Design and application of binary sequences

with optimised lowpass properties

Gordon B. Lockhart

Indexing term: Binary sequences

Abstract: It is shown that binary sequences having prescribed spectral properties may be designed by anoptimisation procedure based on a delta-modulation algorithm. The method is specifically applied to thedesign of lowpass sequences of lengths 128,256 and 512 bits which, after a simple integration process, exhibitspectral properties related to type-2 nonrecursive-digital-filter impulse responses. Highpass sequences can beobtained by a simple transformation. The limitations of the procedure and the implications of quantisationerror are discussed. A theoretical prediction is obtained for the smallest sequence length necessary to achievea given spectral bandwidth. Applications include digital filters, which can be simply implemented by digitalhardware performing primitive arithmetic operations.

1 Introduction

The need for digital sequences having controlled spectralproperties arises in a wide variety of applications1"4

including digital filtering and the generation of test anddata signals. Discrete-time theory may be applied to thedesign of a wide range of discrete sequences with amplitudevalues defined on a continuous scale and prescribed spectralproperties, and this approach leads naturally to imple-mentations which use a p.cm. format for number repre-sentation and conventional binary arithmetic processing.This is a satisfactory approach provided hardware resourcespermit multibit representations both of sample values andof arithmetic operations which are sufficiently accurate toensure that quantisation effects are reduced to tolerablelevels. If a reduction in hardware complexity is paramount,however, favouring simple serial schemes and primitivearithmetic, then there is some advantage in basing design onbinary rather than multibit sequences since requirementsfor storage and manipulation can be considerably simplified.Unfortunately, direct synthesis techniques for binarysequences of a power comparable to those available formultibit sequences do not exist and therefore an approachbased on an optimisation procedure is adopted here.

The design procedure which will be presented employsa delta-modulation algorithm to generate finite-lengthbinary sequences which, after a simple integration process,exhibit lowpass spectral properties with exact linear phase.It will be shown that such sequences are easily generatedand can be usefully applied in simple digital filters.

2 Construction of binary sequences with specifiedspectral properties

If xn = ± S where S is constant for n = 1, 2 , . . . , N thenthe spectrum of such a binary sequence regarded as adiscrete-time signal1 is given by

N

X(f) = Z xn exp [ -

Paper T239C, first received 18th April and in revised form 21stJune 1978Dr. Lockhart is with the Department of Electrical & ElectronicEngineering, University of Leeds, Leeds LS2 9JT, England

where to — 2nf and T is the sampling interval. It is difficultto choose values of xn to suppress higher-frequency com-ponents but if the sequence is integrated then a wider rangeof unquantised sequences may be approximated.5 In thiscase the spectrum becomes

s fY(f) = I ynexp\-i(n-l)uT

n = l Lwhere

yn = ZThe sequence yn is constrained to rise or fall by S units persample. For example, integrating the binary sequence xn inFig. 1 results in the sequence yn as shown. Sequence yn canbe interpreted as an approximation to the unquantisedsequence zn in Fig. 1, which belongs to a class of non-recursive-filter impulse responses with attractive lowpassproperties.6

In general, a sequence yn can be derived as an approxi-mation to any sequence zn of TV samples by means of thefollowing delta-modulation algorithm. Given a value S andan initial value y0 = 0 then for n = 1, 2, 3 , . . . , TV

o

-S,zn -

and

yn = xn +yn-i

The sequence xn can be interpreted as the binary sequencegenerated by passing sequence zn through a linear deltamodulator7 with a step of S units. Sequence yn corre-sponds to samples of the reconstructed input signalobtained by integrating xn. Provided that the slope ofsequence zn is constrained so that \zn — zn_x \ <S theerror sequence (yn — zn) will correspond to granular noise7

introducing a random, but essentially frequency inde-pendent, perturbation to the spectrum of sequence yn. If,

COMPUTERS AND DIGITAL TECHNIQUES, OCTOBER 1978, Vol. 1, No. 4 197

0140-1335/78/040197 + 08 $01-50/0

however, \zn—zn^l\>S then slope-overload distortion7

can occur leading to gross frequency-dependent errors.This latter distortion places a major restriction onspectra realisable by sequence yn. For example, let zn =A cos ncjx T for n = 1, 2 , . . . , TV where T is the samplinginterval and a^ defines the required frequency of a spectralpeak. If slope overload is to be avoided in the approxi-mation process then A should be chosen so that

dn{A sinnco1T) = < S

Since the maximum slope occurs in the vicinity of zn = 0 itis sufficient to ensure that

< S

0)

But A/S is the number of quantisation intervals available torepresent zn in the range from zero to peak value and,therefore, expr. 1 provides a measure of precision obtainablein constructing a response at any frequency Wi. Taking thisas a guide to the general case, where sequence zn is asuperposition of sampled sinusoids, it becomes apparentthat the accuracy of approximating sequences will fall withincreasing bandwidth. It follows that a direct applicationof the algorithm favours the construction of lowpasssequences.

3-50325

300

275

2 50

225

2 00•g 175

| 1 50f i 2 5°100

0750-50

0250 00

-025

-050-075

> = (s.-s.s.-s.s.-s, s, -s, s,-s, s, -s,s, -s, s, s,-s,'-s, s,-s, s, -s,s( -s,-s,s, -s, s, -s,-s,s,-s,

- -s, s . -s .s . -s^-s .s .s , -s.s.s.s, -s,s,s, .s, s, s,s, s,s,s s, s.s, -s, s,s,s, -s , s)

axis ofsymmetry

0 5 10 15 20 25 30 35 40 45 50 55 60 65sample number

Fig. 1 Representation of sequence zn by sequences yn and xn

co T

1 2 3 4 5 6frequency sample number

Fig. 2 Specification oftype-2 spectrum for B — 2

3 Lowpass designs

Type-2 finite-impulse-response (f.i.r.) lowpass nonrecursivefilter designs, introduced by Rabiner,6 provide a usefulsource of unquantised lowpass sequences. Such sequenceshave the advantage of symmetry, and hence a linear phaseresponse, and also converge to zero at their extremities,eliminating difficulties of slope overload due to end dis-continuities. The specification of a type-2 spectrum withone transition coefficient is illustrated in Fig. 2. Passbandand stopband regions are defined at fixed points by uni-formly distributed samples of unity or zero at intervalsof 11 NT Hz where N is the length of the sequence. It shouldbe noted that the first point is situated half a samplinginterval from zero frequency. The bandwidth B is definedas the number of sampling points in the passband region.The value of the transition coefficient T is chosen in anoptimisation procedure to minimise the maximumsidelobe appearing between frequency samples of zerovalue in the stopband region. Values of T for a wide varietyof sequence lengths and bandwidths are available intabulated form.6

For N even the sequence is given by

TV(2)

where

7'h —

1, k < B + 1

T, k = B + 1

Because zn = 0 when n — N the sequence is effectively(TV— 1) samples long, and can be interpreted as an evensequence with the axis of symmetry at n =TV/2. In orderto preserve symmetry the approximating sequence yn mustreturn to zero at n = TV and, therefore, the binary sequencexn will be TV samples long. These features are apparent inFig. 1 for TV = 128 and B — 3. In this case the maximumdifference between consecutive samples of zn is D — 0-235and, with S = 0-25 to avoid slope overload, application ofthe delta-modulation algorithm results in a 128-samplebinary sequence xn and a 127-sample approximatingsequence yn as illustrated. The magnitude spectrum ofsequence zn which is plotted in Fig. 3 exhibits a maximumstopband sidelobe between frequency samples 7 and 8. Itis evident from the spectrum of sequence yn in Fig. 4 thatthe process of quantisation has caused some deviation fromthe sampling point values, and increased the maximumstopband sidelobe to a maximum of approximately— 22-5 dB between frequency samples 63 and 64. Ingeneral, if a different value of S is used in the quantisationprocess, then a different distribution of stopband sidelobeswill result, depending on the exact path taken by sequenceyn in tracking zn. The variation of the value of themaximum stopband sidelobe M with the normalised stepR — S/D, where D is the maximum value of \zn — zn_t I,is plotted in Fig. 5 for the approximation of zn illustratedin Fig. 1. Since the sequence yn is inherently limited to afinite number of possible states, the variation is irregularand particularly difficult to predict in the vicinity of slopeoverload (i.e. R = 1). It will be shown that a 1-dimensional

198 COMPUTERS AND DIGITAL TECHNIQUES, OCTOBER 1978, Vol. 1, No. 4

optimisation procedure can be employed to find the valueof/? corresponding to the global minimum of M.

4 Optimisation procedure

Optimisation of the normalised step height R is illustratedin Fig. 6. First, the sequence zn is constructed using tabu-lated values or optimisation6 to find the transition coef-ficient T. Sequence zn is then scanned to find the maximumdifference D = max \zn — zn_x |, where « = 1,2, . . . ,7Vand z0 —0, and R is set to an initial trial value R = R i. Thefirst value of S is fixed at S = RiD and the delta-modulationalgorithm is applied to samples 1 — 7V/2 to generate thefirst 7V/2 values of sequence yn. If R < 1 then slope-overload distortion can result in an asymmetrical sequenceyn if the delta modulation algorithm is applied to the entireTV samples of sequence zn. Symmetry is therefore imposedon sequence yn at this stage by setting y^-n =yn-Typically the values of the type-2 sequence zn fall below Sfor a number of samples at the extremities, causing sequenceyn to 'idle' (i.e. 0, S, 0, S, 0, S as apparent in Fig. 1). Whenzn < S/2 this is likely to lead to an approximation which isinferior to zero-valued samples. Provision is therefore madefor truncating sequences xn and yn by a fixed number ofsamples at their extremities.

2 3 U 5 6 7 8 9 10 11 12frequency sample number

Fig. 3 Type 2 spectrum (N = 128,B = 3)

Points on the magnitude spectrum | Y(f) | are computedby taking the discrete Fourier transform of sequence yn

using a fast-Fourier-transform (f.f.t.) algorithm. First, zerosare added to sequence yn to increase its total length to atleast 47V samples. This ensures6 that at least 3 interpolatedpoints are available for sidelobe definition between theoriginal stopband fixed points specified on the spectrum ofsequence zn in Fig. 2. Values are converted to a dB scalewith 0 dB set to the average passband value. The value M,and the position, of the maximum stopband sidelobe arethen found by scanning the stopband region up to afrequency U/2T, where U defines the range of optimisedstopband required as a fraction of the 'foldover' frequency

1/2T. At this stage other parameters, such as maximumpassband ripple, can be determined as necessary. Theprocess is repeated to generate a cycle of values of M forRn=Rn_1+ AR wi th n=2,3,. . . , 6 , and Mn theminimum value of M, is found aXR = Rmin. Subsequently,AR is halved, a new value for Ri is calculated so thatRmin falls centrally in the reduced interval Ri — R6, andthe entire cycle is repeated with the 6 new values of/?. Asthe process continues, /?m, n and Mmin are updated only ifa smaller Mmin is found. In this way successive ranges ofRare localised in the region where the global minimum of Mis most likely to occur. The process is terminated when 6identical values of M, and hence identical sequences yn, arefound in the same cycle.

5

0

-5

-10

-15

. -20

S> - 3 0o£ - 3 5

-40

-45

-50 1 2 3 5 6 7 8 9 10 11 12 13 62 63frequency sample number

Fig. 4 Spectrum of sequence yn (N = 128, B = 3, S = 0-25)

-21

-22

-23

-24

-25

-26

-O-27

-29

-30

-31

-32

; yP11080 085 0 90 0 95

||

1_

I • i 1

100 105 1-10R

1-15 120

Fig. 5 Variation of maximum sidelobe value (M) against nor-malised step R(N=128,B = 3)

A typical course of convergence is illustrated in Table 1for TV = 138, B = 3 and U= 0-5. In this case sequences*,,and zn are truncated by 5 samples at their extremities sothat the effective sequence length is 128. Initial values ofRt and AR are set to 0-8 and 0-07 respectively and con-vergence is obtained in 10 cycles. Values of Mmin andRmin are tabulated, in Table 1, for each cycle withassociated values of passband ripple. Mo, the maximumsidelobe value in the nonoptimised stopband from (U/2T)to ( l /2r)Hz, is also given. Although the value of Mmin

appearing in cycle 5 is the global minimax,7l/m,-n reverts toa significantly higher value in cycle 6. This illustrates thedifficulty of applying a less cautious optimisation procedureto a rapidly fluctuating function with numerous localminima. The spectrum of the optimised sequence yn

COMPUTERS AND DIGITAL TECHNIQUES, OCTOBER 1978, Vol. 1, No. 4 199

Table 1: Course of convergence for optimisation of a 128-point sequence

Cycle °min Mn Passbandripple

= 3, t/ = 0-5

dB dB dB

12345678910

0-8000000-9225000-9487500-9706250-9816520-9935930-9963280-9982420-9986520-998857

1-150001 -097501 036251-014371 003441 -004531-001791 000981 000020-99954

1-010000-9925000-9662500-9881250-9990620-9957810-9996090-9987890-998925identicalvalues

-27-69- 28-92- 28-30- 28-92-30-24- 28-92-30-24- 30-24-30-24-30-24

-25-63- 24-28- 24-99- 24-28-2505- 24-28- 2505- 2 5 0 5- 25-05- 2 5 0 5

0-81020-75320-67930-69620-80220-69620-80220-80220-80220-8022

construct z.

find D=max|zn-zn_, |

set R = R1

SrRDgenerate sequence yn

via d m . algorithm

generate interpolatedmagnitude spectrumvia f.f.t. algorithm

find maximum side lobe

M and other parameters

as required

repeatR—R*A R

exit after 6repetitions

update M . and R .mm min

AR—AR/2

compute new R1,R«-R1

stop if 6identical M

Fig. 6 Flowchart for optimisation procedure

200

1 2 5 6 7 8 9 1011 1213 14 5960616263frequency sample number

Fig. 7 Optimised lowpass spectrum (N — 128, B = 3, U — 0-5)

exhibits a maximum stopband sidelobe in the vicinity offrequency sample 7 at — 30-24 dB, as illustrated in Fig. 7.Outside the optimised region the maximum sidelobe is at— 25-05 dB in the vicinity of frequency sample 59. Theinitial values of Rx and AR above were used in mostoptimisations and were found to provide a generally satis-factory compromise between a fast rate of convergence anda sufficiently detailed search through various values ofR. Ina very small number of cases, where the minimax wassuspect (e.g. if a larger minimax was obtained for a smallervalue of U), initialising the optimisation with more care-fully chosen values of R t and AT? was usually sufficient tocorrect convergence.

5 Results and lowpass sequence

A selection of optimised designs for lowpass sequences ispresented in Table 2. All sequences were derived fromtype-2 specifications and, although they are of length 128,256 or 512 bits, optimisation was based on longer sequencesto achieve these standard lengths after truncation, thusavoiding inferior results due to lengthy idling sequencesat the extremities of sequence yn. The use of lengthiersequences in this way causes a downward shift in cutofffrequency relative to the nominal cutoff for the standardsequence. This can be ignored for low values of B duringoptimisation, but an adjustment was made for higher valueswhen truncation is severe and the shift can produce mis-leading values of ripple and cutoff. In general, adjustmentwas made for B > 3 and this was achieved by reducing thenominal cutoff by one point on the interpolated spectrum.Since there are normally 4 points per frequency sample onthe interpolated spectrum this corresponds to a reductionof£byO-25.

COMPUTERS AND DIGITAL TECHNIQUES, OCTOBER 1978, Vol. 1, No. 4

Table 2. Selection of optimised lowpass sequences for N — 128, 256 and 512

SequencelengthN

128sequence:

128sequence:

128sequence:

256sequence:

256sequence:

256sequence:

256sequence:256

sequence:256sequence:

256sequence:

256sequence:

512sequence:

512sequence:

512sequence:

512sequence:

512sequence:

512sequence:

512sequence:

512sequence:

512sequence:

512Sequence:

512sequence:

Bandwidth OptimisedB stopband

U

3 1/2AB54A9253DF3D2. . .

3 1/8A2B2A92956DFDFDB. . .

3-755 M 2

3AD5

3AD5

3ADS

3ABB'

3-75S24A

3-75524/4

1/8D29296DF27>4. . .

10M55292494,456ff3F7DM. . .

1/22As5292494A5AB3FnB26. . .

1/42/455292494y45>*fl3F7fl06. . .

1/16!6^2522929294>455D2F7F3DFff25. . .

1/22ff5M056>4252924952£7flF4fi0/*. . .

1/8205Mfl56/4252924>!l52070F47£6. . .

3-75 1/165*AB5S29252A2B3FDF27B5. . .

4-75A653

3AB57

3AB53

3AB52

1/82056/*05325252ff6£F20Ftf6. . .

1!6/42fl5M752294>4492924/\4/*522056S6£2

1/2,4 3053>4 5524/*94/44/44929294/49526,40£7

1/86>42 B5 M 7 52 294/4 492924/44^ 554556fl6£3

3 1/16A2D526A2B5*6A2532A54A4A49494A52A'iB6B

3AB5-

4-75A2Dl

4-75A3Dl

4-75A3B5

4-75A2Dl

6-75

Optimisedminimum

dB

- 30-24

-33-63

- 3 0 0 7

- 34-63

-37-49

- 38-60

-43-67

-31-47

-32-73

-44-47

-35-51

-41-86FBFDF9EFDE2DA2. . .

-43-103BF7F2BFSBFDE3D6A. .

- 4605FBFDF9DFD3B6A. . .

Non-optimisedminimum" " o

dB

- 2 5 0 5

— 23-76

- 20-45

—

- 33-42

-33-23

-29-18

- 2 7 09

-27-47

- 24-69

- 2 2 0 0

—

-38-79

-41-23

-47-26 -38-88272BEF27F3BF2DFDE2DB5A. . .

1/32 -49-13'6/*20536/*39524,494,44/44949494/4296>4ff5tf73££F2ffF4£F20F7tf2i

1

1/2

1/8

1/32

1/2

6-75 1/16as for U = 112 above

- 32-83

-35-62

-36-17

-45-15

-32-62yB7BF6EDA. . .

— 3500

- 37-625B5. . .

—

- 30-65

-28-16

- 29-98

- 25-67

- 25-67

Passbandripple

dB

0-8022

1-002

0-8520

0-6337

0-6202

0-6179

0-6143

0-5815

0-5655

0-3769

0-4337

0-5961

0-5900

0-5937

0-5705

0-5832

0-5187

0-5229

0-5067

0-5707

0-5674

0-5674

No designs are included for which Mmin is greater than— 30 dB. This eliminates a number of designs for A^= 128where the approximation error is greatest. Also, opti-misations for the same values of B and N, generating thesame sequence yn and value of M as a higher value of U,are omitted as redundant. (This tends to occur when theposition of the maximum sidelobe for U=\ lies at arelatively low frequency so that optimisation at lowervalues of U depends on finding new sidelobe configurationswhich move this sidelobe out of range of the optimisedstopband). The value of maximum stopband sidelobe inboth stopband regions is given for every sequence. Forvalues of U< 1 it is apparent that the average of Mmin andMo is approximately constant and, therefore, for small U,the advantage of increased attentuation in the vicinity of

cutoff must be balanced against a reduction in the high-frequency region.

Since sequence xn has odd symmetry only the firstNI2 bit values are specified in Table 2. For economy ofnotation a hexadecimal code has been adopted to representthe sequence in groups of 4 bits, i.e.:

1 = (-s,-S,-S, +S)

2 = (-S,-S, +S,-S)

F = (+ S, + S, + S, + S)

Repeating groups are indicated by an index, e.g.

B2 = (+ S, -S , + S, + S, + S, -S, + S, + S)

COMPUTERS AND DIGITAL TECHNIQUES, OCTOBER 1978, Vol. 1, No. 4 201

6 Theoretical considerations

As B increases it becomes necessary to increase the step Sto avoid slope overload. This will increase the quantisationerror as indicated by expr. 1, but increasing B will also tendto lift the first quantisation level above zero significantlyabove values of sequence zn at its extremities. If thesevalues fall below 5/2 it becomes reasonable to set them tozero by truncating the sequence. Since the extent of low-level 'tails' of sequence zn increases with B, it follows that,for a given sequence length, a limiting value of B will existabove which no advantage is likely to be gained withouttruncation and a consequent reduction of N.

Since all linear-phase lowpass sequences yn approximatea sampled sinjc/bt sequence this can be conveniently sub-stituted for sequence zn in order to illustrate the abovelimitation. Let zn = sin mrT/n-nT where T is the samplinginterval and « = . . .— 1,0, 1 . . . . The spectrum of thisideal lowpass sequence has a cutoff at 0-5 Hz in the range0 to l/2rHz.

Let Tt =(/

i = 1,2,3,

denote the maximum value of the rth sidelobe. If Tw is themaximum value of the smallest sidelobe i = W whichsatisfies Tt > S/2 then, since . . .Ti.l>Ti>Ti+l. .. , only2W sidelobes can be usefully represented in a truncatedsequence yn with step S and lengthN = 2(W + 1)1 T. Now,the interval between frequency samples for the type-2specification of Fig. 2 is (l//V7)Hz and, if the cutofffrequency of the type-2 spectrum is defined at the firstzero-valued sample, then this occurs at a frequency of(B+1'5)/NT=(B+1-S)/2(W+1), and equating to thecutoff of the ideal sequence

(ft+1-5) _ J_2(W+ 1)~ 2

Therefore

B = W-0-5

If 5" is set to the threshold of slope overload so that S —max \zn —zn-i I then the condition Tt>S/2 implies that

sin max simrnT smn(n— l)T

2-nnT 2ir(n-l)T

This inequality can be used to find W and B for any given Tand, hence, the maximum useful sequence length N =2{W + l)/7! The results, plotted in Fig. 8, were found toprovide a useful prediction of maximum B for a givensequence length. Computed results for maximum B withthe additional constraint of M < — 30 dB are also plottedand fall approximately at 0-75 of the predicted values.

Prediction of the value of M poses difficult theoreticalproblems. If TV is large, so that quantisation error isnegligible, then the value of M must identify with that forsequence zn, but if N is small the spectrum of thequantisation error will predominate in the stopband region.Between these extremes M will depend on a combination

of both effects. Although the spectral properties ofdelta-modulation quantisation noise can be characterisedstatistically for an ensemble of error sequences, given theduration and size of the step7 accurate prediction ofmaximum spectral magnitudes cannot be achieved by thismethod.

A guide to the expected change in M as a result ofdoubling TV for constant B can be obtained by consideringthe unquantised type-2 sequence defined in Eqn. 2. DoublingN doubles the sampling rate but leaves the form of sequencezn unaltered. As a consequence max \zn—zn-x\ will beapproximately halved so that the step S can be halved. Itcan then be argued that, on average, quantisation errorswill be halved leading to a similar reduction in the expectedvalue of the maximum stopband sidelobe. This points to a6dB change in M when N is doubled and, although inpractice (vV<530) the change varied from 0-5 to 8-5 dB,the average was 6-43 dB. Due to the limitations of theminicomputer system employed optimisation was notattempted for TV > 530, but the change can be expectedto tend to zero as N becomes sufficiently large to cause Mto approach the minimax of the source sequence zn.

Although the value of the transition coefficient T isoptimum with respect to sequence zn it is not necessarilyoptimum for the quantised sequence yn. This suggests a2-dimensional optimisation process in which both 5" and Tare varied to find the global Mmin. However, experiments inwhich S was optimised for a range for values of T, centredabout the optimum T for sequence zn, suggest that anyresulting decrease of Mmin greater than 0-5 dB is unlikely.

12 r

predicted values

computed values (M<-30dB)

50 100 150 200 250 300 350 400 450 500 550 600N

Fig. 8 Prediction of bandwidth B for a given sequence length N

7 Applications

Lowpass sequences designed by the methods describedabove can be simply generated by the shift register andcounter arrangements of Fig. 9. (In this case, if the binarysequence xn assumes the values ± 1 rather than ± S, theoutput spectrum becomes Y(f)/S, or a line spectrum withthis envelope, if the sequence is recirculated continuously).If analogue output is required then the counter output canbe applied to a digital-analogue convertor (d.a.c.) andlowpass filter (l.p.f.). If required, the optimisation pro-cedure can be modified to compensate for sample-and-holddistortion and the frequency response of the lowpass filter.

202 COMPUTERS AND DIGITAL TECHNIQUES, OCTOBER 1978, Vol. 1, No. 4

For example, if the d.a.c. outputs flat-topped pulses ofduration Ts equal to the sampling period, and l.p.f. is asimple RC section, the output spectrum becomes

= L(f) S(f) Y(f)

sin nfTnfT

where

S(f) =

and

L{f) = 1/V1 + iflfcffc, the cutoff frequency of l.p.f., can be chosen in thevicinity of B to obtain a significant improvement in stop-band performance. For example, for B = 2-75, TV = 64 andfc chosen midway between frequency samples 3 and 4,optimisation with / / ( / ) replacing Y(f) results in thesequence A292AF2D . . . giving M = — 33-45 dB with apassband ripple of 0-646 dB.

Other modified spectral weightings can be toleratedduring optimisation provided they do not impair stopbandperformance or give rise to unacceptable passband ripples.For example, if the shift register in Fig. 9 is fed to a d.a.c.followed by an analogue integrator then the resultingspectrum will be modified only by sample-and-hold dis-tortion and performance will be comparable to thatindicated in Table 2. Optimisation which compensates forthis distortion and any deviation of the integrator from theideal is quite feasible. (A particularly simple realisationresults if RC integration is employed).

Fig. 9 Generation of digital or analogue lowpass sequences

Ts delaysinput

accumulator(Ts delay)

In digital-filtering applications optimised lowpass-filterresponses can be obtained by discrete convolution of adigital input signal with sequence yn serving as the impulseresponse. This can be achieved by convolving a multibitinput sequence un with the binary sequence xn beforeaccumulation, as in Fig. 10, so that coefficient multipliersbecome ± 1. A serial but much-simplified version of Fig. 10can be obtained by replacing the bank of multipliers by asingle adder/subtractor which successively adds or subtractsun, un _ ! , . . . , un_N to or from the contents of theaccumulator according to the value of JCI , x2,. . . , xN heldin a static register. After N additions/subtractions persampling period the output gn assumes the correct value, anew value of un is accepted by the input shift register, andthe process repeats. This arrangement leads to a straight-forward software description where the adder/subtractor isviewed as a primitive arithmetic processor repetitivelyprogrammed by sequence jcn. It follows that a micro-processor implementation would involve only addition orsubtraction as arithmetic instructions, storage for N signalsamples and N/2 bits for sequence xn since it is sym-metrical.

A binary transversal filter (b.t.f.) with binary coefficientsresults if the input signal in Fig. 10 is binary valued. In thiscase only single-bit multiplication is necessary, and thiscan be implemented by exclusive NOR gates. Such anarrangement8 can be used for direct filtering of any binarysignal such a binary data, pseudorandom sequences or deltamodulation. In the case of delta modulation (d.m.) the useof optimisation techniques to generate optimum binary-valued impulse responses is an important application, sincebinary coefficients lead to simple direct methods for d.m.digital filters operating with delta-modulated inputs andoutputs.9 The lowpass sequences presented here can beused to realise nonrecursive d.m. filters, having lowpasscharacteristics, which are directly comparable with con-ventional types based on p.cm. signal processing. In thecase of recursive d.m. filters it has been shown9 that therequired feedback signal may be derived by a b.t.f. withbinary coefficients which appropriately weight and filterthe d.m. output signal. The procedure which has beendescribed may be applied to optimisation of such ab.t.f. impulse response if the required impulse response isspecified. In principle, optimisation of the overall frequencyresponse of the recursive section could be achieved.

It should be noted that highpass characteristics can bederived by simple transformation of optimised lowpasssequences. The z transform of sequence yn in terms ofsequence xn is given by

X2Z- l .XnZ

N-1

Replacing z above by — z will rotate the unit circle by nradians resulting in a lowpass to highpass transformation,10

i.e. for TV even

- lY{-z) =

X\ X2Z T~ — Y 7N~l

XNZ1 + z- 1

Fig. 10 Digital lowpass filter using optimised sequence as impulseresponse

COMPUTERS AND DIGITAL TECHNIQUES, OCTOBER 1978, Vol. 1, No. 4

This implies negation of every second term of sequence xn

and replacement of the integration operation by a digitalfilter with transfer function 1/(1 + z"1) which correspondsto the difference equation -gn =#„_! + ( - / „ ) where /„and gn are input and output sequences, respectively. Thisfunction can also be simply implemented by accummulation

203

processes provided fn and gn are appropriately negated.Other transformations may be suitable, but for simplicityof implementation the binary format of sequence xn shouldbe preserved.

8 Conclusions

It has been shown that binary sequences of various lengthshaving prescribed spectral properties can be designed by anoptimisation procedure based on a simple delta-modulationalgorithm. The method has been exemplified in the designof binary sequences having lowpass properties related toRabiner's type-2 nonrecursive filter impulse responses, andspecific results are given for sequence lengths of 128, 256and 512 bits. Highpass sequences may be obtained by asimple transformation of the lowpass sequence. In general,the method can be applied to binary sequences having anygiven spectral properties, provided the limitation imposedby the slope-overload constraint of expr. 1 is respected.Applications include nonrecursive and recursive digitalfilters which can be simply implemented by digital hardwareperforming primitive arithmetic operations. Further appli-cation may be found where the advantages of digital-signalprocessing' do not justify the full complexity of multibitnumber representation and arithmetic.

9 Acknowledgment

The use of a machine-coded f.f.t. routine written by W. Al-Salman is acknowledged.

10 References

1 RABINER, L.R., and GOLD, B.: Theory and application ofdigital signal processing' (Prentice-Hall, 1975)

2 BOGNER, R.E., and CONSTANTINIDES, A.C.: 'Introductionto digital filtering' (Wiley, 1975)

3 GIBBS, A.J.: 'Optimisation of binary transversal filters withconstrained sampled impulse responses', Int. J. Circuit TheoryandAppl, 1976, 15, pp. 151-160

4 HILL, F.S., and LEE, W.U.: 'PAM pulse generation using binarytransversal filters', IEEE Trans., 1974, COM-22, pp. 904-913

5 LOCKHART, G.B.: 'Binary transversal filters with quantisedcoefficients', Electron. Lett, 1971, 7, pp. 305-307

6 RABINER, L.R., GOLD, B., and McGONEGAL, L.A.: 'Anapproach to the approximation problem for nonrecursive digitalfilters', IEEE Trans., 1970, AU-18, pp. 83-104

7 STEELE, R.: 'Delta modulation systems' (Pentech Press, 1975)8 LOCKHART, G.B., and BABARY, S.P.: 'Binary transversal

filters using recirculating shift registers', Radio & Electron. Eng.,1973, 43, pp. 224-226

9 LOCKHART, G.B.: 'A recursive section for filtering delta-modulated signals'. Proceedings of the Florence Conference ondigital signal processing, 1975, pp. 213-220

10 CONSTANTINIDES, A.C.: 'Frequency transformation fordigital filers', Electron. Lett., 1967, 3, pp. 487-489

Gordon B. Lockhart was born in Edin-burgh, Scotland, on 13th July 1942.He received the B.Sc. (Eng.) (withhonours) and the M.Sc. degrees fromAberdeen University, in 1965 and1966, respectively. He joined the Com-munication Section of the Departmentof Electrical Engineering, ImperialCollege, London, in 1966 to work onsingle-sideband techniques for broad-casting with the support of a BBC

research scholarship and received the Ph.D. degree from theImperial College in 1970. From 1969 to 1971 he wasemployed as a research fellow at the University of Tech-nology, Loughborough, sponsored by the Joint SpeechResearch Unit (Eastcote London) working on digital encod-ing of speech, particularly delta-modulation techniques. Hehas been employed as a Lecturer at the Department ofElectrical and Electronic Engineering, University of Leeds,since October 1971 and his special interests are in the areasof communication and digital signal processing.

Some papers to be published in future issues:

Compression of binary images by stroke encoding. I.D. Judd

System design to minimise failure. B.H. Corbett

Error detection of polynomial computations. M. Karpovsky

Determination of the set of the irredundant modulo-2 sums of products expressions of a logic function. Z.M. Lofti,D. Aoulad-Syad and A.J. Tosser

A guide to pattern recognition using random access memories. I. Aleksander and T.J. Stonham

Comparison of universal logic gate with nand and nor gates in the realisation of functions of three variables. S.L. Hurst andN.P. Pflaeger

204 COMPUTERS AND DIGITAL TECHNIQUES, OCTOBER 1978, Vol. l,No. 4