Achievability of the Rate ${1/2}\log(1+\es)$ in the Discrete-Time Poisson Channel
aa r X i v : . [ c s . I T ] S e p Achievability of the Rate log(1 + ε s ) in theDiscrete-Time Poisson Channel Alfonso Martinez
Abstract
A simple lower bound to the capacity of the discrete-time Poisson channel with average energy ε s is derived. The rate log(1 + ε s ) is shown to be the generalized mutual information of a modifiedminimum-distance decoder, when the input follows a gamma distribution of parameter / and mean ε s . I. I
NTRODUCTION
Consider a memoryless discrete-time whose output Y is distributed according to a Poisson distri-bution of parameter X , the channel input. By construction, the output is a non-negative integer, andthe input a non-negative real number. The channel transition probability W ( y | x ) is thus given by W ( y | x ) = e − x x y y ! . (1)This model, the discrete-time Poisson (DTP) channel, appears often in the analysis of optical com-munication channels. In this case, one can identify the input with a signal energy and the output withan integer number of quanta of energy.Let P X ( x ) denote the probability density function of the channel input. We assume that the inputenergy is constrained, i. e. E[ X ] ≤ ε s , where E[ · ] denotes the expectation operator and ε s is the averageenergy. Random variables are denoted by capital letters, and their realizations by small letters.An exact formula for the capacity C( ε s ) of the DTP channel is not known. Recently, Lapidoth andMoser [1], derived the following lower bound C( ε s ) ≥ log (cid:18) ε s (cid:19) ε s √ ε s ! − (cid:18) r π ε s (cid:19) . (2)Observe that this bound diverges for vanishing ε s . Capacity is given in nats and the logarithms arein base e . A. Martinez is with Centrum Wiskunde & Informatica, The Netherlands. e-mail: [email protected].
November 21, 2018 DRAFT
A closed-form expression for the mutual information I ( X ; Y ) achieved by an input with a gammadistribution of parameter ν was derived by Martinez in [2], namely I ( X ; Y ) = Z ε s − (cid:18) − ν ν ( ν + ε s (1 − u )) ν (cid:19) u ν − − u ! du log u + ( ε s + ν ) log ε s + νν + ε s (cid:0) ψ ( ν + 1) − (cid:1) , (3)where ψ ( y ) is Euler’s digamma function. For ν = 1 / , numerical evaluation of the mutual informationgives a rate which would seem to exceed log(1+ ε s ) for all values of ε s . In this paper, we prove thatthe rate log(1 + ε s ) is indeed achievable by this input distribution. The analysis uses a suboptimumminimum-distance decoder, similar in spirit to Lapidoth’s analysis of nearest neighbor decoding [3].II. M AIN R ESULT
Let the input X follow a gamma distribution of parameter / and mean ε s , that is, P X ( x ) = 1 √ πε s x e − x εs . (4)This choice led to good lower and upper bounds in [1] and [2] respectively.We consider a maximum-metric decoder; the codeword metric is given by the product of symbolmetrics q ( x, y ) over all channel uses. The optimum maximum-likelihood decoder, for which q ( x, y ) = W ( y | x ) , is somewhat unwieldy to analyze (Eq. (3) gives the exact mutual information). We considerinstead a symbol decoding metric of the form q ( x, y ) = e − ax − y x , (5)where a = 1 + ε s . The reasons for this choice of a will be apparent later.Clearly, the decoder is unchanged if we replace the symbol metric q ( x, y ) by a symbol distance d ( x, y ) = − log q ( x, y ) , and select the codeword with smallest total distance, summed over all channeluses. This alternative formulation is reminiscent of minimum-distance, or nearest-neighbor decoding.Indeed, the metric in Eq. (5) is equivalent to a minimum-distance decoder which uses the distance d ( x, y ) = ( y − √ ax ) x = y x + ax − y √ a. (6)The term − y √ a is common to all symbols x and can be removed, since it does not affect thedecision.For a = 1 , the distance in Eq. (6) naturally arises from a Gaussian approximation to the channeloutput, whereby the channel output is modeled as a Gaussian random variable of mean x and variance x . This approximation is suggested by the fact that a Poisson random variable of mean x approachesa Gaussian random variable of mean and variance x for large x . November 21, 2018 DRAFT
Minimum-distance decoders were considered by Lapidoth [3] in his analysis of additive non-Gaussian-noise channels. For our channel model, even though noise is neither additive (it is signal-dependent), nor Gaussian, similar techniques to the ones used in [3] can be applied. More specifically,since we have a mismatched decoder , we determine the generalized mutual information [4]. For agiven decoding metric q ( x, y ) and a positive number s , it can be proved [4] that the following rate—the generalized mutual information— is achievable I GMI ( s ) = E (cid:20) log q ( X, Y ) s R P X ( x ) q ( x ′ , Y ) s dx ′ (cid:21) . (7)The expectation is carried out according to P X ( x ) W ( y | x ) . This quantity is obviously a lower boundto the channel capacity.Our main result is Theorem 1.
In the discrete-time Poisson channel with average signal energy ε s , the rate log(1+ ε s ) is achievable. This rate is reminiscent of the capacity of a real-value Gaussian channel with average signal-to-noiseratio ε s . Similarly to the situation in this channel, the rate is achieved by a form of minimum-distancedecoding. Differently, the input follows a gamma distribution, rather than a Gaussian. Proof:
We evaluate the generalized mutual information I GMI ( s ) for an input distributed accordingto the gamma density, in Eq. (4). First, we evaluate the expectation in the denominator [5, Eq. 3.471-15] Z ∞ e − x ′ εs − asx ′ − sy x ′ √ πε s x ′ dx ′ = e − y q s (1+2 aεss ) εs √ aε s s . (8)Further, using the expression of the first two moments of the Poisson distribution, namely X y W ( y | x ) y = x, X y W ( y | x ) y = x + x, (9)together with the input constraint R P X ( x ) x dx = ε s , we can explicitly carry out the expectation in The moment generating function of a Poisson random variable of mean x is readily computed to be e x ( e t − . The firsttwo moments are the first two derivatives, evaluated at t = 0 . November 21, 2018 DRAFT
Eq. (7), I GMI ( s ) = Z P X ( x ) X y W ( y | x ) log (cid:0) q ( x, y ) s (cid:1) dx − Z P X ( x ) X y W ( y | x ) log (cid:18)Z P X ( x ′ ) q ( x ′ , y ) s dx ′ (cid:19) dx (10) = s Z P X ( x ) X y W ( y | x ) (cid:18) − ax − y x (cid:19) dx − Z P X ( x ) X y W ( y | x ) − y s s (1 + 2 aε s s ) ε s − log √ aε s s dx (11) = − s (( a + 1) ε s + 1) + p ε s s (1 + 2 aε s s ) + 12 log(1 + 2 aε s s ) . (12)Choosing ˆ s = ε s ( a − ε s +2 ε s ( a +1)+1 , the first two summands cancel out. And for a = 1 + ε s wehave that a ˆ s = 1 , and therefore I GMI (ˆ s ) = 12 log(1 + ε s ) . (13)The same rate, log(1 + ε s ) , is also achievable by a decoder with a = 1 . In this case, we have toreplace the generalized mutual information by the alternative expression I LM [4], given by I LM = E (cid:20) log a ( X ) q ( X, Y ) s R P X ( x ) a ( x ′ ) q ( x ′ , Y ) s dx ′ (cid:21) . (14)As for I GMI , s is a non-negative number; a ( x ) is a weighting function. Setting a ( x ) = e − sεs x we havethat I LM is given by Eq. (11), thus proving the achievability.The bound provided in this paper is simpler and tighter than Eq. (2). It would be interesting toextend Theorem 1 to channel models Y = S ( X )+ Z , where S ( X ) corresponds to the case consideredhere and Z is some additive noise Z , with a Poisson or a geometric distribution. A different inputdistribution and another modified decoding metric are likely required for either case.R EFERENCES [1] A. Lapidoth and S. M. Moser, “Bounds on the capacity of the discrete-time Poisson channel,” in
Proceedings of the41st Allerton Conf. on Communication, Control, and Computing , October 2003.[2] A. Martinez, “Spectral efficiency of optical direct detection,”
J. Opt. Soc. Am. B , vol. 24, no. 4, pp. 739–749, April2007.[3] A. Lapidoth, “Nearest neighbor decoding for additive non-gaussian noise channels,”
IEEE Trans. Inf. Theory , vol. 42,no. 5, pp. 1520–1529, September 1996.[4] A. Ganti, A. Lapidoth, and ˙I. E. Telatar, “Mismatched decoding revisited: general alphabets, channels with memory,and the wide-band limit,”
IEEE Trans. Inf. Theory , vol. 46, no. 7, pp. 2315–2328, November 2000.[5] I. S. Gradshteyn and I. M. Ryzhik,
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