capacity of optical wireless communication channels · 2019-11-28 · optical wireless...

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Research

Article submitted to journal

Subject Areas:

Wireless communications,

information theory

Keywords:

Intensity modulation, capacity

bounds, asymptotic capacity

Author for correspondence:

Anas Chaaban

e-mail: [email protected]

Capacity of Optical WirelessCommunication ChannelsA. Chaaban1 and S. Hranilovic2

1School of Engineering, University of British Columbia,

Kelowna BC, Canada2Department of Electrical & Computer Engineering,

McMaster University, Hamilton ON, Canada

Optical wireless communications is realized bymodulating the intensity of a light source anddetecting intensity fluctuations at the receiver. Thismode of operation, known as intensity-modulationand direct-detection (IM/DD), is simple to implementin practice. However, computing the channel capacityof the underlying channel is not straightforwarddue to the amplitude constraints that arise due toIM/DD operation. In particular, the transmit signalmust be non-negative while the peak and averageamplitudes are constrained due to practical and safetyconsiderations. Though a closed-form for the capacityof IM/DD channels is not known, much work hasbeen done to find capacity bounds and asymptoticcapacity expressions. In this paper, a description ofthe IM/DD channel and its physical constraints ispresented, followed by a review of recent progresspertaining to the capacity of IM/DD channels.Additionally, capacity achieving distributions arediscussed along with simple constructions thatapproach capacity.

1. IntroductionCapacity is the ultimate “speed limit” of informationtransmission over any communications channel. It tellsthe system designer what communication rates arepossible (i.e., reliable) and which are not. Although thecapacity and methods to approach it are known for awide host of channels (e.g., in particular the additivewhite Gaussian channel (AWGN)), there exist channelsfor which the capacity is ill understood.

One such channel is the IM/DD channel whichmodels optical intensity communications. Understandingthe capacity of the IM/DD channel became ever moreimportant recently due to the renewed interest in opticalwireless communications (OWC) in various forms such

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X ⊕ Y

background light thermal noise

Figure 1: A depiction of an OWC communication system.

as Light-Fidelity (LiFi), Visible-Light Communications (VLC), Free-Space Optics (FSO),Underwater Optical Communications (UOC), and non-line-of-sight optical communication [1–5].Motivated by this, the IM/DD channel capacity has been investigated by several research teamsrecently, and significant advances have been achieved which have greatly contributed to ourunderstanding of this channel.

In this work, we provide the reader with a review of these recent advances. We start with adescription of optical wireless channels based on IM/DD. Then, we focus on a common IM/DDchannel model which is the input-independent Gaussian channel model, and we review resultson the capacity of this channel model, and most importantly for communication engineers, wereview the form of distributions which approach capacity. Finally, we discuss the relation of theresults with systems based on uni-polar orthogonal frequency-division multiplexing (OFDM),and discuss recent extensions to systems with multiple apertures, i.e., multiple-input multiple-output (MIMO) IM/DD channels.

2. Channel Model and ConstraintsNotation: Throughout this paper, lower-case font denotes deterministic scalars and upper-caseletters denote random variables. The notation X ∼ fX indicates that X is distributed accordingto a distribution fX (probability density or mass function), EX [·] denotes the expectation withrespect to X , and P{·} denotes the probability of an event. Calligraphic letters denote sets while

the Q-function defined as Q(x) =∫∞x

1√2πe−

u2

2 du. For a random variable X , H(X) denotes itsentropy ifX is discrete and h(X) denotes its differential entropy if X is continuous. Additionally,for X ∼ fX and Y ∼ fY , I(X;Y ) denotes the mutual information between X and Y , andD(fX‖fY ) denotes the relative entropy (Kullback Leibler Divergence) between their distributions.

An optical wireless communication (OWC) system sends data by modulating the intensity of alight source (e.g., a light-emitting diode (LED), laser diode) in discrete time intervals of length t asshown in Fig. 1. Let the sequence of modulating symbols transmitted in each t interval be denotedas {xi}ni=1, for some integer n> 1. The symbol xi physically represents the driving electricalcurrent signal which is applied to the light source while transmitting the ith symbol which isfixed over a the symbol duration t. For both LEDs and lasers, the transmitted light intensity canbe written as ηeoxi where ηeo is the electrical-to-optical conversion efficiency of the light sources.Without loss of generality, ηeo = 1 in the balance of the paper.

Given that the light source is able to modulate only the output intensity or optical powerof the source, rather than the amplitude and phase of the carrier as in radio systems, ∀n, xi ≥ 0.Additionally, for eye-safety and due to the limited dynamic range of the devices, a peak limitationis placed on the emissions from the source, that is xi ≤A. Another unique feature of opticalwireless channels is that for visible light communication (VLC) channels, users may dim or alterthe average brightness of the source. An average optical power constraint constraint EX [X]≤ E

models a dimming level in VLC applications.In the OWC channel, the transmitted signal is attenuated during propagation and it may be

impacted an additive by background light. The attenuation is modelled as a linear factor α andarises due to beam divergence and is dominated by absorption and scattering in longer range

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free-space optical links. Since this factor is simply a scale, to make the discussion more clear andwithout loss of generality, α= 1 for the balance of the paper.

The overall channel model of an OWC system depends on the regime in which the system isoperating. When the optical power at the receiver is very low, the discrete nature of the receivedphotons must be considered. Notice that optical intensity or power is measure of photon flux.Thus, the intensity can also be considered as the rate of photon arrival. In this low power regime,the receiver is a photon counter which outputs a statistic depending on the number of receivedphotons in each t interval. The received signal is composed of both the transmitted opticalintensity signal as well as background light where background light is modelled as a source with aconstant arrival rate of λ photons per second. Conditioned on the transmitted signal, the detectedphoton count yi in interval i is conditionally Poisson, i.e.,

P(yi|xi) =(xi + λ)yi

yi!exi+λ, xi ∈R+, yi ∈Z+ (2.1)

where t= 1 for convenience and without loss of generality. Notice that in this discrete-time Poisson(DTP) channel the detection process for a given transmitted intensity is inherently stochastic anddepends in general both on the transmitted signal as well as the background light. Such DTP linksoccur often in applications such as long range inter-satellite optical links operating at ranges ofmany thousands of kilometres [34,35] .

Definition 2.1 (DTP Channel). Let X be a random variable representing the transmitted intensitydistributed on [0,A] with EX [X]≤ E. The received signal is corrupted by a background light of intensityλ. In the discrete-time Poisson channel, the channel output, Y , which represents a count of the number ofphotons, is conditionally Poisson with rate x+ λ.

When the intensity of the received signal increases, the distribution of the number of receivedphotons in any interval approaches a Gaussian in distribution with mean and variance of xi + λ.In this regime, the channel is termed intensity-modulation with direct-detection (IM/DD). Ignoringthe scaling constants as defined above, the received signal can be modelled as

yi = xi + zi, (2.2)

where zi is Gaussian and has zero mean under the assumption that the bias to the mean dueto the constant background light is removed. Notice that in general, the variance of the addednoise depends on both the background light as well as the signal. This signal-dependent noiseis a significant practical impairment when the signal power received at the detector is largewith respect to the background light and is often termed the conditionally Gaussian IM/DDchannel [6].

Definition 2.2 (Signal-Dependent IM/DD Channel). LetX be a random variable distributed on [0,A]

with EX [X]≤ E, andZ(X) be Gaussian noise with zero mean and forX = x has variance σ2 = σ20 + σ21x

for constant σ0 and σ1. A channel with inputX and output Y =X + Z(X) is termed a signal-dependentIM/DD channel.

In most cases of practical interest for indoor and terrestrial OWC, the background light greatlydominates the intensity of the received signal intensity. In this case, the noise variance is wellmodelled as being independent of the transmitted signal giving rise to the following IM/DDchannel definition.

Definition 2.3 (IM/DD Channel). Let X be a random variable distributed on [0,A] with EX [X]≤ E,and Z be Gaussian noise with zero mean and constant variance σ2. A channel with input X and outputY =X + Z is termed an IM/DD channel.

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Notice that this channel is also appropriate in the case when front-end electronic noise of thereceiver dominates background-induced shot noise. Given the great practical significance of thesignal-independent IM/DD channel for visible light communication channels, this paper willfocus on the channel in Def. 2.3. However, we also present some recent connections betweenthe fundamental DTP channel in Def. 2.1 and IM/DD channels.

3. Channel Capacity

(a) What is Channel Capacity?The information rate is the amount of information sent over a channel per unit time. The task ofthe communication system designer is to try to achieve the highest information rate. However,the rate can not be increased without bound due to the limited ability of the receiver to discernamongst different signals distorted by noise. What is this communications rate limit?

This limit is termed the channel capacity. As the name suggests, it is a property of the channeldefined by inputs, outputs, and a channel law which relates the output to the input (xi, yi, and(2.2), respectively, in the IM/DD channel). It is not a property of the transmission scheme usedto send bits over this channel but is rather a property of the channel itself. The channel capacityis the largest number of bits per transmission that can be sent ‘reliably’ over a channel using anytransmission scheme that respects the imposed constraints [7].

To transmit information over the IM/DD channel using n symbols, the transmitter chooses amessage M uniformly at random from amongst a set of messagesMn = {1, 2, . . . , `n}. Note thateach message can be represented using log2(`n) bits. The transmitter encodes the message Minto a sequence {x}ni=1 using an encoding function φn :M 7→ [0,A]n which satisfies all channelconstraints, and sends {xi}ni=1 over the channel.

The receiver receives {yi}ni=1, and attempts to decode M using a decoding function ψn :

R 7→M. This operation results in a decoded message M . The error probability induced by thisoperation is pn = P{M 6=M}, and each transmitted message carries log2(`n) bits.

The transmission rate R (in bits/transmission) is said to be an achievable rate if there exists asequence of Mn, φn, ψn so that R=

log2(`n)n and pn→ 0 as n→∞ (which defines the meaning

of ‘reliable’ communication). The maximum achievable rate R is the channel capacity C. Since C

depends on the channel constraints A and E, we shall denote it Cρ(A) where ρ= AE is the peak-to-

average optical power ratio.The units of channel capacity can be express in bits/transmission or nats/transmission, where

1nat= 1log(2)

bits. We will use nats/transmission henceforth for convenience.

(b) Memoryless-Channel CapacityThe IM/DD channel belongs to the family of memoryless channels because noise is i.i.d., andhence, its capacity is well defined. In particular, the channel capacity can be written as [7]

Cρ(A) =maxfX

I(X;Y ), (3.1)

where fX is the distribution of X , and the maximization is over the set of feasible distributions(i.e., those that satisfy the channel amplitude constraints). For IM/DD channels, the non-negativity, peak and average amplitude constraints onX make solving this problem cumbersome.

This is in stark contrast with the AWGN channel with real-valued X with power constraintEX [X2]≤ p. For the AWGN channel the optimal input distribution is Gaussian and the capacity

is given by 12 log

(1 + h2p

σ2

)nats/transmission. The elegance and simplicity of this expression has

inspired a wide range of studies, and led to significant developments in wireless communications.Alas, to date, no one has been able to derive such an elegant expression for IM/DD channels!

At this point, the best way to evaluate this capacity is to use a non-linear optimizationframework to obtain the optimal input distribution. For example, the Blahut-Arimoto algorithm

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[8,9] can be used or numerical optimization methods can be applied directly. Though the optimalinput distribution f∗X and channel capacity can be found numerically, insights necessary for thedesign of communication systems are not apparent.

Recent research has led to capacity bounds and asymptotic capacity expressions that can helpto circumvent this problem. Instead of calculating the capacity of the IM/DD channel, one canresort to upper and lower bounds, i.e., find Cρ(A) and Cρ(A) such that Cρ(A)≤ Cρ(A)≤ Cρ(A).To be meaningful, bounds should be tight, especially in practical regimes of operation, and simpleenough to permit analysis. The following section sheds light in recent developments in this area.

4. Capacity Upper BoundsUpper bounds on the capacity of IM/DD channels have been derived using two main approaches:(i) using a dual expression for the channel capacity [10] and (ii) using sphere-packing [11,12].

(a) Duality-Based Upper BoundsApart from (3.1), an alternate expression for the capacity of the IM/DD channel is [17]

Cρ(A) = inffY

supfX

EX[D(fY |X‖fY )

], (4.1)

where the infimum is over all distributions fY on the set of real numbers, the supremum is overall distributions fX satisfying the constraints, and fY |X is the distribution of Y given X (channellaw). Consequently, choosing fY freely leads to a capacity upper bound

Cρ(A)≤ supfX

EX[D(fY |X‖fY )

]. (4.2)

The question is how to choose fY and how to upper bound this supremum. This has been tackledin [10] by choosing fY to be ‘similar’ to the distribution of a max-entropic X , i.e., a distributionthat maximizes h(X) (given in (5.1) and (5.2)). The intuition is that Y and X have ‘similar’distributions when the noise variance is small. Thus, it is expected that the resulting boundsshould be tight asymptotically, i.e., when A,E� σ. The supremum and expectation in (4.2) areupper bounded in [10] using further analysis leading to the following bounds.

Theorem 4.1 ( [10]). The capacity of the IM/DD channel in Def. 2.3 is upper bounded as C≤ Clmwρ (A)

where

Clmwρ (A) =

(1− 2Q

(δ + A

2

σ

))log

(A+ 2δ√

2πσ(1− 2Q( δσ ))

)− 1

2+Q

(δ

σ

)+

δ√2πσ

e−δ2

2σ2 ,

(4.3)

for ρ≤ 2, and

Clmwρ (A) =

(1−Q

(δ + A

ρ

σ

)−Q

(δ + (A− A

ρ )

σ

))log

(A(e

µδA − e−µ(1+

δA )

√2πσµ(1− 2Q( δσ ))

)− 1

2+Q

(δ

σ

)

+δ√2πσ

e−δ2

2σ2 +σµ

A√2π

(e−

δ2

2σ2 − e−(A+δ)2

2σ2

)+µ

ρ

(1− 2Q

(δ + A

2

σ

)), (4.4)

for ρ > 2, where µ and δ are positive free parameters.

Another upper bound can be derived using the dual expression in (4.2), by choosing fY to bea Gaussian distribution. The intuition is that when A� σ, the output distribution fY is ‘similar’to the distribution of noise, which is Gaussian. This leads to the following upper bound.

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Theorem 4.2 ( [10]). The capacity of the IM/DD channel in Def. 2.3 is upper bounded by Cρ(A)≤ C0ρ(A)

where

C0ρ(A) =

1

2log

(1 +

Aρ′ (A−

Aρ′ )

σ2

), (4.5)

where ρ′ =max{2, ρ}.

This upper bound was also derived in [12] using the following approach. First, we notethat a random variable X ∈ [0,A] with E[X]≤ E has a maximum variance of A

ρ′ (A−Aρ′ ) where

ρ′ =max{2, ρ}. This maximum variance is achievable if X takes on values in {0,A}, with P{X =

A}= 1ρ′ and P{X = 0}= 1− 1

ρ′ . Then, Y =X + Z has a maximum variance of Aρ′ (A−

Aρ′ ) + σ2.

Consequently, h(Y ) can be no larger than the differential entropy of a Gaussian random variablewith variance A

ρ′ (A−Aρ′ ) + σ2, since the Gaussian distribution maximizes differential entropy

under a variance constraint [13]. This leads to the upper bound (4.5).In addition to the results above, a bound derived by McKellips [14] for peak-constrained

channels can be adapted to the IM/DD channel with ρ= 2. This bound, presented next, is alsovalid for any ρ, Cρ(A)≤ C2(A) [10].

Theorem 4.3 ( [14]). The capacity of the IM/DD channel in Def. 2.3 is upper bounded by Cρ(A)≤ Cm2 (A)

where

Cm2 (A) = log

(1 +

A√2πeσ2

). (4.6)

This bound in this theorem was originally derived in [14] by maximizing h(Y ) when thesupport of Y is parsed into two regions, [0,A] and (−∞, 0) ∪ (A,∞). It was re-derived and refinedin [15] using the duality approach (4.2).

Example 4.1. For an IM/DD channel with A= 10, E= 4, and σ2 = 1, the bounds in Theorems 4.1, 4.2,and 4.3 evaluate to C

lmwρ (A) = 1.3257, C

0ρ(A) = 1.6094, and C

m2 (A) = 1.2296 nats/transmission. Thus,

the capacity Cρ(A) of this channel is no larger than 1.2296 nats/transmission.

While the duality-based bounds presented here rely on careful analysis, in the next section ageometric view of upper bounding the IM/DD capacity is presented.

(b) Sphere-Packing Upper BoundsThe channel requires a bound on the average optical power, i.e., E[X]≤ E. By the law of largenumbers, the transmitted sequence of symbols {xi}ni=1 must satisfy 1

n

∑ni=1 xi ≤ E almost surely

as n grows. Combining this with the constraint X ∈ [0,A], we conclude that {xi}ni=1 must liewithin the intersection of an n-dimensional hypercube of side length A and an n-simplex withside length nE. This is termed the permissible signal region.

Similarly, since the noise Z satisfies E[Z2] = σ2, then, for large n, the noise sequence {zi}ni=1

must lie almost certainly near the surface of a ‘noise sphere’ in n-dimensions of radius√nσ2 [18].

As a result, for large n, each transmit sequence {xi}ni=1 will be mapped to a point on a sphereof radius

√nσ2 about {xi}ni=1 at the receiver. Intuition suggests that, for reliable communication,

spheres corresponding to different sequences {xi}ni=1 must have negligible overlap at thereceiver. How many such spheres can be fitted with their centers within the permissible signalregion?

One can upper bound this number of spheres by dividing the volume of the√nσ2-

neighbourhood of the permissible region (the set of points in the n-dimensional space whichare at at most a distance

√nσ2 outside the permissible region) by the volume of an individual

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noise sphere. Then, taking the logarithm of this upper bound, dividing by n, and taking the limitas n→∞ leads to a capacity upper bound.1

Finding the volume of the√nσ2-neighbourhood is not straightforward. To simplify the

problem, let us consider the cube or the simplex separately, which still leads to an upper boundsince dropping a constraint can not shrink the permissible region.

One can find the volume of the√nσ2-neighbourhood of a cube (Fig. 2) using the Steiner-

Minkowski theorem for polytopes [21, Proposition 12.3.6]. This theorem basically counts thevolume of portions constituting the

√nσ2-neighbourhood as depicted in Fig. 2b. This approach

was used in [12] to derive a capacity upper bound.The volume of the

√nσ2-neighbourhood of a simplex (Fig. 2c) is more difficult to bound. The

Steiner-Minkowski theorem for polytopes requires calculating the dihedral angles of the simplex,which becomes difficult at higher dimensions. In [11], the authors bounded these angles to obtaina capacity upper bound. Other bounds based on a similar approach were derived in [22,23].Instead of using the Steiner-Minkowski theorem, a different approach for bounding the volumewas developed in [12] leading to a tighter bound.

By taking the minimum of the resulting sphere-packing bounds for a cube and for a simplex,one obtains a capacity upper bound for the IM/DD channel. This is given in the followingtheorem.

Theorem 4.4 ( [12]). The capacity of the IM/DD channel in Def. 2.3 is upper bounded by Cρ(A)≤Ccmaρ (A) where

Ccmaρ (A) =min

{sup

α∈(0,1]α log

(A

√2πeσα(1− α)

3(1−α)2α

), supα∈(0,1]

α log

( √eAρ

√2πσα

32 (1− α)

1−αα

)}.

(4.7)

Example 4.2. For an IM/DD channel with A= 10, E= 4, σ2 = 1, the upper bound in Theorem 4.4evaluates to C

cmaρ (A) = 1.4104 nats/transmission. This is tighter that the bound C

0ρ(A) in Example

4.1, but looser than the bound Clmwρ (A) and C

m2 (A) in the same Example. The main reason is that the

sphere-packing bounds in Theorem 4.4 ignore one of the two constraints (A or E).

To provide a complete picture of the capacity, these upper bounds must be accompanied bycapacity lower bounds. This is the topic of the next section.

5. Capacity Lower BoundsA lower bound on capacity can be found by computing the mutual information for a given inputdistribution. Several capacity lower bounds have been developed for the IM/DD channel usingdifferent approaches. Some of the developed bounds are tight in some regimes of operations,allowing us to characterize capacity in these regimes. We present these lower bounds next.

We focus on lower bounds which have simple analytical expressions. These are namely lowerbounds which use a continuous input distribution fX . Lower bounds based on discrete inputdistributions will be discussed in Sec. 6.

To derive a lower bound, let us assume that X is a continuous random variable. This might berestrictive, but will lead to a simple lower bound expression. Note that I(X;Y ) = h(Y )− h(Y |X).Since h(Y |X) = h(Z) which does not depend on X , one can maximize I(X;Y) by maximizingh(Y ) with respect to fX . Instead of maximizing h(Y ), we make the following observation: Inthe extreme case of zero noise, Y and X have the same distribution. Therefore, one expects thatif noise is relatively small compared to the received signal, then Y has a distribution which is‘similar’ to that of X . In fact, the signal-to-noise ratio in VLC channels can often be high making

1A formal proof of the sphere-packing approach for bounding capacity can be found in [19, Chapter 5] and [20, Appendix B].

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A

noisespheres

λ

(a) A depiction of sphere-packing in a cube.With n= 2, the cube is a square, and thesphere is a circle.

A

λ

(b) The λ-neighborhood of the square. Itconsists of the square itself, four quarter-disks, and four rectangles.

2E

2E

noise spheres

λ

(c) A depiction of sphere-packing in asimplex. With n= 2, the simplex is a triangle,and the sphere is a circle.

λ

2E

2E

(d) The λ-neighborhood of the triangle(simplex). It consists of the triangle itself,three portions of a disks, and three rectangles.

Figure 2: Spheres of radius λ centered within a convex object will certainly be contained withinthe convex set whose boundary is at a distance λ from the object (its λ-neighbourhood).

this observation practically relevant. This means that in this regime, a ‘good’ choice of fX is onewhich maximizes h(X).

To find the maximum h(X), one can use the maximum-entropy theorem [13, Theorem 12.1.1].For a random variable X ∈ [0,A] with EX [X]≤ E and E≥ A

2 , the maximum h(X) is achievedwhen X is uniformly distributed, i.e.,

fuX(x) =1

Afor x∈ [0,A]. (5.1)

If E< A2 , then h(X) is maximized when X follows the truncated-exponential distribution

f teX (x) =µ

A(1− e−µ)e−µxA for x∈ [0,A], (5.2)

where µ is the unique solution of EA = 1

µ −e−µ

1−e−µ . Using the entropy-power inequality [13] andthese distributions, the following simple capacity lower bounds were derived in [10].

Theorem 5.1 ( [10]). The capacity of the IM/DD channel in Def. 2.3 is lower bounded by C≥ Clmwρ (A)

nats/transmission, where

Clmwρ (A) =

12 log

(1 + A2

2πeσ2

), ρ≤ 2

12 log

(1 +

A2e2µρ (1−e−µ)2

2πeσ2µ2

), ρ > 2,

(5.3)

where µ is the unique solution of 1ρ = 1

µ −e−µ

1−e−µ .

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The right hand side in (5.3) is an achievable rate which can be achieved using a uniform fX(5.1) or a truncated-exponential fX (5.2), respectively.

Another lower bound based on a continuous distribution was derived in [12] using atruncated-Gaussian input distribution. This distribution leads to an achievable rate which is ahigher than (5.3) for a wide range of A, but is more complicated to evaluate. We restrict ourattention to (5.3) here, which converges to capacity as A→∞ as we shall see in Sec. 7.

Example 5.1. For an IM/DD channel with A= 10, E= 4, and σ2 = 1, the lower bound in Theorem 5.1evaluates to Clmw

ρ (A) = 0.9111 nats/transmission. Combining this with the Example 4.1, we conclude thatthe capacity of this channel is between 0.9111 and 1.2296 nats/transmission.

6. Approaching CapacityThe lower bounds above are based on a continuous input distribution. This section focuses ondiscrete input distributions.

(a) Discreteness of Optimal Input DistributionIn [24], it was shown that the capacity achieving input distribution for the IM/DD channel isdiscrete, with a finite number of mass points. Thus, the optimal X is distributed according to

fdX(ak) = pk, k= 0, 1, . . . ,K, (6.1)

where (a0, a1, . . . , aK) is the support with ak ∈ [0,A] ∀k= 0, 1, . . . ,K, and (p0, p1, . . . , pK) arethe associated probabilities.

Although this is attractive from an implementation perspective, finding the optimal discretedistribution can be difficult. The optimization problem to be solved is

maxK,ak,pk

I(X;Y )|X∼fdX

subject to K ≥ 2;

K∑k=0

pk = 1;

K∑k=0

pkak ≤ E

ak ∈ [0,A] ∀k= 0, 1, . . . ,K,

(6.2)

where

I(X;Y ) =−∫∞−∞

(K∑k=0

pk√2πσ

e−(y−ak)2

2σ2

)log

(K∑k=0

pk√2πσ

e−(y−ak)2

2σ2

)dy − 1

2log(2πeσ2). (6.3)

This is a nonlinear optimization problem, which can be solved numerically. If K and ak arefixed, then the elegant Blahut-Arimoto algorithm [8,9] can be used to find pk. However, howcan we determine the optimal K and ak? To tackle this problem, [24] provides an algorithmand conditions which enable finding the optimal distribution. First K is initialized to 1 (twomass points). Then, the optimization (6.2) is solved for ak and pk numerically using standardsolvers (numerical integration and maximization). The obtained ak and pk are then tested usingthe conditions in [24, Corollary 3]. If the conditions are satisfied, then the obtained distribution isoptimal. Otherwise, K is incremented and the process is repeated. The result is a discrete inputdistribution which is capacity achieving.

Example 6.1. For an IM/DD channel with A= 10, E= 4, and σ2 = 1, the solution of theoptimization in (6.2) is K = 4 (5 mass points), (a0, . . . , a4) = (0, 2.7752, 5.0525, 7.3046, 10), and(p0, . . . , p4) = (0.3476, 0.1934, 0.1517, 0.1395, 0.1678) as shown in Fig. 3. The capacity is Cρ(A) =

1.1853 nats/transmission.

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0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5Optimal: Capacity 1.1853 nats/trans.Capacity-approaching: Rate 1.1724 nats/trans.

Figure 3: Discrete input distributions for an IM/DD channel with A= 10, E= 4, h= 1, and σ2 = 1.

The calculated capacity Cρ(A) = 1.1853 nats/transmission is between the bounds in Examples

4.1 and 5.1, given by Clmwρ (A) = 1.3257 and Clmw

ρ (A) = 0.9111 nats/transmission. However,neither of the bounds are tight for this example. Nevertheless, we shall see later that this is notthe case asymptotically in the limit of large A,E. Before we delve deeper into asymptotic capacity,let us ask the following question: Is there a way to approach the capacity of the channel withoutsolving problem (6.2)?

The answer is yes, as demonstrated in [16], and is discussed next.

(b) Capacity Approaching DistributionsRather than solving (6.2), let us fix the discrete input distribution to have the support(a0, . . . , aK) = (0, `, 2`, . . . ,A− `,A), where `= A

K . This way, we do not have to find the optimalsupport, but the optimal probabilities pk and number of mass points K + 1. This simplifies theproblem slightly.

For a further simplification, instead of maximizing I(X;Y ) under this input distribution, let usmaximize H(X) and find the max-entropic distribution with the given support. The intuition isthat the max-entropic distribution approaches the maximum h(Y ) when σ2→ 0. Thus, one needsto solve the simpler problem

maxpk

H(X)

subject toK∑k=0

pk = 1,

K∑k=0

pkkA

K≤ E.

(6.4)

The problem can be solved using using the Lagrangian, leading to the following solution

pk =

1

K+1 ρ≤ 2tk0∑Kj=0 t

j0

ρ > 2,(6.5)

where t0 is the unique positive root of∑Kk=0

(1− kA

KE

)tk.

Using this solution, one can initialize K = 1 (i.e., two mass points), calculate pk, and evaluatethe achievable rate I(X;Y ). Then, K is incremented, and the process is repeated until theachievable rate stops increasing. This leads to the best achievable rate using a max-entropicdiscrete input distribution with equally spaced mass points, which we shall denote Cfh

ρ (A).This distribution deviates from the optimal input distribution, but achieves close-to-optimalperformance. The following example illustrates this point.

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Example 6.2. For an IM/DD channel with A= 10, E= 4, and σ2 = 1, the best rate that can beachieved using a max-entropic discrete input distribution with equally spaced mass points is Cfh

ρ (A) =

1.1724 nats/transmission, achieved with K = 4 (5 mass points), (a0, . . . , a4) = (0, 2.5, 5, 7.5, 10), and(p0, . . . , p4) = (0.2884, 0.2353, 0.1920, 0.1566, 0.1278) as shown in Fig. 3.

The achievable rate of the proposed distribution is generally close to capacity for any A,E.Before comparing the capacity bounds with the achievable rates of the discrete input distributionsabove in Section 7, we make the following connection to the underlying DTP channel.

(c) Connection to DTP ChannelsAlthough this review centres on the signal-independent IM/DD channel with Gaussian noise (seeDef. 2.3), the relationship to the underlying DTP channel in Def. 2.1 is less well explored. It is wellknown that the the capacity achieving distribution for the DTP channel under peak and averageconstraints is discrete. In the case of large background light, i.e., large λ, the channel law of theDTP channel with high intensity background light will approach the Gaussian statistics of theIM/DD channel in Def. 2.3. The following theorem has been established for DTP channels withlarge background light.

Theorem 6.1 ( [35]). The capacity-achieving distribution for the DTP channel in Def. 2.1 with ρ > 2 andλ sufficiently large is binary with support at points (0,A) and associated probabilities(

ρ− 1

ρ,1

ρ

). (6.6)

Notice that the distribution (6.6) is identical to the maxentropic source distribution given in(6.5) for K = 1 and it satisfies both peak and average amplitude constraints with equality. Inthe case of high λ the channel law of the DTP channel approaches that of the Gaussian IM/DDchannel and the results on capacity-achieving distributions coincide.

7. Comparison and Asymptotic CapacityFor comparison, we fix the peak-to-average-ratio ρ, and we vary A/σ, which we call the signal-to-noise ratio (SNR). We start with a comparison for ρ= 2. This case represents an IM/DD channelwith a peak constraint only, since the average constraint becomes redundant when ρ= 2 [10].Fig. 4 shows the upper bounds and achievable rates. By comparing the bounds with the capacityCρ(A) evaluated by solving problem 6.2 numerically, we can see that the upper bound C

0ρ(A) is

asymptotically tight at low SNR, and the bound Clmwρ (A) is asymptotically tight at high SNR. The

upper bound Ccmaρ (A) is also fairly tight at high SNR. The upper bound C

m2 (A) is tight at high

SNR when ρ= 2, and nearly tight overall. Moreover, we see that a max-entropic equally-spaceddiscrete input distribution achieves a rate Cfh

ρ (A) which is close to capacity for any SNR. Thelower bound Clmw

ρ (A), which has the advantage of a closed-form expression, is only tight at highSNR. Fig. 5 shows a similar comparison with ρ= 4. In this case, the bound C

cmaρ (A) is slightly

tighter than Clmwρ (A).

The convergence of upper and lower bounds at high and low SNR has been also proved,leading to the asymptotic capacity of the IM/DD channel. This is described next.

Theorem 7.1 (High-SNR capacity [10]). For a fixed ρ= AE , the capacity of the IM/DD channel in Def.

2.3 satisfies

limAσ→∞

(Cρ(A)−

1

2log

(A2

2πeσ2

))= 0, (7.1)

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0 5 10 15 200

1

2

3

4

SNR (dB)

Rat

e(n

ats/

tran

smis

sion

)

C0ρ(A)

Clmwρ (A)

Cm2 (A)

Ccmaρ (A)

Clmwρ (A)

Cfhρ (A)

Cρ(A) (6.2)

−10 −5 0 5 1010−3

10−2

10−1

100

SNR (dB)

Rat

e(n

ats/

tran

smis

sion

)

C0ρ(A)

Clmwρ (A)

Cm2 (A)

Ccmaρ (A)

Clmwρ (A)

Cfhρ (A)

Cρ(A)

Figure 4: Capacity bounds and achievable rates for an IM/DD channel with ρ= 2.

0 5 10 15 200

1

2

3

4

SNR (dB)

Rat

e(n

ats/

tran

smis

sion

)

C0ρ(A)

Clmwρ (A)

Cm2 (A)

Ccmaρ (A)

Clmwρ (A)

Cfhρ (A)

Cρ(A)

−10 −5 0 5 1010−3

10−2

10−1

100

SNR (dB)

Rat

e(n

ats/

tran

smis

sion

)

C0ρ(A)

Clmwρ (A)

Cm2 (A)

Ccmaρ (A)

Clmwρ (A)

Clmwρ (A)

Cρ(A)

Figure 5: Capacity bounds and achievable rates for an IM/DD channel with ρ= 4.

for ρ≤ 2; and

limAσ→∞

(Cρ(A)−

1

2log

(A2e2

µρ (1− e−µ)2

2πeσ2µ2

))= 0, (7.2)

for ρ > 2, where µ is the unique solution of 1ρ = 1

µ −e−µ

1−e−µ .

To simplify this asymptotic capacity expression, [12] showed that the high-SNR capacity

can be well approximated by min{

12 log

(A2

2πeσ2

), 12 log

(eA2

2πσ2ρ2

)}, within a gap ≈ 0.1

nats/transmission. On the other hand, at low SNR, the following statement holds.

Theorem 7.2 (Low-SNR capacity [10]). For a fixed ρ= AE , the capacity of the IM/DD channel in Def.

2.3 satisfies

limAσ→0

Cρ(A)1ρ′ (1−

1ρ′ )

A2

2σ2

= 1, (7.3)

where ρ′ =max{2, ρ}.

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Finally, we note that similar bounds and asymptotic capacity results have been establishedfor the case when A=∞, i.e., when the channel only has an average intensity constraint. Thesebounds can be found in [10,11]. The low-SNR slope in this case was bounded in [10]. However,the low-SNR capacity remains an open problem.

8. Discussion and Extensions

(a) On Uni-Polar OFDM SchemesOrthogonal Frequency-Division Multiplexing (OFDM) is a transmission scheme which enjoyssimple construction. However, it is not directly applicable in IM/DD channels due to the non-negativity constraint of xi. To circumvent this, several uni-polar OFDM schemes have beenproposed in the literature [30,31]. Perhaps the most famous is DC-offset Optical OFDM (DCO-OFDM) [32], which constructs an real-valued OFDM signal by forcing frequency-domain symbolsto be Hermitian-symmetric, and then applies a DC-offset, and clips the resulting signal at 0 andA. This can be described by the following equation

xi =min

{A,max

{0,

1√m

m−1∑k=0

skej 2πikm + d

}}, (8.1)

where m is the OFDM block-length, sk is the frequency domain symbol on the kth subcarrier, j isthe imaginary unit, and d is the DC-offset. This operation involves and inverse-Fourier transform,a DC-offset, and clipping. On the receiver side, a Fourier transform is applied, and detection isapplied in the frequency domain.

This construction shifts the encoding/decoding from the time domain to the frequencydomain, where the frequency-domain channel for each subcarrier is a complex Gaussian channel.Namely, for a given subcarrier k, the transmitter encodes information into a sequence {sk[`]}n`=1

where sk[`]∈C. Then, the operation in (8.1) is applied to {sk[`]}m−1k=0 to generate {xi[`]}m−1i=0 ,which is then transmitted. The receiver receives {yi[`]}m−1i=0 where yi[`] = xi[`] + zi[`], and appliesa Fourier transform to obtain

rk[`] = sk[`] + qk[`] + pk[`], (8.2)

where qk[`] is complex Gaussian noise (the Fourier transform of zi) and pk[`] is clipping distortion.The receiver collects {rk[`]}n`=1 from which it decodes {sk[`]}n`=1. If we assume that the clippingdistortion is Gaussian, and that sk[`] is subjected to a power constraint ESk [|Sk|

2]≤ pk, then weknow exactly how to achieve the capacity of the channel in (8.2); we construct the codewords{sk[`]}n`=1 as i.i.d. complex Gaussian with zero mean and variance pk.

While this is attractive scheme in practice, it has some limitations which prevent it fromachieving the IM/DD channel capacity. One of the limitations is the clipping distortion pk[`].This is introduced due to clipping, and would not be present if xi is chosen to follow any of thedistributions discussed in Sections 5 and 6. To limit the clipping distortion, one has to choose pkto be small so that the transformed signal 1√

m

∑m−1k=0 ske

j2πikm has a small amplitude with high

probability. This results in a decrease in the achievable rate.Even if we neglect this clipping distortion, this construction has other limitations. By

studying (8.1), one can find that Xi follows a clipped-Gaussian distribution [31], and that

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X0, X1, . . . , Xm−1 are dependent. Due to this, we have

h(Y0, Y1, . . . , Ym−1) = h(X0 + Z0, X1 + Z1, . . . , Xm−1 + Zm−1)

= h(X0 + Z0) +

m−1∑i=1

h(Xi + Zi|X0 + Z0, . . . , Xi−1 + Zi−1)

<

m−1∑i=0

h(Xi + Zi)

<

m−1∑i=0

h(Xi + Zi)

(8.3)

where h(·|·) is the conditional differential entropy, and Xi is a random variable distributedon [0,A] following the optimal input distribution (solution of (6.2)). Here, the first inequalityfollows since conditioning ‘strictly’ reduces entropy when variables are dependent, which isthe case when xi is constructed using DCO-OFDM. The second inequality follows since thecapacity achieving distribution for an IM/DD channel is discrete and unique [24], i.e., does notcoincide with a clipped-Gaussian distribution. From this, we conclude that the achievable rate ofDCO-OFDM, even with the clipping distortion neglected, is strictly less than capacity.

A similar discussion applies to other unipolar-OFDM constructions. Reference [30] studiesthe achievable rates of several unipolar-OFDM schemes, and clearly shows a gap between theseachievable rates and capacity bounds from [11,33]. Whether this is a limitation of unipolar-OFDM in general or a limitation of existing constructions of unipolar-OFDM remains to be seen.Nevertheless, [30] shows that some multi-layer uni-polar OFDM schemes achieve rates which arefairly close to capacity at high SNR for the average-constrained IM/DD channel (A=∞).

(b) MIMO IM/DD ChannelsThe results in Sections 4-6 which consider a Single-Input Single-Output (SISO) IM/DD channel,have been recently extended to IM/DD systems with multiple transmitters (multiple channelinputs) and/or multiple receivers (multiple channel outputs). This can be classified into Single-Input Multiple-Output channels (SIMO), Multiple-Input Single-Output (MISO), and Multiple-Input Multiple-Output (MIMO) channels.

(i) SIMO IM/DD Channels

In the SIMO case, the transmitter has one aperture, and the receiver has nr apertures. The receiverreceives a vector Y ∈Rnr with Y =hX +Z, where h= (h1, . . . , hnr)

T is the vector of channel‘attenuation’ from the transmitter to the receivers’ apertures, and Z is a vector of independentGaussian noises with zero mean variance σ2. One can easily express the capacity of this case as afunction of the capacity of a SISO IM/DD channel, and show that maximum-ratio combining isoptimal. If we define Y =UTY where U is an orthonormal matrix with h

‖h‖ as its first column,

then Y will be given by Y = (‖h‖X + Z1, Z2, . . . , Znr)T , where Z has the same distribution

as Z. The channel from X to Y has the same capacity as the channel from X to Y since themultiplication by U is an invertible transformation. Hence, the capacity of the channel is equal toCsimoρ (‖h‖A). The bounds presented in Sections 4-6 can be directly applied.

(ii) MIMO IM/DD Channels

In the MIMO channel, the transmitter has nt apertures and the receiver has nr apertures. Wecan model the channel as one whose input is X ∈ [0,A]nt where nt is the number of transmitter

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apertures, and output

Y =HX +Z, (8.4)

where Z is a vector of independent Gaussian noises with zero mean variance σ2, and H =

(h1, . . . ,hnt) where ht is the vector of channel ‘attenuation’ from transmitter aperture t to thereceivers’ apertures. The MISO channel is a special case of the MIMO channel with nr = 1. Theaverage constraint is expressed as a total average constraint

∑nti=1 E[Xi]≤ E.

For the MISO and MIMO cases, we face another difficulty. Now, the SISO codebooks whosecodewords are scalar sequences {xi}ni=1 do not directly apply. Instead, we need to generatevector codebooks with codewords {xi}ni=1 where xi ∈ [0,A]nt . The optimal codebook is onewhose symbols are independent and identically distributed according to a distribution whichmaximizes I(X;Y ), with respect to fX subject to peak and average constraints. This problem hasbeen investigated in [25–29].

In [25], capacity bounds were derived for two cases: nt ≤ nr and nt >nr. For nt ≤ nr, it wasshown that if ρ≤ 2

nt(peak constraint is dominant) and H has full column rank, then the following

characterizes the asymptotic capacity at high SNR

1

2log

∣∣∣∣ A2

2πeσ2HTH

∣∣∣∣ . (8.5)

This high-SNR capacity is asymptotically achievable as Aσ →∞ with a fixed ρ≤ 2

nt, using a QR-

decomposition scheme. Namely, the transmitter sends nt independently coded streams from itsapertures, where the symbols Xi are uniformly distributed on [0,A]. To resolve crosstalk, thereceiver multiplies Y by Q where H =QR is the QR-decomposition of H , Q is an orthogonalmatrix, and R is upper triangular. This way, the receiver obtains an equivalent channel with uppertriangular structure given by Y =RX + Z. The receiver then starts by decoding Xnt withoutcrosstalk, then it removes the contribution of Xnt from Y to decode Xnt−1 without crosstalk,and so on.

If nt ≤ nr, H has full column rank, and ρ > 2nt

(average constraint is dominant), then theasymptotic capacity at high SNR (Aσ →∞) is within a small gap of

1

2log

∣∣∣∣ e

2πσ2min

{A2

ρ2n2t,A2

e2

}HTH

∣∣∣∣ . (8.6)

This is achievable using a similar QR-decomposition scheme with a truncated-Gaussian inputdistribution for each transmitted stream.

The asymptotic capacity expressions in (8.5) and (8.6) reveal a multiplexing gain equal to nt,when H has rank nt and nt ≤ nr. If the rank of H is less than nt which in turn is less than nr,then we can use the same transmission scheme to achieve a multiplexing gain equal to the rank ofH after deactivating some transmit apertures (switching them off, or on at a constant intensity).

If nt >nr, then [25] provides capacity bounds and scaling laws. In this case, transmissionschemes can use a pre-coding matrix G∈Rnr×nt

+ to reduce the effective MIMO channel toan nr × nr channel, over which the above results can be applied. This case was studied moreextensively in [29], which develops minimum energy signalling schemes, capacity lower bounds,upper bounds, and asymptotic capacity expressions.

The capacity of the MIMO IM/DD channel was characterized in the low-SNR regime (Aσ → 0)in [26], where it was shown to be equal asymptotically to (for both nt ≤ nr and nt >nr)

maxai:∑nti=1 ai≤

1η

A2

2σ2

nt∑i=1

nt∑j=1

hTi hj min{ai, aj}(1−max{ai, aj}). (8.7)

This is achievable using a maximally-correlated nt-ary binary input distribution. In thisdistribution, the transmitted vector symbols can take values in {0,A}nt such that if Xi =A andEi ≤ Ej , thenXj =A. For instance, if nt = 2 and E1 ≥ E2, then X can take only three values, (0, 0),(A, 0) and (A,A).

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(iii) MISO IM/DD Channels

In the MISO channel, the transmitter has nt apertures, and the receiver has one aperture. Theinput-output relation can be written as Y =hTX + Z where Z is Gaussian with zero mean andvariance σ2, h= (h1, . . . , hnt)

T and hi is the channel attenuation from transmitter aperture i tothe receiver aperture. We can assume without loss of generality that h1 ≥ h2 ≥ · · · ≥ hnt .

Transmission schemes for the MISO channel can be extracted from [25] as a special case whennr = 1. However, this paper only provides achievable rates and high-SNR capacity scaling laws.In [27], properties of capacity achieving input distributions for the MISO case were derived.In general, [27] shows that one can restrict the optimization maxfX I(X, Y ) to distributionssatisfying Xi > 0⇒Xj =A∀j < i. In other words, the transmitter modulates LEDs at two levels:(i) modulating the number of fully-ON LEDs (Xj =A) from 0 to nt − 1 so that fully-ON LEDs arethose with the strongest channels, and (ii) modulating the intensity of the next LED (next in termsof channel strength) using some probability distribution. Note that a similar behaviour has beenidentified in [26] for the MIMO channel at low SNR.

Using this result capacity lower bounds were derived in [27], in addition to capacity upperbounds. It shows that when the peak constraint is dominant, i.e. ρ≤ 2

nt, then the MISO channel

capacity is equal to Cρ(∑nt

i=1 hiA). Asymptotic capacity results in Section 7 can be used in this

case. Namely, the high SNR (Aσ →∞) capacity is

1

2log

((∑nti=1 hi

)2A2

2πeσ2

).

The above asymptotic capacity also applies if the average constraint is dominant, i.e., ρ > 2nt

,

and the peak-to-average ratio ρ satisfies ρ≤ ρth ,(12 + 1∑nt

i=1 hi

∑nt

k=1 hk(k − 1))−1

.

On the other hand, if the average constraint is dominant and ρ > ρth, then the high-SNRcapacity is

1

2log

((∑nti=1 hi

)2A2

2πeσ2

)+ ν,

where

ν = sup

λ∈(max{0, 12+

1ρ−

1ρth},min{ 1

2 ,1ρ})(1− log

(µ(λ)

1− e−µ(λ)

)− µ(λ)e−µ(λ)

1− e−µ(λ)− D

(p

∥∥∥∥ h∑nti=1 hi

)),

µ(λ) is the solution of 1µ −

e−µ

1−e−µ = λ, p= (p1, . . . , pnt), pi =hia

i∑ntk=1 hka

k , and a is the unique

positive solution of∑nti=1 ihia

i∑ntk=1 hka

k = 1ρ − λ+ 1.

The low-SNR capacity of the MISO case has been given in [26] and [27], and coincides with(8.7) with nr = 1.

9. ConclusionOptical wireless channels utilize a largely under utilized huge portion of the spectrum whichis available to enable future ultra high speed networks. In spite of ongoing work to developOWC technologies, there exist practical and theoretical open problems to solve. In particular,this review has highlighted a string of results over the previous decade which have endeavouredto characterized the fundamental channel capacity of optical wireless IM/DD channels. Manybounds have been derived which are tight in different regimes. Capacity-approaching and insome cases capacity-achieving distributions for IM/DD channels have been derived. In spite ofthis intense effort, the unique amplitude characteristics of IM/DD optical wireless channels haveresulted in a large number of open areas of research.

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Current challenges exist in extending these fundamental results to practical VLC and opticalwireless implementations. The design of tractable and efficient modulators and encoders isessential to the future progress of OWC technologies. In particular, VLC luminaries requireenergy efficient approaches which can leverage multiple emitters and receivers. In addition, amulti-disciplinary approach is required to understand the capacity of IM/DD channels in thepresence of modulator nonlinearities such as those in high-illuminance LEDs. Many theoreticaland practical challenges exist in these scenarios which are ripe for ongoing investigation.

Authors’ Contributions. Chaaban drafted the primary technical sections of the paper. Hranilovic editedand added additional technical content. Both authors designed the paper structure, edited the paper, jointlydrafted the final manuscript and have read and approved this manuscript.

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