Analysis of Duobinary

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IEEE T R A N S A C T I O N S ON C OM M U N IC A TION TEC HN OLOGY VOL. COM-14, N O . 2 APRIL, 966

REFERENCES[I ] E. N. Gilbert, Capacity of a burstrnoise channel, Bel l Sys .[2] J. J. Metznerand K . C. Morgan, Coded feedback com-Tech . J . , vol. 39, pp . 1253-1265, September 1960.munication ystems, Proc. Natl. Electronics Conf., vol. 16,pp. 250-257, 1960.[3] B. Reiffen, W. G. Schmidt, and H. L. Yudkin, The design ofan error-free da ta transmission system for telephone circuits,

224-231, July 1961.T ra n s . A IE E (C o m m u n ica t io n a n d E lec t ro n ics ) ,ol. 80, pt . 1, pp.[4] R. J. Benice and A. H. Frey, Jr.; An analysis of retransmissionsystems, IEEE Trans. nCommunica t ionTechnology, vol.COM-12, pp. 135-145, December 1964.[5] R. J. Benice and A. H. Frey, Jr., Comparisons of error controltechniques, IEEE T r a n s . on Communicationechnology,

[6] R. C. Bose an d D. K. Ray-Chaudhuri, On a class of errorvol. COM-12, pp. 146-154, December 1964.pp. 68-79, March 1960.correcting binary group codes, Information and Control, vol. 3,[7] A. B. Fontaine and R.G. Gallager, Error sta tist ics and codingforbinary transmission over telephone circuits, P ro c. I R E ,vol. 49, pp., 1059-1065, June 1961.[8] M. Horsteln, Efficient communication through burst-errorchannels by means of error detection, Hughes Aircraft Com-

[9] F. E. Froehlich and R. R. Anderson, Da ta transmission over apany SSD Rept.4286R, May 1964.self-contained errordetection and retransmission channel,[lo] A. B. Fontaine , Queuing characteristics of a elephone da taBell Sys. Tech . J. , vol. 43, pp. 375-398, January 1964.transmission system with feedback, T r a n s . I E E E ( C o m m u -nicationan dElectronics), vol. 82, pp. 449455, Sept. 1963.

An Analysis of the Duobinary TechniqueAbstract-The spectral shaping characteristics of Lenders

duobinary transformation are examined and illustrated by severalexamples. It i s found that the duobinary transformation does notcompress bandwidth in a sense that would be directly applicable tothe Sampling Theorem, but rather shapes the spectrum such that itbecomes possible for samples sent at Nyquist rate through a non-flat filter to convey one bit of information each, even though theyare statistically dependent and restricted to three levels. Thus theduobinary technique might be quite efficient for certain nonflat,band-limited channels which have a poor signal-to-noise ratio.

INTRODUCTIONA ONGST TH E numerous data transmission methodspublished in the last few years, one of considerableinterest is the duobinary technique, described in publica-tions of A. Lender [l] , 2] and F. B. Bramhall [3]. Thistechnique is claimed to double switching speed while re-taining the binaryform of the originalsignal, and naddition, to possess a built-in no-redundancy error de-tection method. These are remarkable properties, sinceconventional systems can reduce bandwidth only at th eexpense of increasing the number of transmission levels(assuming that the input, data st ream is nonredundant),and can detect errors only by transmitting redundantcheck digits in addition to the information digits.

The heoreticalfoundations of the duobinarysystemare not laidown very clearly inhe available publications.The object of this paper is thus to examine the principlesunderlying the duobinary technique, and to try t o answer

1965.Manuscript received February 4, 965; revised December 27,The author is with Bell Telephone Labs., Inc., Holmdel, N. J. Hewhere this work was performed.was formerly with Rank Xerox Ltd., Uxbridge, Middlesex, England,

some of t he questions tha t occur to manywhen first meet-ing it ; i.e.,

1) Does the duobinary transformation always result ina 2-to-1 bandwidth compression?2) Is there a contradiction between he known optimum(exponential) exchange rate betweenbandwidthandnumber of transmission levels, and that claimed for theduobinary?3) Does the duobinary signal utilize channel capacitymore efficiently than the straight binary?A similaranalysis is then applied to he polybinary

scheme [4], which can be regarded as the extension of t heduobinary.

PRELIMINARIESIn a data transmissionsystem the nput and output

consist of a sequence of numbers, while the transmittedintelligence takes the form of a continuous wave. A con-nection between the two is given by the Sampling Theo-rem, concisely stated by Shannon [ 5 ] :If a function f(t)containsno frequencies higher than W c/s, it iscom-pletely determined by giving its ordinates at a series ofpoints spaced l/2W seconds apart. Note that this is es-sentially an interpolation rule, i.e., an instruction for re-generating a continuous wave from discrete samples. Indata transmission one is usually more nterested n theconverse, i.e., in extracting data (samples) from a contin-uous wave. In accordance with the Theorem, if the wave-form contains no frequency componentsigher tha n W c/s,not more than 2W functionally independent samples canbe extracted from it. (Two sets of samples are function-

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ally dependent if the knowledge of one set is sufficient forcomputing the values of the other set.) Thisgives then theupper limit of the number of data (each represented by asample) tha t can be transmitted through a band-limitedchannel.

The requirements of da ta transmission go beyond func-tional independence, however. It is not enough that onesample does not precisely determine the next-it shouldnot even influence it probabilistically. In other words, if atthe input the sample values fluctuate entirely at random,there should be nothing in the channel that prevents theoutput samples rom being equally andom.Thus, forexample, if the input consists of a train of equally prob-able positive and negative unit impulses, samples of theoutput taken at the same rate should be statistically in-dependent. A necessary condition for this is that they areuncorrelated. It can be shown, by using Bennetts [6] re-sults, that theonly network with bandwidth W that yieldsuncorrelated output samples at the ra te W per second isthe rectangular ideal filter. Generally, samples may beuncorrelated yet statistically dependent. However, sincethe impulse response of this network is the (sin .)/x func-tion with zeros a t multiples of 1/2W, samples taken a tthis rate will also be statistically independent. It follows,then, that any other network, limited to W c/s but notflat within the band, yields functionally independent, butstatisticallydependent,output samples at herate2Wsamples per second. Such channels are thus not suitablefor transmitting 2W independent data per second, unlesstheda ta are irst recoded intoacorrelated sequence.(Alternatively,differential decoding process maybeused, but this requires a memory in the decoder leadingto an accumulation of errors.)What has been said above must be somewhat modifiedwhen the transmitted data equire only a few (i.e., 2 or 3)levels, and the channel noise is low in comparison with hespacing between he levels. In these circumstances one canafford to have some distortion in the waveform or, whatamounts to he same hing, some interferencebetweenconsecutivesamples, since the receiver canstill decideeasily which datum was transmitted. The question of howmuch interference can be tolerated, or how many signallevels can be usefully employed, cannot be answered ingeneral, but a few special cases have been examined byHill [7].

THE DUOBINARY TRANSFORMATIONLet us now investigate the properties of the duobinary

transformation. Following Lender [l we take as the start-ing point a binary sequence ,, in which the values a , = 1and a, = - are assumed with equal probabilities p(1) =p (- ) = l/2. From this, a duobinary sequence , (values

The departure from Lenders use of the values 1 and 0 in th ea , sequence is intentional. Bennett [6] shows that unless the meanvalue of th e a sequence is zero, its spectrum contains, in addition tothe continuous part, also several Dirac impulses, and these give riseto unnecessary complications.

+1, 0, - 1) is generated by the following rules:If a, = -1, b, = 0.If a , = +1, the polarity of b, depends on the polarity

of b,-k corresponding to the last n-k = 1.If a , and anPkare separated by an even numberf zeros,b, = +b,-,; otherwise b, = - ,+

An equivalent method is to convert the a, sequence firstinto another differential sequenced,, using the ruled , = d,-l if a , = +1d , = -dn-l if a , = -1,

the possible values of d, being +1/2 and - /2. The duo-binary sequence b, can then be obtained by adding alge-braically the last two members of the d, sequence, i.e.,

These sequences are merely numbers representing themessage values. To obtain a waveform suitable for trans-mission, such numbers may be regarded as the intensitiesof impulse trains, applied o a realizable network.The out-put can then be-written as

b, = d, + d,-I.

where g ( t ) is the impulse response of the network, c , anarbitrary random time series of integral numbers, and Tthe spacing of the train of impulses.

Let ~ ( t ) enote a wave generated romabinary se-quence, and y ( t ) one generated from the duobinary trans-form of the same sequence:

and let GV) be the Fourier-transform of g ( t ) . The powerspectra of x ( t ) and y ( t ) are then obtained from Bennetts[6 ] formula:

1

On combining these one obtains

I n other words, the power spectrum of the duobinarywave is the product of the spectrum of the two-level waveand the weighting function cos2nTf.

The significance of this operation is best illustrated bysome typical examples.I ) The sinc puke. Let the individual pulses be shaped inaccordance with the sinc x interpolating function:

g l ( t ) = A- A sinc t/T.sinat/Tat/ T


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The spectral expressions are as follows:Gl(f) = AT, If I 5 1/2T W,(')(f) = A2T, f I 5 1/2T

= 0 elsewhere = 0 elsewhereW,")(f) = A2T cos2aTFf I 5 1/2T

= o elsewhereThese functions are shown in Fig. 1.

2) The raised cosine spectrum.sin 2at/T

g 2 ( t ) = A T 2 d [ 1 - 2t/T)2]Gz(f) = +AT(l +'COSaTf) = ATcos2$aTf

for If I 5 1/T= o elsewhere

W z ( 2 ) ( f ) A2T c0s4afT/2 I f 1 6 1/T= o elsewhere

Wv ( 2 ) ( j ) A2T (~0s~aTf/2)(cos?rTf)~If I 6 1/T. .= o elsewhere.3) The Gaussian pulse.

g 3 ( t ) = A exp(- t2/T2)G3@)= AT exp(- apT 2)

W,(3) ( f ) A2T exp(- 2af2T2);W,U) = A2Texp(-2af2T2) cos2aTf.

4)The rectangular pulse.

= 0 elsewhere

[= Wz'4'(2f)].The effect of the duobinary conversion on signal band-

widths can now be assessed for the above examples.Thespectrum of the sinc unction, GI@), is strictly

limited to bandwidth1/2T.Theduobinary pectrumW,(l)(f) has the same bandwidth, bu t some of the energyis shifted towards the low-frequency region.

Gz(f) is band-limited to 1/T, and so is W,@)(f),but thelatter has a further zero a t 1/2T, and most of i ts energylies below this value.

The Gaussian response g3(t) is not band-limited at all,nor is W,(3) ( f ) ,lthough i t has zeros at f = (2k + 1)/2T,k = 0, 1, 2, . . .. The duobinary spectrum thus ooks quitedifferent from the original.

The originalspectrum of the rectangularpulse g4(t)has zeros at themultiples of 1/T, and is not band-limited.Bya remarkable coincidence, however, theduobinarytransformation causes the spectrum to be compressed tohalf its widthon the frequency axis, since W,(4)cf) =

IW:"tf)

Fig. 1. The sinc pulse g l ( t ) and its spectrum.

T ZT I L I L2T 2TFig. 2 . The raised cosine spectrum Gz(f).

Fig. 3. The Gaussian pulse g3 ( t ) and its spectrum.

Fig. 4. The rectangular pulse g4(t) and its spectrum.

W,(4)(2f). Naturally,both hese pectraextend to i n -finity.

Consider now the question of bandwidth compression.It is clear th at if the binary wave has frequency compo-nents above f = 1/2T, the duobinary train, too, will havefinite components there, except a t discrete points wherecos2aTf = 0. Thus the bandwidth as defined in the Sam-pling Theorem is not compressed at all. On the other hand,it is noted tha t in the special case of the rectangular pulseg4(t), the spectrum is compressed by 2-to-1 along the fre-quency axis. This means that certain parameters of t hespectrum, which are sometimes used t o define ((bandwidth"in a loose sense, are also halved. These are, e.g.,

1) the frequency at which the spectrumfirst falls tozero,2 ) the frequency below which lies a specified propor-tion (e.g., 90 percent) of the spectral energy, and

3) the frequencyabove which allenergymay be re-moved while keeping the ensuing waveform distor-tion within specified limits, etc.

The last of these serves as the basis for Lender's practi-cal scheme, especially as described in [ 2 ] .The rectangular


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duobinaryrain isand-limited by low-pass filterwhich cuts of f at 1/2T C/ S.~ Th e esultingwave is nolonger rectangular,but etains enough amplitude dis-crimination at th e sampling points to enable the receiverto recover the individual digits.

An alternative echnique orobtaining hiswave isdescribed in [ l ] nd [3], and it is instructive to examinethis. Here he rectangular duobinary wave s not generatedat all. Ins tead, he differential binary pulse train d, isapplied directly to a transmission filter which represents,to aroughapproximation, the conversion characteristiccos2aTf combined with band-limiting. For a digit spacingT the filter hasa cutoff frequency bandwidth) W =1/2T and ahalf-amplitudefrequency f1 = 1/4T. Fromthe preceding discussion there is clearly no objection, intheory, to recovering functionally independent samples atthe rate 2W = 1/T per second, but these cannot be sta-tistically independent. (In fact, themaximum rate for thismode is W = 2f i samples per second.) Thus if t he band-widthcriterion is to be satisfied for this nonflat ilter,some correlation must be permitted to exist between thesamples. This is exactly what the duobinary transforma-tion produces: for example, whenever 6 +1 is received, itis certain tha t the next sample will be +1 or 0, but not- . Similarly, the patt ern - ,0, can be followed only by0 r +1,not by - 1, and even a0 can be followed only byeither another 0 or one of +1 and -1, depending on thenumber of 0s since the last 1. (Note, however, that eachsample still represents one bit of information, since thereare two equally probable alternatives.)

The answers to the first two questions posed in the I n-troduction are now clear: the duobinary transformation isessentially a means for makingt possible for samples sentat the Nyquist rate through a nonflat filter to convey onebit of information each, even though they are statisticallydependent. This is achieved by shaping the spectrum ofthe wave train to suit the filter, and no t by compressingthe bandwidth, and thus the uestion of optimal exchangebetweenbandwidthand the number of signaling levelsdoes not arise.

UT~LIZATIONF CHANNELAPACITYThe efficiency of an encoding technique can be objec-

tively measured with reference to theultimate capacity ofthe communication channelemployed. Figures of efficiencyso computed can, however, be rather meaningless, in viewof the difficulty of realizing coding systems which evenapproach capacity. More useful comparisons can be madebetween different encoding methods utilizing essentiallysimilar techniques. Thus the efficiency of the duobinarysystemmightbe assessed in elation to ts generalizedform, the ternary code,3 on the grounds that if the signal-

quoted in the paper as thealf-amplikrde point.* Readers should not confrlse this with the figure f~ = 1/4T,duobinary technique preserves the binary form of the original sig-3 Readers should no t be misled by Lenders stat ement that thenal, which certainly does not apply to thecommunication channel.

to-noise ratio in the channel is high enough to allow trans-mission with, say, a lop6 error probability, it may be re-garded as a noise-free three-level discrete channel. Such achannel has a log23= 1.58bits/sample capacity, providedthe samples are statistically independent. Compared to ththe duobinary with itsbit/sample information rate showsan efficiency of only 0.62.This is not a fair omparison, how-ever, for the duobinary does not require a flat pass band,whereas the general ternary code does.

In order oarrive a t a more appropriate figure, onewould have to ind the information rate of the best ternarycode tha t still satisfies the same spectral constraint as heduobinary. (Such codes will be based on either statisticaldependence of the transmitted samples or on differentialdecoding of originally independent samples hat have beensubjected to intersymbol interferenceowing to the spectralshaping.) This is not easy, for the spectral shape specifiesonly the autocorrelation function of the signal, but notthe transition probabilities of its samples, which are alsorequired for calculating the information rate. Thus, givena channel used in a duobinary system, one cannot directlyderive a superior code, though one maybe ound em-pirically.

The last introductory question is thus seen to be mis-directed. If the channel allows ternary transmission withacceptable error rate, this will be preferable to a binarycode. For higher noise levels only the binary will do, un-less an error-correcting ternary code is developed. (Theduobinary scheme has onlya rudimentary error detectingfeature.) As an alternative, the systemknownassup-pressed singles in long distancedc elegraphy may beused, whose performance is broadly equivalent to th at ofthe duobinary. THEPOLYBINARYYSTEM

The polybinary ransformationconvertsa two-levelsequence to a K-level sequence, where K > 3. The spec-trum of the wave y ( t ) generated from the multilevel se-quence is the product of the spectrum of z(t) , the originalwave, and a weighting factor:

For the special case of rectangular pulses, the spectrumof the transformed signal is again compressed by a factorof ( K - 1) . The transformation can be carried out by apassive filter or, in a very similar scheme [8],by a digitaltechnique involving delay lines and an algebraic adder.

Lender [4] claims that such a signal is capable of trans-mitting (K - 1) binary channels over a bandwidth whichnormallyaccommodatesonlya single binarychannel.This is not a veryprecise statement, but rom the examplesgiven, the system seems to work as follows. Rectangularbinary pulses occurring at the rate (K - l)fi re fed intoa passiveL-C filter H&) with ominal half-ampli-tude point at f1 c/s. The output is a K-level wave fromwhich the original data stream may be reconstructed by


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IEEE TRANSACTIONSONCOMMUNICATION TECHNOLOGY

sampling at thepulse rate. Each sample yields one binarydatum qui te independently of the adjacent samples. Inthe examplegiven by Lender, fi = 600 c/s, andwithK = 5, the input data rate s quoted as 4800 bps.

One ni gh t initially assume that th e system has beendesigned to work into a band-limited channel, and tha tH&) cuts off completely at, say, 2f I c/s. If so, i t is dif-ficult to see how the system could possibly operate, sincesampl-es aken a t the ra te 4f1c/s (2400 c/s) will, by virtueof the Sampling Theorem, completely determine the out-pu t wave and hence the other 2400 samples per second.Since each sample represents one datum, the knowledgeof only half the da ta wouldseem to be enough for re-constructing the entire set. This is possible only if in theoriginal set half the dat a are functionally dependent onthe other half, but then they should not have been trans-mitted in the first place. Thus one is led to conclude thatthe polybinary scheme can work. only in an open bandchannel, and its sole advantage lies in concentrating thespectral energy in he low-frequency regions.-- -- --_-_._ ___

VOL. c O M - ~ ~ ,O. 2 APRIL, 1966

ACKNOWLEDGMENTThe author is very grateful to b. Bingham of the Post

Office Research Station, London, England, for his pene-trating criticism of the original version of this paper.

REFERENCES[ l ] A. Lender, The duobinaryechnique for high-speed da ta

vol. 82,pp. 214-218, Ma y 1963.transmission, IEEE Trans. on Communication and Electronics,[ 2 ] -, Fas ter digital ommunicationwith uobinary tech-niques,Electronics,vol. 36, pp. 61-65, March 22, 1963.[3] F. B. Bramhall, Duohinary coding equipment permits doubledtransmission speeds, Autom. Elec. Tech. J . , vol. 8, pp. 252-256,July 1963.[4] A . Lender, Correlative igital ommunicationechniques,[5] C . E. Shannon,Communication n th e presence of noise,IE EE Internatl Conv.Rec. , vol. 12, pt. 5, pp. 45-53, March 1964.Proc. IRE, vol. 37, pp. 10-21, January 1949.[G I W. R. enne tt, Statistics of regenerative digital transmission,[7] L. W. Hill, Transmlsslon of multi-level digital signals throughBell Sys . Tech. J. , vol. 37 , pp. 1501-1543, November 1958.noise-free distorting channels, Proc . ZEE (London),vol. 109,pt . B, pp. 165-173; March 1962.[8] J. K. Wolf, On the application of somedigital sequences to

vol. 11 , pp. 422-427, December 1963.communication, ZEEE Trans. on Communications Systems,

The Maximum Capacity of Tandem LinkMultichannel Troposcatter Systems

Abstract-This analysis expresseshe maximum erformanceof a multichannel system, as repr esented by the maximum channelcapacity for a specified noise level, in terms of all known significantdesign variables. Such factors a s channel loading, propagation loss,multipath, cosmic noise, and feeder echo are included. More than 60independen t variables associated with system design are t aken intoaccount.It is believed that the result s are unique, since this is the firstknown approach in which all significant noise components arebrought into a single embracing expression.

Maximum performance is found to be associated both with anoptimum deviation and with an optimum carrier frequency. Contraryto previous opinion, the latt er isound to increasewith distance.

The analysis indicates that bandwidths suitable for carrying TVsignals are feasib1e;under suitable conditions. However, in ord er todevelop general design procedures, which will achieve the maximumpossible performance, considerably more knowledge is required con-cerning certain basic aspects of behavior, particularly with respectto medium aperture oss, multipath, feeder echo, and antenna beam-width.at the 1965 IE EE MIL-E-CON Symposium, Washington, D. C.Div., New Rochelle, N. Y.

Manuscript received September 17, 1965. This pape rwas presentedThe auth or s with Litto n Systems, Inc., Westrex Communications

T INTRODUCTIONHE PR ES EN T demand for high-quality circuits hasbrought with it th e need to design circuits for opti-mum performance-usually defined as the conditions pro-viding maximum channel capacity for specified per chan-nel noise level. To a considerableextent, the design oftroposcatter systems, thus far, haseen predicated on olu-tion of the thermal noise problem alone. However, exami-nation of the matter indicates that the system design oranalysis engineer must manipulate some thirty significantindependent variables, and deal witht least twentymore,to achieve a compromise design, which takes into accountall significant noise sources. Theoptimum solution isan extremely complex matter ndhemost uitablechoice of parameters is not necessarily clear cut. In mostcases, a solution by normalprocesses is almost impossible.It is the purpose of this paper toexamine this problem,and indicate a means for arriving at a single and all em-bracing result. Of necessity, because of space limitationsas well as certain accepted facets f the ar t, th e erivationof these results will be reduced to a minimum.

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Analysis of Duobinary

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