Suboptimal filtering in a zero-guard, dicode PPMsystem operating over dispersive optical channels

M.J.N. Sibley

Abstract: Optimal filters for use in digital pulse position modulation (PPM) systems consist of anoise-whitened matched filter followed by a proportional-derivative-delay (PDD) network.Although the PDD network can be removed, with a small loss in sensitivity, the practicalimplementation of the remaining matched filter is complex. An alternative is to use suboptimalfiltering and the use of a third-order Butterworth filter in a zero-guard interval dicode PPM systemoperating over a dispersive optical channel is examined. Gaussian-shape received pulses areassumed, and a bandwidth-limited PIN–bipolar receiver with both frequency invariant and variantnoise is considered. This original analysis shows that the bandwidth of the Butterworth filter isrelatively independent of the channel. It is also shown that the performance of such a filter issuperior to that of a noise-whitened matched filter. It is concluded that a dicode PPM system can usea simple third-order Butterworth filter. The best predicted sensitivities when coding 1 Gbit/s PCMdata are 237.48 dBm in a high-bandwidth link and 232.24 dBm with a link bandwidth equal to 1.2times the data rate.

1 Introduction

Several pulse position modulation (PPM) schemes have beenproposed to utilise the available bandwidth of optical channels[1–15]. Such schemes give a greater sensitivity than standardPCM at the expense of a large bandwidth expansion factor,thus the final data rate can be prohibitively high [16]. Anexception is multiple PPM [12] which, although it can run atslower speeds than digital PPM, does require complexdecoding logic and this limits its appeal.

Dicode PPM is an alternative coding scheme that is verysimple to implement, as it uses only two data slots to codethe data transitions in a PCM signal [17, 18]. A 0-to-1 datatransition gives a pulse in the first time-slot, slot S for set(symbol S), while a 1-to-0 change gives a pulse in thesecond time-slot, slot R for reset (symbol R). Constant PCMdata, either 00 or 11, results in no pulses, symbol N. Incommon with other digital transmission systems line codingof the original data is used to increase timing content [19].Such coding also limits error propagation in dicode PPMbecause it restricts the maximum number of consecutivelike symbols [17]. At the decoder a decoding flip-flop iseither set or reset according to the position of the receivedpulse in a frame [17].

Early work on digital PPM considered the use of aclassical matched filter as the predetection filter [1–9].Although such a filter results in the best sensitivity thepractical implementation is difficult due to two things.The first is the requirement to whiten the noise using anoise-whitening filter before the signal is applied to thematched filter. This noise-whitening filter is physically

unrealisable due to the frequency characteristic of thepreamplifier noise given by (6). Secondly, the matched filtersection is matched to the shape of the received pulse whichdepends on the optical channel. Thus filters have to beconstructed for every link. In view of these difficulties it hasbeen proposed to replace the noise-whitened matched filterby a suboptimum filter based on a third-order Butterworthfilter [10] which is easily realised. Such filters are alreadywidely used in optical PCM links.

Previous analyses considered the use of a classicalmatched filter in an optical dicode PPM link. It wasshown that such a scheme offers a comparable, andsometimes better, sensitivity than digital PPM [17]. In thispaper the performance of a zero guard interval, dicode PPMsystem that uses a receiver with frequency variant noise anda third-order Butterworth filter is analysed. The operatingbit-rate chosen is 1 Gbit=s. The use of a zero guard intervalmeans the dicode PPM system operates at twice the speed ofthe original PCM and so bandwidth expansion effects arereduced. The use of a Butterworth filter greatly simplifiesthe design of the predetection filter. The performance of thesystem is compared with that of one using an ideal noise-whitened matched filter and it is shown that the Butterworthreceiver can operate over channels with bandwidths as lowas 1.2 times the bit-rate. Thus it can be used in links whichsuffer from high levels of dispersion. The effects ofintersymbol interference (ISI) are taken into account.

2 Receiver system

The receiver system used in the simulations, Fig. 1, consistsof a bandwidth-limited PIN–bipolar (PINBJT) transimpe-dance optical receiver, with frequency-variant noise [20].This receiver is followed by the predetection filter based ona simple third-order Butterworth filter. A classicalmatched filter is also considered.

Let the received pulse shape have the following property:Z 1

�1hpðtÞdt ¼ 1 ð1Þ

q IEE, 2004

IEE Proceedings online no. 20040769

doi: 10.1049/ip-opt:20040769

The author is with the Department of Engineering, Huddersfield University,Queensgate, Huddersfield, West Yorkshire, HD1 3DH, UK

Paper first received 31st July 2003 and in revised form 11th May 2004.Originally published online 26th July 2004

IEE Proc.-Optoelectron., Vol. 151, No. 4, August 2004 237

and be gaussian in shape, so that

hpðtÞ ¼1ffiffiffiffiffiffiffiffiffiffi

2pa2p e

� t2

2a2 ð2Þ

Thus the Fourier transform of the pulses is

HpðoÞ ¼ e�a2o2

2 ð3Þ

The pulse variance a is linked to the fibre bandwidth by [21]

a ¼ffiffiffiffiffiffiffiffiffiffiffi2 ln 2

pTb

2pfn

ð4Þ

where Tb is the PCM bit-time and fn is the channelbandwidth normalised to the PCM data-rate. The pulseshape presented to the threshold detector is given by

voðtÞ ¼b�q

2p

Z 1

�1HpðoÞZTðoÞHf ðoÞe jotdo ð5Þ

where b is the number of photons per pulse, � is the quantumefficiency of the detector, q is the electronic charge, ZTðoÞ isthe frequency dependent transimpedance of the receiver,and Hf ðoÞ is the transfer function of the filter.

In wide bandwidth systems the noise from the receiverusually has a double-sided, frequency-invariant term So anda component that rises with the square of the frequency [20].Thus the equivalent input noise current spectral densitySeqðoÞ can be written as [1]

SeqðoÞ ¼ So 1 þ o2

o2n

� �ð6Þ

where on is the frequency at which the o2 noise has thesame value as So: Thus the noise at the filter output is

hn2oi ¼

1

2p

Z 1

�1So 1 þ o2

o2n

� �jZTðoÞHf ðoÞj2do ð7Þ

Two different predetection filters were considered, a third-order Butterworth filter with variable bandwidth and a noise-whitened matched filter. For the Butterworth receiver, thelimited preamplifier bandwidth was accounted for. Taking asingle-pole frequency response for the preamplifier

ZTðoÞ ¼RT

1 þ joop

ð8Þ

where RT is the midband transimpedance and op is the�3 dB bandwidth of the receiver.

For the matched filter a compensating network wasintroduced so that the preamplifier bandwidth was largeenough so as not to affect the pulse shape. Thus the filter ismatched to the shape of the received pulse.

2.1 Butterworth filter

The frequency response of a third-order Butterworth filter is

Hf ðoÞ ¼1

ð joÞ3 þ 2ð joÞ2oB þ 2joo2B þ o3

B

ð9Þ

where oB is the �3 dB bandwidth of the filter given by

oB ¼ffiffiffiffiffiffiffiffiffiffiffi2 ln 2

p

akB

ð10Þ

and kB is the bandwidth factor. The filter bandwidthnormalised to the PCM bit-rate fBn is

fBn ¼ fnkB

ð11Þ

Analytical solutions to (5) and (7) are difficult to obtain andso it is preferable to solve them using numerical methods.

2.2 Matched filter

A matched filter requires white noise at its input and so thepreamplifier noise given by (6) must be whitened first. Thusthe frequency response of the noise-whitened matched filteris given by

Hf ðoÞ ¼1

1 þ o2

o2n

� �HpðoÞ� ð12Þ

where HpðoÞ� is the complex conjugate of the receivedpulse. The first part of (12) is due to the noise-whiteningfilter while the second part is due to the matched filter. Asthe received pulse is gaussian in shape, voðtÞ is given by [10]

voðtÞ ¼ b�qRT

on

4ea

2o2n 2 coshðontÞ � e�onterf aon �

t

2a

� �h�eonterf aon þ

t

2a

� �ið13Þ

with the noise given by [10]

n2o

¼ R2

T So

on

2exp a2o2

n

� �erfcðaonÞ ð14Þ

where erf(x) and erfc(x) are the error function andcomplementary error function of argument x.

3 Error probabilities

Dicode PPM uses four equiprobable symbols: S, R and twoN-symbols for 00 and 11 transitions. If an S-pulse has beentransmitted, only symbols R or N can follow it both withprobabilities of 1=2: The use of line coding means that theprobability of the next pulse being an R is 1 if the maximum

Fig. 1 Block diagram of dicode PPM receiver used in simulations

IEE Proc.-Optoelectron., Vol. 151, No. 4, August 2004238

run of like symbols has been reached. A similar argumentapplies if an R-pulse is originally transmitted.

In common with digital PPM, wrong-slot, erasure andfalse alarm errors can affect the detection of pulses [2, 3].If ISI is present, the precise pulse sequence will affect theerror probability.

3.1 Effects of ISI on error probability

When operating with ISI, the pulse shape and errorprobability is pattern dependent and so a general, repetitivesequence such as S xN R yN S must be considered. Herex and y are the number of N-symbols following an S- or anR-symbol, respectively. With reference to Fig. 2, the pulseshapes for the S- and R-symbols are found from

voSðx;y; tÞ ¼ voðtÞS pulse

þvoðtþð1þ 2yÞTsÞprior R pulse

þvoðt�ð3þ xÞTsÞpost R pulse

ð15Þ

voRðx;y; tÞ ¼ voðtÞR pulse

þvoðtþð3þ xÞTsÞprior S pulse

þvoðt�ð1þ 2yÞTsÞpost S pulse

ð16ÞOnly ISI between adjacent pulses has been considered dueto the time separation between pulses. The equivalent PCMerror probability is

Pe ¼Xn�1

y

Xn�1

x

1

2

� �xþ2 1

2

� �yþ2

Px;yErrorx;y

� �

þ 1

2

� �nþ1 1

2

� �yþ2

Pn;yErrorn;y

�

þXn�1

x

1

2

� �xþ2 1

2

� �nþ1

Px;nErrorx;n

� �

þ 1

2

� �nþ1 1

2

� �nþ1

Pn;nErrorn;n ð17Þwhere Px;y is the probability of a particular type of erroroccurring, and Errorx;y is the number of PCM errors theevent generates. The x and y limits depend on the type oferror being considered, while n is the maximum number ofconsecutive like symbols in the PCM data. This is limiteddue to the line coding of the PCM data.

The following Sections discuss the various detectionerrors and Table 1 lists the parameters needed to predict theequivalent PCM error-rate using (17).

3.2 Wrong-slot errors

Wrong-slot errors are generated by noise on the rising edgeof a pulse causing it to appear in the time-slot immediately

either before or after the current slot, Fig. 3a. This error isminimised if the detection time is in the centre of the timeslot of width Ts: Thus the probability of a wrong-slot errorPs is given by [1]

Ps ¼ 0:5erfc Qs=ffiffiffi2

p� �ð18Þ

where

Qs ¼Ts

2

slopeðtdÞffiffiffiffiffiffiffiffiffihn2

oip ð19Þ

and slopeðtdÞ is the slope of the filtered pulse at the thresholdcrossing instant td: The presence of ISI means the slopes ofthe S- and R-pulses are different and so the respective pulseshapes given by (15) and (16) must be differentiated withrespect to time. If the S-pulse appears in the preceding time-slot R this will not cause an error as the decoder is already ina reset state. If the false pulse appears in the followingR-slot, an immediate decoding error results and allfollowing PCM bits will be in error until a correct R-pulseis received. If the S pulse is followed by x N-symbols, thenumber of PCM errors is ðx þ 1Þ:

Noise could cause a pulse in slot R to appear in thepreceding or the following S-slot. In both cases ðy þ 1ÞPCM errors are generated.

3.3 Erasure errors

Erasure errors are generated if the amplitude of the pulse atthe decision time falls below the threshold level, Fig. 3b.The probability Per is given by

Per ¼ 0:5erfcQerffiffiffi

2p� �

ð20Þ

with Qer given by

Qer ¼vpk � vdffiffiffiffiffiffiffiffiffi

hn2oi

p ð21Þ

where vpk is the peak signal voltage after the filter and vd isthe pulse voltage at the decision time: the threshold voltage.The value of vpk depends on the precise pulse sequence andmust be obtained from (15) and (16). This error has the sameeffect on both S- and R-pulses and causes ðx þ 1Þ or ðy þ 1ÞPCM errors, respectively.Fig. 2 Dicode PPM pulse sequence set=reset=set=reset

Table 1: Parameter listing for predicting error probabilityfor wrong-slot, erasure and false alarm errors in zeroguard dicode PPM system with ISI

Type of error x y Errorx ;y

Wrong slot S! R 0 0 x þ 1

S R 0 0 y þ 1

R! S 0 0 y þ 1

Erasure S! N 0 0 x þ 1

R! N 0 0 y þ 1

False alarm R � 2Ts 1 0 1

R � Ts 0 0 y þ 1

Rþ Ts 0 1 y

S � 2Ts 0 2 1

Sþ 3Ts 2 0 x

N! S see text see text see text

N! R see text see text see text

R � 2Ts refers to the pulse slot that is two slots before the R-pulse; R � Ts

refers to the pulse slot that is one slot before the R-pulse, etc

IEE Proc.-Optoelectron., Vol. 151, No. 4, August 2004 239

3.4 False alarm errors

False-alarm errors occur when noise causes a thresholdviolation in a slot in which no pulse is present, Fig. 3b.The probability of this happening in a given time-slot is

Pt ¼Ts

tR

0:5erfcQfffiffiffi

2p� �

ð22Þ

where Ts=tR is the number of uncorrelated samples pertime-slot and tR is the time at which the autocorrelationfunction of the filter has become small; Qf is given by

Qf ¼vd � voISIffiffiffiffiffiffiffiffiffi

hn2oi

p ð23Þ

where voISI is the ISI-induced voltage present in the time-slot. This depends on the pulse sequence as well as the typeof false-alarm error being considered. The followingSections discuss the various false-alarm errors generated.

3.4.1 Pulse in slot R: A pulse in slot R couldcontribute to false alarm errors in slot R of the precedingframe two slots earlier at a time td � 2Ts: If this framealready contains an S-pulse, i.e. x ¼ 0; no decoding error is

generated because the S-pulse stops the decoder before thefalse alarm. However, a single PCM error results if theframe is empty, i.e. x � 1: The probability of this erroris found from (23) and (22) with the ISI-induced voltagegiven by

voðR�2TSÞ ¼ voRðtd � 2TsÞ ð24Þ

The R-pulse can also contribute to a false alarm in the S-slotimmediately before it one slot earlier at a time td � Ts: Thisresults in ðy þ 1Þ PCM errors with a probability found from(23) and (22) with an ISI-induced voltage given by

voðR�TsÞ ¼ voRðtd � TsÞ ð25Þ

ISI from the R-pulse can also generate following frame falsealarms but the false alarm must occur in slot S (one slot laterat time td þ Ts) of an empty frame, i.e. y � 1; to generate yPCM errors. The probability is found from (23) and (22)with an ISI-induced voltage of

voðRþTsÞ ¼ voRðtd þ TsÞ ð26Þ

3.4.2 Pulse in slot S: A pulse in slot S can producefalse alarms in frames before and after the pulse. With priorframe false alarms, errors will only occur if a SET signal isgenerated two slots earlier at time td � 2Ts: The sequenceR N S ðy ¼ 1Þ has already been accounted for in theprevious Section and so the lower limit for y is 2. The ISI-induced voltage is

voðS�2TsÞ ¼ voSðtd � 2TsÞ ð27Þ

The S-pulse can also cause false alarms in slot R ofthe following frame three slots later at a time td þ 2Ts: Thesequence S N R has already been accounted for in theprevious Section and so the lower limit for x is 2. The ISI-induced voltage is

voðSþ3TsÞ ¼ voSðtd þ 3TsÞ ð28Þ

3.4.3 No pulses: False alarms can occur in theinterval between pulses where there is no ISI. The decodingerror depends on where the false alarm occurs in a run ofN-symbols. If the sequence is S xN R, ðx þ 1 � kÞ PCMerrors result if the false alarm occurs on the kth N-symbol.A similar argument applies to the R yN S sequence. Thus theequivalent PCM error probability is

PeN ¼ 2Xn�1

x¼3

Xx�1

k¼2

1

2

� �xþ3

PNðx þ 1 � kÞ

þ 2Xn�1

k¼2

1

2

� �nþ2

PNðn þ 1 � kÞ ð29Þ

where PN is the probability of a false-alarm error in anempty slot without ISI obtained using (23) and (22) withvoISI ¼ 0: The factor of two appears in (29) because a falsealarm between S-and R-pulses gives the same errorprobability as between R- and S-pulses. The lower limit inthe summation is three because pulse spreading into thefirst and last frames of the sequence has already beenaccounted for.

4 System modelling

A variable bandwidth PINBJT receiver was considered,with a low-frequency input equivalent noise current spectral

Fig. 3 Occurrence of errors

a Wrong-slot errorb False-alarm and erasure errors

IEE Proc.-Optoelectron., Vol. 151, No. 4, August 2004240

density of 16 10�24 A2=Hz (double-sided) and an initialnoise corner frequency of 6 109 rad=s: These are typicalvalues obtained from a variety of manufacturer’s datasheets. An operating wavelength of 1:55 mm and aphotodiode quantum efficiency of 100% were taken andsimulations were carried out using a data rate of 1 Gbit=swith line coding that resulted in n ¼ 10:

The total equivalent PCM error probability is obtainedby adding together the individual probabilities and shouldbe the same as for a PCM system, taken as 1 error in109 pulses. The time at which the autocorrelation functionof the Butterworth filter becomes small is 8=oB; while forthe matched filter it is a: A threshold variable v wasdefined as

v ¼ vd

vpk

ð30Þ

The received pulse shape, derivative and noise can befound for a given set of parameters. With v as the variable,the decision time can be determined and the number ofphotons per bit b found. The required optical power isgiven by

P ¼ bh�n þ 1

8nB ð31Þ

where B is the bit-rate.

5 Results and discussion

Figure 4 shows the variation with fn in the number ofphotons per pulse required for a system with optimumreceiver and Butterworth-filter bandwidths. The results for amatched filter are also shown.

At fn ¼ 100 the Butterworth receiver requires 6,488photons per pulse whereas the matched filter requires6,587 photons, sensitivities of �36:40 and �36:33 dBm;respectively. Wrong-slot errors are minimal and so falsealarms and erasures determine the error-rate. For theButterworth filter the false alarms are mainly due to apulse in R spreading into the preceding and following Stime-slots (R� Ts and Rþ Ts; respectively), although theother sources are still significant. For the matched filterthe false alarms are mainly due to Rþ Ts: The variationin the threshold variable v is shown in Fig. 5. As seen, vis higher for the matched filter due to the increased ISI.

A high threshold increases erasure errors and reduces thesensitivity.

At a bandwidth of three times the bit-rate Fig. 4 showsthat the Butterworth receiver requires 7,373 photons perpulse ð�35:85 dBmÞ whereas the matched filter receiverrequires 14,021 photons ð�33:05 dBmÞ: With the matchedfilter the threshold rises to 0.705, due to ISI, and the errorrate is determined by wrong-slots and erasures, the latterbeing the largest due to the high value of v. For theButterworth filter the dominant error sources are erasuresand false alarms caused by Rþ Ts:

Figure 6 shows the normalised waveforms prior tothreshold detection for the two systems at a bandwidth of1.2 times the bit-rate. As seen, ISI makes it impossible to seta threshold level with the matched filter. The Butterworthsystem can still be used and Fig. 4 shows that it requires22,825 photons ð�30:94 dBmÞ with v ¼ 0:679: The error-rate is now determined by wrong-slots and the dominanterasure errors. Operation below this bandwidth is imposs-ible due to ISI.

Figure 7 shows the variation in optimum preamplifier andfilter bandwidth with fn: The receiver and filter bandwidthsare relatively constant at 1.25 and 2 times the PCM rate,respectively, until fn ¼ 3: Thus the optimum pulse shapepresented to the threshold detector is due to the recei-ver=filter combination because the optical pulses are nothighly dispersed. Below fn ¼ 3 the filter bandwidthincreases only slightly, whereas that of the receiver risessharply to 6.1 GHz at fn ¼ 1:2: This is due to the opticalpulse being highly dispersed and so the receiver bandwidth

Fig. 4 Variation in number of photons per pulse with normalisedchannel bandwidth fn

Fig. 6 Normalised received pulse shape for pulse sequence RSand normalised bandwidth of three times the bit-rate

Fig. 5 Variation in threshold parameter v with normalisedchannel bandwidth fn

IEE Proc.-Optoelectron., Vol. 151, No. 4, August 2004 241

must be increased to introduce less ISI, so as to try andmaintain the optimum pulse shape.

Figure 8 shows the effect of variations in the receiver andfilter bandwidths for fn ¼ 100 and 1.2. The presence ofvarious optima can be seen. With fn ¼ 100; operation awayfrom the optimum incurs little penalty. However, the filterbandwidth is critical due to the large amount of ISI presentin the link when fn ¼ 1:2:

Figure 9 shows the variation in optimum receiver andfilter bandwidths as a function of noise corner frequency on

for fn of 100 and 1.2. The bandwidths are highly dependenton on for fn ¼ 100 because there is little ISI in the linkand so minimising the noise is important. Conversely,minimising the ISI is more important for fn ¼ 1:2:Also shown in Fig. 9 is the required number of photonsper bit. For fn ¼ 100; the lowest number of photons is5,066 ð�37:48 dBmÞ whereas for fn ¼ 1:2 it is 16,913ð�32:24 dBmÞ:

6 Conclusions

This paper has analysed the performance of a dicode PPMsystem that uses a third-order Butterworth filter as thepredetection filter. The original analysis presented has takeninto account inter-symbol interference effects. It has beenshown that a receiver using the Butterworth filter requiresless photons per pulse than one using a noise-whitenedmatched filter. The Butterworth receiver can also operatewith lower channel bandwidths (1.2 times the PCM rate).

Simulations show that the bandwidths of the receiver andthe Butterworth filter are independent of the channelbandwidth, for normalised bandwidths greater than three,and this greatly simplifies the design of a link. It has alsobeen shown that, with a low bandwidth link, the receiver andfilter bandwidths are relatively insensitive to changes innoise corner frequency because the minimisation of ISI isthe criteria. With a high bandwidth link, ISI is low and sothe receiver and filter bandwidths must be adjusted tominimise the noise.

The best predicted sensitivities, when using a Butterworthfilter and 1 Gbit=s PCM data, are �37:48 dBm for fn ¼ 100;and �32:24 dBm for fn ¼ 1:2:

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a 100b 1.2

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IEE Proc.-Optoelectron., Vol. 151, No. 4, August 2004242

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