Roundoff noise prediction in

short - wordlength

fixed-point digital filtersA.J. McWilliam and B.J. Starrier

Index ing term: Digital filters

Abstract: Consideration is given to recursive fixed-point digital filters employing a signal wordlength in therange of 4 to 8 bits. Experimentally determined values of roundoff-noise variance are found to deviatemarkedly from those predicted in the normal manner. Such deviation is caused mainly by correlations be-tween signal and error sequences. A method is proposed whereby improved predictions of noise variance maybe obtained in the case of first order filters. Finally, it is demonstrated that, when the 'coefficient slicing'realisation is used to realise filters with sufficiently quantised coefficients, such filters yield both a very lowlevel of roundoff noise, and also excellent agreement between the predictions of a simple model and measurednoise levels.

1 Introduction

In this paper consideration is given to roundoff-noiseprediction in recursive fixed-point digital filters employinga signal wordlength in the range of 4 to 8 bits, such asmight be available in a hard-wired or microprocessor-basedsystem. The standard statistical model1 of roundoff-errorsequences generated at constant multipliers is well estab-lished and gives rise to a relatively simple method of round-off-noise prediction,2 as set forth in Section 2. This model,however, was not formulated with a particular view tosystems using very short wordlengths. This paper summar-ises the experimental results and theoretical conclusions ofa study of the effects of using very short wordlengths whichis reported at greater length elsewhere.3

The consequence of the first assumption is that thevariance of the roundoff-error sequence at any multipliercan be calculated to be A2/12, where A is the quantisationwidth. That is, the variance is assumed to be independentof the input signal and of the multiplier value. The infer-ence drawn from the second assumption is that all round-off-error sequences generated in a filter are statisticallyindependent of one another, in which case if two or moreerror sequences are summed sample-by-sample the varianceof the resulting sequence is the sum of the variances of thecomponent sequences.

Consider the example of a second-order direct formfilter having m nonintegral coefficients defining the zerosof the transfer function H(z) and p nonintegral coefficientsdefining its poles. The variance of the roundoff-errorsequence at the filter output is given by

2 Roundoff-error variance prediction

The standard method of roundoff-error variance predictionin digital filters employing fixed-point arithmetic is foundedupon the following two assumptions about the statisticalproperties of roundoff error sequences generated at con-stant multipliers.1 (The input sequence to any such multi-plier is assumed to be wide-sense-stationary.)

(a) All error values within the interval allowed by thequantisation width have an equal probability of occurrence.

(b) Roundoff errors, created at any particular multiplier,form a random sequence which is correlated with neitherthe input nor the output signal sequence at the multiplier.Knowles and Edwards1 impose the restriction that thequantisation width be small in comparison to the signalamplitude. Liu2 suggests that these assumptions are 'quitesatisfactory at least down to a wordlength of 8 bits'. Thesecond of the above assumptions clearly should not beapplied to the sign-magnitude-truncation roundoff process.7

Paper T123 E, received 15th July 1977Dr. Stanier is, and Dr. McWilliam was formerly, with the Departmentof Applied Physics & Electronics, University of Durham, ScienceLaboratories, South Road, Durham DH1 3LE, England. Dr. Mc-William is now with the Department of Electronic & ElectricalEngineering, University of Technology, Loughborough, Leics. LEll3TU, England

ELECTRONIC CIRCUITS AND SYSTEMS, JANVAR Y1978, Vol. 2, No.l

2m J

S(z)dz

Ul-iB(z)B(z~1)z 0)

where B(z) is the denominator of the filter transfer functionH(z), and S(z) is the power spectral density of the sequence{en} formed by the sample-by-sample summation of the(m + p) independent roundoff error sequences.

The power spectral density is defined by

S(z) = Ree{i)Z (2)

where Ree(i) is the autocorrelation function of the sequence{en} and o\ is its variance. Because {en} is assumed to be alinear superposition of independent random sequences, italso is considered to be random. Therefore the auto-correlation function is taken to be unity for zero-lag, andzero elsewhere. This leads to a major simplification of eqn.1 to give

°2e2nf Jdz

|z |=lB(z)B(z~l)z

(3)

0308-6984/78/123E-0009 $1-50/0

where, from the first basic assumption of the model,

i A2

a\ = (m + p) — (4)

The contour integral of eqn. 3 can easily be evaluated fora given set of filter coefficients,8 yielding a prediction ofthe output roundoff-noise variance. Appropriate equationsfor other filter forms can be derived in a similar manner.

00 02 0-4 06 08 0-85 09 0-95 10pole radius

Fig. 1 Deviation of experimental noise variance from model pre-dictions 6 V

First-order filters, wordlength = 8 bits• uniform error-distribution modelo nonuniform error-distribution model

3 First-order filters

3.1 Experimental results

Measurements on a computer simulation model3 show thatthe level of roundoff noise practically obtained at the out-put of a first order filter can exceed the predicted level by avery significant factor. Many first-order highpass filters weretested, each being designed to have its pole at a particularradius from the centre of the unit circle in the Z-plane andto have a theoretical gain of unity at the half samplingfrequency. The actual roundoff-noise variance measuredVc, was compared with that predicted using eqns. 3 and 4,Vs. A measure of the agreement between the two is givenby5K=|101og l 0 (Ks /Fe) |dB.

Fig. 1, a plot of 5 V for filters employing an 8-bit signalwordlength and the rounding process, is typical of theresults obtained. The predictive formula is seen to workwell for filters with a pole at \z\ <0-95, but as the filterpole approaches the unit circle the discrepancy betweentheory and experiment increases severely. For a filter withits pole at \z I = 0-99, Ve exceeds Vs by over 20 dB. As thesignal wordlength decreases, the pole radius at which 8 Vbegins to deviate markedly from zero also diminishes; fora 4-bit signal wordlength this threshold pole radius is| z |=0-7 . The gradient of 5 V in the region above thethreshold also decreases with a reduction in signal word-length such that the maximum deviation obtained remainssteady at around 20 dP. Note that the theoretical pre-dictions are most misleading in filters designed to have ahigh degree of frequency selectivity, that is those with apole near the unit circle in the Z-plane.

3.2 Error distribution prediction

The general difference equation for an ideal digital filter

r syn = 5 > * n - i - Zb&n-t (5)

1=0 1=1

indicates the statistical dependence of the output sampleyn on current and previous input samples xn _,- and previousoutput samples yn-{. For a first-order filter (r = 0, s — 1),yn is dependent only on xn and yn-x. The delayed out-put yn-\ is dependent on jcn_1} but if {xn} is a randomsequence then xn and yn-x should be statistically inde-pendent. Such independence of the samples from whichthe current filter output.is formed is unique to the first-order case and leads to a method of predicting signalamplitude distributions which is not applicable to higher-order filters.

In a practical filter a given value of yn may be formedfrom a finite number of combinations of xn and yn.tvalues. The probability of occurrence of any given outputis the sum of a number of joint probabilities, each referringto a particular combination of xn and yn-x, which yieldsthe required output. That is,

= E I iPxmPyk) (6)

where py. is the probability of occurrence of output level/,and pXm is the probability of input level m. Such anequation can be written for each of the N allowed outputlevels. Assuming that the amplitude distribution of theinput sequence {xn} is known the px terms can be givenactual values, leaving the pys as the only unknowns.

Furthermore, it is clear that

N

I Pyj = (7)

Replacing'any one of the N equations of the form of eqn.6 by eqn. 7 yields a set of Af simultaneous equations whichcan be solved to give Pyx . . . PyN- The amplitude distrib-ution of the output sequence {yn } may thus be predictedaccurately.

Clearly the amplitude distribution of {yn } must also bethat of {yn-i}> the input sequence at the multiplier b\.Given the amplitude distributions of both {xn}and {yn-\}and particular values for the filter multipliers a0 and by, theamplitude distributions of the error sequences generated atthese multipliers can be determined. This information per-mits the evaluation of the variances of the two errorsequences. That is, the first of the two assumptions of thesimple model is no longer necessary.

This modification to the noise prediction method hasbeen tested experimentally for the first-order highpassfilter designs described earlier. Fig. 1 permits a typical com-parison of the simple uniform error amplitude distributionmodel with the model which does not rely upon such anassumption. Experiment shows3 that the variance of theequivalent input error sequence {en} can be predicted veryreliably. However, Fig. 1 indicates that the inclusion of thisfeature in the model yields no improvement over the simplemodel which calculates the variance of {en} from eqn. 4.

3.3 Signal to noise correlation

The fact that the use of eqn. 3 for roundoff-error varianceprediction is unreliable, even when the expected value of

10 ELECTRONIC CIRCUITS AND SYSTEMS, JANUARY 1978, Vol. 2,No.l

a | can be accurately determined as described above, indi-cates that the simplification of eqn. 1 which yields eqn. 3is often invalid. This simplification follows from theassumption that the error sequence {en} is purely random,that is its autocorrelation function is identically zero for allnon-zero lags. The sequence {en} is formed by the sample-by-sample summation of the two error sequences beinggenerated, {ean} at the multiplier a0 and {ebn} at themultiplier b\. Hence {en} can only be purely random ifboth {<?<,,„} and {eb n} are purely random.

The experimental results summarised above are obtainedfor the situation where the filter input sequence {xn} ispurely random, in which case there is no possible mech-anism whereby the sequence {ea>n} can be nonrandom.Therefore, if there are situations where {en} can be non-random, this must be caused by nonrandomness of thesequence {ebn}. Since the filter output sequence {yn} isnonrandom, any correlation between the signal sequence{yn-i} and the error sequence {ebiT1} at the multiplier b\would result in the nonrandomness of {ebn}. Experimentshows3 that a significant degree of signal/error correlationexists at the multiplier b\ in those situations where thereare large discrepancies between the levels of error variancepredicted and those actually measured.

3.4 An improved predictive noise model

At first sight the only way of taking such signal/error cor-relation into account in a predictive model might seem tobe by incorporating the evaluation of eqns. 1 and 2. Inaddition to the mathematical complexity of such a calcu-lation, it is not at all clear how the autocorrelation functionof {en} could be predicted. An alternative approach to theproblem is therefore adopted.

Consider the sample x{ of an input sequence {xn} to amultiplier a; the ideal output is yt but on rounding y'{ isproduced and a roundoff error e,- is created. That is,

y\ - (8)

The correlation coefficient r between xn and en is definedas

r = ( 9 )

For finite sequences of length N this may be written

N

r . 1 <

where

Nox ay Nax oy

N

' = I xtet

(10)

(ID

Zero correlation is only achieved when r is zero. Usingeqn. 8 it is possible to rewrite eqn. 11

N

' = Z Xiiyl-axi (12)

By postulating a change in the theoretical multipliervalue to a + 5a, the error sequence {en} is redefined. Hencethe correlation coefficient is altered, and, if 5a is chosencorrectly, can be set to zero. Replacing a by a + 5a in eqn.12, and equating the r.h.s. to zero, defines the required

modification. That is,

N

and therefore

N

5a = r'l Z x}1=1

= r = 5a £ x? = 0 (13)

(14)

Since, for any given practical multiplier a and signal word-length, there is a unique, known relationship between xt

and e,- for all possible values of xh r is dependent only onthe amplitude distribution of {xn}. Hence calculation of therequired theoretical coefficient, a + 5a, for zero signalerror correlation needs only the value of the practicalcoefficient a, the signal wordlength, and the amplitude dis-tribution of {*„ }. A method of predicting this informationin the case of first-order filters has been described above.It is thus possible to predict the required changes in thetheoretical values of the filter multipliers a0 and 6t suchthat no signal/error correlation should occur.

No change in the practical filter coefficients or actualsignal sequences is involved in the proposed modificationfor zero correlation. By altering the theoretical coefficientsthe ideal filter response is redefined. Because the practicalsignal sequences remain unaltered the roundoff-error se-quences become modified such that zero signal/errorcorrelation is achieved.

5-5

50

c 30

I 2-0

10

0000 02 04 0-6 0-8 085 09 0 95 10

pole radius

Fig. 2 Deviation of experimental error variance gain from modelpredictions 8G

o rounding• sign-magnitude truncation

Once the required theoretical filter coefficient modifi-cations have been predicted it is a simple matter to cal-culate the variances of the redefined roundoff-errorsequences at a0 and b\, sum them to give the variance of{en}, and, finally, compute the output error variance byevaluating the simplified eqn. 3 using the modifiedtheoretical value of the coefficient b\. It should be obviousthat this approach may be applied directly to the problemof error-variance prediction in filters employing the sign-magnitude-truncation roundoff process.7

Fig. 2 displays typical results of the performance of thedescribed method of error-variance prediction. It is a plotof the deviation 8G of the predicted filter error variancegain Gp from that actually measured, Ge, for the same filter

ELECTRONIC CIRCUITS AND SYSTEMS, JANUARY 1978, Vol. 2,No.l 11

realisations as considered in Fig. 1. Gp is found by evalu-ating

1 rt7

(15)1

B(z)B(z'l)z

using the modified theoretical value of b\. 8G is given by8G= |101og10(Gp/Ge)|dB. Comparison of Fig. 2 withFig. 1 reveals a most significant improvement in the per-formance of the predictive model. It also indicates that thismethod deals almost as successfully with the sign-magni-tude-truncation process as it does with rounding.

The remaining discrepancies between the predicted andactual performances are most probably caused by nonzerolagged correlations of the general form E(xnen-i), i ¥= 0, atthe filter multiplier b\. However, it is not possible to re-move such correlations at the same time as reducing thenonlagged correlation to zero. It is conceivable that an opti-mum prediction could be achieved by reducing, but not re-moving, some or all of the possible lagged and nonlaggedsignal/error correlations. However, the prediction of thelagged signal/error correlation coefficients would requireknowledge of the precise ordering of the filter signal se-quences, not merely their amplitude distributions. Hencethe proposed method may represent the best that can beachieved by this approach.

4 Higher-order filters

4.1 Signal/noise correlation

The problem of correlation between the signal and round-off-error sequences at a filter multiplier is not confined tothe first-order case. Experiments performed on second-order filters show that this correlation once again leads tothe nonrandomness of the roundoff-error sequences and tothe consequent inaccuracy of the noise-variance pre-dictions.3

As stated previously, the method employed for first-order filters to predict the amplitude distributions of thesignal sequences, and hence the signal/noise correlationcoefficients, cannot be applied to higher-order filters be-cause of the statistical interdependence of the signal se-quences at the various multiplier inputs. However, pre-liminary investigation on second-order filters3 indicatesthat, given a random input sequence with a uniform ampli-tude distribution over the allowed amplitude range, theamplitude distribution of the output sequence is generallyapproximately Gaussian.

The variance of the assumed Gaussian distribution canbe calculated from the variance of the filter input sequenceand the theoretical white-noise gain of the filter.

This would lead to a prediction of the variance of theerror sequence generated at each multiplier, and the re-quired modifications to the theoretical filter coefficientsto yield zero nonlagged correlation between the signal androundoff error sequences at each of the filter multipliers.

a first-order filter.) For example, a particular filter mayhave N roundoff error sources giving rise to the sequences{ex,n\ {^2,n}, • • • fev.n)- Consider that there is partial cor-relation between {e l n} and {e2>n}, that is, E(eltne2>n) ^ 0.If these two sequences are summed sample-by-sample toform {e'n} = {eln + e2 n], then, because of the correlationbetween {el>n} and {e2n} the variance of {e'n} is not equalto o\ + o\. Hence one of the basic assumptions of thepredictive noise model is invalid. Consider now that alagged correlation exists between {e2>n}and {e^^-t}, thatis, E(e2tn

e3,n-i) ^ 0- These two sequences may effectivelybecome summed sample by sample in the filter to yield asequence {en"}= {e2n + e3M.t}. Because of the laggedcorrelation between {e2 „} and {e3>n_j }the autocorrelationfunction of the composite sequence {e^} does not go tozero for all nonzero lags, that is, it is not random. Thus thesecond of the basic assumptions of the predictive noisemodel is seen to be violated.

Parker and Girard4 suggest that these correlations can beincorporated into a predictive model by forming the variouscovariance matrices, both lagged and non-lagged, of theroundoff error sources. Note that this method takes noaccount of the correlations which exist between signal anderror sequences at the filter multipliers.

4.3 Coefficients/icing implementations

The conclusion drawn from the foregoing discussion mustbe that for conventional filters of higher than first orderthe problem of roundoff noise prediction is somewhat in-tractable in practice. It is, therefore, very interesting tocompare the behaviour of the class of so-called 'coefficient-slicing' implementations6 as set forth, for example, byPeled and Uu.s In a conventional filter the individualsignal by coefficient multiplications are performed first,then these products are summed according to the filterdifference equation to produce the required output. Thecoefficient slicing implementation replaces the discretefilter multipliers with an r.o.m. which stores the sums ofcoefficient by signal partial products.

Consider the example of a second-order differenceequation

2

(16)

Assuming that all signal samples are represented in 2scomplement fixed-point arithmetic, having quantisationwidth A, and that the wordlength is (B + 1) bits, then, forexample, xn may be written

J 3 - 1

XnJ2J\A (17)

where xnj is the /th bit of xn and xn0 is the l.s.b. of xn.Hence eqn. 16 may be rewritten

4.2 Correlation between error sequences

Parker and Girard4 have recently demonstrated that cor-relations can arise between the roundoff-error sequencesgenerated at the various filter multipliers. This occurs whentwo or more of the filter multipliers have the same inputsequence, either simultaneously or after a delay of somenumber of samples. (Note that this situation cannot arise in

yn - -2BA{aoxnt QiXn-I.B

a2xn.2j

(18)

12 ELECTRONIC CIRCUITS AND SYSTEMS, JANUARY 1978, Vol. 2, No.l

Defining a function \p of five binary arguments rx . .. r5 as

^{ri,r2,r3ir4trs} = A2B{fl0>-i + «i '2 +^2^3

-b1r^-b2r5} (19)

where n . . . r5 are each either 1 or 0, eqn. 18 may bewritten

B - l

;=o

As the function \p arising from the difference eqn. 16 hasfive parameters each of which may have one of two values,1// has 2s = 32 distinct values defined by eqn. 19. Each ofthese 32 values is a sum of the possible partial products ofthe five coefficient-by-signal multiplications. Eqn. 20 indi-cates that the filter output yn may be formed by anadditive combination of (B + 1) values of \p, each scaled bythe appropriate power of 2. Fig. 3 shows a block diagramof this realisation, in which the r.o.m. permanently storesthe 2s distinct values of i//.

The roundoff-error analysis presented by Peled and Liu5

appears to be inaccurate, and the following correction istherefore proposed. As the range of values which can bestored in the r.o.m. is bounded by - 2 B A and (2B - 1)A,it may be necessary to scale down the stored values of \p bya factor of 2q. Moreover, the theoretical \p values cannotgenerally be precisely represented in the available _signalwordlength. Hence the values stored in the r.o.m. 1// willcontain roundoff errors as defined by

${ri,r2,r3,r4,rs} = 2~q\p{r1 ,r2,r3,u,r5} + < ;(21)

where e'^j is the roundoff error, and q = 0 indicates that noscaling is necessary.

Let vn be the actual computed filter output after anynecessary rescaling. Then, unlike Peled and Liu, neglectingquantisation of the input sequence {xn}

vn = 2qB-l

j=o

(22)

where e^ is the roundoff error generated in the accumu-lation of the (possibly scaled down) filter output. Anynecessary scaling up of this output has to be carried outafter the creation of this error. Using eqns. 20,21 and 22, itis possible to express the roundoff error at the filter outputen thus:

cn ~ un ynB-l

+)-B ii

J - en,B; = 0

Comparison of this eqation with that given by Peled andLiu5 shows a significant difference in the term 2Q account-ing for the rescaling of the filter output.

The error sequences {e'nj} and {e^"}are considered to berandom and to have a variance of A2/12, when rounding isbeing employed. Hence combining the roundoff errorsources yields a random sequence {en} which is taken tohave variance

(24)

Performing the summation yields

1 3 I 112

where the approximation is good for B > « 3.

(25)

This variance can be substituted into eqn. 3 in order to givean estimate of the variance of the error sequence at thefilter output.

Table 1 presents the results of noise variance measure-ments taken on a typical second-order bandpass filter. Itpermits both a comparison of a coefficient slicing imple-mentation with a conventional direct realisation, and alsoan assessment of the accuracy of the predictive noise model

location contents

Fig. 3 Coefficient-slicingfilter implementation

ELECTRONIC CIRCUITS AND SYSTEMS, JANUAR Y1978, Vol. 2, No.l 13

in the two realisations. The variance values are absolute andthe quantisation width A is unity. All errors are created bythe rounding process. Results are presented for 6-bit, 7-bitand 8-bit wordlengths. Table 1 shows that the coefficientslicing implementation yields significantly less noise thanthe conventional form.

It can also be seen that the predictive noise model doesnot work particularly reliably for the coefficient slicingform because the model assumes that each of the roundofferror sequences {e'nj} has a variance of A2/12. However,the variances of these sequences depend both on therelationship between the set of theoretical i// values andthose actually stored, and also on the relative probabilityof any particular i// value being addressed.

Table 1: Error variance measurements: second-order bandpass filter

Realisation

Direct-form:experimental

Noise model

Coefficient slicingform: experimental

Noise model

Reduced-noisecoefficient slicingform: experimentalNoise model

Wordlength, bits6

2-63±0 042-2467

0-85+0 021 -0485

0-357±

00050-3592

Table 2: Transfer function poles and zerosfilter

Realisation

Ideal

Zeros

0-07868020

yO-69727700

7

2-44±0-042-2467

1-01+0 021 0485

0-481±0 0060-4717

8

2-33±0-042-2467

1 0 5±0 021 0485

0-442±0 0060-4372

: second-order bandpass

Poles

0-10009766

y'0-94411209

Reduced noisecoefficient slicingform

6 bits 008333333±

yO-70217915

009375±

yO-93070454

7 bits 008333333+

y'0-70217915

0-1015625±

jO -94653846

8 bits 0-08+

y 0-68818602

0-1015625+

yO-94240255

4.4 A coefficient slicing form yielding reduced noise

If the theoretical i// values can be chosen so that they canbe stored in the r.ojn. without error, then all the {e^j} se-quences will have zero variance and the filter will yieldsignificantly less roundoff noise. For an r.ojn. with a word-length of (B + 1) bits, this can be achieved if filter co-etiicients are selected which can be precisely represented ina maximum of (B — q) bits, excluding any sign bit. Co-efficient quantisation, and the consequent approximationof the desired filter response are, of course, common in thedesign of practical conventional filters. However, coefficientquantisation in coefficient-slicing realisations can have a

desirable effect on roundoff noise. An appropriate noisemodel can be formed by setting all the e'^j terms to zero ineqn. 23, which results in the following simplification ofeqn.25:

(26)

Noise variance results for the 'reduced-noise' coefficientslicing implementations of the ideal second-order bandpassfilter previously considered are presented in Table 1, whichindicates the reduction in roundoff-noise level achieved bythe introduction of a sufficient degree of coefficientquantisation. Extremely good agreement between the pre-dicted noise levels and those measured is apparent for these'reduced-noise' implementations. Table 2 allows an assess-ment of the deviation from the ideal filter transfer functioncaused by the coefficient quantisation, as indicated by poleand zero positions in the Z-plane. Even with a 6-bit word-length it is possible to find a reasonable approximation tothe ideal transfer function. Once again these results aretypical of the various filters tested.3

It should be clear from the results presented that this'reduced-noise' coefficient slicing form not only yields verylow roundoff noise, but also behaves in exactly the wayassumed by the simple predictive noise model. The onlysource of roundoff noise is at the final accumulation of thefilter output. Clearly the roundoff sequence at this pointdoes in practice have a variance extremely close to A2/12,and is also purely random. Indeed there is no mechanismwhereby this sequence could become correlated either withitself or with any of the signal sequences. This filter form isthe only one whose output noise variance can be predictedaccurately with a simple noise model, which, together withthe low noise levels obtained, justifies a strong recommend-ation for its use.

5 Conclusion

The experimental results3 which have been summarisedabove indicate that, when very short signal wordlengths inthe range of 4 to 8 bits are in use, the standard statisticalmodel1 cannot with any certainty of accuracy be employedto predict roundoff-noise levels. For first-order filters amethod has been proposed whereby the correlation ofsignal and error sequences at the filter multipliers may bereduced by a redefinition of the ideal filter coefficients.This yields a significant improvement in the accuracy of thepredictive noise model. For conventional filters of higherthan first order this method does not apply and the prob-lem of precise roundoff-noise prediction appears to beintractable. However, use of a coefficient slicing imple-mentation, employing sufficiently quantised filter co-efficients, yields a filter with properties of very low round-off noise, and excellent agreement between the predictionsof a simple model and measured noise levels. This form isstrongly recommended.

References

1 KNOWLES, J.B., and EDWARDS, R.: 'Effect of a finite-word-length computer in a sampled-data feedback system', Proc. IEE,1965,112, (6), pp. 1197-1207

14 ELECTRONIC CIRCUITS AND SYSTEMS, JANUARY 1978, Vol. 2,No.l

2 LIU, B.: 'Effect of finite word length on the accuracy of digitalfilters - a review', IEEE Trans., 1971, CT-18, pp. 670-677

3 McWILLIAM, A.J.: 'Roundoff error sequences in fixed-pointarithmetic digital processors'. Ph.D Thesis, University ofDurham, 1976

4 PARKER, S.R., and GIRARD, P.E.: 'Correlated noise due toroundoff in fixed point digital filters', IEEE Trans., 1976,CAS-23, pp. 204-211

5 PELED, A., and LIU, B.: 'A new hardware realisation of digital

filters', ibid. 1974, ASSP-22, pp. 456-4636 PELED, A, LIU, B., and STEIGLITZ, K.: 'A note on implement-

ation of digital filters', ibid., ASSP-23, pp. 387-3897 LIU, B., and VAN VALKENBURG, M.E.: 'On roundoff error of

fixed-point digital filters using sign-magnitude truncation', ibid.1972, CT-19, pp. 536-537

8 ASTROM, K.J., JURY, E.I., and AGNIEL, R.G.: 'A numericalmethod for the evaluation of complex integrals', ibid. 1970,AC-13, pp. 468-471

1978 IEEE International Symposium onCircuits and Systems

The 1978 IEEE International Symposium on Circuits and Systems will be held on 17th—19th May, 1978, at theRoosevelt Hotel in New York City, U.S.A., (one week preceding Electro 78). This symposium, sponsored by theIEEE Circuits and Systems Society, is the eleventh annual international conference devoted to all aspects oftheory, design and applications of circuits and systems.

Areas of interestAn expanded technical program is planned, with emphasis on the links between theory and practice and on newlyemerging technologies. Papers have been solicited in, but not restricted to, the following areas:

New concepts and novel approaches for the analysis and design of circuits and systems, such as large-scale circuitsand systems, digital and analogue solid-state circuits including integrated circuits, passive and active filters,distributed and microwave networks, nonlinear and time-varying circuits and systems, graph theory and appli-cations, analogue and digital signal-processing systems, and multidimensional filters.

Computer-aided techniques for the analysis, design, layout, testing and manufacturing of circuits and systems,including new algorithms and user-oriented languages.

New devices and circuits including modelling, analysis, design, and applications in signal processing, communi-cations, instrumentation, and control.

Symposium CommitteeGeneral Chairman: H.E. Meadows, Dept. of EECS, Columbia University, New York, NY 10027, 212-280-3112,

3104

Technical Programme: Bede Liu, Dept. of EECS, Princeton University, Princeton, NJ 08540, 609-452-4628, 4630

Special Programmes: J.F. Kaiser, Bell Laboratories, Murray Hill, NJ 07 07974, 201-582-2058

Publicity: Kenneth R. Laker, Bell Laboratories, Holmdel, NJ 07733, 212-979-9034

Finance: T.G. Marshall, Dept. of EE, Rutgers University, Piscataway, NJ 08854, 201-932-2577

Local Arrangements: S.J. Oh, Dept. of EE, City College of New York, New York, NY 10031, 212-690-6685

Publications: Hing So, Bell Laboratories, Whippany, NJ 07981,201-386-3211

Exhibits: S.J. Vahaviolos, Western Electric Co., Princeton, NJ 08540, 609-639-2426

15