Synthesis of new canonical digital ladderfilters by continued fractions

M. Ismail. M.Sc, Ph.D., P.E., Sen.Mem.I.E.E.E., and Prof. H.K. Kim. M.Sc,Ph.D.. Sen.Mem.I.E.E.E.

Indexing terms: Filters and filtering, Networks, Mathematical techniques

Abstract: Synthesis of digital filters by continued fractions is generally investigated. Several new canonic ladderrealisations of a digital transfer function are presented, and the conditions for realisation are discussed. The newrealisations are based on the applications of continued fraction expansions that proceed in terms of At + Btzand BjZ~l + Ai on an alternating basis. The resulting structures avoid the presence of delay-free loops andachieve a relatively small range of multiplier values. They are more versatile than the existing structures, allow-ing one to synthesise functions more readily. Furthermore, it will be shown that existing structures can begenerated as special cases of the presented structures. Illustrative examples are provided.

1 Introduction

The use of the continued fraction method in analoguepassive circuit analysis and synthesis is well established inthe literature of classical network theory [1, 2]. There hasrecently been a renewed interest in the subject of continuedfractions [3-7] and its applications to both continuous(analogue) and discrete (digital) systems [7]. This is due inpart to the advent of computers and the resulting impor-tance of the algorithmic character of continued fractions. Itis also due to the automatic order reduction associatedwith continued fraction expansions.

One of the recent applications, which started in theearly and mid 1970s [8-11], is the synthesis of digitaltransfer functions by continued fractions. In this respect,several digital ladder filter structures have been reported[8-11]. These structures are canonic with respect to bothdelays and multipliers and well suited for implementationusing large-scale integration [8]. They also perform well inthe presence of arithmetical round-off errors [6]. Unfor-tunately, some of the resulting structures are not imple-mentable due to the presence of delay-free loops [8-11].

Generally, a broader knowledge of all possible networkrealisations of a digital transfer function may have practi-cal, as well as theoretical, interests. In this paper, severalnew canonic ladder realisations of digital transfer functionsare presented. The realisations are based on applications ofa mixture of the first and second Cauer continued fractionforms. It turns out that these mixed Cauer forms proceedin terms of A-x + B(z and BjZ'1 + A-} on an alternatingbasis. The new structures avoid the presence of delay-freeloops and compare favourably with those reported in Ref-erences 8-11, in so far as the range of values of multipliersis concerned. They are also more versatile than the existingstructures, allowing one to synthesise functions morereadily and obtain designs that cannot be obtained withthe older structures. In fact, it will be shown that existingrealisations are special cases of the presented realisations.Additional new structures may also be generated by com-bining the proposed structures with existing ones. In thisrespect, design rules have been established to avoid thepresence of delay-free loops. Illustrative examples areincluded.

Paper 3603G (E10), received 27th February 1984

Dr. Ismail is with the Department of Electrical Engineering, University ofNebraska-Lincoln, W194 Nebraska Hall, Lincoln, NE 68588-0511, USA, and Prof.Kim is with the Department of Electrical Engineering, University of Manitoba,Winnipeg, Manitoba R3T 2N2, Canada

2 New canonic realisations

The digital transfer function is expressed as

anzn + an_xz"-x + . . . + fllz + fl

G(z) = ; —: ; ~ (1)

The realisations reported so far [8-11] are obtained byexpanding G(z), or its reciprocal, into one of the followingcontinued fraction expansions:

(i) Cauer first form(ii) Cauer second form(Hi) Stieltjes [12] forms.*

The resulting realisations are classified according to Mitraet al. [8, 9] and are summarised in Table 1. The aboveforms have one property in common which is the fact thatthe division cycles of the expansion are carried out fromone end of the function G(z) (either the left end or the rightend) and the remainder functions.

G(z) can also be expanded into one of the followingforms, each of which proceeds from both ends of the func-tion simultaneously.

Assuming even n, we have the following expansionforms:

Case I A:

G(z) = A0 +

(2a)

Table 1: Classification of canonic ladder realisations of G(z)

Expansioncarriedout onG(z) [8]Expansioncarriedout on1/G(z) [9]

Cauerfirst

Type IA

Type IMA

Stieltjes*first form

Type HA

Type IVA

Cauersecond

Type IB

Type 1MB

Stieltjes*second form

Type MB

Type IVB

* Stieltjes forms can also be classified into the Stieltjes-first form (eqn. 8 in Reference8) in which poles at infinity are removed and the Stieltjes-second form (eqn. 13 inReference 8) in which poles at the origin are removed

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where A{ and J5, are real constants. In this case,t if an = 0then Ao = 0.

Case IB: In this case we carry out the expansion on thereciprocal of G{z):

1

blz

B2z~l + A

Bnz-' + An

(2b)

where A{ and £, are not the same as those of eqn. 2a. Inthis case Ao = 0 if bn — 0.

Case HA: G(z) is expanded as follows:

1G(z) = A0

(2c)

In this case, the first division cycle is carried out from theright end of G(z), i.e., Ao = ao/bo.

Case I IB: The expansion is carried out on the reciprocalof G(z) as follows:

1

ilz~l+Al

A, + B7z+

(2d)

where A{ and J3, are not the same as those in (2c). In thiscase, Ao is given by bo/ao.

t The expansion 2«, with Ao = 0, is called the 'mixed-Cauer' form [13] or the'Cauer-third' form [14].

2.1 ImplementationsIn order to implement any of the above four expansions,we need building blocks that realise the two functions

1

G2(z) =

Bz+ T{z)

11 + A + T(z)

Fig. 1 shows realisations of GY(z) and G2(z).

(3a)

(36)

Section realising Gx{z)

Fig. 1 B Section realising G2(z)

Using a step-by-step procedure, the realisations of G(z)as given by eqns. 2a, 2b, 2c and 2d are obtained as shownin Figs. 2A, 2B, 2C and 2D, respectively. The structures areindicated for n even. If n is odd, the procedure holds,except that the last stage in both Figs. 2A and 2B wouldemploy the section in Fig. 1A and the last stage in bothFigs. 2C and 2D would adopt the section shown in Fig.IB. All the structures are canonic with respect to both

Fig. 2A Case IA ladder realisation

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delays (n delays) and multipliers (2n + 1 multipliers). One order to implement digital transfer functions using eqn. Id,can see that no delay-free loops exist in each of the struc- one must have either Ao = 0 (b0 = 0) or Ay — 0 in eqn. Id.

o - O

Fig. 2B Case IB ladder realisation

z"1

Fig. 2D Case IIB ladder realisation

For instance, if Ao = 0, eqn. 2d is written as

1G(z) = B.z-1 + Ai

+1

Fig. 2C Case IIA ladder realisation

tures of Figs. 2A, 2B and 2C; hence they are implement-able as digital filters. In Fig. 2D, one may notice thepresence of a delay-free loop through the multipliers \/A0

and l/zlj. This does not matter if the structure is to beused in realising a continuous transfer function where thedelays are replaced by integrators, the adders by summing

amplifiers and the multipliers by amplifiers. However, in Fig. 3 An implementable Case IIB ladder realisation obtained if Ao = 0

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^ .

which is implementable as shown in Fig. 3. On the otherhand, if Ax = 0 , two realisations are possible as shown inFigs. 4a and 4b.

a ° o t) o

Fig. 4 Implementable Case IIB ladder realisations obtained if At = 0

It is worth noting that existing ladder realisations aresimply special cases of the new ladder realisations. Forinstance, Type IA, presented by Mitra and Sherwood [8],is obtainable as a special case of Case IA ladder realisationof Fig. 2A by letting Ax = B2 = A2 = BA • • • = 0. Also,each of the new structures is virtually its own transpose[17]. This can be seen if the new structures are redrawnwith the multipliers of (— 1) absorbed into the other multi-plier values.

In a way analogous to the two-pair extraction method![15, 16] where conditions are established for each iter-ation, necessary and sufficient conditions for the existenceof the expansions 2a-2d can be established for each divi-sion cycle [7]. For instance, in the first division cycle ofeach of the four expansions 2a-2d one needs, for realisabil-ity, ajbn =/= ao/bo. In expansion 2c, if Ao = 0, one needs,for realisation, that an ^ 0 and bx — Ax ax =/= 0 in the firstdivision cycle. Similar conditions have to apply at eachdivision cycle. A complete set of necessary and sufficientconditions for the existence of the above expansions havebeen developed and can be found in Reference 7.

3 Illustrative examples

3.1 Example 1Consider the digital transfer function

G(z) =1200z4 + 320z3 + 270z2 + 45z +

600z4+ 100z3 + 129z2 + lOzThis function has been considered in Reference 8 where ithas been pointed out that only one realisation (Type IB) ispossible as an implementable digital filter. In the following,however, it will be shown that two more implementablerealisations for G(z) are possible using the proposed newladder structures.

Case I A: Expand G(z) according to expression 2a:

G(z) = 2 +

5z + 0.0 +

3+O.2Z"1 +

- 5 z +0.5435 +

Case IIB: Since b0 = 0, use the form of expression 2d':

1

10.1472 +0.1693Z"1

1 + 2

0.8264 + lOz +

2.2 Constraints on realisationAs with continued fractions and most other realisationmethods, there are some degenerate cases when the realis-ations discussed here do not exist. Generally, necessaryand sufficient conditions must be satisfied for a rationalfunction to have a certain continued fraction expansion.Since the expansion forms developed are basically mix-tures of the first and second Cauer forms, one may intu-itively assume that if both Cauer first and second formsexist* for G(z) or 1/G(z), a certain mixed-Cauer form wouldalso exist for G(z) or 1/G(z), respectively. However, one canshow [7] that if G(z) (or its reciprocal) is expandable intoCauer first only or Cauer second only, a mixed Cauer formmay or may not exist.

* In classical network theory [18], if G(z) for z = s is interpreted as a driving-pointfunction, then the existence of one Cauer form implies the existence of the othertwo. In this respect, it is noted that well known text books, written by distinguishedauthors [1, 2], overlooked, or did not mention, the mixed-Cauer form.

1.9064Z"1 + 1.9724 +1

-0.0928-0.4447z

which is implementable according to Fig. 3.The new structures may have a relatively small range of

multiplier values. The multiplier constants for Case IIB are(Bx, A,, \/B2, -A2, I/A,, -B2, 1/B4, -A4) = (0.1, 2,0.1, -0.8264, 0.5070, - 1.9064, -2.2489, 0.0928). The ratioof the largest to the smallest multiplier constant is 2.2489/0.0928 or 24.2338. The corresponding ratio for Type IB [8]is 2498 which is more than 100 times as high as the ratiofor Case IIB.

3.2 Example 2Consider the following digital transfer function

G ( )

One may notice that a3/b2 = ao/bo = 1. As pointed out

t Recently [19], it has been pointed out that the ladder realisation based on thetwo-pair extraction method are, in fact, obtainable from those of Reference 8.

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earlier, this makes the existence of any of the expansionforms 2a-2d impossible. To see that, let us perform theexpansion 2a of Case IA:

5z2 + 6z

+ 11

z3

J3 +

+ 4z2

5z2 +- z 2 -

•f z + 1

6z + 1

-5z z 3 -

z* A

— z

l-5z2 +

h 5 z 2

6z + 1

6z + 1

Now, if one proceeds according to expression 2a, a con-stant term cannot be removed. However, one can remove az~x term followed by the desired constant term as follows:

-z2 - 5z |6z+

Divide divisor and dividend by z and proceed in the samecycle:

Now proceed to the next cycle as follows:

12129

— z

- 5

00

The final form of the expansion is

1G(z) = 1 +

7 19 1 - 1~z ~ 25 ~ 5Z

1- 1

The realisation of the above expansion is shown in Fig. 5

Fig. 5 Realisation ofG(z) of example 2 in the modified Case IA ladder

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which takes the form of Case IA realisation of Fig. 2A witha slight modification; that is a delay and a multiplier areadded in the feedback of the second stage to implement(—^z"1 of the second cycle of the expansion. The multi-plier constants are (Ao, \/Bx, —Ax, \/A2, —B2, +•£) =(1, - 1 , §§, f§, —W. +l)- T h e r a t i o o f t h e largest to thesmallest multiplier value is 21.557.

It is worth noting that the same transfer function hasbeen considered in Reference 9. Referring to Table 1, onecan show [7] that, out of the realisations indicated in thefirst row, only Type IB is possible, which has a multiplierratio of 37.706.* This ratio is approximately 1.75 timesthat of the modified Case IA ladder realisation. Anotheradvantage of the new structure over that of Type IB is thefact that it uses one multiplier less.

3.3 Example 3In this example, it is shown that additional new structuresmay be generated by combining the proposed structureswith those of Table 1. In so doing, and to avoid the pre-sence of delay-free loops, one should avoid the followingsituations [7]:

(i) Two consecutive division cycles that belong to eitherType IIB or IVB of Table 1.

(»") Two consecutive division cycles, while proceeding inthe expansion, that are a constant followed by a cycle ofStieltjes-second form or vice-versa.

(Hi) Situations similar to Type IIIB when an ± 0 [9].

Now consider the following digital transfer function:

720z2 + 240z + 12G(z) =

720z3 + 600z2 + 72z

which has been considered in Reference 8.One may carry out the first two division cycles accord-

ing to Type IA of Table 1 as follows:

G(z) =1

z +1

2 + T(z)

where

T(z) =120z + 10

360z2 + 60z + 1

T(z) is then expanded according to Case IA as follows:

1T(z) =

3z + 0.1 +1

O.5556Z"1 +6.666

The final realisation of G(z) is shown in Fig. 6 which hasno delay-free loops.

4 Conclusions

Several new canonic digital ladder structures are devel-oped and their readability conditions are discussed. Thestructures are based on the applications of a mixture ofCauer-first and Cauer-second continued fraction expan-sions, thus the division cycles are carried out on both endsof the transfer function (and its remainders) simulta-neously. As a consequence, the resulting structures are ver-satile with a relatively small range of multiplier values.

* It has been shown [9] that Type 111A is possible. In this case, the multiplier ratiois found to be 30.8642 which is 1.432 times that of the modified Case IA

The new structures are general in the sense that existingladder realisations can be generated as special cases of thepresented realisations.

Fig. 6 Realisation of G(z) of example 3

Also, it has been shown that the number of multiplierelements needed in the synthesis may be reduced with thenew structures.

The availability of numerous implementable structuresis an advantage since it may provide a vehicle for a designthat meets certain prescribed error criteria. A completecomparison between the available canonic ladder struc-tures is a useful topic for additional research work.

Finally, it is worth noting that continued fractions havealso been applied [20] to synthesis of two-dimensionaldigital filters. In this respect, the expansion forms used arethe extensions of one-dimensional expansions. Conse-quently, new two-dimensional ladder structures can beobtained as extensions of the one-dimensional ladderstructures presented herein. The realisations of such two-dimensional structures have been discussed in details inReference 7.

5 Acknowledgments

The authors wish to thank the reviewers for their construc-tive criticism and valuable suggestions, and David Haighfor his encouragement.

6 References

1 WEINBERG, L.: 'Network analysis and synthesis' (McGraw-Hill,New York, 1962)

2 GUILLEMIN, E.A.: 'Synthesis of passive networks' (John Wiley &Sons, New York, 1957)

3 JONES, W.B., and THRON, W.J.: 'Continued fractions: analytictheory and applications' (Addison Wesley (Advanced book program),Reading, Mass., 1980)

4 DAVIS, A.M.: 'A new Z-domain continued fraction expansion', IEEETrans., 1982, CAS-29, pp. 658-662

5 ISMAIL, M., and KIM, H.K.: 'A simplified stability test for discretesystems using a new Z-domain continued fraction method', ibid.,1983, CAS-30, pp. 505-507

6 JONES, W.B., and STEINHARDT, A.: 'Digital filters and continuedfractions', Lecture Notes in Mathematics, 1982, 932, pp. 129-151. Alsopresented at a seminar-workshop held at Leon, Norway, 1981

7 ISMAIL, M.: 'Some new results on the continued fraction methodand its applications to continuous and discrete systems'. Ph.D. disser-tation, Dept. of Electrical Engineering, University of Manitoba, Win-nipeg, Canada,1983

8 MITRA, S.K., and SHERWOOD, R. J.: 'Canonic realizations ofdigital filters using the continued fraction expansion', IEEE Trans.,1972, AU-20, pp. 185-194

9 MITRA, S.K., and SAGAR, A.D.: 'Additional canonic realizations ofdigital filters using the continued fraction expansion', ibid., 1974,CAS-21, pp. 135-136

10 HWANG, S.Y.: 'Realization of canonical digital networks', ibid., 1974,ASSP-22, pp. 27-39

11 CONSTANTINIDES, A.G.: 'Some new digital filter structures basedon continued fraction expansion' in SKWIRZYNSKI, J.K., andSCANLAN, J.O. (Eds.): 'Network and signal theory' (Peregrinus,London, 1973)

12 WALL, H.S.: 'Analytic theory of continued fractions' (New York,Chelsea, 1948 and 1967)

13 SHIEH, L.S., and GOLDMAN, M.J.: 'A mixed Cauer form for linearsystem reduction', IEEE Trans., 1974, SMC-4, pp. 584-588

14 SHIEH, L.S., and GOLDMAN, M.J.: 'Continued fraction expansionand inversion of the Cauer third form', ibid., 1974, CAS-21, pp.341-345

15 ACHA, J.I., and ROBLES-DIAZ, R.: 'Two new two-pair structuresfor the realization of digital filters in the pure ladder form', Proc.IEEE, 1980, 68, pp. 1342-1343

16 NEUVO, Y., and MITRA, S.K.: 'Canonic ladder realization of IIRdigital filters', ibid., 1982, 70, pp. 763-764

17 JACKSON, L.B.: 'On the interaction of round off noise and dynamicrange in digital filters', Bell Syst. Tech. J., 1970, 49, pp. 159-184

18 VAN VALKENBURG, M.E.: 'Modern network synthesis' (JohnWiley & Sons, New York, 1962)

19 ACHA, J.I., AYERBE, A., ROBLES-DIAZ, R., and YAN, G.T.:'Comment and reply: Modified canonic ladder realisations of IIRdigital filters', Electron. Lett., 1983, 19, (10), pp. 381-382

20 MITRA, S.K, SAGAR, A.D., and PENDERGRAS, M.A.: Realiza-tion of two-dimensional recursive digital filters', IEEE Trans., 1975,CAS-22, pp. 177-184

Mohammed Ismail was born in Egypt on26th August 1951. He received the B.S. andM.S. degrees from Cairo University, Cairo,Egypt, in 1974 and 1978, respectively, andthe Ph.D. degree from the University ofManitoba, Winnipeg, Canada in 1983, allin electrical engineering.

From 1974 to March 1978 he was aResearch Engineer in the ElectronicIndustries Research and DevelopmentCenter, Cairo, Egypt, where he did research

in the areas of active networks and filter design, as well as devel-opment of solid-state TV tuners, IF and video TV circuits. FromMarch 1978 to March 1980 he was a Communication SystemsEngineer in the Arab Organisation for Industrialisation, Cairo,Egypt, where he was responsible for the development of micro-wave and PCM multiplexing systems. In March 1980 he joinedthe Department of Electrical Engineering, University of Mani-toba, Canada, as a Research and Teaching Assistant, and sinceAugust 1983 he has been with the Department of ElectricalEngineering, University of Nebraska-Lincoln, as an AssistantProfessor. His main research interests are active, digital andswitched-capacitor filters, analogue VLSI, digital signal pro-cessing and general network and systems theory.

Hyong Kap Kim was born in Chungup,Korea, on 29th May 1930. He received theB.S. degree in electrical engineering fromthe Chonpuk National University, Korea,in 1955, and the M.S. and Ph.D. degreesfrom the University of Pennsylvania, Phila-delphia, both in electrical engineering, in1960 and 1964, respectively.

From 1960 to 1961, he was a DesignEngineer at the General Electric Company,Philadelphia.

From 1961 to 1964 he was an Instructor at the Moore Schoolof Electrical Engineering, University of Pennsylvania, and from1964 to 1965 he was on the Technical staff at the Institute forCo-operative Research, University of Pennsylvania. Since 1965 hehas been on the staff of the Department of Electrical Engineering,University of Manitoba, Winnipeg, Canada, where he is currentlya Professor. His teaching and research interests are mainly in thetheory and design of passive and active networks. He gave invitedlectures at the Korea Advanced Institute of Science during 1975—1976 while on sabbatical leave from the University of Manitoba.

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