Roundoff noise upper bounds for cascadedrecursive digital filter structures

K. Mondal, Ph.D., Mem. I.E.E.E., and S.K. Mitra, Ph.D., Fel. I.E.E.E., Fel. A.A.A.S.

Indexing terms: Filters and filtering, Mathematical techniques, Noise and interference

Abstract: Two upper bounds on the variance of roundoff noise at the output of a cascade digital filterstructure are established. Some of these bounds are easier to compute than the exact roundoff-noise variance.Algorithms utilising these bounds for the generation of low-noise cascade realisation (by proper pairing ofpoles and zeros and ordering of the 2nd- and/or lst-order sections) are suggested. An exhaustive search toobtain such a realisation becomes economically feasible because of the reduced complexity of computations.Illustrative examples are given, showing the application of one of these algorithms.

1 Introduction

The realisation of recursive digital filters by cascading 2nd-and/or lst-order sections has many desirable properties, suchas better noise performance than the direct form realisation[1], highly modular realisation of high-order digital filtersand the feasibility of time sharing in the actual implemen-tation of the filter.

When a fixed-point digital filter is realised under dynamicrange constraints by cascading 2nd-order sections, the resultingroundoff error caused by the use of finite word length ishighly dependent on the pole-zero pairing and the orderingof sections. Jackson [1] has derived expressions for the round-off error for such filters when the individual sections arerealised by direct form structures and has proposed some'rules of thumb' for determining a good cascade realisation.Lee [2] has suggested an optimisation procedure of suchcascade realisations based on a 'minimax noise principle'.Dynamic programming [3] has been suggested as an optimis-ation method to produce an optimal cascade realisation. Liuand Peled [4] proposed a heuristic optimisation procedurethat will produce a 'near optimal' assignment. Several otherauthors [5, 6] have also investigated this problem.

In this paper, we establish two new upper bounds onnoise variance for a cascade realisation and use these to obtaina low-noise cascade realisation. These upper bounds areeasier to calculate and involve the noise variance expressionsof the individual sections. The algorithms to be described laterobtain assignments for minimising these upper bounds (bypole-zero pairing and ordering). These lead to at least onesuboptimal cascade realisation with low roundoff noise. Itshould be noted that, recently, Jackson [7] has derived lowerbounds on roundoff-noise variance derived from the coef-ficient sensitivities. The method of obtaining the upper boundson noise variance used in this paper also follows a similartechnique. Although Jackson's bounds are quite tight, theyare not easy to calculate when the individual sections in thecascade are not in direct forms.

2 Upper bounds on variance of noise

We first develop an expression for the output roundoff-noisevariance for a general scaled digital filter. A digital filterrealisation may be characterised by a signal flow graph inwhich: (a) every branch is either a constant multiplier or adelay (z"1), and (b) no delay-free loops exist. In general, bysimple signal flow graph manipulations (such as by shiftingthe constant multiplier through branches), all the constant

Paper 1956G, first received 25th June 1981 and in revised form 7thApril 1982Dr. Mitra is, and Dr. Mondal was formerly, with the Department ofElectrical & Computer Engineering, University of California, SantaBarbara, CA 93106, USA. Dr. Mondal is now with the Bell Labora-tories, Allentown, PA 18103, USA and the Department of Electrical &Computer Engineering, Lehigh University, Bethlehem, PA 18015, USA.

multiplier branches can be made to be incident only on thosenodes where more than one branch is incident. This isequivalent to shifting the multipliers in the filter, so that theyare only at the input of the adder elements. The filter needsto be Lp-scaled, so that, at each such node,

P (1)

where Hm (z) is the transfer function at that node with respectto the input of the filter. If there are k'm noninteger multi-plier branches incident on the mth node, then there is anequivalent white-noise input at the mth node of variancek'mq2/12, where q is the quantisation step size. This noiseis communicated to the output through a filter with transferfunction Gm (z). Therefore, the contribution to the output-noise variance caused by the noise source at the mth node isk'mq2 \\Gm(z)\\\l\2. If eqn. 1 is not satisfied at each node,the filter structure then needs to be scaled by dividing themultiplier coefficients in branches incident on the mth nodeby the scale factor sm = \\Hm{z)\\p and multiplying thecoefficients in branches leaving the mth node by sm. The totaloutput-noise variance for the Z,p-scaled filter in the generalcase is therefore:

(2)

where a\ =q2/l2.The noise-variance expression in eqn. 2 for a scaled filter

is solely in terms of the quantities of the correspondingunsealed filter, except for the fact that, in general, km =£ k'mbecause, on scaling, some new noninteger multiplier coef-ficients may get introduced into the structure.

The summation in eqn. 2 is over the number of summingnodes and any contribution resulting from multiplier branchesincident on the output has also been included in eqn. 2. Thisis possible, because the usual assumption of uncorrelatednoise can be made in this case.

In the case of cascade realisation of a digital transferfunction, Hm(z) and Gm(z) are not, in general, low-orderfunctions for all m and cannot be evaluated easily, unlessthe constituent sections in the cascade are all 2nd-orderand/or lst-order direct form realisations. Let the numberof summing nodes in the z'th section of the cascade be Ni,and let

= M (3)

250 0143-7089/82/050250 + 07 $01.50/0

where n represents the total number of sections. Also, let

IEEPROC, Vol. 129, Pt. G, No. 5, OCTOBER 1982

ky denote the number of noninteger multiplier branchesincident at the node / of the /th section of the cascade.The corresponding scale factor and the noise-transfer function(for the unsealed version of the cascade realisation) will bedenoted by sijiP and G{j(z), respectively. The situation isillustrated by the block diagram depicted in Fig. 1, whereXt and Yt are the input and the output of the /th section,and X and Y are the corresponding variables of the overallcascaded structure. In this Figure, Ey denote the noise-sourceinputs and Wy are the node outputs. In terms of thesenotations, the noise-variance expression in eqn. 2 can bebroken up into the sum of variances due to individual sections,as follows:

N;

(4)

roundoff-noise variance can be varied. But the bound obtainedin eqn. 9 is not affected by such orderings if the individualsections in the cascade are kept invariant.

However, if the pole-zero pairings in the overall transferfunction are changed, the individual section transfer functions7ft (z) get changed, thereby changing Pk. The individualsection noise variances o\ also get modified, and this modifies°max\ • These observations suggest an algorithm for realisingany transfer function by a cascade structure, so that a ^ ^ jis minimised when the individual sections are realised by aparticular type of configuration (e.g., direct form, lattice,ladder etc.).

The evaluation of the o^axi u s m g ecln- 9 is simple becauseall the transfer functions involved are, at most, 2nd-orderif the individual sections in the cascade are of order two orless.

where

andX =

(5)

sUtP = \\Hu(z)\\}

The basic inequality used for deriving the upper bounds isgiven below:

\\F1(z)F2(z)I \\F2{z)II, (6)

where Fx(z) and F2(z) are two real rational functions of thecomplex variable z.

x=x.

1stSECTION

SUMMINGNODES

Y| Xj

i - thSECTION

N,

SUMMING

NODES

J 1

n- thSECTION

SUMMINGNODES

Fig. 1 Cascade realisation block diagram

First boundLet the transfer function of the ith section be:

(7)

Let us denote the noise variance at the ouput of the /thsection by the symbol of, when the /th section is consideredindividually and is scaled. Also, let

Pi = I I ^ ) L i = 1 ,2 , . . . , / !

The first bound with the notations above is:

o 2m a x l =

fe=i[af/Pf]

(8)

(9)

°maxi decreases if the P,s are kept fixed and the of value isdecreased. By keeping the individual section transfer functionsinvariant, Pts can be kept fixed for all /; and then, realisingeach section using low-noise structures, of of each sectioncan be decreased. This will lead to a lower value of o^axi •

For any cascade structure, it is known [1] that, by chang-ing the ordering of the sections in the cascade, the output

Second boundWe define

Yt-X

= ff Th(z)(10)

and

(11)

which is the L^-norm of the transfer function between the(/— l)th section output and the overall input of the cascade.Also, we define

l\f — II11 \Z) II \ -1 A/

Rt is the norm of the transfer function between the overalloutput of the cascade and the /th section input.

In terms of these, the second bound is given by:

}Rl°max2 - X Qi-l°i^i + i

where

(13)

(14)

It can be shown that the bound obtained in eqn. 9 can beobtained as a special case from eqn. 13, and therefore eqn.13 is the more general form of the noise bound. The maindrawback of the bound in eqn. 13 is its computational aspect.The computation of a£,ax2 involves the calculation of theZ^-norms of higher-order transfer functions, in contrast tothat of most 2nd-order transfer functions for the case of

°moxl-Some observations are in order now. The bound in eqn.

13 is dependent on the individual section noise variances(with scaled sections) of and also on the ordering of thesections. The ordering of the sections will affect the values ofQi and Rh and hence affect a2,ox2 • All the three parametersQt, of and Rt get changed by the change in the pole-zeroordering. However, Qt and /?,• are dependent on the input/output transfer functions, and therefore can be evaluatedwithout bothering about digital networks realising the individ-ual sections.

Thus, the calculation of a ^ x (and 0 ^ 2 ) i s much easier

IEEPROC, Vol. 129, Pt. G, No. 5, OCTOBER 1982 251

than the calculation of the exact noise variance given ineqn. 4 for obtaining a low-noise cascade realisation of adigital transfer function.

3 Simple algorithms to obtain low-noise cascaderealisations of recursive digital filters

In this Section, we propose two simple algorithms to obtaina cascade realisation of an IIR digital transfer function G(z),so that the upper bound on noise variance at the output isminimised, and the structure obtained is in the Lp-scaledform. The main computations are the calculation of Z^-normof various rational functions and also the noise variance atthe output of any scaled section considered individually.

First algorithmThis algorithm is based on the use of the bound expressionin eqn. 9. Let

G(z) = . . .Nn(z)/D1(z)D2(z) . ..Dn(z)

(15)

be the given transfer function to be realised where Nt(z)•and Di(z) are (for all /) 2nd- and/or lst-order polynomials.The basic philosophy of the algorithm to be presented now isthat first we form all possible n\ pole-zero pairings from eqn.15 and calculate the a ^ ^ i in each case using eqn. 9. To that,we assume that the individual sections are realised by someknown structure, for which closed-form expressions of noisevariance and scale factors are known. One possible choicecould be the direct-form structure. The pole-pairs showingthe minimum 0^0x1 a r e chosen. If the a^,^! for this pairingis less than, or equal to, the noise-variance specification,no further optimisation is necessary. Otherwise, the individualsections in the cascade could be replaced by structures withlower noise variance. The algorithms to be described willbe very successful if a bank of 2nd- and/or lst-order structuresrealising a general IIR digital transfer function is available.Also, closed-form expressions of a/2 for these structures asa function of transfer function (input/output) coefficientsshould preferably be known, so that all the computationscan be done without any simulation of the structures on adigital computer.

The algorithm is now described in the following text.The input to this algorithm comprises the numerator and thedenominator polynomial factors of the transfer functionG(z) to be realised.

Step 1Initialise the numerator factor index / «- 1.

Step 2Initialise the denominator factor index/ <- 1.

Step 3Calculate the Z^-norm of the transfer function Ni(z)/Dj(z)and denote it

Step 4Evaluate the output-noise variance for the scaled direct-formrealisation of the transfer function Ni(z)/Dj(z) and denoteit by ofj.

Step 5Increment /•<-/ + 1. Check the value of/. If / < « , go to step3 above.

Step 6Increment i2 above.

Step 7Form all possible n\ permutations of the «-set {1, 2, . . . , « }and denote them by n1,n2,. . . ,nnl. The elements/ belongingto a permutation nk correspond to the indices of somedenominator polynomials Dj(z).

Step 8Initialise / •«- 1.

Step 9Consider the permutation TT,- obtained in step 7. Let

The cascade now consists of the sections with input/outputtransfer functions Nl(z)/Dk.1(z), N2(z)/DkQ(z),. . . ,Nn(z)/Dkin(z). The required norms Plkn , P2k{2,. .. ,Pnkin, and thenoise variances af^.j, affe.2,. . . , Onii^ were all obtained insteps 3 and 4. The upper bound expression in eqn. 9 is usedto obtain a^axi f° r t m s cascade.

Step 10Increment /•*-/+ 1. If / < n\, go to step 9.

The cascade with minimum a^axi is selected as the requiredone. If this Omaxi is less than, or equal to, the tolerable limiton noise variance, this particular cascade is acceptable. Other-wise, the individual sections in the cascade can be replaced bylower noise-variance structures (which could be different fromthe direct-form structures). If the requirements are stillunfulfilled, one needs to change the ordering of the sections inthe cascade (which can be done effectively by using the secondalgorithm to be described next).

Second algorithmLet us assume that a particular pole-zero pairing is alreadygiven. Thus, the inputs to the algorithm given are the transferfunctions Tx{z), T2(z),. . . , Tn{z) of the n sections consti-tuting the overall transfer function G(z).

Step (i)Form all possible n\ permutations of the n-set {1, 2 , . . . , « }and denote them by n1,n2,... ,nn\. The elements k belong-ing to a permutation, say 7r,-, correspond to the indices of sometransfer functions Tk(z).

Step (ii)Initialise /«- 1.

Step (Hi)Consider the permutation 7rf obtained in step (/). Let

The cascade now consists of the transfer functions Tk

k.2 Tk. in order. Evaluate the required norms to

i + 1. Check the value of i. If / < n, go to step

obtain 0^0x2 a s given by eqn. 13.

Step (iv)Increment i<-i + 1. If / < n\, go to step (iii).

At the completion of step (iv), we are left with n\ upperbounds o^noxi and we can choose the one which is minimumamong them. The particular ordering associated with thispole-zero pairing is the desired ordering.

Example 1We shall consider the 6th-order type-2 Chebyshev band-rejection filter used by Jackson [ 1 ] . The specifications forthe filter are 2.26 dB passband ripple, 25 dB stopband attenu-ation, and a transition ratio of 0.53. We are neglecting theconstant 0.76091619 from the 6th-order transfer function.

Six different 2nd-order pole-zero pairings are possible,

252 IEEPROC, Vol. 129, Pt. G, No. 5, OCTOBER 1982

owing to the fact that the overall transfer function is oforder six. The relevent coefficients are shown in Table 1.Each section is'assumed to be implemented by using direct-form 2 structure without any loss of generality.

Table 1: 6th-order Chebyshev-2 band-rejection filter

Transfer function = 11

Case Section (i)

III

IIIIVV

VI

021

0.1

<*!<*.

<*•

« i

0.90352914- 1.7636952

1.0— 1.8118373-1.8118373— 1.6545862— 1.7442502- 1.7442502- 1.6545862

0.84506679- 1.44227789

1.0-1.6545862-1.7442502-1.8118373- 1.6545862— 1.8118373— 1.7442502

0.75829007— 1.5334490

1.0-1.7442502- 1.6545862- 1.7442502-1.8118373- 1.6545862-1.8118373

Actual noise variances op are calculated from the following'expression:

j =oj =

where p = 1 or °°, and

slltP = 111/(1 +0uz-1+02 1z-2)llp

Sii.p = 1(1+an*"1 +a2 1z-2)/(l+01 1z-1

(16)

(17)

02iz"2)llp

(18)

Table 2:

Case

1

II

II I

IV

V

VI

Comparison

Scalingtype

*-,oo

LI

oo

between ofnBCX and actual output-noise variancefor example 1

arnax\ l°o ' n dB

28.0795539.52891

33.3696145.30581

41.4903451.57554

32.652443.1981

38.9491650.0608

39.713849.84804

a2IOQ in dB

23.97109836.21487

23.9606536.18024

25.3771237.97997

24.8510837.29943

24.2172336.47658

25.8637538.55769

Table 3: Coefficients of an 8th-order bandpass filter

0u

1.402660.8263911.073470.941754

0.01021571.324430.63181.25358

- 0.88866- 1.04661— 0.804891-1.16031

0.9268670.9306060.9738540.976782

ao = 0.002939

n

(22)

•21 14

a2lSll,pS2I,P

Fig. 2 Grcuit for computing noise variance

U

..,-lln3

q22Sl2,p

'22,p

a23S,3,pS23,p

ifZ^+folZ"2)]

(19)

(20)

(21)

a2iS2i,pS l i , P

Fig. 3 Grcuit for computing noise-variance bound

IEEPROC, Vol. 129, Pt. G, No. 5, OCTOBER 1982 253

Table 4: Numerator coefficients for all possible pole-zero pairings of the 8th-order bandpass filter transferfunction

Case

1

II

III

IV

V

VI

VII

VIM

IX

X

XI

XII

XIII

XIV

XV

XVI

XVII

XVIII

XIX

XX

XXI

XXII

XXIII

XXIV

"if

"of"fl

"of"if

"of"if

"of"if

"of"if

"of"if

"of"if

"of"if

"of"if

"of"if

"of"if

"of"if

"of"if

"of"if

"of"if

"if

"of"if

"of"if

"of"if

"of"if

"of"if

"of"if

"of"if

/ • = 1

1.40266— 0.0102157

1.40266— 0.0102157

1.40266-0.0102157

1.40266-0.0102157

1.40266-0.0102157

1.40266-0.0102157

0.826391- 1.32443

0.826391- 1.32443

0.826391- 1.32443

0.826391- 1.32443

0.826391— 1.32443

0.826391— 1.32443

1.07347-0.6318

1.07347-0.6318

1.07347-0.6318

1.07347-0.6318

1.07347-0.6318

1.07347-0.6318

0.941754- 1.25358

0.941754-1.25358

0.941754- 1.25358

0.941754— 1.25358

0.941754- 1.25358

0.941754- 1.25358

/= 2

0.826391- 1.32443

0.826391- 1.32443

1.07347-0.6318

1.07347-0.6318

0.941754-1.25358

0.941754- 1.25358

1.40266-0.0102157

1.40266-0.0102157

1.07347-0.6318

1.07347— 0.6318

0.941754— 1.25358

0.941754— 1.25358

1.40266— 0.0102157

1.40266-0.0102157

0.826391- 1.32443

0.826391— 1.32443

0.941754-1.25358

0.941754-1.25358

1.40266-0.0102157

1.40266-0.0102157

0.826391— 1.32443

0.826391- 1.32443

1.07347-0.6318

1.07347— 0.6318

/=3

1.07347-0.6318

0.941754-1.25358

0.826391

— 1.32443

0.941754-1.25358

0.826391- 1.32443

1.07347-0.6318

1.07347-0.6318

0.941754-1.25358

1.40266-0.0102157

0.941754— 1.25358

1.07347-0.6318

1.40266— 0.0102157

0.826391— 1.32443

0.941754-1.25358

1.40266-0.0102157

0.941754-1.25358

0.826391- 1.32443

1.40266-0.0102157

0.826391-1.32443

1.07347-0.6318

1.40266-0.0102157

1.07347-0.6318

0.826391- 1.32443

1.40266-0.0102157

; = 4

0.941754-1.25358

1.07347-0.6318

0.941754

-1.25358

0.826391- 1.32443

1.07347-0.6318

0.826391— 1.32443

0.941754-1.25358

1.07347-0.6318

0.941754-1.25358

1.40266-0.0102157

1.40266-0.0102157

1.07347-0.6318

0.941754- 1.25358

0.826391-1.32443

0.941754-1.25358

1.40266-0.0102157

1.40266— 0.0102157

0.826391- 1.32443

1.07347-0.6318

0.826391- 1.32443

1.07347-0.6318

1.40266-0.0102157

1.40266-0.0102157

0.826391- 1.32443

U2 =

n

The upper bound on noise variance

254

(23)

/

(24)

is computed from

°maxl,p

+ Pjwhere, from Fig. 3:

_ n2 p2 n2 i p2~ *2,<x>r3,°°o\,r

2 p2 n2 (25)

H z - 1 +/32,z-2)llp

(26)

+ 3PlP + 3QtPP?2] (27)

IEEPROC, Vol. 129, Pt. G, No. 5, OCTOBER 1982

i = 1,2,3Table 5: Comparison between o?naxi and a 2 for example 2

(28)

Table 2 shows the upper bounds and also the actual noisevariances at the output for the six different cases of pole-zeropairings.

It can be noted from the contents of Table 2 that minimum°maxi is obtained for the pole-zero pairing pertaining tocase I, but actual calculated noise variance for the pole-zeropairing of case II shows the least magnitude. It is also evidentthat the pole-zero pairings corresponding to the cases I and IIdiffer in output-noise variance only by a very small value.Thus, by choosing the pole-zero pairing pertaining to case I,we still can obtain a low-noise cascade realisation of a giventransfer function. The output-noise variance can further bereduced by using structures other than direct-form 2 toimplement the 2nd-order transfer functions in the cascadeand also by proper ordering of sections in the cascade.

Example 2The transfer function for an 8th-order elliptic bandpassfilter [10] with passband ripple of 1 dB, minimum stopbandattenuation of 45 dB and a passband extending between900-1100 rad/s is given by:

• — i • — 2_ , . _5_ &r>; •+" OLtiZ ~r OLniZ

where the coefficients are summarised in Table 3. The sam-pling frequency for the filter is 6000 rad/s and the stopbandedge frequencies are 800 and 1200 rad/s.

There are 24 different pole-zero pairings possible for thisfilter. Table 4 lists all the 24 pairings. Table 5 shows the°maxi a nd °2 a t the output of these 24 different cascaderealisations.

Again, in this case we are considering only direct-form2 realisation for the constituent sections in each cascade.Note that we are only performing pole-zero pairing, andordering of sections is kept invariant. Thus, the noise variancesshown in Table 5 are not the optimum values.

Observations similar to those in example 1 can also be madehere. Less discrepancy between the actual noise varianceo2lao and the bound o?naxilao might be possible withsections in cascade realised in forms other than direct-formII. Also, a tighter upper bound that can be computed is

4 Conclusions

In this paper, we have developed two upper bounds on thenoise variance at the output of a cascade realisation of adigital transfer function. These bounds are much simplerto calculate, compared with the calculation of the actualnoise variance. Two algorithms were developed, based on thesebound expressions, which result in a cascade having leastupper bound on the output-noise variance. This will at least

Case

1II

I I IIVV

VIV I I

VIIIIXX

XIX I I

XIIIXIVXV

XVIXVII

XVIIIXIXXX

XXIXXII

XXIIIXXIV

amaxi /°<

L2 -scaling

55.6747674.09371466.07256971.05337771.5067358.0685766.52077684.9631470.82578384.27812371.26901276.25903874.98217279.97866568.88043582.33496579.72980671.27176987.48762187.3428781.38593689.69919786.8021778.3441 58

o in dB

L -scaling

59.50122176.88695369.58687673.90716174.50066961.46677467.56746385.65159171.7758284.97212172.14909377.0332175.42151180.31272169.32840682.66264680.08861871.64569787.83475287.69892481.74229890.04812887.15219378.711191

a2IOQ in

L7 -scaling

11.00719514.45396711.31512611.13559614.70593711.37599111.3576814.61596511.19883612.6285815.4792416.57539211.40439611.22859410.93432412.43990612.57463210.93587315.19493212.36783316.80450915.77196513.29030111.943929

dB

/.^-scaling

26.61823330.01794825.985124.82158630.45481226.73970326.82506230.11371726.27979226.48611432.16303833.36458526.11629324.99230326.17396826.38525826.74367625.54562630.80611527.52497733.50472732.34691227.43391326.434001

lead to a low-noise cascade realisation of a digital transferfunction.

5 Acknowledgment

This work was supported by the National Science Foundationunder grants ENG 77-20622 and ECS 79-18028.

6 References

1 JACKSON, L.B.: 'An analysis of roundoff noise in digital filters.Sc.D. thesis, Stevens Institute of Technology, Hoboken, NJ, 1969

2 LEE, W.S.: 'Optimization of digital filters for low roundoff noise'.Proceedings of IEEE international symposium on circuit theory,Toronto, Ont., Canada, 1973, pp. 381-383

3 HWANG, S.Y.: 'On optimization of cascade fixed-point digitalfilters', IEEE Trans., 1974, CAS-21, pp. 163-166

4 LIU, B., and PELED, A.: 'Heuristic optimization of the cascaderealization of fixed-point digital filters', ibid., 1975, ASSP-23,pp. 464-473

5 STEIGLITZ, K., and LIU, B.: 'An improved algorithm for orderingpoles and zeros of fixed-point recursive digital filters', ibid., 1976,ASSP-24, pp. 341-343

6 RAHMAN, M.H., and FAHMY, M.M.: 'A roundoff-noise minimiz-ation technique for cascade realization of digital filters'. ProceedingsIEEE ISCAS, Phoenix, AZ, Apr. 1977, pp. 45-48

7 JACKSON, L.B.: 'Roundoff noise bounds derived from coefficientsensitivities for digital filters', IEEE Trans., 1976, CAS-23, pp.481-485

8 MULLIS, C.T., and ROBERTS, R.A.: 'Roundoff noise in digitalfilters: frequency transformations and invariants', ibid., 1976,ASSP-24, pp. 538-550

9 MITRA, S.K., KAMAT, P.S., and HUEY, D.C.: 'Cascaded latticerealization of digital filters', Int. J. Circuit Theory & Appl., 1977,5, pp. 3-11

10 ANTONIOU, A.: 'Digital filters: analysis and design' (McGraw-Hill,NY, 1979)

IEEPROC, Vol. 129, Pt. G, No. 5, OCTOBER 1982 255

Sanjit K. Mitra received the M.S. andPh.D. degrees in electrical engineeringfrom the University of California,Berkeley, in 1960 and 1962, respectively.He joined the University of California,Santa Barbara in 1977 as a Professor ofElectrical & Computer Engineering. He isa consultant to Lawrence LivermoreNational Laboratory and is a consultingeditor for the Van Nostrand ReinholdCompany, New York. He has held visiting

appointments in India, Japan and West Germany.

Kalyan Mondal received his B.S. (Hons.)in physics, B. Tech. and M. Tech. degreesin radio physics and electronics from theUniversity of Calcutta, Calcutta, Indiain 1969, 1972, and 1974, respectively.He graduated from the University ofCalifornia, Santa Barbara in 1978 witha Ph.D. degree in electrical and computerengineering. Dr. Mondal is currently amember of the technical staff at theBell Laboratories, Allentown, PA and an

Adjunct Assistant Professor in the Department of Electrical &Computer Engineering at the Lehigh University, Bethlehem,PA. His research interests lie in the areas of digital signalprocessing and VLSI system design.

CorrespondenceCOMPUTER-AIDED DESIGN OF ALL-PASSDIGITAL FILTERS : APPLICATION TOTHE PHASE LINEARISATION OF RECURSIVEDIFFERENTIATORS

Indexing terms: Computer-aided design, Filters and filtering

Abstract: Reference is made to a paper published in 1970 andapplying non-linear optimisation by the Fletcher Powell methodto the linearisation of the phase response of recursive digitalfilters. Wideband recursive differentiators are especially con-sidered.

In recent issue of IEE Proc. G, Electron. Circuits & Syst. [1],Prof. C. Charalambous and Prof. A. Antoniou have presentedan all-pass synthesis method based on a minimax algorithmand aimed at the linearisation of the group delay of a digitalrecursive filter.

The aim of this letter is to bring to the attention a similarwork published in 1975 by W. Strepenne and myself [2].

Since linearisation of recursive differentiators is not dealtwith in Reference 1 or in other similar works, I think that thissubject, treated in our paper, might be of interest to yourreaders, despite the fact that the modern algorithm of Refer-ence 1 is more efficient.

In Reference 2, we applied Fletcher and Powell's optimi-sation algorithm to synthesise all-pass filters linearising Butter-worth bilinear filters (an easy application) and wideband re-cursive differentiators given by Steiglitz [3] and presenting a

strong phase nonlinearity. Using a least-square criterion wehave linearised the phase of these differentiators by all-passwith, respectively, one real pole, two real poles, one pair ofcomplex conjugate poles, and one pair of complex conjugatepoles plus one real pole. Linearisation is excellent in 80% ofthe Nyquist band. Since Steiglitz's differentiators have a goodmodulus in 90% of this Nyquist band, our linearisation oftheir phase may be useful when group delay of signals is of in-terest (of course, some additional linear delay is unavoidable).

16th April 1982 J. LEFEVRE

Universite Catholique de Louvain,Departement de Physiologie,Avenue Hippocrate 55,B - 1200 Brussels, Belgium

References

1 CHARALAMBOUS, C , and ANTONIOU, A.: 'Equalization ofrecursive digital filters', IEE Proc. G. Electron. Circuits & Syst.,1980,127, (5) pp. 219-225

2 LEFEVRE, J., and STREPENNE, W.: 'Computer-aided designof all-pass digital filters : application to the phase linearizationof recursive filters', Acta Tech. Belgica (Revue H F), 1971, pp.123-134

3 STEIGLITZ, K.: 'Computer-aided design of recursive digitalfilters', IEEE Trans., 1970 AU-18,pp.l23-129

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