M ay 2 00 8 On the Capacity of Free-Space Optical Intensity Channels

New upper and lower bounds are presented on the capacity of the free-space optical intensity channel. Thischannel is characterized by inputs that are nonnegative (represent ing the transmitted optical intensity) and by outputs that are corrupted by additive white Gaussian noise (because in free space the d isturbances arise from many independent sources). Due to battery and safety reasons the inputs are simultaneously constrained i n both their average and peak power. For a fixed ratio of the average power to the peak power the difference between the upper and the lower bounds tends to zero as the average power tends to infinity, and the ratio of the upper and lower bounds tends to one as the average power tends to zero. The case where only an average-power constraint is imposed on the input is treated separately. In this case, the differe nce of the upper and lower bound tends to 0 as the average power tends to infinity, and their ratio tends to a constant as the power tends to zero. I. I NTRODUCTION We consider a channel model for short optical communications in free space such as the communication between a remote control and a TV. We assume a channel model based on intensity modulation where the signal is modulated onto the optical intensity of the emitted light. Thus, the channe l input is proportional to the light intensity and is therefor e nonnegative. We further assume that the receiver directly measures the incident optical intensity of the incoming sig nal, i.e., it produces an electrical current at the output which is proportional to the detected intensity. Since in ambient li ght conditions the received signal is disturbed by a high number of independent sources, we model the noise as Gaussian. Moreover, we assume that the line-of-sight component is dominan t and ignore any effects due to multiple-path propagation lik e fading or inter-symbol interference. Optical communication is restricted not only by battery power, but also for safety reasons by the maximum allowed peak power. We therefore assume simultaneously two constraints: an average-power constraint E and a maximum allowed peak power A. The situation where only a peak-power constraint is imposed corresponds to E = A. The case of only an average-power constraint is treated separately. In this work we study the channel capacity of such an optical communication channel and present new upper and lower bounds. The maximum gap between upper and lower bound never exceeds 1 nat when the ratio of the averagepower constraint to the peak-power constraint is larger tha n 0.03 or when only the average power is constrained but not the peak power. Asymptotically when the available average and peak power tend to infinity with their ratio held fixed, the upper and lower bounds coincide, i.e., their difference tends to 0. We also present the asymptotic behavior of the channel capacity in the limit when the power tends to 0. The channel model has been studied before, e.g. in [1] and is described in detail in the following. The received signal is corrupted by additive noise due to strong ambient light that causes high-intensity shot noise in the electrical output s ignal. In a first approximation this shot noise can be assumed to be independent of the signal itself, and since the noise is caus ed by many independent sources it is reasonable to model it as an independent and identically distributed (IID) Gaussian pr ocess. Also, without loss of generality we assume the noise to be zero-mean, since the receiver can always subtract or add any constant signal. Hence, the channel output Yk at timek, modeling a sample of the electrical output signal, is given by Yk = xk + Zk, (1) where xk ∈ R0 denotes the timek channel input and represents a sample of the electrical input current that is proportional to the optical intensity and therefore nonneg ative, and where the random process {Zk} modeling the additive noise is given by {Zk} ∼ IID NR ( 0, σ ) . (2) It is important to note that, unlike the input xk, the outputYk may be negative since the noise introduced at the receiver ca n be negative. Since the optical intensity is proportional to the optical power, in such a system the instantaneous optical power is proportional to the electrical input current [2]. This is in contrast to radio communication where usually the power is proportional to the square of the input current. Therefore, in addition to the implicit nonnegativity constraint on the in put, Xk ≥ 0, (3) we assume constraints both on the peak and the average power, i.e., Pr[Xk > A] = 0, (4) E[Xk] ≤ E . (5) We shall denote the ratio between the average power and the peak power byα, α , E A , (6) where we assume 0 < α ≤ 1. Note thatα = 1 corresponds to the case with only a peak-power constraint. Similarly, α ≪ 1 corresponds to a dominant average-power constraint and onl y a very weak peak-power constraint. We denote the capacity of the described channel with peak-power constraint A and average-power constraint E by C(A, E). The capacity is given by [3] C(A, E) = sup I(X ;Y ) (7) whereI(X ;Y ) stands for the mutual information between the channel inputX and the channel output Y , where conditional on the inputx the outputY is Gaussian∼ N ( x, σ ) ; and where the supremum is over all laws on X ≥ 0 satisfying Pr[X ≥ A] = 0 andE[X ] ≤ E . In the case of only an average-power constraint the capacity is denoted byC(E). It is given as in (7) except that the supremum is taken over all laws on X ≥ 0 satisfyingE[X ] ≤ E . The derivation of the upper bounds is based on a technique introduced in [4]. There a dual expression of mutual informa tion is used to show that for any channel law W (·|·) and for an arbitrary distributionR(·) over the channel output alphabet, the channel capacity is upper-bounded by C ≤ EQ∗ [ D ( W (·|X) ∥ ∥R(·) )] . (8) Here,D(·‖·) stands for the relative entropy [5, Ch. 2], and Q∗(·) denotes the capacity-achieving input distribution. For more details about this technique and for a proof of (8), see [4, Sec. V], [6, Ch. 2]. The challenge of using (8) lies in a clever choice of the arbitrary lawR(·) that will lead to a good upper bound. Moreover, note that the bound (8) still contain s an expectation over the (unknown) capacity-achieving inpu t distributionQ∗(·). To handle this expectation we will need to resort to some further bounding like, . g. , Jensen’s inequality [5, Ch. 2.6]. The derivation of the firm lower bounds relies on the entropy power inequality [5, Th. 17.7. 3]. The asymptotic results at high power follow directly by evaluation of the firm upper and lower bounds. For the low power regime we introduce an additional lower bound which does not rely on the entropy power inequality. This lower bound is obtained by choosing a binary input distribution, a choice which was inspired by [7], and by then evaluating the corresponding mutual information. For the cases involving a peak-power constraint we further resort to the results on th e asymptotic expression of mutual information for weak signa ls in [8]. The results of this paper are partially based on the results in [9] and [6, Ch. 3]. The remainder is structured as follows. In the subsequent section we state our results and in Section III we give a brief outline of some of the derivations. II. RESULTS We start with an auxiliary lemma which is based on the symmetry of the channel law and the concavity of channel capacity in the input distribution. Lemma 1: Consider a peak-power constraint A and an average-power constraint E such thatα = AE > 12 . Then the optimal input distribution1 in (7) has an average power equal to half the peak power EQ∗ [X ] A = 1 2 , (9) irrespective ofα. I.e., the average-power constraint is inactive for all α ∈ ( 1 2 , 1 ] , and in particular C(A, αA) = C (