information processing of systems with optical amplifiers and photon-counting receivers

Information processing of systems with opticalamplifiers and photon-counting receivers

Cherif Bendjaballah

For optical channels transmitting signals of weak power, several information criteria, such as the channelcapacity, the cutoff rate and the rate distortion, are analyzed. These performances are calculated forvarious formats of modulation and for systems that include optical amplifiers and utilize photon-countingreceivers. Comparison with the results that are derived from the Gaussian approximation of theintensity distribution is made. © 2001 Optical Society of America

OCIS codes: 060.1660, 060.4510, 270.5290.

1. Introduction

Many uses of laser communications that involve thedetection of highly attenuated signals require theamplification of the laser signal radiation. Such alight amplification introduces noises that limit theperformance of the optical communications system.However, the recent progress in the design of theerbium-doped optical fiber amplifiers has shown thatthe receiver sensitivity of such a system can be greatlyimproved. However, owing to noises, such as thenoise resulting from the beating of this field with itselfand with the signal, there are severe limitations in allsystems that utilize such an amplifier. To account forthese fluctuations, a Gaussian approximation for theintensity distribution seems justified as a result of theapplication of the central limit theorem.

In fact, recent papers1–3 showed that this is not re-ally valid. Consequently, the detection performance,calculated for a optical communication system includ-ing such an amplifier, needs to be re-examined. Thisresearch has actually been done for a system that usesa binary source of symbols, and a significant deviationhas been noticed between the exact probability of errorin detection and the results derived from the Gaussianapproximation. Moreover, the measurements re-cently achieved on the probability density functions

clearly confirm that the dominant part of the noises isnon-Gaussian.4 It is therefore instructive to seekwhether such conclusions can be extended to othercriteria of signal processing, namely, those derivedfrom the information theory: channel capacity, cutoffrate, and rate distortion.

The present study is limited to the systems workingat low signal intensity. The approach that we adoptis a semiclassical one in which the light beam is clas-sically described and the process of photon detection isquantum mechanically treated. Hence such a chan-nel is a map of continuous-input variable �intensity� todiscrete-output variable �number of photons�. Thephysics of the amplification processes from the point ofview of quantum noise5 are not considered becausesome of their effects on the transmission performancehave already been discussed.6

In Section 2 the analytical expressions of the pho-ton number probability densities are derived underthe assumption that the noise is contributed by opti-cal amplification. In Subsection 3.A the measures ofinformation such as the channel capacity and thecutoff rate are examined for the uncoded pulse-position-modulation �PPM� format of the input sig-nal. These measures of information have alreadybeen analyzed7,8 for the quantum-limited channel�i.e., a detector system receiving a coherent lightsource with no extra noise added�, proving to be animprovement of the performance. A survey of theseresults from both classical and quantum points ofview has been published.9 The limitation of the per-formance, owing to the presence of noise photons anderasures, has been evaluated for devices operatingwith a soft decision10,11 and a hard decision12 pro-cessing. These results are adapted to the presentcontext of optical amplification. Hence the maximi-

The author �e-mail address: [email protected]� is with theLaboratoire des Signaux et Systemes du Centre National de laRecherche Scientifique, Ecole Superieure d’Electricite, 3 rue Joliot-Curie, 91190, Gif sur Yvette, France.

Received 2 November 2000; revised manuscript received 3 July2001.

0003-6935�01�295153-09$15.00�0© 2001 Optical Society of America

10 October 2001 � Vol. 40, No. 29 � APPLIED OPTICS 5153

zation of these normalized measures of informationto the signal average photon number yields the opti-mum length of the PPM, which is determined interms of the amplifier characteristics: the noisephoton and the number of modes.

Making use of the continuous modulation of theintensity, we analyzed another measure of informa-tion, the rate distortion, in Subsection 3.B. It char-acterizes the minimum-expected distortion achievableat a particular rate, given a source distribution and adistance between a random variable �the modulatedintensity� and its representation �the photon number�.For a noiseless channel, it has been proved thatphoton-counting processing is a useful technique totransmit and reproduce the statistical properties of alaser source in the domain of moderate level of inten-sity.13,14 In Subsection 3.B we show that this con-clusion can be extended to amplified signals in thepresence of noise photons. Finally, in Section 4some comments are given on the basis of the analyt-ical and numerical results of Section 3.

2. Statistics of Light Amplification

We briefly present the main results of the theory oflight amplification that are needed to establish thephoton-counting distributions. The light field is de-picted by a collection of M independent modes2:

e0�� k�1

M

ck exp�i�k��. (1)

Each mode of amplitude ck of zero mean and equalvariance �k

2 � �2, @k, oscillating at frequency �k, is ofGaussian distribution p��ck� � 1��2�� exp��ck�2�2�2�. This field is directed to an ideal photon multi-plier of quantum efficiency � � 1, which measures thestationary integrated intensity J�t0, T� � �t0

T �e0��2d�.

The light field can be written as e0�� ¥k�1M �ck�2.

Assuming that the real and the imaginary parts ofthe field are independent and Gaussian, the inten-sity distribution follows as

p0�J� � � 1J0�M JM�1

�M � 1�!exp�� J

J0� , (2)

where the mean intensity is J0 � 2�2. Now let acoherent field es of constant intensity Js, called thesignal, be superimposed on the field of distributionEq. �2� . The total intensity J � �t0

T �e0�� es�2d� is

of distribution

p1�J� �1J0

� JJs�

M�12

exp�� J � Js

J0�IM�1�2�JJs

J0� , (3)

where IM�1�z� is the modified Bessel function of orderM � 1 for the argument z. The first and secondmoments of the total intensity J are �J� � J1 � Js �MJ0 and �J2� � M�M � 1�J0

2 � 2�M � 1�JsJ0 � Js2,

and the variance is �J2 � MJ0

2 � 2J0JS. The in-tensity is here supposed to be weak so that the outputprocessing requires a photon-counting receiver. Wealso assume that the photon detector can measure

the noise alone and can detect any signal superim-posed on the noise.

The photon number distribution of registering nphotons within the time interval 0, T is obtainedfrom the intensity distributions Eqs. �2� and �3� byassuming that the random-point process of detectionis a doubly stochastic one, i.e., a Poisson process ofparameter �0

T J��d�15:

pi�n� �1n! �

0

�

dJ��0

T

J��d��n

� exp��0

T

J��d��pi�J�, i � 0, 1. (4)

The time duration 0, T is supposed to be very shortcompared with the correlation time of the fluctua-tions of the light intensity. When the signal is ab-sent, the corresponding intensity distribution p0�J� isgiven by Eq. �2� and substituted into Eq. �4� to yield

p0�n� ��n � M � 1�!n!�M � 1�!

N0n

�1 � N0�n�M , N0 � J0 T. (5)

This can also be obtained by the substitution of Eq.�3� into Eq. �4� in the case where Js � 0 and IM�1�z�� z�2�M�1��M � 1�!. The average value of the pho-ton number n is �n0� � MN0, and its variance is �0

2 ��n0��1 � N0�. When the signal is present �Ns � JsT�,Eq. �4� yields

p1�n� �1

�n � M � 1�!N0

n

�1 � N0�n�M

� exp�� Ns

1 � N0�Ln

M�1�� Ns

N0�1 � N0�� . (6)

In this case, the average value of the photon numberis �n1� � �n0� � Ns, and the variance is here given by�1

2 � �02 � Ns�1 � 2N0�. The term in Ns is due to

the shot noise and that in 2N0Ns is due to the signal-noise beating.

The generalized Laguerre polynomials LnM�1�z�, of

order �M � 1�, for the argument z, are functions of n.They obey the following relations16,17 that are neededin the sequel,

�i� ��M�j

�l�1

M 1�nl � �l�!

Lnl

�l � zl�

�1

� j � �M�!Lj

�M��l�1

M

zl� ,

�ii� L0M�1� z� � �M � 1�!, @ z,

�iii� LnM�1�0� �

�n � M � 1�! 2

n!�M � 1�!,

�iv� z3 �, 2LnM�1��z� �

�n � M � 1�!n!

zn, (7)

5154 APPLIED OPTICS � Vol. 40, No. 29 � 10 October 2001

where �M � ¥l�1M nl, and �M � ¥l�1

M �l � M � 1.With these it is proved that when M 3 �,

p1�n�3 P�n, �n1��– �n1�

n

n!exp��n1��,

which can also be seen from the Gaussian approxima-tion of Eq. �3�,

p1�J� 1��J�2� exp��J2�2�J2�O¡

�J3 0

��J � J1�

that leads to the Poisson distribution of parameter�n1� � J1T by use of Eq. �4�.

3. Information Criteria

The communication system we deal with is of directdetection utilizing two types of the modulation ofthe signal: continuous intensity modulation andPPM. For the the first one, the source of messages�x that modulates the intensity J of an ideal laseris followed by an amplifier whose properties havebeen briefly recalled in Section 2. The resultingintensity is detected by a photon multiplier consid-ered as an ideal device with no extra noise. Themeasured signal is then composed of a set of timeinstants corresponding to the times of photon de-tector interactions. Thus the received signal con-stitutes a random-point process for which thephoton-counting technique, i.e., measurement of�n, the number of photons registered within thetime duration 0, T , is employed for recovering theavailable information of the intensity. This com-munication system can be studied from two pointsof view: detection and information.

From the first point of view, the performance can becharacterized by the probability of error of statingerroneously that the signal �x is present in the mea-sured outcome �n. The reliability of the receivedmessages is not considered.

From the second point of view, one can be inter-ested in evaluating how close the measured outcome�n is to the input message �x. For this purpose, totransmit the message through the best channel, onecan study the channel performance by maximizing I��x, �n�, the mutual information between �x and �nwith respect to the source properties yielding thechannel capacity C.

Another interesting function is the cutoff rate,which provides a useful criterion of reliability be-cause the Per

d probability of error in decoding can bemade such that Per

d � � exp��LrR0�, where Lr is thelength of the coding alphabet of the source messagesand � is a function that can here be considered asconstant. For a given Per

d and R0, the Lr is thereforea characterization of the efficiency of the communi-cation system.

However, recent research18 demonstrated that theperformance of the Turbo-encoded optical channelscan be substantially improved without increasing theneeded bandwidth, as was the case for the Reed–

Solomon encoding. The merits of such systems are(i) the performances approach the channel capacity;(ii) the coding length required for such achievement islow; and (iii) the complexity of the decoder is reason-able. Thus such a system of modulation is quiteuseful. Therefore the “practical” aspect of the cutoffrate seems no longer interesting. In fact, the inter-est in studying the cutoff rate is rather theoretical.

However, one can also study the reproduction, upto a certain error d, of the statistical properties of thesource �x whose distribution p�x� is given, with re-spect to all possible channels. The results are thenappropriately described by the rate distortion19 R�d�.This is again obtained by optimizing I��x, �n� undera constrained criterion whose average is d. The di-rect theorem establishes that it is possible to deter-mine a coding scheme with a rate that is arbitrarilyclose to R�d� and an average distortion that is arbi-trarily close to d. The converse theorem proves thatthere is no code achieving reliable reproduction withd at a rate below R�d�.

Before analyzing the results derived from the con-tinuous modulation of the intensity, we focus ourattention on the PPM format of modulation.

A. Uncoded Pulse-Position-Modulation Format

Recently systems with more complicated types ofmodulation including coded PPM are described andtheir performance calculated.19,20 We restrict ourstudy to a simplified system, the uncoded PPM, inwhich the problem of the synchronization with thelight pulse during its propagation inside the amplifieris not taken into account.

Let us briefly recall that the PPM frame is usuallydivided into L slots and that the signal light ispresent only during the lth time slot corresponding tothe lth symbol taken from the alphabet of length L.Here the direct detection receiver, synchronized withthe PPM frame, delivers L random variables n1,n2, . . . , nL within the L slots, such that11

p�n1, n2, . . . , nL�l � � p1�nl� �i�1,i�l

i�L

p0�ni�. (8)

When the signal is present �respectively, absent� in agiven slot, the distribution is p1�n� respectively,p0�n� . Each input message is emitted with the apriori probability �i. Now, because the differencebetween the optimized capacity �or the cutoff rate�and the one attained with the equiprobable distribu-tion is small,20 it is assumed that �i � 1�L, @i �1, . . . , L.

However, an erasure symbol of probability � is ingeneral necessary. In fact, this symbol is needed totake account of the case in which no decision can bemade in decoding the output message. This is thecase, for example, when the counts for two or more�but not all� slots are the same, which leads to the


choice of an incorrect decision. The expression of thenormalized channel capacity is known as

� �C�L

�1�L ��1 � ��log� L

1 � �� p log�p�

� �L � 1�q log�q�� , (9)

where

Ns � �L,

� � �n�0

�

p1�n�p0�n� L�1,

p � �n�1

�

p1�n��k�0

n�1

p0�k��L�1

,

q �1 � p � �

L � 1. (10)

A good approximation of Eq. �9�, p log L, is muchbetter than the conventional log L for large values ofL.

Let us denote � � 1��1 � N0� and set � � �M

exp��L�. From Eq. �10�, taking into account thatp0�n� and p1�n� are respectively given by Eqs. �5� and�6�, we can easily derive �. We therefore evaluateLc*, the length of the PPM format that corresponds tothe maximum of � for the most interesting case N030. This provides the optimum alphabet size, in thesense of the most efficient use of the available power.The maximum of the capacity subject to the averagepower constraint can be calculated approximately, forexample, for q3 0, p3 1 � �. By retaining only thefirst two terms in the expansion of � in Eq. �10�, onegets � � ��1 � �LM�1 � �� . Then making �� L � 0 leads to

log Lc* ��1 � Ns M�1 � �� 1

�1 � �Ns � Ns2M�1 � �� 1

O¡�3 1

exp�Ns� � 1exp�Ns� � Ns � 1

, (11)

�* �1 � �

Nslog Lc*O¡

N0 � 0

1 � exp��Ns� 2

Ns exp��Ns� exp�Ns� � 1 � Ns . (12)

Note that for �3 0, Ns3 0 and Lc*3 �. To obtaina more explicit evaluation of Lc*, we eliminate Nsbetween these two equations to get Lc*��*�. Thusfor a moderate value of N0, we find

Lc*O¡Ns3 0

exp�*�1 � MN0� . (13)

This is plotted in Fig. 1 for a weak value of N0 �N0 �0.02� versus �*, for several values of M � 1, 2, . . . ,

10. The results of the approximation yielding rela-tion �13� are only in qualitative accordance with thecomputed results.

The curves corresponding to p0�n� and p1�n� of bothPoisson distributions with the parameters N0, N1 �Ns � MN0, where M � 1 and M � 10, are plotted indashed curves in the same figure. These curves arecalled a and b, respectively. The deviations from theprevious curves are clearly minor although noticeable.

However, as pointed out above, instead of studyingthe capacity, the cutoff rate R0 can characterize theperformance attainable by a digital memoryless chan-nel with a practical format of coding and decoding.

To establish the expression of R0 for a memorylessdiscrete PPM-modulated channel, we now supposethat the channel has L input symbols with relativefrequencies �l and the decoder delivers N symbols.Let p��n�l � be the transition probability that theemitted symbol l is decoded as n, then the cutoff rateis

R0 � �logmin�

��n�!N

��l�1

L

�l�p��n�l ��2� , (14)

where the minimization is carried over the set �:��l � 0, ¥l �l � 1. Note that the output alphabet is!N � �L, the symbol ¥�n denotes ¥n1�0

� . . . ¥nL � 0

� , and�n stands for the vector �n1, n2, . . . , nL�.

To calculate Eq. �14� for the PPM soft decodingprocessing that reads n � 0, �, we assume that �l �1�L, @l � 1, . . . , L. Then we arrive at

" �R0

s

�L�

1�L (log L � log1 � �L � 1�

� ��n�0

�

�p0�n�p1�n��2�) , (15)

Fig. 1. Plot of the optimized length Lc* of the PPM format versusthe average photon number constrained capacity �* with M � 1, 3,5, 7, 10 and N0 � 0.02, the average value of the noise photonnumber. In dashed curve �a�, MN0 � 0.02 and in dashed curve�b�, MN0 � 0.2; the curves correspond to the cases in which thenoise and signal distributions are Poisson.


where p0�n� and p1�n� are here given by Eqs. �5� and�6�, respectively. Here again we can see that an up-per bound to R0

s is log L, as expected.In fact, to compare the results of the cutoff rate

with those given by the capacity, we need to calculateR0

h, the cutoff rate for hard decoding:

�1� For n � nth �where nth is the threshold�, thesymbol 1 is selected, and the corresponding transitionprobability is p1�n� with n � nth, �;

�2� For n � 0, nth, the symbol 0 is selected, and thecorresponding transition probability is p0�n�.

However, because this threshold should be opti-mized, a simplification of the calculations is to use thefact that R0

h � R0s � �,9 and we limit ourselves to the

study of R0s and then of ".

As above, only approximate calculations can lead toresults in closed forms, for instance, for N0 3 0 andfor a signal of weak level Ns 3 0. We set � �exp��NsL� �1 � 2�LN0Ns� and keep Ns � �L fixed.We seek "� L � 0, so that the equation to solve in Lis given by

1 � ��L � 1�1 � �

log� L1 � �L � 1��

� 1 ��Ns L�L � 1�

1 � � �1 ��LNsN0

Ns L�1 � �LNsN0�� ,

(16)

from which it follows

Lr* �2Ns

�1 � 2�MN0�, "* � 1 � Ns �Ns

2

4. (17)

Therefore Lr*�"*� is derived from relations �17� toread

Lr*O¡Ns3 0

21 � "*

�1 � 2�MN0�, (18)

which is plotted versus "* in Fig. 2 for a weak valueof N0 �N0 � 0.02� and for several values of M � 1,2, . . . , 10.

Here again the results of the approximation yield-ing relation �18� for "* ## 1 are in accordance with thecomputed results. In the same figure the corre-sponding curves are plotted in dashed curves in thecase in which both p0�n� and p1�n� are Poisson distri-butions with the parameters N0, N1 � Ns � MN0,where M � 1 and M � 10 for curves a and b, respec-tively. In this case, the deviations are clearly signif-icant.

Figure 3 displays the optimized length L* �Lc* andLr*� for M � 5 versus $, which is either the optimizedcapacity �* for curve C or the optimized cutoff rate "*for the curve R0. Comparing relation �18� with re-lation �13�, we can conclude that Lr* � Lc*, @M.Also note that Lc*��*� strongly depends on M. Thisis less true for the cutoff rate. In both cases, we have

L* � 3 because for Ns %% 1 we basically need tocalculate � L �log L�L� � 0, which is verified for L*� e.

The important conclusion of this section is that, fora given average power constrained capacity �or cutoffrate�, the receiver using the capacity as the criterionof performance requires a smaller length for the for-mat of the PPM at least for $ � 0.5 and MN0 � 0.10;this is not the case for MN0 � 0.06. This is despitethe fact that @Ns and @M, we have R0�Ns� � C�Ns�.However, by use of the same constraint, it can beshown that Lr*�Ns� � Lc*�Ns�.

As seen just before, the PPM format of modulation

Fig. 2. Plot of the optimized length Lr* of the PPM format versusthe average photon number constrained cutoff rate "* with M � 1,3, 5, 7, 10 and N0 � 0.02, the average value of the noise photonnumber. In dashed curve �a�, MN0 � 0.02 and in dashed curve�b�, MN0 � 0.2; the curves correspond to the cases in which thenoise and signal distributions are Poisson.

Fig. 3. Comparison of the optimized length L* of the PPM formatversus $, which is either the average photon number constrainedcapacity �* for curve C or the average photon number constrainedcutoff rate "* for curve R0, with M � 5 and the average value of thenoise photon number N0 � 0.02.


is well adapted for practical communication systemsfor which the complete statistics of the signal areneeded. In the case one is interested in only a specificsignal statistic, e.g., the first moment, it can be shownthat a continuous modulation is more appropriate.

B. Continuous Intensity Modulation

As proposed some time ago,21 the continuous modu-lation of the source intensity is a useful method forthe transmission of information. The statistics ofthe source of messages are given by the probabilitydensity denoted pa�x�, which is here normalized sothat pa�x� � exp��ax�, for x � 0, 1; pa�x� � 1 ��0

1 pa�u�du, for x � 1; and pa�x� � 0, for x % 1. Thismodulation scheme is chosen with a % 0 to transmitthe source intensity for the maximum value of thetransmission coefficient of the modulator. The mu-tual information IN relevant to the channel x � n isgiven by15,21

IN�a, Ns� � �n �

0

1

dxP�x, n�log�q�n�x�

�0

1

d&P�&, n� , (19)

where the transition probability is q�n�x� and P�x, n�� q�n�x�pa�x�. The channel capacity and the cutoffrate Eq. �14� are defined by

C � maxpa� x�

IN � maxa

IN, (20)

R0 � maxa �log �

n��

0

1

dxpa�x��q�n�x��2� . (21)

To estimate the value of a � am that yields approx-imations �20� and �21�, the expansion up to the secondorder of pa�x� ' c�1 � ax � a2x2�2� leads to c � 3�2and am � 1, in accordance with the results of Ref. 21for Ns � 1. Actually, the values deduced from thenumerical computation are such that am � 2, for Ns �2. Although it seems difficult to explain the differ-ence, it is possible to show that there exists an am, thesolution of approximation �20�. To do so, we calcu-late Eq. �19� in the specific case in which q�n�x� is thePoisson distribution. We then use the property ofconvexity � of Eq. �19� with respect to pa�x�. Wetherefore evaluate IN�a, Ns�, assumed to be indepen-dent of N0 and M for the present purpose. We finallyshow that

IN�a, Ns��O¡a � 0

Ns3 0

Ns2

24,

O¡a3 0

Ns3 0

a3

Ns � f �a�Ns

2

24,

O¡a3 �

@Ns

0,

(22)

where f �a� ' �1 � a � a2�2� for a � 1. Then wederive

IN�0, Ns� � IN�a � am, Ns�, (23)

IN�0, Ns� � IN�a � am, Ns�, (24)

which proves that IN�a, Ns� is maximum for a � am.To derive more explicit expressions for C and R0, a

simple method of approximation of Eq. �19�, based onthe discretization of pa�x�, turned out to be conve-nient. This consists in considering the system tooperate as a binary source coded so that the messageof 0 is emitted over the channel with the a prioriprobability �0. The corresponding transition proba-bility and the average photon number are p0�n� and�n0� � N0M, respectively. The message of 1 is emit-ted with the a priori probability �1, ��0 � �1 � 1�.The transition probability is p1�n�, and the averagephoton number is �n1� � �n0� � Ns. Hence the chan-nel capacity is

C � max�1

�1 H�p1� � �0 H�p0� � H�P� , (25)

where H�pi� � ¥n pi�n� logpi�n� , i � 0, 1, and P ��1p1�n� � �0p0�n�.

The cutoff rate R0* that is the upper bound for theactual cutoff rate is derived from Eq. �14� for thebinary channel

R0 � R0* � �log(1 � 2�0�1�1 � exp

� ��n1� � ��n0��2 ). (26)

Analysis of Eq. �26� in various situations and for dif-ferent constraints, such as the average power con-straint or the bandwidth constraint, has beenachieved in Ref. 15. When there is no constraint, itis seen that R0*�Ns� � �0�1Ns for weak values of N0and Ns. In fact, one finds that Eq. �26� is ratherappropriate for the approximation of the capacitygiven in relation �25�. In the following, we take �0 ��1 � 1�2, although for some cases, a better choicewould be 1�2 � �1 � 1 � 1�e, �0 � �1 � 1.

With respect to the cutoff rate R0, a reasonablygood approximation for Ns 3 0 and for a weak N0 ismade below. Starting with Eq. �21� and using thesame method of calculation that led to relation �26�,we derive

R0 � log2

1 � �n�0

�

p1�n�p0�n� 1�2

. (27)

The p1�n� given by Eq. �6� is then expanded up to Ns2.

With the help of Eqs. �7�, we use the approximation

�p1�n�p0�n� � �1 �Ns�1 � (�

2�

Ns2�1 � (�2

8 ��

np0�n��1 �

znM

�z2�n � 1�n2M�M � 1��

1�2

,


where z � Ns�1 � (�2 �( and ( � N0��1 � N0�, forM %% 1, such that �M � 1�( ## 1. Noting that¥np0�n�n � N0M, one finally obtains

R0O¡Ns3 0 Ns

2�1 � (�2

16M(�

Ns2

16MN0, (28)

which shows that R0 decreases with M. However,this is only in qualitative accordance with the numer-ical results, which, in fact, exhibit a slower variationwith M.

Now we briefly deal with the questions that arerelevant to the rate distortion. This function is con-venient to establish a link between the informationperformance with the functions derived from the de-tection criteria.18 Specific applications to photon-counting systems have recently been proposed.13,14

The intensity J of distribution �3� is now modulatedsuch that Js � Jsx, the noise being unchanged. Themutual information and the average distortion dNare

IN�a, Ns� � �n �

0

1

q�n�x�pa�x�

� log�q�n�x�

�0

1

q�n�&�pa�&�d& dx, (29)

dN�a, Ns� � �n �

0

1

c�x, n�q�n�x�pa�x�dx, (30)

c�x, n� � �x � xn�, (31)

where c�x, n� is the fidelity criterion. The estimatorxn can be determined for the Poisson approximationq�n�x� � �Nsx�n�n! exp��Nsx� under the maximum aposteriori MAP estimation for which it is enough toset pa�x� � cst. exp��ax� �0 � x # 1� because theconstants are unimportant. Taking the logarithm of

Q�x�n� �q�n�x�pa�x�

r�n�, r�n� � �

0

1

q�n�&�pa(&)d&,

differentiating with respect to x, equalizing the resultto zero, and solving it, we obtain xn � n��Ns � a�.The exact values of xn achieved by numerical compu-tation that used exact q�n�x� given by Eq. �6� do notnoticeably differ from this expression. So Eq. �30�becomes

dN�a, Ns� � �n �

0

1 �x �n

Ns � a�q�n�x�pa�x�dx. (32)

By eliminating Ns between Eqs. �29� and �32�, thecurve RN�d� for a given a3 0 can readily be obtained.The curves are normalized so that R�dmax� � 0 fordmax � 0.5 and where R1�d, N0� � RM�d, N0�M�. Inthe following we set dN – d.

It should be emphasized that the RN�d� is not the

rate distortion because Eq. �29� has not been opti-mized. However, it is a good approach in the sensethat is pointed out below. As in Refs. 13 and 18, onthe basis of the Gaussian approximation of pa�x� forx � !X � ��, �, Rl�d� � h�pa� � h�g�, the Shannonlower bound, is expressed in terms of the differentialentropy h� f � � ��!X

f �x�log f �x� dx and

g�s; x� �exp�sc�x�

�!X

exp�sc� z� dz

�exp��s�x��

��

�

exp ��s� z��

dz.

One easily finds

Rl�d� �12

log��X2

2ed2� for 0 � d � ��X2

2e �1�2

, (33)

where �X is the variance of x.Figure 4 shows the RN�d� plotted together with the

lower bound Eq. �33� for a few values of M and forN0 � 0.05. The unsmoothed structure of the curvesresults in the evaluation of d, which depends on theinteger values of the threshold in the estimation of xn.The RN�d� curves decrease with M.

For M � 1, the curve differs significantly from thecorresponding Poisson distribution plotted in adashed curve. For M � 20, the curve approaches theRl for low values of d. Because this lower bound isknown to be close to the actual rate distortion,18 Rl�d�� R�d�, and because RN�d� � Rl�d� for a d that issmall enough, we can state that RN�d� � R�d� for d �0.15. This clearly means that reproducing thesource with an amplifier is better when a large valueof M is used, at least for small values of N0. How-ever, this requires the construction of a device withthe length of the PPM format higher than the oneneeded for a lower M.

Fig. 4. Information rate RN for the amplified coherent-source in-tensity modulated versus d, the average value of absolute differencefidelity criterion, using the photon number decoding for differentvalues of M � 1, 5, 10, 20. The RN for the coherent source with theaverage value of the noise photon number N0 � 0 is plotted in adashed curve. Curve Rl is the Shannon lower bound.


4. Concluding Remarks

First, it is verified that the inequality C�Ns� � RN�d�18

holds for N0 ## 1 in the whole range of Ns. It is alsoseen that for 0 � Ns � 2, in the case of the continuousmodulation, the channel capacity is approximated byR0* given by Eq. �26�, the pseudo-rate distortion be-ing approximated by relation �22�.

Therefore the cutoff rate is a lower bound to boththe channel capacity and the rate distortion R0 �RN � C, inequalities that hold for another choice ofthe source statistics, e.g., the binary-source messageswith a PPM format. For instance, it is seen in Fig. 3,that for M � 5 and N0 � 0.02 and a rate of 0.55nats�photons, the optimized length must be such that12 � L* � 17. Furthermore, the above inequalitieswill allow us to better bound the distortion. In fact,by use of the property RN�d� ' Rl�d� and Eq. �33�, itis deduced that

� �

24e�1�2

exp��C� � d � � �

24e�1�2

exp��R0�,

so that dmin % 0, in accordance with the numericalresults displayed in Fig. 4.

As it can be seen from Eq. �6�, the Gaussian ap-proximation of p1�J� leads to the Poisson distributionof the parameter �n1� for p1�n�. Recall that such anapproximation was found inappropriate, for the in-tensity detection processing as already demonstrat-ed.3 This is clearly seen by calculation of theprobability of error Per for the on–off keying �OOK�modulation that reads Per � 1

2 �Pe � Pf �, where Pc �1 � Pe, the probability of correct detection, is Pc � �0

Jth

p1�J�dJ and Pf, the probability of false alarm, is Pf ��Jth

� p0�J�dJ. Jth is the intensity threshold such thatJ � Jth3 p�J� � p1�J� and J # Jth3 p�J� � p0�J�.The threshold is deduced from )�J� � p1�J��p0�J�,

the likelihood ratio such that )�Jth� % 1. The prob-ability distributions p0�J� and p1�J� are those givenby Eqs. �2� and �3�, respectively. To conclude, thephoton number communication is demonstrated to bea useful technique to transmit and to reproduce thestatistical properties of a light source in the domain ofa rather low level of intensity.

It is also observed that the Gaussian approxima-tion of the intensity distribution is justified for thecapacity performance, as shown in Fig. 1, and isinappropriate for the cutoff rate performance, asshown in Fig. 2. Nevertheless, in the case in whichthe input intensity is continuously modulated, theGaussian approximation leads to accurate re-sults, at least for M % 2 �see the dashed curvesversus the number M, in Fig. 5�. This has beenproved for a weak N0, a moderate value of Ns, and1 � M � 20.

The Laboratoire des Signaux et Systemes, is ajoint laboratory of the Centre National de la Re-cherche Scientifique and the Ecole Superieure d’E-lectricite, associated with the University of Paris,Orsay. The author thanks E. Ademovic for hiskind help in performing some computations.

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Fig. 5. Continuous modulation of the signal intensity. The av-erage number of signal photons is Ns � 5, and the average numberof noise photons is N0 � 0.05. Comparison of the channel capacityC and the cutoff rate R0 versus M, the number of independentmodes, with the results of the Gaussian approximation that areplotted in dashed curves.


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information processing of systems with optical amplifiers and photon-counting receivers

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