MATLAB

• The M-file poly2rc can be used to realizean allpass transfer function in the cascadedlattice form

• To this end Program 6_3 can be employed

basic iir digital filter structures - signal and image...

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