For system bandwidths less than the inverse-baud rate. it is not possible to satisfy all these criteria simultaneously; tradeoffs that have to be made are illustrated.
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Abstract-This paper studies the problem of designing a suitable pulse shape for teletext data transmission. The following four criteria are used: 1) Nyquist I criterion, 2) Nyquist I1 criterion, 3) degree of overshoots in the channel signal, and 4) robustness to sampling phase jitter.For systembandwidthslessthantheinverse-baudrate. it is not possible to satisfy all these criteria simultaneously; tradeoffs that have to be made are illustrated. Several candidate pulse shapes are A pulse shape,which givenanda compositecriteriondeveloped. satisfies the Nyquist I criterion and is closest to satisfying the Nyquist I1 criterion, in a sum-of-squares-of-deviations sense, is recommended.
1. INTRODUCTION ELETEXT is abroadcastsystemofdatatransmission Twhich uses some of the unused verticalblankinginterval (VBI) lines of a television signal as the channel [ 1] - [4] . Systemparameterssuch as suitabledatarates, pulse shapes, etc.,are in the processof being standardized [ S I . Inthis paper, we consider theproblemofchoosinga pulse shape suitable for teletext. Previous works [6]-[8] have suggested several pulse shapes; however, these were mainly arrived at by starting with the familyofcosinerollofffiltersandoptimizing over the rolloffparameter.Inthispaper,we will not imposesucha restriction on the set of allowable solutions: we shall develop a set of reasonable criteria first and then find the pulse which meets the set of criteria.
11. SYSTEM MODEL AND CONSTRAINTS
TRANSMITTER
CHANNEL
RECEIVER
Baseband system model. H ( f ) = H ~ ( f ) f i ~ (ak f ) =; ? 1 .
Fig. 1 .
A. Basebarld Model If the transmitted pulse is h T ( t ) ,then the data bar level is defined as bar =
x
h,(t - kT)
k
that is, the responseofthetransmitterfilter to an impulse [6], [9] that in order train. It has been found experimentally to avoid visibility during vertical retrace and a "buzz" on the sound channel, the bar level should be restricted to approximately 70 IRE units (70 percent of white level) and the signal overshoots above this level should not be excessive. Thus, we have an amplitude limited channel and, in designing a pulse, the overshoots shouldbe restricted. A m o r ei m p o r t a n st y s t e mc o n s t r a i n t is t h e bandwidth. Forthe NTSC TVsystem(theNorth American standard) thebasebandcomposite video bandwidth B is 4.2 MHz; thus, the inserted pulses will have to be band limited to this value. Finally, the system disturbances taken into account will be additivewhite Gaussian noise andmultipathinterferenceor "ghosts." Thus, we assume the baseband system model shown signal a hasbeen inFig. 1. Theattenuationofthedelayed taken to be in the range 0 < a < 0.3 [6], [7], but is generally unknown. The delay T,. has also been observed to have a wide range of values. In designing a signaling pulse we will assume that a = 0, but will, at the same time, constrain the overshoots so that when reflections arise the amount of eye closure will be minimized.
The existing TV system has been optimized t o carry analog pictureinformation.Certainnonidealitiessuch as nonlinear phase characteristics of the IF filter in the TV receiver, multipath propagation, echoes in CATV cables, and distortion due to envelope detection of the vestigial sideband signal,which have had negligible effectsonthe video signal, will hamper the data signals. Some of these problems will be solved by the improvement in future TV receivers, such as the use of SAW devices for IF filters and the use of IC synchronous detectors. It is difficult to include these nonidealities in the system model since they depend on the make of the receiver used.MultiB. Excess Bandwidth pathpropagationorechoesin CATV cablesarechannel According to the Nyquist I criterion [lo, p. 451, [ 11, p. problems,hence easier tomodel;thus, we will takeinto (ISI), for signaling without intersymbol interference account the effect of multipath propagation in the design of 811 if R = l/Tis the baud rate and T is the signaling period, then a pulse shape for teletext. theminimumbandwidthrequired is Bmin = R/2. If B is is referred t o as the excess Paper approved by the Editor for Communication Theory of the the system bandwidth, ( B - & i n ) IEEE Communications Society for publication after presentation at the bandwidth and is usually expressed as a percentage or fraction CanadianCommunicationsandEnergyConference,Montreal, P.Q., Canada, October 1982. Manuscript received October 28, 1982; revised of Bmin. We shall denote the excess bandwidth by e(0 < 6 < 1); thus, 0 = 1 (100 percent excess bandwidth) implies R = March 1, 1983. This work was supported in part by a Strategic Grant from the Natural Sciences and Engineering Research Council (NSERC) B = 2 Bmin. Fornotationalconvenience,wenormalizethe of Canada. frequency variable so thatthe signaling period T is unity; TheauthorsarewiththeDepartmentofElectricalEngineering, University of Toronto, Toronto, Ont., Canada M5S 1A4. thus, B = (1 + 0 ) / 2 . 0090-6778/83/0700-0871$01.00 0 1983 IEEE
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-31, NO. 7,JULY 1983
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Fig. 3.
(g) (h) Fig. 2. Transferfunctionsofvariousfilterswithspecialproperties. (a)Raisedcosine;satisfiesNyquist I and 11. (b)Triangular;minimum bandwidth, positive pulse satisfying Nyquist I. (c) Cosine rolloff;satisfiesNyquist I. (d)Linearrolloff;"minimizes"peak.(e) Modified raised cosine; minimizes a measure of closeness to Nyquist I1 subject to Nyquist I. (f) Modified triangular; minimizes a robustness t o phasejittermeasuresubjecttoNyquist I. (9) Truncated raised cosine; minimizes a measure of closeness to Nyquist I and 11. (A chosen so that h ( 0 ) = 1.) (h) Composite filter; a filter designed using the composite criteria with (a,0, y) = ( 1 , 1, l/n2).
Eye diagram for the truncated raisedcosinefilter[Fig.2(g)] and definitions of eye parameters.0 = 0.47.
role; an example, plotted for a sequence of 256 pseudorandom data bits, is shown in Fig. 3 . From the eye diagram, the following fourquantities canbe estimated:eyeheight,eyewidth, degree of overshoots, and flatness of the eye at the sampling point. The eye height is the most important of these since it gives the noisemarginat the optimum sampling point. The eye width is related to timing recovery; the narrower the eye, themorescatteredarethezero crossingsand the greater is the phase jitter in the derived clock. As we shall see shortly, so as to minimize the theovershootsshouldberestricted effectofmultipathpropagationand, if the pulseshaping is (HR(f) is anideallow-pass donetotallyatthetransmitter filter), they should be minimized since the channel is amplitudelimited.Finally,any real system will have asampling clock which has some phase jitter; thus the eye diagram should have a small rate of closure with respect to offsets in sampling time. The pulseshape featureswhichdeterminethe aboveeye diagram criteria will now be discussed.
A. Eye Height Variousbaudrateshavebeenproposedforteletext;the rate of 5.727 MBd,which is (815) of the colorsubcarrier frequency and 364 times the horizontal synchronization frequency,'seems to be a popular choice for the NTSC system [ 5 ] . For the NTSC system bandwidth of 4.2 MHz, this baud rate yields a 0 of 0.47. Thus, we will design our pulses for the entire range of 0 ; but will illustrate our results for 0 = 0.47. By considering different criteria of optimization,we will arrive at various pulses with special properties. For convenience incomparison,thespectra of all theresulting pulses h(t) (the transfer functions H ( f ) of the corresponding filters) are shown in Fig. 2 . (All the filters are assumed to be zero-phase; hence, Hdf) is real and even about zero frequency.) 111. PERFORMANCE CRITERIA
Assuming the sampling time to = 0 (and recalling that T = l), the eye height is defined as
For maximum eye height, we require that h(k) = 6kO
k=
*2, ":
(3)
The above is the Nyquist I criterion for zero ISI. In the frequency domain, (3)is equivalent to
x
H ( f - k ) = 1. Theultimateperformancecriterion is theprobability of k error;however, sincethis is difficult to deal with,other related criteria will be used. In evaluating the performance of a The unique minimum bandwidth solution to (3) is given pulse,theeye diagramof the signal &Ukh(t - k ) , h(t) = sin nt/nt. y ( t ) (Fig. 1) if H R ( f ) is an ideal low-pass filter,' plays a key
(5)
For 0 # 0, the solution is not unique; the only requirement, assuming 0 < 1, is that H ( f ) have vestigial symmetry at f =
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SOUSA AND PASUPATHY: PULSE SHAPE DESIGN
where A ( f ) is even about f = 1/2 and limited to (1 - 0)/2
= elsewhere Every practical sampling clock will have some phase jitter; thus, one of our criteria is that the eye diagram have a slow rate ofclosurewithoffsets in sampling time. We shall now try to relate this desired property to propertiesof the signaling pulse. -1 < f < O (9) The ideal pulse, according to thiscriterion,wouldbea elsewhere squarepulse, thus requiringalargebandwidth.Sincethe
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-31, NO. 7 , JULY 1 9 8 3
3) Smoothness of the eye at the sampling point can be increased by controlling eJ2 (1 2). Ideally, we would like a pulse to satisfy all of the above conditions; however, since the bandwidth is limited, it is not possible to satisfythem all simultaneously.Thus,acompromise betweenthedifferentcriteriamust be made; we shall explore this in subsequent sections.
IV. NYQUIST I SOLUTIONS We have already stated that the Nyquist I criterion is the most important. Thus, in this section, wewill find solutions which satisfy the Nyquist I criterion exactly and are as close is to satisfying the remaining criteria aspossible.Closeness defined by a sum-of-squares measure which results in tractable mathematical expressions.
A. Nyquist I and II Constraints TheNyquist I1 criteriondetermineseyewidth,and, to some extent, overshoots.ThecombinedNyquist I and I1 criterion is h(k/2)=6kOfi(6k,-1
Fig. 5.
Eye diagram for the linear rolloff filter [Fig. 2(d)].
e
= 0.47.
bandwidth B < 1 and for mathematical tractability, we solve the following problem: find an h(t) which minimizes
eJ2 =
+6kl)
k=0,?1,*2,”‘.
(15)
The ,minimum bandwidth solution to (15) is the raised cosine filter of Fig. 2(a) [ 141 . For 0 < 1, there is no solution which satisfies (15); thus, we minimize an error in the Nyquist I1 criterion (6), defined as
(/~?‘(k))~ k
subject to a constraint on the bandwidth of h ( t ) , and some subject to the Nyquist I criterion (3). Keeping in mind that normalization constraint. h ( t ) shouldsatisfy (3), the above optimization is equivalent of variations to minimizing the quantity UsingParseval’s theoremandthecalculus [13], the solution to the above problem can be shown [17] to satisfy
If1H(f)= (1 - IflW(1 - Ifl).
(1 3)
subject to the NyquistI criterion. Thus, H ( f ) = 0 for allfexcept for I f 1 E [(l - 0)/2, (1 + 0)/2] Using Parseval’s theorem,the inverse Fouriertransform, wherethesolution is nonunique. If we choose H(f) inthe and appending the normalization constrant JH(f)df = 1, with above interval such that H ( f ) + H( 1 - I f ] ) = 1, then the solu- the Lagrange multiplier A, (17) becomes tion is (1+0)/2 JC
[H2(f)-H(f)
=
COS
nf+ M ( f ) ]df. (18)
-(1+0)/2
The result (14) is as close as possible to satisfying the Nyquist I criterion and still achieving f?J2 = 0. As 6 increases towards 1, (14) becomes the triangular filter [Fig.2(b)]whoseeyediagramcanbeshown to be indeed flat at the sampling point. So far we have statedfourcriteriathattheeye diagram shouldideallysatisfyand subsequently we haverelatedeach criterion to properties of thepulse. In summary, we have the following. 1) For maximum eye height and eye width, the pulse must satisfy the Nyquist I and I1 criteria, respectively. 2) To minimizeovershoots,the pulseshouldbe as “positive” as possible.Imposing theNyquist I1 criterionhelpsto limit overshoots.
Since H ( f ) is limited to B = (1 + 0)/2 and satisfies the Nyquist I , it is fullyspecified for If1 E { [0, (1 - 0)/2] U [(l + 0)/2, l]}. For I f 1 E [(l - 0)/2, (1 + 8)/2] , H ( f ) is obtained by minimizing J , (18) using the calculus of variations. The end result is the modified raised cosine filter of Fig. 2(e), with the eye diagram shown in Fig. 6. In comparison to the eye diagrams for the cosine rolloff (Fig. 4) and linear rolloff (Fig. 5) filters, it has better zero crossing properties (on theaverage), smaller overshoots, and a slightly smallerrateofeyeclosure withrespect to samplingoffsets.Themodifiedraisedcosine filter is discontinuous; hence, the peak is theoretically infinite [ 11, p. 841. The peak, however,occursonlyforaspecific infinitesequenceandhencedoesnotappear in theeye diagram of Fig. 6, plotted using only 256 data bits. Nevertheless,
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SOUSA A N D PASUPATHY: PULSE SHAPE DESIGN
Fig. 6 .
Eyediagramforthemodifiedraised 8 = 0.47.
cosine filter [Fig. 2(e)]. nodifil:d triangular filter [Fig. 2(f)]. 0.47
8 =
most of the overshoots are slightly smaller for this filter than the linear rolloff filter which was “optimum” according to the peakcriterion. (Also notethatalthoughzero crossing and, hence,eyewidth is controlledbymeasure(17),minimizing (17) does not necessarily minimize deviations in the eye width, defined in the worst case, mean squares, or any other sense.)
e I 1 2 ,the error in Nyquist 11,is defined by (16) and e J 2 ,the e2 fordifferent smoothnessterm,by(12).Minimizationof sets of weight coefficients (a, p, r) will yield different pulses withdifferent emphasis on an eyediagramcharacteristic y = 0 and according to the correspondingweight.Thus,if a > 0, then the optimization reduces to that in Section IV-A; B. Nyquist I and Smoothness at Sampling Point if 0 = 0 and a > 7,then it reduces to that in Section IV-B. If, in addition to the restriction on bandwidth, the Nyquist to keep in mind in theminimization Theonlyconstraint I criterion is alsoimposed in theminimizationof eJ2 (12), of e2 is that h(r) mustbebandlimited to B = (1 + 0)/2 then the solution is the modified triangular filter of Fig. 2(f), and normalized such that with the eye diagram in Fig. 7. The minimization of (12) subject to the Nyquist I criterion was also suggested in [15] as a way of making use of the excess bandwidth in pulse design. The modified triangular filter Minimization of (19) (see Appendix B for details) gives the was also obtained by Franks [16], using a different method, [17], as the solution which is opti- filter and later by Cattermole mum with respect to samplingphase jitter. From Fig. 7, we see thattheimprovement in robustness to phase jitter does Hc?,p.y(f; X) not seem sufficient towarrantthe worse zero crossingsand ‘ pcosnf-h 1-e overshoots. If1 a + p 4n2yf2 V. COMPOSITE CRITERION
0.8. In other words, according to these three (8 = 1); measures, there is no need for a raised cosine filter 8 = 0.8 will be sufficient,thus saving 20 percent in bandwidth. The cosinerollofffilter, as alreadynoted, is notgood; itsperformance is worse thanthatforthemodified raised cosine for all 8. Depending on which of the two errors, eII or eJ', is worse,thecosinerolloffmay beworse thanthe linear rolloff and the modified triangular filters for all 8.
.4
.3 0 . L
0
a
.2
01 0
.2
.4
.8
.6
Excess Bandwidth
1
e
(C) raised Fig. 8 . Performance versus excess bandwidth. (a) -Truncated cosine, (a,p, y) = (1, 1, 0). --- Cosine rolloff. (bj -Linear rolloff. --- Modified raised cosine, (a,@, y) = (-, 1, 0). (c) -Composite, (a,@, y) = (1, 1, l/n2). --- Modified triangular, (a,@, y) = (-, 0, 1).
Themodified raisedcosinefilter satisfies theNyquist I criterion at the expense of more error in the Nyquist I1 criterion when compared to the truncated raised cosine. Among all the filters investigated, the linear rolloff filter minimizes the peak. Also, if 0 E 0.7, this filter is comparable to the modified raised cosine in terms of the elI and eJ' errors. e,' error Themodifiedtriangularfilterminimizesthe I criterion. When comparedtothe subject totheNyquist
SOUSA PULSE AND PASUPATHY:
877
DESIGN
truncated raised cosine and the modified raised cosine for high filter trades off erroreJ’ for error eII. Finally,thecompositefilter is aninstanceofacompromise on allcriteria. For three largeexample, 8none , for of the errors eI, eII, eJ’ are greater than 0.075.
By Parseval’s theorem (25) becomes
e , this
112
e12 2
[ q f )+ H(l
-f)l
I’
Based on our results, we recommend the pulse correspondmodified theing to [Fig.filter cosine raised 2(e)] teletext. for The pulse satisfies theNyquistIcriterionand minimizes a sum of deviations in the Nyquist I1 criterion; it is also close to the one that maximizes the robustness to samplingphase jitter [Fig. 2( f)] .
- [h(-3)
APPENDIX A
If h ( t ) satisfies the Nyquist I criterion, then the transform H ( f ) has vestigial symmetry at I f 1 = 1/2 and can be written as
+ h(3)] + ;.
(27)
Using Parseval’s theorem on the first summation, (25) and (26) on the second summation, and the inverse Fourier transform, (27) becomes
- 2 [MY) + W 1 - f ) n e (1
where n(f)is the ideal low-pass filter with B = 1 / 2 , G ( f ) is an even function restricted to I f 1 = 0/2, and * denotes convolution. In the time domain, (23) becomes
-4[H(f)-H(l
-f)n,(l
-a]
COS
~ f df. }
(28)
Lastly, we solve for f?J2
sin nt h(t) = __ g(t). 7rt
NOW,
(26)
Now, for eII we have
VII. CONCLUSION
MINIMUM BANDWIDTH POSITIVE NYQUIST I PULSE
df- 1.
eJ2 =
for h j t ) 2 0,we require sign (sin 7rt/7rt) = sign (g(t)).
[h’(k)] k
Since g ( t ) is continuous (otherwise the bandwidth would be infinite), we require that g(k) = 0 for k # 0 ; that is, g ( t ) itself mustsatisfytheNyquistIcriterion.Now,thebandwidthof H ( f ) is equal to the sumof bandwidths of n(f) Again assuming H(f)real, (29) can be written in the form and G Q ;thus, to minimize the bandwidth ofH(f), the bandC ( f ) = n(f) widthof G ( f ) has to beminimized.Hence, and h( t ) = (sin nt/nt)’. APPENDIX B
ne (1 -f)] sin 27rkf df and by Parseval’s theorem, we have
We now substitute (26), (28), and (30) in (19) and obtain where
0
elsewhere
Since the integral of (3 1) is over the range 0 < f < 1/2, H ( f ) and H(l - f ) are independentandmay be thought of as different functions. The contraint, written over the same integration limits is
2 [I2
[H(f) + H ( l - f)II,(l -f)]
df = 1 .
(32)
Appending the integrand of (32)tothe integrandof(31) with the Lagrange multiplier A, and differentiating the resultant integrand with respect to H ( f ) and H(l - f ) and setting when eachresult to zero, we obtaintwoequationswhich, solved simultaneously, give (22). The solution for H ( f ) gives the overall solution for f in the range 0 < I f 1 < 1/2 and the solution for H( 1 - f) gives the result for f in the range 1/2 < I ~ I +0 ~ 2 .
[I41 0.B. P. Rikert de Koe and P. Van der Wurf, “On some extensions of Nyquist’stelegraphtransmissiontheory,” Proc.IEEE, vol.57,pp.701-702,Apr.1969. [I51 S . Fan.“Numericaldesignofdatatransmissionfilters.”Ph.D. dissertation, Dep. Syst. Sci., Univ. California, Los Angeles, 1979. [I61 L. E.Franks,“Further resultsonNyquist’sprobleminpulse transmission,” IEEE Trans. Commun. Technol.,vol. COM-16, pp. 337-340,Apr.1968. [I71K.W. Cattermole,“Channelcharacteristics for digitaltransmission,” NATO Adv. Study Inst. New Directions in Signal Processing, Darlington, England, Aug. 1974.
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ACKNOWLEDGMENT The authors would like to thank Dr. M. S. Sablatash at the Communications Research Centre, Ottawa, Ont., Canada, for suggesting the research topic and for providing many technical reports and other literature in thegeneral area of teletext.
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REFERENCES SpecialIssueonConsumerTextDisplaySystems(Teletextand Viewdata), IEEE Trans.ConsumerElectron., vol.CE-25,July 1979. SpecialSection on TeletextandViewdata, Proc. IEE, vol.126, pp.1349-1428,Dec.1979. J . R . Storey, A. Vincent, and R . FitzGerald, “A description of the broadcast Telidon system,” IEEE Trans. Consumer Electron..vol. CE-26, pp. 578-586, Aug. 1980. Y. Numaguchi, S . Harada. and T. Uehara, “Experimental studies of transmission bit rate for Teletext signal in the 525-line television system.” IEEE Trans. Broadcast.. vol. BC-25, pp. 137-142, Dec. 1979. Dep. Commun. Canada, Broadcast Specifications BS-14, Issue 1, June 1981. M. J . Kallaway and W. A. Mahadeva. “CEEFAX: Optimum transmitted pulse shape.” BBC Res. Rep. RD 1977/15. V.P.Fasshauer,“OptimumsendsignalzurUbertragungVon Videotext,” Mittlungen, vol. 22, 1978. C . EilersandP.Fockens,“Teletexttransmissionpulseshape optimization,” IEEE Trans. Consumer Electron., vol. CE-27, pp. 551-554, Aug.1981. P. R. Hutt, “Oracle-A fourth dimension in broadcasting,” IBA Tech. Rev., London, England, Sept. 1976. R. W. Lucky, J . Salz, and E. J. Weldon, Jr., Principles of Data Communications. NewYork:McGraw-Hill,1968. W . R . Btnnett, IntroductiontoSignalTransmission. New York: McGraw-Hill,1970. F. Amoroso and M. Montagnana, “Distortionless data transmission with minimum peak voltage,” IEEE Trans. Inform. Theory, vol. IT-13, pp. 4 7 0 4 7 7 . July 1967. L. E.Franks, SignalTheory. EnglewoodCliffs, NJ: PrenticeHall, 1969, ch. 6.
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