iir direct form realization

8/12/2019 IIR Direct Form Realization

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Copyright 2001, S. K. Mitra

Basic IIR Digital Filter

Structures The causal IIR digital filters we are

concerned with in this course are

characterized 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

1z


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Basic IIR Digital Filter

Structures AnN-th order IIR digital transfer function is

characterized by2N+1 unique coefficients,

and in general, requires2N+1 multipliersand 2Ntwo-input adders for implementation

Direct form IIR filters:Filter structures in

which the multiplier coefficients areprecisely the coefficients of the transfer

function


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Direct Form IIR Digital Filter

Structures Consider for simplicity a3rd-order IIR filter

with a transfer function

We can implementH(z)as a cascade of two

filter sections as shown on the next slide

33

22

11

33

22

110

1

zdzdzd

zpzpzpp

zD

zPzH

)(

)()(


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Structures

where

33

22

1101

zpzpzppzP

zX

zWzH )(

)(

)()(

)(zH1

)(zH2

)(zW)(zX )(zY

33

22

11

21

11

zdzdzdzDzW

zYzH

)()(

)()(


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Structures The filter section can be seen to be

an FIR filter and can be realized as shown

below]3[]2[]1[][][ 3210 nxpnxpnxpnxpnw

)(zH1


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Structures The time-domain representation of is

given by

Realization of

follows from the

above equationand is shown on

the right

)(zH2

][][][][][ 321 321 nydnydnydnwny

)(zH2


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Structures A cascade of the two structures realizing

and leads to the realization of

shown below and is known as thedirectform Istructure

)(zH2

)(zH1)(zH


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Structures Note:The direct form Istructure is

noncanonic as it employs6 delays to realize

a 3rd-order transfer function

A transpose of the

direct form Istructure

is shown on the rightand is called thedirect

form I structuret


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Structures

Various other noncanonic direct form

structures can be derived by simple block

diagram manipulations as shown below


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Structures

Observe in the direct form structure shown

below, the signal variable at nodes and

are the same, and hence the two top delayscan be shared

1 '1


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Structures Likewise, the signal variables at nodes

and are the same, permitting the sharing

of the middle two delays Following the same argument, the bottom

two delays can be shared

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

2

'2


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Structures

Direct form realizations of anN-th order IIR

transfer function should be evident


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Cascade Form IIR Digital

Filter Structures Examples of cascade realizations obtained

by different pole-zero pairings are shown

below


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Filter Structures Examples of cascade realizations obtained

by different ordering of sections are shown

below


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Filter Structures There are altogether a total of 36different

cascade realizations of

based on pole-zero-pairings and ordering

Due to finite wordlength effects, each such

cascade realization behaves differently from

others

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

zPzPzPzH321

221


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Filter Structures Usually, the polynomials are factored into a

product of 1st-order and 2nd-order

polynomials:

In the above, for a first-order factor

k kk

kk

zz

zzpzH

22

11

22

11

01

1

)(

022 kk


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Filter Structures Consider the 3rd-order transfer function

One possible realization is shown below

2

221

12

222

112

111

111

1

1

1

1

0 zzpzH

zz

z

z

)(


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Filter Structures

Direct form II Cascade form


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Parallel Form IIR Digital Filter

Structures A partial-fraction expansion of the transfer

function in leads to the parallel form I

structure Assuming simple poles, the transfer function

H(z) can be expressed as

In the above for a real pole

k zz

z

kk

kkzH2

21

1

110

10

)(

012 kk

1z


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Structures A direct partial-fraction expansion of the

transfer function inzleads to the parallel

form IIstructure Assuming simple poles, the transfer function

H(z) can be expressed as

In the above for a real pole

k zz

zz

kk

kkzH2

21

1

2210

10

)(

022 kk


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Structures The two basic parallel realizations of a 3rd-

order IIR transfer function are shown below

Parallel form I Parallel form II


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Structures Example - A partial-fraction expansion of

in yields

321

321

20180401

0203620440

zzzzzz

zH ...

...

)(

21

1

1 50801

2050

401

60

10

zzz

zzH ..

..

.

.

.)(

1z


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Structures

The correspondingparallel form Irealization

is shown below


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Structures Likewise,a partial-fraction expansion of

H(z)inzyields

The corresponding

parallel form II

realization is shown

on the right

21

11

1

1

50801

25020

401

240

zz

zz

z

zzH

..

..

.

.)(


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Realization Using MATLAB

The cascade form requires the factorization

of the transfer function which can bedeveloped using the M-filezp2sos

The statementsos = zp2sos(z,p,k)

generates a matrixsoscontaining the

coefficients of each2nd-order section of the

equivalent transfer functionH(z) determined

from its pole-zero form


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sosis an matrix of the form

whosei-th row contains the coefficientsand , of the the numerator and

denominator polynomials of thei-th2nd-

order section

6L

2L1L0L2L1L0L

221202221202

211101211101

dddppp

dddppp

dddppp

sos

}{ ip}{ id


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Ldenotes the number of sections

The form of the overall transfer function is

given by

Program 6_1can be used to factorize an

FIR and an IIR transfer function

L

i iii

iiiL

ii

zdzdd

zpzppzHzH

12

21

10

22

110

1

)()(


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Note:An FIR transfer function can be

treated as an IIR transfer function with a

constant numerator of unity and adenominator which is the polynomial

describing the FIR transfer function


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Parallel forms I and II can be developedusing the functionsresiduezand

residue, respectively Program 6_2 uses these two functions


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Realization of Allpass Filters

AnM-th order real-coefficient allpass

transfer function is characterized by

Munique coefficients as here the numerator

is the mirror-image polynomial of thedenominator

A direct form realization of requires

2Mmultipliers

Objective -Develop realizations of

requiring onlyMmultipliers

)(zAM

)(zAM

)(zAM


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Realization Using Multiplier

Extraction Approach

Now, an arbitrary allpass transfer function

can be expressed as a product of 2nd-orderand/or 1st-order allpass transfer functions

We consider first the minimum multiplier

realization of a 1st-order and a 2nd-orderallpass transfer functions


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First-Order Allpass Structures

Consider first the1st-order allpass transfe

function given by

We shall realize the above transfer function

in the form a structure containing a single

multiplier as shown below1d

121122211122111 ttttztzt ,,

1d

1X

2X1

Y

2Y


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We express the transfer function

in terms of the transfer parameters of the

two-pair as

A comparison of the above with

yields1

1

11

11

zd

zdzA )(

221

21122211111

221

1211211 111 td

ttttdt

td

dtttzA

)()(

111 XYzA /)(

1211222111

221

11 ttttztzt ,,


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Substituting and in

we get

There are 4possible solutions to the above

equation:

Type 1A:

Type 1B:

121122211 tttt

111

zt 1

22 zt

22112 1 ztt

11 21212122111 tztztzt ,,,

121

112

122

111 11

ztztztzt ,,,


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Type 1A :

Type 1B :

We now develop the two-pair structure for

the Type 1Aallpass transfer function

22112

122

111 11

zttztzt ,,,

121

112

122

111 11

ztztztzt ,,,

t

t


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From the transfer parameters of this allpass

we arrive at the input-output relations:

A realization of the above two-pair is

sketched below

2112 XzXY

221

22

11

1 1 XYzXzXzY )(


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In a similar fashion, the other three single-multiplier first-order allpass filter structures

can be developed as shown below

Type 1B Type 1At

Type 1Bt


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Second-Order Allpass

Structures A 2nd-order allpass transfer function is

characterized by 2unique coefficients

Hence, it can be realized using only 2multipliers

Type 2allpass transfer function:

221

11

21121

21

zddzd

zzdddzA )(


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Type 2 Allpass Structures


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Type 3 Allpass Structures

R li ti U i M lti li


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Realization Using MultiplierExtraction Approach

Example -Realize

A3-multiplier realization of the aboveallpass transfer function is shown below

321

321

20180401

40180203

zzz

zzzzA

...

...)(

)..)(.()..)(.(

211

211

50801401805040

zzzzzz


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Realization Using Two-Pair

Extraction Approach Let

We use the recursion

where

It has been shown earlier that is

stable if and only if

mm

mm

mmmmm

zdzdzdzd

zzdzdzddm zA

)(

)(

...

...)(

11

22

11

11

22

11

1

11 ,...,, MMm],[)()(

)(

zAk

kzAm

mm

mmzzA

11

mmm

dAk )(

)(zAM

12

mk for 11 ,...,, MMm


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Extraction Approach If the allpass transfer function is

expressed in the form

then the coefficients of are simply

related to the coefficients of through

)()(

)()(

''...'

'...'')( 1

12

21

1

121121

11

mm

mm

mmmm

zdzdzd

zzdzddm zA

)(zAm 1)(zAm

111

2

mi

d

dddd

m

immii ,'

)(zAm 1


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Extraction Approach To develop the realization method we

express in terms of :

We realize in the form shown below

)(zAm 1)(zAm

)(zAm

)(

)()(zAzk

zAzkm

mm

mmzA1

111

1

)(zAm

)(zAm 11

X

1Y

2X

2Y

2221

1211

tt

tt


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Extraction Approach

The transfer function of the

constrained two-pair can be expressed as

Comparing the above with

we arrive at the two-pair transfer parameters

)()()()(

zAtzAtttttm

m

mzA122

121122211111

11 XYzAm /)(

)(

)()(

zAzk

zAzkm

mm

mmzA1

111

1


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Extraction Approach

Substituting and in the

equation 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


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Extraction Approach Some possible solutions are given below:

11 2112

121

2211 tzktzktkt mmm ,)(,,

mmmm ktzktzktkt 11 21

112

12211 ,)(,,

2

21

12

12

1

2211 11 mmmm ktzktzktkt ,,,

2

21

1

12

1

2211 1

mmm ktztzktkt ,,,


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Extraction Approach Consider the solution

Corresponding input-output relations are

A direct realization of the above equations

leads to the 3-multiplier two-pair shown on

the next slide

11 2112

121

2211 tzktzktkt mmm ,)(,,

212

11 1 XzkXkY mm )(

21

12 XzkXY m

R li ti U i T P i


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Extraction Approach

The transfer parameters

lead to the 4-multiplier two-pair structure

shown below


112

12211 ,)(,,


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Extraction Approach Likewise, the transfer parameters

lead to the 4-multiplier two-pair structure

shown below

2

21

12

12

1

2211 11 mmmm ktzktzktkt ,,,


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Extraction Approach A2-multiplier realization can be derived by

manipulating the input-output relations:

Making use of the second equation, we can

rewrite the first equation as

212

11 1 XzkXkY mm

)(

21

12 XzkXY m

21

21 XzYkY m


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Extraction Approach A direct realization of

lead to the2-multiplier two-pair structure,

known as thelattice structure, shown below

21

21 XzYkY m

2112 XzkXY m


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Extraction Approach Consider the two-pair described by

Its input-output relations are given by

Define


112

12211 ,)(,,

21

11 1 XzkXkY mm )(

2

1

12 1 XzkXkY

mm

)(

21

11 XzXkV m )(


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Extraction Approach We can then rewrite the input-output

relations as and

The corresponding 1-multiplier realizationis shown below

21

11 XzVY 112 VXY

1V

1V


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Extraction Approach Anmth-order allpass transfer function

is then realized by constraining any one of

the two-pairs developed earlier by the

th-order allpass transfer function)( 1m

)(zAm

)(zAm 1


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Extraction Approach The process is repeated until the

constraining transfer function is

The complete realization of based onthe extraction of the two-pair latttice is

shown below

)(zAM

10 )(zA


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Extraction Approach 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 structure

requires2Mmultipliers A realization withMmultipliers is obtained if

instead the single multiplier two-pair is used

1|| mk 11 ,...,, MMm

)(zAM


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Extraction Approach Example -Realize

332211

321

121

1

zdzdzd

zzdzdd

321

321

3 2.018.04.01

4.018.02.0)(

zzz

zzzzA


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Extraction Approach We first realize in the form of a

lattice two-pair characterized by the

multiplier coefficientand constrained by a 2nd-order allpass

as indicated below

)(3 zA

)(2 zA2.033 dk

2.03 k


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Extraction Approach The allpass transfer function is of the

form

Its coefficients are given by

)(2 zA

22

11

2112

''1

''

2 )(

zdzdzzdd

zA

4541667.0223

231

)2.0(1

)18.0)(2.0(4.0

1

'1

d

dddd

2708333.022

3

132

)2.0(1

)4.0)(2.0(18.0

1

'2

d

dddd


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Extraction Approach Next, the allpass is realized as a


multiplier coefficientand constrained by an allpass as

indicated below

)(2 zA

)(1 zA2708333.0

'22 dk

,2.03 k 2708333.02k


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Extraction Approach The allpass transfer function is of the

form

It coefficient is given by

)(1 zA

11

11

"1

"

1 )(

zd

zd

zA

3573771.02708333.1

4541667.0

1)(1

"

1 '2

'1

2'2

'1

'2

'1

d

d

d

dddd


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Extraction Approach Finally, the allpass is realized as a


multiplier coefficientand constrained by an allpass as

indicated below

)(1 zA

1)(0 zA3573771.0

"11 dk

,2.03 k 2708333.02k

3573771.01 k


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Cascaded Lattice Realization

Using MATLAB The M-filepoly2rccan be used to realize

an allpass transfer function in the cascaded

lattice form To this end Program 6_3can be employed

iir direct form realization

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