Upper and Lower Bounds on the Capacity of Wireless Optical Intensity Channels

Transcription

1 ISIT7, Nice, France, June 4 June 9, 7 Upper and Lower Bounds on the Capacity of Wireless Optical Intensity Channels Ahed A. Farid and Steve Hranilovic Dept. Electrical and Coputer Engineering McMaster University Hailton, ON, Canada {faridaa,hranilovic}@caster.ca Abstract Iproved upper and lower bounds on the capacity of wireless optical intensity channels under non-negativity and average optical power constraints are derived. We consider intensity odulated/direct detection (IM/DD) channels with pulse aplitude odulation (PAM). Utilizing the signal space geoetry and a sphere packing arguent, an upper bound is derived. Copared to previous work, the derived upper bound is tighter at low signal-to-noise ratios. In addition, a lower bound is derived based on source entropy axiization over discrete distributions. The proposed distribution provides a tighter lower bound copared to previous continuous distributions. The derived bounds asyptotically describe the capacity of PAM optical intensity channels at both low and high SNR. I. INTRODUCTION In this paper, we study the capacity of wireless optical intensity odulated/direct detection (IM/DD) channels. In these channels, inforation is odulated as the instantaneous optical intensity and hence the inforation bearing signal is restricted to be non-negative. An average aplitude, i.e., average optical power, constraint is iposed to ensure eyesafety. The direct application of techniques fro electrical channels to this channel is thus not straight forward due to the aplitude constraints. Here, we present iproved upper and lower bounds for wireless optical channels which take the aplitude constraints into account explicitly. Wireless optical channels can be well odelled as conditionally Gaussian channels with signal independent noise []. For conditionally Gaussian channels with bounded-input and power constraints the capacity-achieving distribution, under certain conditions, is shown to be discrete with a finite nuber of probability ass points [], [3]. Siilar results were obtained for optical photon counting channels, i.e. Poisson channels, with optical power constraints [4]. Since the channel capacity is the axiu utual inforation between transitter and receiver over all possible input distributions, any input distribution results in a lower bound for the channel capacity. Based on this reasoning, a lower bound for the capacity of wireless optical IM/DD channels was coputed using the axentropic continuous exponential distribution satisfying the aplitude constraints [5]. The channel capacity of wireless optical intensity channels can be upper bounded by applying a siilar sphere-packing arguent presented by Shannon [6] in a region which guarantees that the aplitude constraints are et. You and Kahn utilized sphere-packing to derive an upper bound for the optical IM/DD channel capacity with ultiple-subcarrier odulation [7]. Results for band and power-liited optical intensity channels were presented in [5] where the total volue is approxiated by a generalized n-cone. As a result, the derived bound is only tight at high signal-to-noise ratio (SNR) and loose at low SNR. In this work, tight upper and low bounds on the capacity of pulse aplitude odulated (PAM) wireless optical IM/DD channels are derived. Using the intuition fro previous studies, a tight lower bound is derived using a faily of entropy axiizing discrete distributions. Although not necessarily capacity achieving, these distributions are shown to provide a tight lower bound for the capacity of wireless optical IM/DD channels at both low and high SNRs. Copared to previous bounds based on continuous distributions, the presented bound has approxiately double the channel capacity at SNR= db. In addition, an analytic upper bound to the channel capacity is derived using a sphere packing arguent. Unlike previous work [5], the Minkowski su of convex bodies is utilized to obtain the exact volue of the outer parallel body at fixed distance fro a regular n-siplex. As a result, the derived upper bound is tighter than previous bounds [5] at low signalto-noise ratios. Since ost wireless optical links typically operate at low SNRs, the tightness of the derived bounds at low SNR provides a useful benchark for counication syste design. II. SYSTEM MODEL Wireless optical counication links transit data by odulating the transitted optical power of a laser. In practical links, only the optical intensity is odulated and detected. In the following analysis, we consider pulse aplitude odulation (PAM). The transitted optical signal is constrained to be non-negative due to physical constraints. Due to eye safety concerns, a constraint is also iposed on the average optical power transitted P, i.e., the average aplitude. The output electrical signal is related to the incident power by the detector responsivity coefficient R. Without loss of generality, we consider R =. A good statistical channel odel for this channel is [], y = x + z /7/$5. c 7 IEEE 46

2 ISIT7, Nice, France, June 4 June 9, 7 where x is the transitted optical signal with average optical power E{x} P, y is the output electrical signal and z is theral noise which is well odelled as zero-ean, signal independent, Gaussian distributed with variance σ. We define the optical signal-to-noise ratio as SNR = P/σ as in previous work [3], [5]. III. LOWER BOUND ON CHANNEL CAPACITY The capacity of the wireless optical IM/DD channel is defined as the axiu utual inforation between channel input and output over all possible input distributions satisfying the non-negativity and average optical power constraints. Consequently, the utual inforation obtained by any input distribution satisfying the aplitude constraints is a lower bound for the channel capacity. Since the capacity achieving distribution of conditional Gaussian channels under aplitude and average power constraints was shown to be discrete, a discrete distribution is proposed and a lower bound on channel capacity is derived. The proposed distribution is obtained through input source entropy axiization. Consider a discrete distribution for x over the alphabet l Z+, where Z + is the set of non-negative integers and /l > is the spacing between ass points. A probability ass of p x (k; l), k Z +, is assigned to each point such that p x (k; l) = and k l p x(k; l) = P. () Thus, p x (k; l) satisfies both the non-negativity and average aplitude constraints of wireless optical IM/DD channels. The entropy of the source is defined as, H l (x) = p x (k; l) log p x (k; l). Although any pf p x (k; l) is sufficient to provide a lower bound, we propose selecting the axentropic distribution subject to () under the intuition that it will be close to the capacity at high SNRs. In other words, for a given l >, p x(k; l) = arg ax p x (k;l) s.t H l (x) Eqn. is satisfied. Applying the ethod of Lagrange ultipliers, the solution is given by, k p lp x(k; l) =. + lp + lp Notice that although p x(k; l) is a faily of distributions paraeterized in l, P is independent of l. Using this faily of distributions, the axiu utual inforation obtained over this set will be a function of both l and σ. For a given σ there is an optiu value for l that axiizes the utual inforation. This results in a lower bound, C L, for the channel capacity and can be forulated as, C L = ax l s.t f y (y) = I l (x; y) = h(y) log πeσ, p x(k; l)δ(y k/l) f z (y), l > where is the convolution operator and δ( ) is the Dirac delta functional. Substituting the discrete distribution p x(k; l) results in the utual inforation, I l (x; y) = log ( [ p x(k; l) e (y k/l) /σ πσ p /σ x(k; l) e (y k/l) πσ )] dy log (πeσ ). Notice that a relation between noise standard deviation σ and the spacing between successive points /l ust exist. Let () lσ = β, w = l y. (3) Rearranging () with respect to l and substituting (3) yields, [ I β (x; y) = p /β x(k; β) e (w k) πβ log ( where p x(k; β) e (w k) /β πβ p X (k; β) = ( + β P σ )] dw log (πeβ ), ) k β P σ + β P. σ Thus, for a given P/σ, I β (x; y) is a function of β which quantifies the ratio between the noise variance σ and ass point separation /l. A lower bound for the capacity of wireless optical IM/DD channels can be obtained as, C L = ax β I β (x; y). Note that, the optiu value for β is a function of P/σ. For a given P/σ, this axiization can be solved nuerically using the bisection ethod over wide range of β to find C L. The lower bound, C L, obtained fro the proposed discrete distribution p x(k; β) is tighter at both low and high SNR than the previously reported bounds based on continuous distributions. Although no analytical for is provided, the bound can be coputed efficiently through nuerical integration. An advantage of this approach, however, is that it avoids a costly search procedure to find the capacity achieving distribution. In addition, it provides a closed for for the input distribution p x(k; β). (4) 47

3 ISIT7, Nice, France, June 4 June 9, 7 IV. CHANNEL CAPACITY UPPER BOUND Due to the non-negativity and average optical power constraints, any sequence of n transitted PAM sybols can be represented geoetrically as the set of points contained inside a regular n-siplex [8]. For conditionally Gaussian channels, the set of the received codewords approaches the parallel body to this regular n-siplex for large n. However, the axiu achievable rate can be upper bounded by the axiu asyptotic nuber of non-overlapping spheres packed in this volue, i.e. via a sphere packing arguent. Unlike previous approaches, we use an exact expression for the volue of the set of received codewords to copute the bound, yielding greater accuracy at low SNRs. A. Set of Received Codewords and Volues Consider transitting a sequence of n independent PAM sybols to for the codeword x = (x, x,..., x n ). The adissible set of transitted codewords, Ψ(P ), is defined as, Ψ(P ) = {x R n : x i, n x i P, i =,,..., n}. The set Ψ is a regular n-siplex of equal side lengths np located at the origin as shown in Fig.. According to the Gaussian noise odel presented, the received vector y has a Gaussian distribution with ean x as follows, i y = x + z, where z has i.i.d. Gaussian coponents. Define B n as the n- diensional ball. In the signal space representation, for large enough n, y will, with high probability, be on the surface of ρb n, centered on x where ρ = nσ. Define the set Φ(P, ρ) as the outer parallel body to Ψ(P ) at distance ρ which results as the Minkowski su of Ψ(P ) and ρb n. Forally, Φ(P, ρ) = {y R n : y = x + b, x Ψ(P ), b ρb n }. The regions defined by Ψ(P ) and Φ(P, ρ) are illustrated in Fig. for the two-diensional case. An upper bound for the wireless optical IM/DD channel capacity can be obtained by applying a sphere-packing arguent and finding the axiu nuber of non-overlapping spheres that can be packed in Φ(P, ρ) as n. Let V ( ) denote the volue of a closed set. The volue of ρb n is given by, V (ρb n ) = κ n ρ n = πn/ (n/)! ρn (5) where κ n denotes the volue of the n-diensional unit ball. The axiu rate can be expressed in ters of the asyptotic nuber of transissible signals as, C li n log. (6) V (ρb n ) Fig.. Two-diensional representation of an n-siplex and its parallel body at distance ρ. B. Volue Approxiation Since Φ(P, ρ) results as the Minkowski su of two sets, analytic expressions exist to copute its volue. The volue can be expressed in ters of the intrinsic volues V (P ) of an n-siplex as [9], = The V (P ) are given as, n V (P )κ n ρ n. (7) = (np ) V (P ) = γ, (8)! where γ n = and when n [9], n γ = n + n + + π e (+)v [ erfc(v) ] n dv. (9) The ratio of the outer volue to the volue of the n-diensional ball V (ρb n ) is thus, n κ n γ np = () V (ρb n ) κ n! ρ = In order to find a siple analytic upper bound for the channel capacity, γ needs to be siplified to a ore copact for. Since ( erfc(v)) for v it follows that, + π e (+)v [ erfc(v) ] n dv n, and (9) can be bounded as, n + + γ < + π e (+)v [ erfc(v) ] n dv. 48

4 ISIT7, Nice, France, June 4 June 9, 7 Consider the substitution erfc(v) = u and applying the bound e (erfc (u)) eu, u, then γ can be upper bounded as follows, n + γ < ( + )( / e) + u ( u) n du, n + < ( + )( e) + u ( u) n du = } {{ } B( +,n ) n + + ( + )( e) ( )!(n )! (n )! = λ where B( +, n ) is the beta function. Substituting into () results in n κ n λ np n < = ψ. V (ρb n ) κ = n! ρ }{{} = ψ The capacity can be upper bounded as follows, C li n log < li = li = ax V (ρb n ) n log [n ax ψ ] n log [ax ψ ] li n log [ψ ] () where the last inequality is due to the onotonic increase of the log ( ) function. Let = αn α, then the capacity can be expressed as, C < ax α li n log [ κ( α)n λ αn κ n (αn)! ( np ρ ) αn ] Bounding the factorial using Stirling s approxiation ( n ) n ( πn e ( n+) n ) n < n! < πn e ( n) e e results in the following upper bound [ α e C < ax log (P α 4π σ where, Θ(α) = α 3α ( α) ( α) ) α ] Θ(α) ( α ) ( α ) () In the following we will show that there is a unique root α < that axiizes the capacity upper bound given in () and explicitly derive a closed for expression for α. C. Uniqueness of α Consider the right hand side of (). To siplify the analysis, the natural logarithic function is considered instead of log where the optiu value for α will not be affected. Denote this ter by J where, e P J = α ln K ln [Θ(α)], K = 4π σ. In order to axiize J with respect to α, let J/ α =, resulting in the cubic equation, Λ(α) = α 3 aα + 3aα a =, (3) where a = exp( ln K ). Proposition : For all a >, there exists a unique root for Λ(α), denoted α, lying in the interval α <. Proof: Note that, Λ() = a and Λ() =. As a result, there exists at least one root of Λ(α) in [, ). The extrea, α + and α, of Λ(α) are, α ± = ( a ± ) a 3 9a The existence of these extrea over R depends on a as follows, No extrea, a < 9; α ± One extriu at α = 3, a = 9; Two extrea, a > 9. To prove the existence of a unique root for Λ(α) in [, ), it is sufficient to prove that α ± > for all values of a >. For a < 9, there are no extrea, and the single root in [, ) is unique. For a > 9, consider the following lea. Lea : If a > 9 then α ± >. Proof: It is clear that α + > 3 when a > 9. In addition, α is decreasing in a and its asyptotic value as a is greater than one as follows, dα da = 3 ( li α = 3 a >. 9/a ) <, when a > 9, As a result α ± > whenever a > 9. Therefore, for all values of a > there is a unique real root for Λ(α) in [, ) and is denoted as α. D. Expression for α To find the unique root α, define Λ o = Λ(a/3), δ = (a 9a)/9, h = δ 3. The following three cases are considered [] to solve (3). (i) Λ o > h : One root exists for Λ(α) and is given by, where Ω ± = α = a 3 + Ω + + Ω, (4) [ ( Λ o ± ) ] /3 Λ o h. 49

5 ISIT7, Nice, France, June 4 June 9, 7 (ii) Λ o = h : Two roots coincide and α depends on the sign of Λ o, { /3 α a = in 3 + Λo, a } /3 3 Λo. (iii) Λ o < h : Three distinct roots, and α is obtained as, { } a πi α = in + δ cos i=,, θ, where cos(3θ) = Λ o h. The upper bound for the optical channel capacity can be explicitly written as follows, α C U = log e α P 4π σ Θ(α. (5) ) V. RESULTS Fig. presents the lower bound, C L, derived using the discrete source distribution p x(k; l) and the upper bound, C U in (5). The proposed lower bound is tight at both low and high SNRs and asyptotically describes the channel capacity. In addition, the utual inforation is shown for one-sided continuous exponential and Gaussian input distributions which satisfy non-negativity and average optical power constraints. The lower bound proposed here, indicates that a significant increase in rate is possible at low SNRs, where ost wireless optical links operate. An SNR argin of 3.7 and db can be noticed between C L and the bound obtained fro the onesided exponential distribution, proposed in [5], at a channel capacity of.5 and bits/channel use respectively. In addition, the presented bound has approxiately double the channel capacity (.85 and.45 respectively) at SNR= db. For the sake of coparison, the utual inforation with unifor M-ary source distributions are also presented. Notice that a significant gap of 3.5 db exists between C L and the lower bound fro a unifor (-PAM) discrete source distribution at C =.5 bits/channel use. The derived upper bound C U is tight at low SNRs and asyptotically incurs an increase of log ( e/) in channel capacity at high SNRs. Copared to the previous upper bound, CU o [5, Eq. ], C U is a significantly better representation for the channel capacity at low SNRs. As a result, C U is a better etric for coparison at low SNRs (SNR <-3.5 db) over CU o, since a ajority of wireless optical IM/DD channels operate in this low SNR regie. Note that, the unique root α depends on SNR through a. Nuerical siulations indicate that (4) is utilized to find α when SNR<9.9 db. As a result, the upper bound can be defined by (4) and (5) at low and oderate SNRs. VI. CONCLUSION Lower and upper bounds for the capacity of PAM wireless optical IM/DD channels are derived. The proposed lower bound is tight at both low and high SNRs. Although no Capacity bits/channel use C U o C U C L Exponential Gaussian 8 PAM 4 PAM PAM P/σ Fig.. Capacity bounds and utual inforation for continuous one-sided exponential, Gaussian and discrete unifor PAM. analytical for is provided, the bound can be efficiently coputed nuerically avoiding a search procedure to find the capacity achieving distribution. The proposed discrete distribution achieves higher utual inforation than the continuous one-sided exponential and Gaussian distributions or discrete unifor M-ary distributions. In addition, an analytical expression for a tight upper bound at low SNRs is derived based on a sphere packing arguent. The asyptotic behavior of the upper bound at high SNRs incurs a constant increase over the actual channel capacity. Since ost wireless optical links operate at relatively low SNRs, the tightness of the derived lower and upper bounds at low SNRs provides a useful benchark for odulation and coding design. REFERENCES [] J. M. Kahn and J. R. Barry, Wireless infrared counications, Proc. IEEE, vol. 85, pp , Feb [] J. G. Sith, The inforation capacity of aplitude and varianceconstrained scalar Gaussian channels, Inf. Contr., vol. 8, pp. 3 9, 97. [3] T. H. Chan, S. Hranilovic, and F. R. Kschischang, Capacity-achieving probability easure for conditionally Gaussian channels with bounded inputs, IEEE Trans. Infor. Theory, vol. 5, pp , June 5. [4] S. Shaai, Capacity of a pulse aplitude odulated direct detection photon channel, Proc IEE Coun. Speech and Vission, vol. 37, pp , Dec. 99. [5] S. Hranilovic and F. R. Kschischang, Capacity bounds for power- and band-liited optical intensity channels corrupted by Gaussian noise, IEEE Trans. Infor. Theory, vol. 5, pp , May 4. [6] C. E. Shannon, Counication in the presence of noise, Proc. IRE, vol. 37, no., pp., Jan [7] R. You and J. M. Kahn, Upper-bounding the capacity of optical IM/DD channels with ultiple-subcarrier odulation and fixed bias using trigonoetric oentspace ethod, IEEE Trans. Infor. Theory, vol. 48, pp , Feb.. [8] S. Hranilovic and F. R. Kschischang, Optical intensity-odulated direct detection channels: Signal space and lattice codes, IEEE Trans. Infor. Theory, vol. 49, no. 6, pp , June 3. [9] U. Betke and M. Henk, Intrinsic volues and lattice points of crosspolytope, Monatsh. Matheatik, vol. 5, pp. 7 33, 993. [] R. W. D. Nickalls, A new approch to solving the cubic: Cardan s solution revealed, The Math. Gazette, vol. 77, pp ,