Overflow oscillation in digital lattice filtersD.T. Nguyen, B.E., Ph.D. M.N.Z.I.E.

Indexing terms: Digital circuits, Lattice filters

Abstract: The overflow oscillation problem in fixed-point arithmetic digital lattice filters is considered.Conditions for overflow stability are formulated in the filter ^-parameter space for the 2-multiplier and the1-multiplier lattice filters, using their state-space description. It is found that the 2-multiplier lattice model isoverflow stable over a much wider range of ^-parameters than the canonical 1-multiplier model, and that theoptimal choice of the sign parameters em of the latter as proposed by Markel and Gray sometimes makesthe filter overflow unstable. However, both lattices are more stable than the direct form realisation.

List of symbols

x{n)x(n)

A\t

F(z)

H{z)IkiMe{

(1 +a,;f1 +a2z~2)

= state vector= nonlinear state vector after an

overflow= system matrix= eigenvalues of system matrix= state transfer matrix (from input to

state variables)= system transfer matrix= unit matrix= multiplier coefficient of ith lattice= order of filter= sign parameter of/th 1-multiplier

lattice= denominator of direct-form 2nd-

order filter

1 Introduction

Despite the many advantages offered by recursive digital filters,there are two main serious oscillation problems inherent in thefixed-point arithmetic representation and operation of datawords and coefficients, using finite register length. One ofthese nonlinear autonomous oscillations is the result of thewrap-around effect of the two's complement overflow in theadders and the other is caused by the coherent effect of therounding of the multiplication product in the recursive part ofthe network. The overflow oscillations usually have very largeamplitudes overriding the input, and cannot be tolerated.

Several scaling techniques to scale down the inputs to theadders have been discussed by Jackson [1]. However, too con-servative a scaling of the signal would mean an inefficient useof register length and would increase the quantisation noise.Most of the practical scaling techniques used are thereforebased on some statistical signal norms, and overflow oscillationis not entirely avoided but is only kept to a low probability. Asimple saturation arithmetic may then be applied to freeze thesignal at the maximum magnitude level whenever overflowoccurs to prevent oscillation, but this results in a signaldistortion.

Barnes and Fam [2], Mills etal. [3], have pointed out that,over a certain region within the unit circle, recursive digitalfilters have decreasing state norms property and are thereforeoverflow stable, i.e.

\\x{n+\)\\2<\\x{n)\\2

where ||JC||2 = xTx denotes the quadratic norm.If all poles are located in this region, overflow oscillation if

it occurs will decay rapidly within a few sampling clocks. In

Paper 1527G, first received 17th December 1980 and in revised form7th April 1981The author is with the Department of Electrical Engineering, Universityof Auckland, New Zealand

particular, the state-space minimum-norm filters [2], thosehaving the norm of their system matrix A equal to the magni-tude of the largest eigenvalue of .4, i.e.

= max

are overflow stable over the entire unit circle.This paper discusses the overflow stability of the digital

lattice filters which have dominated the field of speechresearch since its introduction by Itakura and Saito [4] in1971, and later have been developed by Markel and Gray [5]for the synthesis of any linear transfer function of the generalform PM(z)/AM(z). Both the numerator and the denominatorare in the polynomial form. The stability of the digital latticefilters has been widely discussed [5, 6] but only strictly in theHurwitz sense, i.e. having all poles inside the unit circle. Thispaper attempts to determine the subregion within the unitcircle in which the lattices are overflow stable. Explicit formu-lation is given for the 2nd-order lattices, and comparison ismade between the well known direct form, the 2-multiplierlattice, and the 1-multiplier lattice digital filters.

2 State-space description of digital lattice filters

In the digital lattice filters, the denominator polynomialAM{z) is realised by cascadingMlattices together, as shown inFig. 1, and the numerator polynomial PM( Z ) 'S obtained from

u(n)

k2

lattice

y(n)

Fig. 1 Cascaded implementation of digital lattices

the weighted sum of the outputs of the lower (backward)branch of these lattices. Each of these lattices is either a2-multiplier type (Fig. 2A) or a 1-multiplier type (Fig. 2B). Inthe latter case, the sign parameters em (m = 1, 2, . . .M) maybe + 1 or — 1, which may be decided by an optimal algorithmproposed by Gray and Markel. The filter parameters km (m =1 ,2 , . . . M) and the tap parameters, vm (m = 1, 2, . . . M + 1)can be obtained efficiently from the coefficients of the poly-nomials AM(z) and PM(z), by a recursive procedure [5].

A linear recursive digital network can be described by thefamiliar state equations, using standard notations:

x(n+ 1) = Ax(n) + Bu(n)

and

y(n) = Cx(n) + Du(n) 0*)

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From eqn. la, the vector transfer function from the input tothe state variables in the Z-domain is therefore

F(z) = (zI-A)'lB

and from eqn. 16, the associated transfer function is

H(z) = Y(z)/U(z) = C(zI-AylB + D

(2)

whose poles are the eigenvalues of the system matrix A.By assigning the states of the system to the outputs of the

z"1 blocks in the lower branch of the filter (Fig. 2), it can beshown that, for the 2-multiplier lattice realisation,

kQkx —k0k2 ... ~kok]\i

A =

(\-k\) -kxk2

0 (l-k22)

0 0

— k 1c

~k2kM

KM-l) KM-1KM

B1 = [*0,*,, . . .*M]

in which k0 = 1, and for the 1-multiplier lattice filter

A —

—k0k2(\ + e

-kxk2

Xm(n.l)

Fig. 2A 2-multiplier lattice model

xm(">

Fig. 2B 1-multiplier lattice model em — ± I

270

XJn-1)

For the purpose of this paper, other system parameters, C andD, are of no interest, since we are concerned only with therecursive portion (denominator) of the system transferfunction.

To appreciate the role of the coefficients km in the deter-mination of the pole locations and hence stability, let us con-sider a 2nd-order transfer function whose denominator has thegeneral form (1 + axz~x + a2z~2).

Using eqn. 2 or the recursive procedure, as outlined inReference 5, it is a simple matter to show that both latticeconfigurations under study have the same eigenvalues, whichare the roots of the equation

+k2 = 0 ,

and that

and

k2 = a2

(3a)

(3b)

Without having to find the system eigenvalues in terms of theparameters kv and k2 , we can obtain the stability square in the

KQK

M-\

M

'-l^M

^-parameter space of the 2nd-order lattice filter directly fromthe well known stability triangle (Fig. 3 A) of the direct formfilter in the a-parameter space, by a linear mapping of eqns. 3aand b. It may then be seen from this stability square (Fig. 3B)that for poles to lie within the unit circle, it is sufficient that

\ki\<; I and \k2 | < 1

and that condition, for underdamped poles, is

(4)

In fact, eqn. 4 is true also for higher-order filters [5 ] , i.e.k2

m<l(m=l,2,...M).

3 Overflow stability of digital lattice filters

3.1 Overflow-stability formulationThe stability discussed in the preceding Section is strictly inthe Hurwitz sense; that is, all the system poles lie within theunit circle and any natural mode oscillations will decay withtime. However, stability theories show that, with certain inputconditions, even if all poles lie inside the unit circle, a norm ofthe state vector may still be increasing at first for a few sampleclocks before it starts to decay. Only in a subset of the unitcircle can the decay be guaranteed immediately.

Consider the overflow phenomenon in a digital filter asbelonging to a class of nonlinear systems which can be simply

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modelled as

* ( / i + l ) = G[Ax(n)]

Mills etal. [3] have separated eqn. 5 into two consecutiveoperations: a linear operation

(5)

(6)

which is followed by a nonlinear operation owing to overflow

x(n + 1) = G[x(n+l)) (7)

From the two's complement overflow characteristic, it isalways true that

\G(x)\<\x\

and hence eqn. 7 results in a decreasing norm, i.e.

«+Oil*It is therefore sufficient to require only eqn. 6 to have adecreasing norm for the filter to be free of overflow oscillation,i.e.

(Ax)T(Ax)<xTxor

xT[I-ATA]x>0 (8)

To satisfy eqn. 8, it is necessary and sufficient that the matrix(/— ATA) be positive definite, i.e. all eigenvalues of ATA havea magnitude less than 1. Sometimes it is more convenient touse the equivalent condition [2] ||v41|2 < 1.

3.2 2nd-order lattice filtersIt can be shown that the eigenvalues o{ATA are roots of

\2-[k2 +k\ +k\k\ +(\-k2)2)\ + k22 = 0

for the 2-multiplier lattice case, and of

X2-[k21+k]kl+(l-€lk1)

2 +k\{\ +elk1)2\ + k2

2 = 0

for the 1-multiplier lattice realisation.a.

Therefore, in the ^-parameter space, the condition foreqn. 8 to be satisfied is

k\ + k\ < 1 (9)

for the 2-multiplier case, and

00a)

together with

(10b)

for the 1-multiplier lattice filter.The shaded areas of Figs. 4A and B show the overflow-

stability zones in which eqns. 9 and 10 are satisfied. Twointeresting conclusions which may be drawn from theseFigures are:

(a) The overflow stability of the 1-multiplier lattice con-figuration is inferior (having much narrower stable zone) tothat of the 2-multiplier model. This is a serious disadvantage inspeech synthesis application in which the ^-parameters aretime variant and kx and k2 are usually large during a voicedsegment.

(b)\n the 1-multiplier lattice filter, condition \0b dictates

Fig. 4A Overflow-stability subregion in k-parameter space for2-multiplier lattice

Fig. 3A Stability triangle in a-parameter space for direct form F'9# ̂ B Overflow stability subregion in k-parameter space for2nd-order filter 1 -multiplier lattice

-1 -

Fig. 3B Stability square in k-parameter space for digital lattice 2nd- Fig. 4C Overflow free subregion in 'equivalent' k-parameter space fororder JUter direct-form filter

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the sign parameter ex; this, in certain situations, contradictsthe optimal choice of this parameter as proposed by Gray andMarkel [5] based on the principle of minimisation of thedynamic range of norms at various nodes in the filters.

For example, consider the case \k21 > I AT, |. Using the optimalalgorithm in Reference 5, since

2 i>

we choose

The system has one eigenvalue of ATA always greater than 1and therefore does not have the decreasing state normsproperty. However, Ebert etal. [7] have pointed out that if

k i l + | f l 2 l<i (12)

overflow conditions will never occur.If we map expr. 12 from the a-parameter space into the k-

parameter space, for comparison purpose, this results in

- 1 * ! I(13)

which makes the filter overflow unstable.

3.3 3rd-order lattice filtersA test for positive definiteness for (/ — ATA) in order tosatisfy eqn. 8 can be carried out for a 3rd-order lattice filterwith some tedious arithmetic. Using identities

det [ATA] = k\

and

det [I-X] = 1 - det [X] -Tr[X] + (Xn + X22 + X33)

where Xit is the cofactor of the diagonal element xit of matrixX, and the requirement that all the m x m principal minors(m = 1 to 3) in the top left-hand corner of (/—ATA) havepositive determinants, we arrive at the condition

k\ < k] + k\ < 1

00a)

3.4 Higher-order lattice filtersSufficient conditions for overflow stability of higher-orderlattice filters cannot be obtained explicitly. However, somenecessary conditions may be formulated and these shouldprovide some further insight into the problem. It can beshown that the diagonal elements of the matrix (/— ATA)for the 2-multiplier case is

- X kl

and the last element is

/ = 1 . 2 . . . . J V - 1

^NN — 1m= 1

It is obvious that to satisfy eqn. 8, from the test of positivedefiniteness, it is at least necessary that

N-l

I k2m<l

and

J V - l

(11*)

{Ub)

Expr. lib is very stringent when \kN\-* 1, as in the case ofa speech synthesiser of assumed ideal glottal excitation.

3.5 Direct-form filtersFor comparison between the overflow stability of lattice filtersand direct-form filters, this Section is included. It is wellknown that, for the direct-form filters, the system matrices are

A = B =

which represents a narrower subregion of overflow stabilitythan that of the 1-multiplier lattice model (Fig. 4C).

4 Conclusion

A state-space description of digital lattice filters has beenpresented for both the 2-multiplier and the 1-multiplier latticerealisations. Conditions for overflow stability of these filtershave been established explicitly for the 2nd- and 3rd-ordercases. An attempt has been made to obtain some necessaryconditions for overflow stability for higher-order lattice filters.

For the 2nd-order case, it was found that the 2-multiplierlattice model is overflow stable over a considerably widerrange of ^-parameters than the 1-multiplier model. This factgives the 2-multiplier model a definite advantage over thelatter one in synthesising time-varying coefficient filters. It wasalso found that the sign parameter et of the 1-multiplierlattice must have the same sign as k^ for the filter to be over-flow stable, and this, in some circumstances, contradicts theoptimal procedure for the choice of e suggested by Gray andMarkel. Both lattice configurations, however, are overflowstable over a wider parameter range than the direct-form filters.

5 References

1 JACKSON, L.B.: 'Roundoff-noise analysis for fixed-point digitalfilters realized in cascade or parallel form', ibid., 1979, AU-18, pp.107-122

2 BARNES, C.W., and FAM, A.T.: 'Minimum norm recursive digitalfilters that are free of overflow limit cycles', ibid., 1977, CAS-24,pp. 569-574

3 MILLS, W.L., MULLIS, C.T., and ROBERTS, R.A.: 'Digital filterrealisations without overflow oscillations', ibid., 1978, ASSP-26,pp. 334-338

4 ITAKURA, F., and SAITO, S.: 'Digital filtering techniques forspeech analysis and synthesis'. Paper 25C-1, 7th internationalcongress on acoustics, Budapest, 1971

5 GRAY, A.H., and MARKEL, J.D: 'Digital lattice and ladder filtersynthesis', IEEE Trans., 1973, AU-21, pp. 491-500

6 MARKEL, J.D., and GRAY, A.H.: 'Linear prediction of speech'(Springer-Verlag, 1976), Chap. 5

7 EBERT, P.M., MAZO, J.E., and TAYLOR, M.G.: 'Overflow oscil-lations in digital filters', Bell Syst. Tech. J., 1969, 48, pp. 2999-3020

D. Thong Nguyen was born in Ha-Tinh,Vietnam, on 1 st January 1941. He receivedhis B.E. degree in electrical engineeringwith first-class honours in 1965 from theUniversity of Canterbury, New Zealand,and his Ph.D. degree in antenna theory in1969 from the University of Auckland,New Zealand. Dr. Nguyen joined theteaching staff at the National Institute ofPoly technique, Saigon, Vietnam, in 1969,and became Associate Professor in 1974.

In 1975, he was appointed senior lecturer in the Departmentof Electrical Engineering, University of Auckland. His currentresearch interests are in the fields of digital systems, data trans-mission, bandwidth compression and speech research.

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