Performance and optimal presynchronisation filtering for direct-detection optical fibre PPM

J .M.H. Elmirg hani

Indexing terms: Direct detection systems, PPM jitter, Self synchronisation, Passive optical network

Abstract: A thorough performance analysis is presented for a self-synchronised direct-detection optical fibre pulse position modulation (PPM) system. A new hybrid erbium-doped fibre amplifier (EDFA) PPM system is analysed. Sensitivity results are presented for a system operating at 622Mbitis and h = 1.53p.m. These results indicate that EDFA PPM offers 7.9dB sensitivity improvement over a comparable PCM system and is beyond the PCM fundamental limit. An original expression is presented for the PPM power spectral density (PSD) under jitter. Original results are given for the extracted-slot clock timing variance and the resultant jitter wrong-slot error probability. The sensitivity penalty associated with self synchronisation is investigated and shown to vary between 0.15dB and 1.9dB for the range of parameters considered. An original presynchronisation filtering strategy is proposed and results are presented demonstrating its effect. The sensitivity penalty was bound to about O.ldB when the proposed presynchron-isation filter was utilised.

1 Introduction

Pulse position modulation (PPM) was proposed by Pierce [l] as a format that can significantly improve the photon-counting channel capacity. It was later studied over the nonideal satellite link [2,3] and over the disper- sive optical fibre link [4,5]. In the case of the optical fibre link, PPM can be used to exchange the abundant monomode fibre bandwidth for improved sensitivity [6-IO]. The sensitivity benefit achieved by using PPM rather than PCM in electrically preamplified optical fibre direct-detection links is typically of the order of 5-12dB. Over an optical fibre link this receiver sensitiv- ity improvement can be interpreted as an increase in the unrepeatered transmission length of the order of 60km assuming a fibre attenuation of 0.2dB/km, which is a typical present day figure [1 11. This increase in the unrepeatered transmission span is very attractive espe- cially in view of the present optical fibre link parame-

0 IEE, 1995 IEE Proceedings online no. 19952243 Paper first received 21st February 1995 and in revised form 14th July 1995

The author is with the Department of Electrical, Electronic Engineering & Physics, University of Northumbria at Newcastle, Ellison Building, Elli- son Place, Newcastle NE1 SST, UK

ters. For example, a link utilising PCM at a bit rate of 565 Mbitis will typically have a 95 km unrepeatered transmission span [ 121. This span generally decreases with an increase in the operating bit rate. The increased sensitivity also has important implications on the mul- tiuser environment. In this case the number of users served over a passive optical network (PON) can be significantly increased by utilising PPM.

Recently there has been an increased interest in using optical amplifiers to replace the electrical pre-amplifier in the optical fibre link. The advent of the erbium- doped fibre amplifier (EDFA) made this proposal more attractive owing to the desirable EDFA properties such as its high gain, good noise characteristics and the fact that it is bit rate and format transparent [13,14]. Sev- eral studies considered PCM with an EDFA receiver front end and have shown that the resultant direct- detection system offers sensitivities that can rival the coherent systems [15-181. In [19] a study was presented for an EDFA PPM system, however, no account was made of the system synchronisation requirements and its performance under jitter.

The PPM system synchronisation is more crucial and demanding than the equivalent PCM system synchroni- sation. This is because of the PPM format temporal nature where information is transmitted by the discrete pulse position in the time frame. The performance studies reported for optical fibre PPM [4-IO] and hence the sensitivities predicted were all based on the assump- tion of perfect transmitter-receiver synchronisation. Synchronisation for PPM deployed over the satellite link was considered in the literature [20-231, however, the satellite and the optical fibre PPM systems have different structures and are subjected to different con- straints and error sources. Methods of achieving slot and frame synchronisation in optical fibre PPM were previously reported [24-26], however, no study was conducted on the implications of self synchronisation on the optical fibre PPM system.

In this contribution an original performance analysis of a self-synchronised EDFA PPM system is presented. The system is shown to offer 7.9dB sensitivity improve- ment over a comparable PCM system. An original expression is derived for the PPM power spectral den- sity under jitter. Use is made of the format cyclosta- tionary properties in deriving the spectral characterisation. A linearised phase-lock loop (PLL) model is utilised to evaluate the timing variance of the extracted-slot synchronisation. Original results are pre- sented for the PPM jitter wrong slot error (JWSE) probability associated with self timing for both pattern- dependent and receiver-noise-generated jitter. This new error source is added to the known optical fibre PPM

259 IEE Proc.-Optoelectron., Vol. 142, No. 6, December 1995

error sources and hence the performance degradation owing to self timing is assessed. The performance deg- radation is interpreted in terms of the two major sys- tem parameters namely the system sensitivity and bit error rate (BER). The sensitivity penalty is shown to vary between 1.9 and 0.15dB for a system with a con- figurable coding level in the range of parameters con- sidered. In the case of fixed coders and decoders, the penalty is shown to be as high as 15dB. A novel pre- synchronisation filtering strategy is proposed and ana- lysed in the context of the self-synchronised PPM system. Using the proposed filtering strategy is shown to limit the jitter-associated penalty to about 0.1 dB for a configurable system and to about 1 dB for a fixed sys- tem.

t erasure wrong slot

error \ ideal pulse

siot ' guard ' t w band

I- I-

allowed band I _ 4 -I PPM frame

PPM signal and error sources Fig. 1

PPM model and error sources

PPM encodes A4 bits of information as the discrete time-position of a single light pulse placed in one of n = 2M possible time slots in a frame of duration Ty A guard band is left at the end of each frame to cater for pulse dispersion and hence avoid interframe interfer- ence (IFI). The ratio of the allowed transmission time to the overall frame duration defines the modulation depth m, where 0 < m < 1. The PPM signal structure is shown in Fig. 1. At the receiver the PPM pulses are optically preamplified and then detected using a PIN photodetector which produces equivalent electrical pulses. The electrical pulses are amplified and then threshold detected to determine their position. At an instant td the received current io(t) crosses the threshold level id with positive slope. Comparison of the time t d with the extracted timing determines which slot con- tains the PPM pulse. The decoder decision (assuming perfect timing signals) can be in error due to one of three error sources: false alarms, erasures and wrong slot errors

2.7 False-alarm errors A false-alarm error occurs when noise in the leading part of the frame (i.e. before the pulse arrival) crosses the threshold level. The event is shown in Fig. 1. The probability of a threshold violation for a sample of the receiver current, assuming that the receiver noise is a Gaussian random variable, is

pt = -erfc 1 { $} 2

where erfc{.} is the complimentary error function and

in which id = io(td) is the receiver output current at the threshold crossing instant td, and < no(t)2 > is the mean square receiver output noise current. The number of uncorrelated samples per time slot can be estimated in terms of the time T~ at which the receiver autocorrela- tion function has become small as {mTflnz,}. The prob- ability per time slot of a false alarm error is then approximated by [27]

2.2 Erasures An erasure error occurs when noise destroys the PPM pulse, thus rendering detection impossible. The event of an erasure error is shown in Fig. 1. The probability of an erasure error Pr is

where

(4)

and ip = io(tp) is the peak output receiver current and occurs at tp. Eqn. 4 gives the probability that a noise spike causes the pulse to fall below the threshold level. This probability applies to a single instant in time and the results over estimate the erasure errors. The results are, however, satisfactory since estimates of the PPM sensitivity will always be on the conservative side.

2.3 Wrong-slot errors A wrong-slot error occurs when noise on the leading edge of the pulse causes a threshold crossing in the time slot immediately before or after the proper slot. A wrong-slot error is illustrated in Fig. 1. The probability of such an error P, is

where

2.4 Performance criterion To compare the PPM system with an equivalent PCM system, an equivalent binary error probability P, is specified. In the case of PPM, assuming that all the PPM symbols are equally likely, then the probability of a PPM symbol error is

( 8 ) n - 1

2 Pes 2 P, + Ps + - Pf The average binary error probability is related to the average symbol error probability by [28]

(9) n

2(n - 1) Pe = ~ Pes

and so the performance criterion becomes

In general, P, is specified for optical fibre communica- tion systems as Pe =

IEE Proc -Optoelectron., Vol. 142, No. 6, December 1555 260

3 Receiver model and sensitivity

The self-synchronised EDFA PPM system is shown in Fig. 2. The system consists of two main subsystems, a detection subsystem and a synchronisation subsystem.

matched threshold binary ’f i l ter - detection - decoder -code

device

Fig.2 Self-synchronised EDFA PPM system

The details of the synchronisation subsystem are dis- cussed in Sections 4 and 5. The detection subsystem consists of an EDFA front end of gain G and sponta- neous emission parameter nsp followed by an optical bandpass filter (OBPF) of bandwidth B centred at the signal frequency v. The detector is a PIN photodiode of quantum efficiency q. This is followed by an electrical preamplifier, a matched filter, threshold detector and finally the PPM to PCM decoder. The amplified spon- taneous emission (ASE) noise PSD output from the optical amplifier can be written as

SASE 1 mtn,,hv(G - 1 ) ( 1 1 ) and the optical noise power incident on the photodetec- tor can be written as

PN = mtn,,hv(G - l ) B (12) where mt is the number of orthogonally polarised modes which is two and h is Planck’s constant. Follow- ing the approach in [19], the noise power spectral den- sity in the electrical domain can be written as

SN = -{2PS,,n,,(G - 1 ) + mtnz,hv(G - 1)2B v2q2

+ P,,, + mtn,,hv(G - 1 ) B ) + Spa hv

(13) where q is the electronic charge and q was taken to be unity. Spa is the electrical preamplifier noise PSD and Pslg is the average signal power incident on the photo- detector:

bhv p . -- ,til - m

1 s

where T, = mTfn is the PPM slot duration and b is the number of photons in the pulse. Assuming the received pulse shape can be approximated by a Gaussian pulse shape, the output current from the photodetector can be written as

t 2

where a is (15)

in which f represents the normalised fibre bandwidth with f = 1 indicating that the PPM system is consuming the same bandwidth as the PCM system cf = 2 twice, etc..). A4 is the number of information bits transmitted per PPM frame. The EDFA PPM system has a noise that depends on the signal as is clear from eqn. 13. The

IEE Pvoc -0ptoelectron , Vol 142, No 6, December 1995

system false alarm errors occur when the signal is absent, the value of SN under this condition is denoted by S$, where S$ is

+ mtn,,hv(G - 1)B) + Spa (17) and the equivalent noise at the output of the matched filter N$ is

N4 = - s4 27r Jm IG,(4I2d-d (18) -03

where GJo) is the matched filter transfer function. The false alarm error probability can then be evaluated from eqns. 1-3 using a conservative estimate of zr as zr = a [8]. Erasure errors occur when the PPM signal is present. Under this condition the noise is as given by eqn. 13, this will be denoted by Ss,g. The noise at the output of the matched filter in this case Nslg is

N,,, = 27r 7 IGP(w)l2h (20) -CO

thus the erasure error probability can be evaluated from eqns. 4 and 5. As wrong-slot errors (WSE) occur in the slots which are directly before or after the slot containing the pulse, a conservative estimate of the WSE probability is given by using the noise when a pulse is present. The WSE probability can be written as

P, = erfc { $} where

The derivative of the output current can be written as

the peak receiver current is bq

2a f i a p = -

and the current at the detection instant A2

the detection time td is

(25)

td is

Under perfect synchronisation the receiver sensitivity was evaluated by using the following algorithm: (1) From knowledge of the received pulse shape and the receiver noise PSD, the system parameters ip, id, t d , i’(td), iV+ and Nsig are evaluated for a given pulse energy b and threshold detection time td or current id.

26 1

(2) A numeric optimisation routine then searches in the b > 0 and 0 < id < ip space for the smallest value of b that meets the performance criterion in eqn. 10. (3) The receiver sensitivity in photons per information bit can be obtained by dividing b by M . For sensitivity evaluation the system was specified to operate at an error rate of To enable comparison with an EDFA PCM system, the practical parameters reported in [15] were adopted in this study. The rele- vant parameters are shown in Table 1.

Table 1: EDFA PPM receiver parameters

Wave I en gt h Bit rate 622Mbit/s EDFA gain 27dB OBPF bandwidth B 76.14GHz Spontaneous emmission parameter nsp 1.3 Electrical preamplifier noise S,,

1.53pm

1 . 2 5 ~ 1 0 - ~ ~ A2/Hz Quantum efficiency 1

“a 101)

information bits

Fig. 3 PPM sensitivity against coding level with ideal synchronisation ____ J = 1

f = 3 -....... f = 5

f= IO

_ _ _ _

~~

photons per bit

Fig.4 ____ PCM

- - _ _ PPM, f = IO

EDFA PPM and EDFA PCM systems BER

PPM, f = 5 -~

The sensitivity of the EDFA PPM system in the absence of jitter is shown in Fig. 3 as a function of the coding level M . As the coding level is increased, the single PPM pulse is used to convey more bits of infor- mation ( M bits), thus the sensitivity of the system improves. As the PPM coding level is increased, the PPM frame is continuously subdivided into finer subdi- visions (n = 2M slots). Accordingly the time slot dura-

262

tion decreases and the PPM probability of wrong slot errors increases. As a result the sensitivity degrades at large values of M. The system sensitivity improves as the normalised fibre bandwidth increases as depicted in Fig. 3. This is because with the large fibre bandwidth values less dispersed PPM pulses can be accommo- dated. The system bit error rate performance is evalu- ated in Fig. 4 at f = 5 and at f = 10. The equivalent optically preamplified PCM system BER is also shown in Fig. 4. The PCM system performance was evaluated using the results in [17].

The performance improvement achieved by using PPM rather than PCM is best illustrated by comparing the results of Fig. 3 with the widely available EDFA PCM results. The best predicted sensitivity as depicted in Fig. 3 (in the absence of jitter) is 17.9 photons per bit cf= 10, M = 5) . The sensitivity predicted for EDFA PCM using the same parameters and the closed solu- tion in [17] is 110 photons per bit to which the experi- mental 152 photons per bit result in 1151 can be compared. The EDFA PPM system thus offers a 7.9dB sensitivity improvement over the EDFA PCM system. The best predicted sensitivity for ideal optically pream- plified PCM comes from a study by Henry [18] who predicted 38 photons per bit. The EDFA PPM hybrid offers a sensitivity that outperforms this fundamental limit for PCM.

guard band

slot H

I- allowed band I< 4 PPM frame

Fig. 5 Jittered PPM signal

4 Synchronisation and jitter

The jittered PPM signal is shown in Fig. 5. Decoding the PPM message at the receiver consists mainly of identifying which PPM slot (out of the n slots) contains the pulse. A threshold level is set and the instant td at which the rising edge of the pulse crosses the threshold level is used to identify the pulse position. This is done by comparing td with the extracted timing signal. In the presence of receiver noise the pulse can be distorted and can hence cross the threshold before or after the ideal instant td. The timing error at the threshold cross- ing at the kth PPM lrame is denoted by el,. This jittered pulse position modulation format can be expressed mathematically as

w

k = - w where tk is a stochastic wide-sense stationary variable that represents the PPM data. Using the notation

TI, = t k + e k

results in w

k T f - I,=-m

IEE Pvoc.-Optoelectuon., Vol. 142, No. 6. December 1995

Under the absence of line coding and memory effects, it can be assumed that the PPM data tk and the jitter ek are each a set of independent identically distributed (iid) random variables. Further, the jitter ek is associ- ated with the PPM pulse. The PPM pulses are always present (ignoring decoding errors) and are transmitted in a single-pulse-per-frame fashion regardless of the data tk being conveyed. Therefore it is reasonable to assume that under PPM signalling, ek and tk are inde- pendent of each other. Accordingly, the digital PPM pulse train in eqn. 30 can be viewed as a stationary process subjected to a repetitive periodic processing operation. This can be treated as a cyclostationary process [29-3 11. According to the Wiener-Kintchine theorem, the power spectral density (PSD) of such a process can be evaluated by first of all determining the autocorrelation function. Rather than the classic time averaging, statistics of cyclostationary processes can be derived by forming a related stationary process from which time invariant statistics can be obtained directly [32]. The related stationary process can be obtained by randomising the time reference for the individual reali- sations of eqn. 30, then averaging the required statistics over the ensemble formed by the set of realisations.

Introducing the phase randomising variable 0 uni- formly distributed in (0, Tf}; the equivalent stationary process will be

00

k = - m

The autocorrelation, the statistic of interest, can then be evaluated by averaging over the ensemble in the period {0, Tj-}. Further, due to the time invariant prop- erties the general correlation R(z) can be expressed as

00

k = - w

where Rk(7) is the correlation between two frames, k frames apart, and is given by

N = nim being the total length of the frame in slots. Eqn. 32 represents frame decomposition while eqn. 33 represents slot decomposition. The partial autocorrela- tion &(7) is

= (g(t0 + o ) g ( t k + T + d))To,Tb,O (34) 8 is random and uniform in {0, Tf}, To represents the jittered pulse position in frame 0 and Tk the jittered pulse position in frame k . Making the ensemble aver- ages explicit gives

&(7) = J'd0 J' dTo J' d T k /U dt p ( T ) * p(To, T k )

Tf Tf Tf

T f o 0 0 -00

x g ( t - To + 0 ) g ( t - T k + 7 + 0) (35 )

in which * denotes convolution, p(T) is the threshold crossing variation probability, and p(To, Tk) is the joint PPM frame distribution. Assuming that to and tk are a set of independent identically distributed random varia- bles, and also that eo and ek are, then

dT0 > T k ) = d T 0 ) d T k ) (36)

For digital PPM the probability is discrete by virtue of the allowed pulse positions, hence

(37) where ai and al are array elements that represent the probability distribution in the frames.

( 3 8 ) The pulse shape g(t) can be represented by an indicator function:

g ( t ) = X ( - + , + ) (4 (39) where

1 i f t E I

0 otherwise X ( I ) ( t ) = {

The correlation integral becomes 00 1 dt g( t - TO + 0 ) g ( t - Tk + T + 0)

T f CO

00

-00

00

= - J ' d t 1

-00 Tf

\ O otherwise (43)

G R,(T + T k - T o ) (44) 0 vanishes and the corresponding integral is reduced. Therefore

Tf Tf

R ~ ( ~ ) = 1 1 d~~ 1 d~~ R J ~ + T~ - T,,) 0 0

T f

taking the Fourier transform of eqn. 45 and invoking the relation in eqn. 32

k = - m where

00

W,(f) 5 .I' e-jzTfTRk(T)dT (47) -00

so that the signal power is

and for k # 0 W O ( f ) = W S ( f ) (48)

IEE Proc.-Optoelectron , Vol. 142, No. 6, December 1995 263

(49)

J (50)

x ,-na.S(To-Tle)Wg(f)dTgdTk

1 (52)

(53)

W , ( f ) = IG(f)12 (54)

G ( f ) .F, d t ) (55)

= -w,(f)Pp(-27rf, 27rf) T f

where

and in general

Fi2)(<1, (3) = / dTo J' dTkpz(T0, Tk)e3(E1To+E2Tk)

Eo Ek (56)

where Fk(2)(c1, t2) is the two-dimensional characteristic function of the joint probability distribution P2(To, Tk), Eo and Ek are the sample spaces for To and Tk, respec- tively. Under the conditions: (i) PPM pulse positions are independent and identically distributed (ii) Time jitter in the different PPM frames is independ- ent and identically distributed (iii) PPM pulse positions and the time jitter are inde- pendent then

and Pz(To,Ti,) = Pl(To)Pl(Tk) (57)

(58) Fp(-2nf , 2nf) = / F ( ' ) ( 2 n f ) l 2 With no line coding and when the receiver noise is the source of timing jitter conditions eqns. 1-3 hold. When k = 0, Ro(z) can be evaluated separately in a similar manner

" 0 0 -cc

x g ( t - To + T f e ) (59)

which gives . N-1

where the sum of the a, probability terms is unity. Therefore at k = 0

Rg(7) A Wdf) (62) The PSD is then given by

adding and subtracting the term k = 0

( 6 6 )

Iww)12 = IB(2;.f)121J(2~f)12 (67)

where

IB(27~j)1~ is the characteristic function due to the data probability distribution, 1J(27~fl1~ is the characteristic function due to the jitter probability distribution.

B ( 2 r f ) = E eJ2Tf'G (68)

~ ( 2 n f ) = c a,eJ2Tf2$ (69)

i * I where E{ } denotes expectation. Therefore

N-1

,=O

~ ( 2 . f ) = p ( ~ ) ~ - ~ ~ ~ f ~ d ~ (70) -00 sa

where it is assumed that p(T) is continuous. In particu- lar, if p(T) is Gaussian,

J ( a n f ) = e - $ & 2 T f ) 2 (71) 264 IEE Puoc.-Optoelectron., Vol. 142, No. 6, December 1555

02 is the threshold crossing timing variance and can be derived from knowledge of the receiver noise.

The PPM PSD as in eqn. 66 consists of two terms, a continuum which is the first term and discrete frame- rate components represented by the second term. If N = n/m is the overall frame duration in slots, then the Nth frame-rate discrete component (i.e. N/Tf> is the slot-rate discrete component. As such the second term of eqn. 66 is the desired discrete component observed by the slot phase-lock loop (PLL) while the first term represents the PPM modulation noise B(27cj) and the receiver jitter effect J(2nj). Both effects i.e. B(2nj) and J(2nj) appear as noise for the slot PLL. The input dis- turbance to the PLL is cyclostationary. As such, an exact analysis for the timing variance at the output of the PLL can be carried out using the method in [33]. Simplification can be introduced by utilising a line- arised PLL model [34], such that the timing variance is expressed as

where BL is the PLL bandwidth, N is the noise power and P, is the discrete power at the PLL input. N and P, are as given in eqn. 66 and discussed previously. The results obtained from eqn. 72 have been shown to over- estimate the timing variance [33]. However, it should be noted that the aim of this and the following Sections is to evaluate the timing impaired performance, thus selecting the PPM system and PLL parameters that will minimise the impairment. The simplification introduced by utilising eqn. 72 is seen to be justified in view of the fact that the results will be on the conservative side. Previously several authors [20-231 have utilised the model in eqn. 72 in conjunction with the timing analy- sis of free space avalanche photodiode (APD) PPM systems. Good agreement was obtained between this model and practical measurements [20,21] and simula- tion [23]. From eqns. 66 and 72, the timing variance can be written as

(73) F 3 - 7 BL

where F, is the slot frequency and the noise N (given by CO) is not assumed white. The function Cy> is the continuum part of the spectrum and the power P, is the discrete slot rate power. These are defined as follows

and

(75)

where B(2nFJ is dropped from the last equation as it is unity.

The following analysis assumes that the pulse shape transform exhibits no singularity at the slot frequency, i.e. is not rectangular full slot width. Typically, the optical fibre PPM received pulse shape is dispersed and is Gaussian type [SI with no singularity at the slot rate

To evaluate o:, knowledge of the receiver noise is essential to evaluate the jitter characteristic function J(2nj). The effect of the receiver noise on the pulse is shown in Fig. 6. The output is a combination of the

1241.

IEE Proc -0ptoelectron , Vol 142, No 6, December 1995

signal y(t) and the noise n(t). Assuming a linear slope in the neighbourhood of the crossing instant td , the timing error Atd owing to the noise n(td) is

Y'(td) being the slope near the instant td. Accordingly the statistics of At, (ek) correspond to those of n(td) scaled by Y'(td). In particular, the variance of Atd (ek) is given by

(77)

where the noise when a signal is present NslS is assumed in the case of jitter. The jittered extracted slot clock can amount to decoding errors. These errors arise if the clock jitter is large enough to cause an ideal pulse to be interpreted as being in the slot before or after the proper slot. This new PPM error source is referred to as the jitter wrong-slot error (JWSE).

threshold level received pulse -/

I td

Fig.6 Threshold crossing timing error

To evaluate the JWSE probability (which is due to the slot synchronisation impairment), the following steps must be followed: (1) From knowledge of the receiver, the signal and noise are evaluated. (2) The threshold crossing timing variance 04 is evalu- ated from eqn. 77. (3) 04, the PPM number of slots, the modulation index and the data probability distribution should then be used with the jittered PPM PSD in eqn. 66 and the lin- earised PLL model in eqn. 73 to yield the timing vari- ance at the PLL output 0:. (4) The deterioration in the PPM bit error rate can then be expressed as an additional error probability Pss:

Ps, = erfc { $} where

Q?, = { ") (79)

When the PPM JWSE probability is taken into consid- eration, the new performance criterion to be met is

The receiver-sensitivity evaluation algorithm of Section 3 has now to meet the new performance criterion. By including the new error source Pss, the system perform- ance in the presence of slot clock jitter can be assessed. The algorithm now searches for the smallest b that meets the new criterion. The receiver sensitivity being

265

evaluated as was done in the case when jitter was ignored. Before evaluating the system performance under jitter, it is instructive to determine the variance of the extracted slot clock and the JWSE probability. It is also of great significance to determine the depend- ence of the variance and the JWSE probability on the system parameters. Throughout the analysis, the PPM data probability distribution was assumed uniform (i.e. equally likely PPM slots were assumed) and the modu- lation index was selected to be m = 0.8.

0

hl TJ e aJ U

a $ 0 0 .- E c

Y

0 U - 2 10

t m

,A / /

,/' / /

/ 1 ,/' / / ,'

/ / ,/'

5 0 ~ 1 0 - ~ normalised PLL bandwidth

Fig.7 generated jitter ~ n = 4

n = 8 n = 16

E pattern-dependent jitter A pattern and receiver jitter,f& = 5

Extracted slot clock timing variance with pattern and receiver

~~~~

~~

The evaluated timing variance for the extracted slot clock is shown in Fig. 7 for various PPM orders for pattern and pattern and receiver jitter. Pattern jitter is that due to the PPM self noise (the data characteristic function). This was modelled by setting the threshold crossing timing variance 0; to zero hence J(2xA = 1. The importance of pattern-dependent jitter is that it is inherent to PPM and sets the bound to the best expected performance even in the absence of receiver noise. In Fig. 7, as the PLL bandwidth is reduced the timing variance is reduced. Similarly, reducing the PPM order (i.e. improving the mark-to-space ratio in the PPM train) reduces the timing variance. In all instances the receiver noise adds to increase the timing variance at a given PLL bandwidth and PPM order.

10-8 I

1x10-3 50x10-3 1 oox 10-3 normalised PLL bandwidth

Fig.8 bandwidth as parameters

Slot variance against P L L bandwidth with PPM order andfibre

~ f 3 d B = f 3 d B = f 0 h d B =

~~~~

~-

The effect of the fibre bandwidth v> on the extracted slot clock timing variance is illustrated in Fig. 8. As

266

the fibre bandwidth is increased, the variance o€ the extracted slot timing is reduced. This can be explained by observing that as the fibre bandwidth is increased, sharper PPM pulse rising edges can be realised. Accordingly, the threshold crossing timing variance is reduced resulting in reduction in the extracted clock timing jitter. Fig. 8 illustrates the impact of the fibre bandwidth as this is increased from 1 to 10. The net benefit obtained by operating at larger fibre bandwidth is not very large. This is to be expected since operating at large fibre bandwidths has two effects. First, as the fibre bandwidth is increased sharper pulses can be accommodated and hence a reduction in the threshold crossing timing variance results. The second effect which is not desirable is that of admitting more noise as the fibre bandwidth is increased. As Fig. 8 illus- trates, there is a net benefit in operating at the larger fibre bandwidth values.

Fig.9 Wrong-slot error probability against P L L bandwidth atSjdB = 5 ~ n = 1 6

n = 64 n = 286

E pattern-dependent jitter A pattern and receiver jitter&, = 5

~~~~

_ _

1 0 - ~ 10-1 IO-' normalised PPL bandwidth

Fig. 10 bandwidths

Wrong-slot error probability at various PPM orders andfibre

~ p:; _ _ _ _ _ _ A:= 10

Fig. 9 shows the jitter wrong-slot error (JWSE) probability variation with the synchronising PLL band- width for various PPM orders under pattern jitter only and pattern and receiver noise generated jitter. Reduc- ing the PLL bandwidth results in reduction in the jitter WSE probability as does reduction in the PPM order.

IEE Proc -0ptoelectron , Vol 142, No 6, December 1995

The case of pattern and receiver noise always demon- strating a higher error rate as anticipated from the results of Fig. 7.

The effect of the fibre bandwidth on the JWSE prob- ability is demonstrated in Fig. 10. Increase in the fibre bandwidth, reduction in the PPM order and PLL bandwidth all lead to reduction in the JWSE probabil- ity. The results also follow from the timing variance values in Fig. 8.

I

/ /

"'1 2 3 4 5 6 7 8 9 PPM coding level,M

Fig. 11 ~ no jitter

PPM system sensitivity under jitter

B J ~ = 4 x 10-3 ~ ~ _ _ B , T ~ = 20 x 10-3 -~

B;T; = 60 x 10-3 ~ .. .. BLTs = 10''

photons ber bit

Fig. 12 ~ no jitter

BER ofjittered PPM system

B,T, = 12 x 10-3 -~~~ B,T, = 17 x 10-3

B ~ T ; = 9 x 10-3 B ~ T ~ = 15 x 10-3 B,T, = 20 x 10-3

~- ~. ~.

The results of Figs. 7-10 are interpreted in Figs. 1 and 12 in terms of the system sensitivity and BER, respectively. As observed in Figs. 11 and 12 the PPM system performance deteriorates significantly under jit- ter. The mechanism involved can be understood by observing that the increase in error probability (due to the additional JWSE probability) is compensated for by reducing the value of the three inherent PPM error sources previously mentioned (eqn. 10). This requires more signal power and hence a degradation in sensitiv- ity results.

At low PPM coding levels (values of M), the frame is subdivided into few subdivisions (n = 2M), hence the PPM slots are wide and the JWSEs are not significant in this region. The result is clear in Fig. 11 and appears as a very low sensitivity penalty at low values of M. As the coding level is increased, the PPM slots get nar- rower and start to be of durations comparable to the

IEE Proc.-Optoelectron., Vol. 142, No 6, December 1995

magnitude of the slot jitter. In this region, the JWSEs become significant. This is clearly seen in Fig. 11 on observing the jittered f = 10 curve. The JWSEs can become significant (even at lower coding levels) if the PPM and PLL parameters are such as to make the jit- ter magnitude large and comparable to the slot dura- tion. This effect is shown in Fig. 11 in the f = 3 jittered region. From this it is apparent that attention should be given to selecting the EDFA PPM and PLL param- eters to minimise the jitter penalty.

The BER penalty due to jitter is shown in Fig. 12 at A4 = 5, f = 5 and at A4 = 5, f = 10. The jitter penalty is reduced by reducing the PLL bandwidth and by oper- ating at larger fibre bandwidths. As an example (with reference to Fig. 11) under slot clock jitter and at f = 10 and BLT, = 4 x the best predicted sensitivity is 18.5 photons per bit. This represents a sensitivity pen- alty due to jitter of 0.15dB which is very small. How- ever at f = 3 and BLT, = lo-' the sensitivity dropped from 48.5 photons per bit to 14.3 photons per bit, a penalty of 1.9dB. Therefore it is of great importance to select the proper combination of phase-lock loop band- width and normalised fibre bandwidth at a given PPM order if the jitter penalty is to remain bound. In many practical situations the available fibre bandwidth may be limited and a wide PLL bandwidth may have to be selected to meet the dynamic performance require- ments, e.g. lock range and acquisition range [34]. Under these conditions it is desirable to realise a self- synchronisation subsystem whose performance can be optimised independent of the fibre bandwidth and PLL bandwidth. The following Section proposes a presyn- chronisation filtering technique that can be utilised under these conditions.

5 Optimum presynchronisation filtering

The topology of the EDFA PPM system receiver is shown in Fig. 2. The power at the output of the electri- cal preamplifier is split among the decoding and the synchronisation subsystems. The front end of the syn- chronisation subsystem is a filter whose role is to mini- mise the threshold crossing timing variance. This minimises the jitter characteristic function J(2~cf) and accordingly the effect of timing jitter on the PPM system. The threshold crossing timing variance 02 depends on both the receiver noise O? and the pulse slope y'(td). Therefore the optimisation network design should include a filter that minimises the effect of both 02 and y'(td). The input to the filter is the PPM signal x(t) , the filter transfer function and output signal are h(t) and y(t) , respectively. Now

Y W a m f Y ( f ) (81) therefore

CO

Putting t d = 0, without loss of generality, it suffices to minimise the integral function 1, given by

1 2 = .i" Sn(f)lH(f)l2df

(85) -cc

+ 7 j2nf Ix( f ) l ,H( f ) le '~( f )d f --w

where

X ( f ) = IX( f ) l e "X( f ) (86)

H ( f ) = I H ( f ) / & H ( f ) (87)

d f ) = $x ( f ) + $H (f) (88)

(89)

and h is a Lagrange multiplier. I2 is minimised only if

~ s , ( ~ ) ~ H ( s ) I + ~j2nf1x( f ) l ey@(f ) = o

10-4-

ai c

2 2 L

ai c n

The result of eqn. 92 is valid to within a scalar multi- plier. Putting t+rHcf> = - vXv) serves to ensure a linear phase characteristic for the output spectrum. Thus

(93)

which for white noise reduces to H ( f ) = j 2 T f X * ( f ) @ h(t) = d - t ) (94)

Therefore the optimum filter that minimises the thresh- old crossing variance hence optimising slot synchroni- sation is a filter matched to the derivative of the input signal and an inverse weighting (in the frequency domain) by the noise spectral density function is required for a nonwhite noise environment.

The noise PSD of the EDFA and the electrical preamplifier considered is white in the frequency range of interest. Therefore the presynchronisation filter is reduced to a matched filter in cascade with a derivative network. Further simplification in the receiver topology is possible if the synchronisation subsystem is caused to derive its input from the output of the matched filter of the decoding branch. In this case the presynchronisa- tion filter is reduced to a derivative network. The pulse slope at the output of the presynchronisation filter is given by

Therefore the threshold crossing timing variance 05 when the filter is included is

where < no(t)2 > is the noise PSD at the output of the presynchronisation filter. When the presynchronisation filter is included, the receiver sensitivity can be evalu- ated by first determining the threshold crossing timing variance 0 4 ~ Following this, the timing variance of the extracted slot clock is evaluated as in eqn. 73. This ena- bles the evaluation of the extra error source P,, as in

268

eqn. 78. The performance criterion to be met under jit- ter is then that given in eqn. 80.

104

103 - .- I n ai : 102 c 0

L

c

a 10'

7 2 3 L 5 6 7 8 9 PPM coding level, M

Fig. 13 ~ no jitter

Jittered PPM sensitivity with and wirhout optimaZfiltering

ELTs = IO-' jittered .. .. BLTs = IO-' opt. filtered

BrTs = 60 x loe3 jittered BLTS = 60 x BLTs = 4 x jittered

opt. filtered -~ _ _ _ _ BLTS = 4 x opt. filtered

10-12 i 100

pho tons ber bit Fig. 14 - no jitter

Jittered and optimally filtered PPM BER

BLFs = 15 x jittered BrTs = 15 x

_ _ B,T, = 20 x iittered optimally filtered

~- BLTi = 20 x IOF3 bpt. filtered

The filter was included in the analysis and new values were evaluated for the threshold crossing timing vari- ance, the PLL timing variance, the error probability owing to the slot clock jitter Pss and the PPM system sensitivity. The PPM system sensitivity under jitter with and without the optimal filtering is shown in Fig. 13 at various PPM orders and PLL bandwidths. The per- formance is seen to improve when the proposed presyn- chronisation filter is included. The BER of the jittered PPM system and of the optimally filtered system is shown in Fig. 14. The optimal presynchronisation filter is seen to amount to a more significant performance improvement when the fibre bandwidth is low. This is depicted in Fig 14 round the f = 5 PPM operating region. At a PLL bandwidth of 1 5 ~ 1 0 - ~ normalised to the slot, the performance improvement obtained by using the presynchronisation filter is more than that obtained when operating at the f = 10 region with a PLL bandwidth of 20x10T3 normalised to the slot. This can be explained in that the filter effect of sharpening the pulse rising edges is more required at the lower fibre bandwidth values i.e. a t f = 3 more than at f = 10.

In Fig. 13 at f = 3 and BLT, = IO-' the sensitivity under jitter dropped from 48.5 photons per bit ( M = 4)

IEE Proc -0proelectvon , lfol 142, No 6. December 1995

to 74.3 photons per bit (A4 = 2), a penalty of 1.9dB. When the presynchronisation filter is included, at the same set of PPM and PLL parameters, the sensitivity improves from 74.3 to 50.2 photons per bit which rep- resents a penalty of only 0.14dB. Hence the proposed filter is seen to offer an improved performance under jitter. As another example, if the f = 10 and B,T, = 4x1 0-3 operating region is considered, the penalty owing to jitter from Fig. 13 and Section 4 is 0.15dB. When the presynchronisation filter is included, no sen- sitivity penalty or performance degradation was appar- ent. These results are further emphasised by the data of Figs. 15 and 16. In Fig. 15 the evaluated jitter sensitiv- ity penalty was as high as 15dB for a system operating at a fixed coding level. With the same PPM and PLL parameters, the jitter sensitivity penalty when the pre- synchronisation filter is included is shown in Fig. 16, this is limited to a maximum of 1dB. In summary, it can be established that the proposed presynchronisa- tion strategy is effective in reducing the effect of timing jitter on PPM. Its use is more desirable at the low fibre bandwidth operating regions, but in general the presyn- chronisation filter can offer an additional parameter that can be utilised in the performance optimisation of the self-synchronised PPM system under jitter condi- tions.

10-3 10-2 normalised PLL bandwidth

Fig. 15 ~ n = 1 6 n = 128 ~- n = 3 2 n = 256

n = 512

Jitter sensitivity penalty without filtering

_ _ _ _ n = 64 ~ .. ~

10-1

I

I I I I I I I I 1 I I I I 1

normalised PLL bandwidth

Fig. 16 ~ n = l 6 n = 128 _ _ n = 3 2 n = 256

- .. ~ n = 512

Jitter sensitivity penalty under optimal filtering

n = 64 ~~~~

The analysis and results presented enable the selec- tion of the system parameters for a given required per-

IEE Psoc.-Optoelectron., Vol. 142, No. 6, December 1995

fonnance when the impairment is due to the extracted slot timing. The performance in the presence of frame clock jitter is yet to be studied and specially when uti- lising the novel techniques proposed in [26].

6 Conclusions

An analysis and results were presented for the self-syn- chronised PPM system under jitter. With ideal synchro- nisation, the EDFA PPM system offered 7.9dB sensitivity improvement over a comparable EDFA PCM system and resulted in a performance beyond the PCM fundamental limit. An original expression was derived for the jittered optical fibre PPM power spec- tral density using the format cyclostationary properties. The extracted slot clock timing variance and the result- ant jitter wrong slot error probability were evaluated. Original results were presented for the system sensitiv- ity penalty under jitter. This was shown to be in the range of 0.15 to 1.9dB for a configurable system in the range of parameters considered. An original presyn- chronisation filtering strategy was proposed and its desirable effects were demonstrated. Jitter sensitivity penalties as high as 15dB were shown to be typical for a system with a fixed coding level. The use of the pro- posed presynchronisation filtering technique resulted in reduction in the penalty to about 1 dB.

7 References

1 PIERCE, J.R.: ‘Optical channels: Practical limits on photon counting’, ZEEE Trans., 1978, COM-26, (12), pp. 1819-1821

2 KATZ, J.: ‘2.5 bit per detected photon demonstration program: Phase ii and iii experimental results’, TDA progress report, Jet Propulsion Laboratory, Pasadena, CA, 1981, pp. 4247 SUN, X., DAVIDSON, F.M., and FIELD, C.: ‘50 Mbps free space direct detection laser diode optical communication system with Q=4PPM signalling’, Proceedings of the SPIE on Free-space laser communication technologies, 1990, Vol. 1218, pp. 385-398 PIRES, J.J.O., and DA ROCHA, J.R.F.: ‘Digital pulse position modulation over optical fibres with avalanche photodiode receiv- ers’, ZEE Proc. J , 1986, 133, (5), pp. 309-313

5 CALVERT, N.M., SIBLEY, M.J.N., and UNWIN, R.T.: ‘Exper- imental optical fibre digital pulse position modulation’, Elect. Lett., 1988, 24, pp. 129-131 GARRETT, I.: ‘Digital pulse position modulation for transmis- sion over optical fibre channels with direct and heterodyne detec- tion’, ZEEE Trans., 1983, COM-31, pp. 518-527

7 GARRETT, I.: ‘Digital pulse position over dispersive optical fibre channels’, International workshop on Digital communica- tions, Tirrenia, Italy, 1983

LEY, M.J.N., and CALVERT, N.M.: ‘Optical fibre digital pulse- position-modulation assuming a Gaussian received pulse shape’, ZEE Proc. J , 1990, 137, (2), pp. 89-96

9 CRYAN, R.A., and UNWIN, R.T.: ‘Reed-Solomon coded homodyne digital pulse position modulation’, ZEE Proc. Z, 1992,

10 CRYAN, R.A., and UNWIN, R.T.: ‘Optimal and suboptimal detection of optical fibre digital PPM’, ZEE Proc. J , 1993, 140, (6), pp. 367-375

1 1 SENIOR, J.M.: ‘Optical fibre communications: Principles and practice’ (Prentice Hall, 1992, 2nd edn.), Chap. 3

12 SENIOR, J.M.: ‘Optical fibre communications: Principles and practice’ (Prentice Hall, 1992, 2nd edn.), Chap. 14

13 URQUHART, P.: ‘Review of rare earth doped fibre lasers and amplifiers’, ZEE Proc. J, 1988, 135, (6), pp. 385407

14 AYRE, R.W.: ‘Explicit expressions for the receiver sensitivity and system penalties of optically preamplified direct-detection sys- tems’, ZEEE Photon Technol. Lett., 1993, 5, (23), pp. 358-360

15 SMYTH, P., WYATT, R., FIDLER, A., EARDLEY, P., SAYLES, A., and CRAIG-RYAN, S.: ‘152 photons per bit detection at 622 Mbit/s to 2.5 Gbitk using an erbium fibre ampli- fier’, Electron. Lett., 1990, 26, (19), pp. 160&1605

16 YAMAMOTO, Y.: ‘Noise and error rate performance of semi- conductor laser amplifiers in PCM-IM optical transmission sys- tems’, ZEEE J. Quantum Electron., 1980, QG16, (lo), pp. 1073- 1081

17 TONGUZ, O.K., and KAZOVSKY, L.G.: ‘Theory of direct- detection lightwave receivers using optical amplifiers’, J. Light- wave Technol., 1991, 9, (2), pp. 174-181

3

4

6

8 CRYAN, R.A., UNWIN, R.T., GARRETT, I., SIB-

139, (2), pp. 140-146

269

18 HENRY, P.S.: ‘Error rate performance of optical amplifiers’, OFC ’89 Technical Digest, Houston, TX, USA, February 1989, Paper THK3

19 PHILLIPS, A.J., CRYAN, R.A., and SENIOR, J.M.: ‘Optical fibre PPM with optical preamplification’, IEE 5th Bangor Com- munication Symposium, Bangor, UK, June 1993, pp. 284-287

20 DAVIDSON, F.M., and SUN, X.: ‘Slot clock recovery in optical PPM communication systems with avalanche photodiode detec- tors’, IEEE Trans., 1989, COM-37, (11), pp. 11641172

21 SUN, X., and DAVIDSON, F.M.: ‘Word timing recovery in direct detection optical PPM communication systems with ava- lanche photodiodes using a phase lock loop’, IEEE Trans., 1990, COM-38, (5), pp. 666-673

22 LING, G., and GAGLIARDI, R.M.: ‘Slot synchronisation in optical PPM communications’, IEEE Trans., 1986, COM-34, (12), pp. 1202-1208

23 CHEN, C.C., and GARDNER, C.S.: ‘Performance of PLL syn- chronised optical PPM communication systems’, IEEE Trans., 1986, COM-34, (lo), pp. 988-994

24 ELMIRGHANI, J.M.H., CRYAN, R.A., and CLAY- TON, F.M.: ‘Slot and frame synchronisation in optical fibre PPM systems with dispersed pulse shapes’, J. Opt. Commun., 1994, 15; (11, PP. 5-10

25 ELMIRGHANI, J.M.H., CRYAN, R.A., and CLAY- TON, F.M.: ‘Spectral characterisation and frame synchronisation of optical fibre digital PPM’, EZect. Lett. , 1992, 28, (16), pp. 1482-1483

26 ELMIRGHANI, J.M.H., CRYAN, R.A., and CLAY- TON, F.M.: ‘PPM phase bearing-events for direct frame phase extraction’, Elect. Lett. , 1993, 29, (9), pp. 775-777

27 HELSTROM, C.W.: ‘Statistical theory of signal detection’ (Per- gamon, 1986, 2nd edn.), p. 309

28 PROAKIS, J.J.: ‘Digital communications’ (McGraw-Hill, 1983), p. 152

29 GARDNER, W.A., and FRANKS, L.E.: ‘Characterisation of cyclostationary random processes’, IEEE Trans., 1975, IT-21, pp. 4-14

30 OGURA, H.: ‘Spectral representation of a periodic nonstationary random process’, IEEE Trans., 1971, IT-17, (2), pp. 143-149

31 WURF, V.D.: ‘On the spectral density of a cyclostationary proc- ess‘, IEEE Trans., 1974, COM-22, (lo), pp. 1727-1730

32 HURD, H.L.: ‘Stationarising properties of random shifts’, SIAM J. Appl. Math., 1974, 26, pp. 203-212

33 MOENECLAEY, M.: ‘Linear phase-locked loop theory for cyclostationary input disturbances’, IEEE Trans., 1982, COM-30, (lo), pp. 2253-2259

34 GARDNER, F.M.: ‘Phase lock techniques’ (Wiley, New York, 1979), Chaps. 3 and 11

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