Copyright © 2001, S. K. Mitra Tunable IIR Digital Filters We have described earlier two 1st-order and two 2nd-order IIR digital transfer functions with

Copyright © 2001, S. K. Mitra

Tunable IIR Digital FiltersTunable IIR Digital Filters

• We have described earlier two 1st-order and two 2nd-order IIR digital transfer functions with tunable frequency response characteristics

• We shall show now that these transfer functions can be realized easily using allpass structures providing independent tuning of the filter parameters


Tunable Lowpass and Tunable Lowpass and Highpass Digital FiltersHighpass Digital Filters

• We have shown earlier that the 1st-order lowpass transfer function

and the 1st-order highpass transfer function

are doubly-complementary pair

1

1

11

21

)(z

zzHLP

1

1

1

12

1)(

z

zzHHP



• Moreover, they can be expressed as

where

is a 1st-order allpass transfer function1

1

1 1)(

z

zzA

)](1[)( 121 zAzHLP

)](1[)( 121 zAzHHP



• A realization of and based on the allpass-based decomposition is shown below

• The 1st-order allpass filter can be realized using any one of the 4 single-multiplier allpass structures described earlier

)(zHHP

)(zHLP )(zHHP



• One such realization is shown below in which the 3-dB cutoff frequency of both lowpass and highpass filters can be varied simultaneously by changing the multiplier coefficient



• Figure below shows the composite magnitude responses of the two filters for two different values of

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

w/p

Mag

nitu

de

= 0.4 = 0.05


Tunable Bandpass and Tunable Bandpass and Bandstop Digital FiltersBandstop Digital Filters

• The 2nd-order bandpass transfer function

and the 2nd-order bandstop transfer function

also form a doubly-complementary pair

21

2

)1(1

12

1)(

zz

zzHBP

21

21

)1(1

12

1)(

zz

zzzHBS



• Thus, they can be expressed in the form

where

is a 2nd-order allpass transfer function

)](1[)( 221 zAzHBS

)](1[)( 221 zAzHBP

21

21

2 )1(1

)1()(

zz

zzzA



• A realization of and based on the allpass-based decomposition is shown below

• The 2nd-order allpass filter is realized using a cascaded single-multiplier lattice structure

)(zHBP )(zHBS



• The final structure is as shown below

• In the above structure, the multiplier controls the center frequency and the multiplier controls the 3-dB bandwidth



• Figure below illustrates the parametric tuning property of the overall structure

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

w/p

Mag

nitu

de

= 0.4 = 0.05

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

w/p

Mag

nitu

de

= 0.8 = 0.1

= 0.5 = 0.8


IIR Tapped Cascaded Lattice IIR Tapped Cascaded Lattice StructuresStructures

Realization of an All-pole IIR Transfer Function

• Consider the cascaded lattice structure derived earlier for the realization of an allpass transfer function



• A typical lattice two-pair here is as shown below

• Its input-output relations are given by

)()()( 11 zSzkzWzW mmmi

)()()( 11 zSzzWkzS mmmm

)(1 zWm

)(1 zSm

)(zWm

)(zSm



• From the input-output relations we derive the chain matrix description of the two-pair:

• The chain matrix description of the cascaded lattice structure is therefore

)()(1

)()(

1

1

1

1zSzW

zkzk

zSzW

i

i

i

i

i

i

)()(111

)()(

1

11

1

11

12

12

13

13

1

1zSzW

zkzk

zkzk

zkzk

zYzX



• From the above equation we arrive at

using the relation and the relations

132211 ])1([1{)( zkkkkzX

)(})]1([ 12

32

2212 zWzkzkkkk

)()( 11 zWzS )()1( 1

33

22

11 zWzdzdzd

33'22

"11 ,, dkdkdk



• The transfer function is thus an all-pole function with the same denominator as that of the 3rd-order allpass function :

)(/)( 11 zXzW

)(3 zA

dzXzW

11

)()(

1

1



Gray-Markel Method

• A two-step method to realize an Mth-order arbitrary IIR transfer function

• Step 1: An intermediate allpass transfer function is realized in the form of a cascaded lattice structure

)(/)()( zDzPzH MM

)(/)()( zDzDzzA MMM

M1



• Step 2: A set of independent variables are summed with appropriate weights to yield the desired numerator

• To illustrate the method, consider the realization of a 3rd-order transfer function

)(zPM

33

22

11

33

22

110

3

3

1

zdzdzd

zpzpzppzDzP

zH)()(

)(



• In the first step, we form a 3rd-order allpass transfer function

• Realization of has been illustrated earlier resulting in the structure shown below

)(/)()(/)()( zDzDzzXzYzA 31

33

113

)(zA3



• Objective: Sum the independent signal variables , , , and with weights as shown below to realize the desired numerator

1Y 1S 2S 3S }{ i

)(zP3



• To this end, we first analyze the cascaded lattice structure realizing and determine the transfer functions , , and

• We have already shown

)(/)( zXzS 11 )(/)( zXzS 12)(/)( zXzS 13

1Y

1X

)()()(

zDzXzS

31

1 1



• From the figure it follows that

and hence

)()()()()( " zSzdzSzkzS 11

111

12

)()()( "

zDzd

zXzS

3

11

1

2



• In a similar manner it can be shown that

• Thus,

• Note: The numerator of is precisely the numerator of the allpass transfer function

)()()( '' zSzzddzS 121

123

)()()( ''

zDzzdd

zXzS

3

2112

1

3

)(/)( zXzSi 1

)(/)()( zWzSzA iii



• We now form

)()(

)()(

)()(

)()(

)()(

zXzS

zXzS

zXzS

zXzY

zXzYo

1

1

1

2

1

3

1

1

14321



• Substituting the expressions for the various transfer functions in the above equation we arrive at

)(

)(

)()(

zD

zzdzdd

zXzYo

3

321

1231

1

4

113

21122 )()( "'' zdzzdd



• Comparing the numerator of with the desired numerator and equating like powers of we obtain

)(/)( zXzYo 1)(zP3

1z

04132231 pddd "'

131221 pdd '

2211 pd 31 p



• Solving the above equations we arrive at

31 p

"'13223104 dddp

'122113 ddp

1122 dp



• Example - Consider

• The corresponding intermediate allpass transfer function is given by

32

321

3

3

2.018.04.01

02.0362.044.0)()(

)(1

zzz

zzzzDzP

zH

321

321

3

13

3

3 2.018.04.01

4.0.018.02.0)(

)()(

zzz

zzzzDzDz

zA



• The allpass transfer function was realized earlier in the cascaded lattice form as shown below

• In the figure,

)(3 zA

1Y

1X

2708333.0,2.0 '2233 dkdk

3573771.0"11 dk



• Other pertinent coefficients are:

• Substituting these coefficients in

4541667.0,2.0,18.0,4.0 '1321 dddd

,02.0,36.0,44.0,0 3210 pppp

31 p

"'13223104 dddp

'122113 ddp

1122 dp



• The final realization is as shown below

352.0,02.0 21 19016.0,2765333.0 43

2.0,2708333.0,3573771.0 321 kkk


Tapped Cascaded Lattice Tapped Cascaded Lattice Realization Using MATLABRealization Using MATLAB

• Both the pole-zero and the all-pole IIR cascaded lattice structures can be developed from their prescribed transfer functions using the M-file tf2latc

• To this end, Program 6_4 can be employed


Tapped Cascaded Lattice Tapped Cascaded Lattice Realization Using MATLABRealization Using MATLAB

• The M-file latc2tf implements the reverse process and can be used to verify the structure developed using tf2latc

• To this end, Program 6_5 can be employed


FIR Cascaded Lattice FIR Cascaded Lattice StructuresStructures

• An arbitrary Nth-order FIR transfer function of the form

can be realized as a cascaded lattice structure as shown below

N

nn

nN zpzH 11)(



• From figure, it follows that

• In matrix form the above equations can be written as

where

)()()( 11

1 zYzkzXzX mmmm

)()()( 1

11 zYzzXkzY mmmm

)()(1

)()(

1

11

1

zYzX

zkzk

zYzX

m

m

m

m

m

m

Nm ,...,2,1



• Denote

• Then it follows from the input-output relations of the m-th two-pair that

)()(

)(,)()(

)(00 zXzY

zGzXzX

zH mm

mm

)()()( 11

1 zGzkzHzH mmmm

)()()( 1

11 zGzzHkzG mmmm



• From the previous equation we observe

where we have used the facts

• It follows from the above that

• is the mirror-image of

,1)( 111

zkzH 111 )( zkzG

1)(/)()( 000 zXzXzH

1)(/)()(/)()( 00000 zXzXzXzYzG

)()1()( 11

11

11

zHzkzzzG

)(1 zG )(1 zH



• From the input-output relations of the m-th two-pair we obtain for m = 2:

• Since and are 1st-order polynomials, it follows from the above that and are 2nd-order polynomials

)()()( 11

212 zGzkzHzH )()()( 1

1122 zGzzHkzG

)(1 zG)(1 zH

)(2 zH )(2 zG



• Substituting in the two previous equations we get

• Now we can write


)()()( 11

2122

zHzzHkzG

)()]()([ 12

2111

22

2 zHzzHzHzkz

)()( 11

11

zHzzG

)()()( 11

2212

zHzkzHzH

)()()( 11

2122

zHzzHkzG

)(2 zG )(2 zH



• In the general case, from the input-output relations of the m-th two-pair we obtain

• It can be easily shown by induction that


)()()( 11

1 zGzkzHzH mmmm

)()()( 1

11 zGzzHkzG mmmm

NNmzHzzG mm

m ,1.,..,2,1),()( 1

)(zGm )(zHm



• To develop the synthesis algorithm, we express and in terms of and for arriving at

)(zHm

)(1 zHm )(1 zGm )(zGm 1,2,...,1, NNm

)}()({)()1(

11 2 zGkzHzH NNNkN

N

)}()({)( 12 )1(

11 zGzHkzG NNNzkN

N



• Substituting the expressions for

and

in the first equation we get

N

nn

nN zpzH 11)(

10

1)()( Nn

NnnN

NN zzpzHzzG

1

121 )()1{(1

1)( N

nn

nNnnNNN

N zpkppkk

zH

})( NNN zkp



• If we choose , then reduces to an FIR transfer function of order and can be written in the form

where

• Continuing the above recursion algorithm, all multiplier coefficients of the cascaded lattice structure can be computed

NN pk )(1 zHN

1N

11

'1 1)( Nn

nnN zpzH

11,21'

Nnp

N

nNNn

k

pkpn



• Example - Consider

• From the above, we observe

• Using

we determine the coefficients of :

43214 08.012.012.12.11)( zzzzzH

08.044 pk

31,24

44

1'

npk

pkpn

nn

)(3 zH2173913.1,2173913.0 '

2'3 pp

2173913.1'1 p



• As a result,

• Thus,

• Using

we determine the coefficients of :

213 2173913.12173913.11)( zzzH

32173913.0 z

2173913.0'33 pk

)(2 zH

21,23

'23'

1"

npk

pkpn

nn

0.1,0.1 "1

"2 pp



• As a result,

• From the above, we get

• The final recursion yields the last multiplier coefficient

• The complete realization is shown below

212 1)( zzzH

1"22 pk

5.0)1/( 2"11 kpk

08.0,2173913.0,1,5.0 4321 kkkk


FIR Cascaded Lattice FIR Cascaded Lattice Realization Using MATLABRealization Using MATLAB

• The M-file tf2latc can be used to compute the multiplier coefficients of the FIR cascaded lattice structure

• To this end Program 6_6 can be employed

• The multiplier coefficients can also be determined using the M-file poly2rc



Documents

Copyright © 2001, S. K. Mitra Tunable IIR Digital Filters We have described earlier two 1st-order and two 2nd-order IIR digital transfer functions with