# Bounds on Capacity Region of Optical Intensity Multiple Access Channel

This paper provides new inner and outer bounds on the capacity region of optical intensity multiple access channel (OIMAC) with per-user average or peak power constraint. For the average power constrained OIMAC, our bounds at high power are asymptotically tight, thereby characterizing the asymptotic capacity region. The bounds are extended to K-user OIMAC with average power constraint without loss of asymptotic optimality. For the peak power constrained OIMAC, at high power, we bound the asymptotic capacity region to within 0.09 bits, and determine the asymptotic capacity region in the symmetric case. At moderate power, for both types of constraints, the capacity regions are bounded to within fairly small gaps. Our results are established based on capacity results of single-user channels, including 1) capacity bounds for optical intensity channels and peak power constrained Gaussian channels, and 2) capacity results of two specific additive noise channels where the noises obey certain maxentropic distributions.

## Authors

• 12 publications
• 8 publications
• ### Capacity bounds for bandlimited Gaussian channels with peak-to-average-power-ratio constraint

We revisit Shannon's problem of bounding the capacity of bandlimited Gau...
05/30/2018 ∙ by Yizhu Wang, et al. ∙ 0

• ### On the Capacity of Intensity-Modulation Direct-Detection Gaussian Optical Wireless Communication Channels: A Tutorial

Optical wireless communication (OWC) using intensity-modulation and dire...
11/16/2020 ∙ by Anas Chaaban, et al. ∙ 0

• ### Tight Capacity Bounds for Indoor Visible Light Communications With Signal-Dependent Noise

Channel capacity bounds are derived for a point-to-point indoor visible ...
10/31/2020 ∙ by Jin-Yuan Wang, et al. ∙ 0

• ### The Capacity Region of the Arbitrarily Varying MAC: With and Without Constraints

We determine both the random code capacity region and the deterministic ...
01/03/2019 ∙ by Uzi Pereg, et al. ∙ 0

• ### A Connection between Feedback Capacity and Kalman Filter for Colored Gaussian Noises

In this paper, we establish a connection between the feedback capacity o...
01/09/2020 ∙ by Song Fang, et al. ∙ 0

• ### List Decoding for Oblivious Arbitrarily Varying MACs: Constrained and Gaussian

This paper provides upper and lower bounds on list sizes of list decodin...
01/09/2020 ∙ by Yihan Zhang, et al. ∙ 0

• ### Simple Bounds for the Symmetric Capacity of the Rayleigh Fading Multiple Access Channel

Communication over the i.i.d. Rayleigh slow-fading MAC is considered, wh...
05/23/2019 ∙ by Elad Domanovitz, et al. ∙ 0

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## I Introduction

Intensity modulation and direct detection (IM/DD) based optical wireless communications (OWC), such as visible light communications (VLC), has received increasing attention in recent years [1, 2]. In IM/DD systems, information is transmitted by varying the intensity of emitted light, i.e., the optical power transmitted per unit area. A widely accepted channel model for IM/DD based indoor OWC is the Gaussian optical intensity channel [3, 4], which captures key properties including nonnegativity of optical intensity, input-independent additive Gaussian noise,111This is an accurate model when the noise is dominated by shot noise from ambient light and/or thermal noise at the receiver [3]. and practical constraints such as limited average and/or peak optical power. The Gaussian optical intensity channel has been used in studies on coding and modulation design [5, 6, 7] as well as channel capacity [8, 9, 10, 11, 12].

In many indoor OWC applications there are multiple non-cooperative users (or devices) transmitting data simultaneously [2]. To explore fundamental limits of multiuser indoor OWC, capacities of several multiuser optical intensity channels including parallel channel [13], multiple access channel [14], broadcast channel [15], etc., have been studied. These channels are building blocks of more complex systems of multiuser indoor OWC.

We consider discrete-time optical intensity multiple access channel (OIMAC) with Gaussian noise. In [14], several bounds on the capacity region of OIMAC have been established where the input of each user is constrained in both its average and its peak power. Specifically, the inner bounds were obtained by using truncated Gaussian input and uniformly-spaced discrete input for each user, respectively; the outer bounds were obtained by known results for single-user optical intensity channels. By optimizing both types of input distributions numerically with respect to signal-to-noise ratio (SNR), in [14], the low SNR capacity region of OIMAC is determined. However, at moderate to high SNR, the gaps left in [14] are still evident, and the high-SNR capacity region of OIMAC is still unknown.

In this paper, by introducing new input distributions and bounding techniques, we provide new inner and outer bounds on the capacity region of OIMAC with per-user average power constraint or peak power constraint. For the average power constrained OIMAC, we derive asymptotically tight inner and outer bounds at high SNR, thereby determining the high-SNR capacity region. At moderate SNR the bounds are also fairy tight. Moreover, we extend the bounds to the -user case without loss of asymptotic optimality, and provide some discussions related to system design. For the peak power constrained case, at high peak-to-noise ratio (PNR), the asymptotic capacity region of OIMAC is bounded to within 0.09 bits, and this gap vanishes in the symmetric case (in other words, the high-PNR capacity region for symmetric case is obtained); at moderate PNR, by combining our outer bound and the inner bound based on discrete input in [14]

, the capacity region is bounded to within a small gap. A key step to our main results is utilizing capacity results of two additive noise channels where the noises obey certain maxentropic distributions, namely, exponential distribution for the average power constrained case and uniform distribution for the peak power constrained case.

The remaining part of this paper is organized as follows. In Sec. II we introduce the OIMAC with power constrains and some useful notations. Our results for average power constrained and peak power constrained OIMAC are provided in Sec. III and Sec. IV, respectively. Finally, the paper is concluded in Sec. V.

Notation

: We use uppercase and lowercase Roman letters to denote random variables and their realizations, respectively. A few exceptions include:

stands for capacity, and stand for entropy and differential entropy, respectively, stands for mutual information, and and stand for the mutual information between and , , with respect to SNR and PNR, respectively. We use and to denote upper and lower bounds on a quantity , respectively. For , we use to denote the other element in . The convex closure (or convex hull) of a set of points is denoted by . The asymptotic relationship

 limt→∞[A(t)−B(t)]=0 (1)

is denoted as .

## Ii Optical Intensity Multiple Access Channel

A discrete-time single-user optical intensity channel with Gaussian noise is given by [9, 10]

 Y=X+Z, (2)

where and . A discrete-time two-user OIMAC has two transmitters and one receiver, and the received signal is the linear superposition of the inputs and a noise[3, 11]:

 Y=X1+X2+Z, (3)

where , and . This paper considers two types of input power constraints, namely, the per-user average power constraint as

 E[X1]≤E1,E[X2]≤E2, (4)

and the per-user peak power constraint as

 0≤X1≤A1,0≤X2≤A2. (5)

We define the optical SNR and PNR as and , respectively, and denote the SNR and PNR of user as and , respectively. These notations can be extended directly to a -user OIMAC

 Y=K∑i=1Xi+Z. (6)

Further consideration on a combined constraint of average and peak power is left to future study.

Throughout the paper, in high-power analysis, we let all or increase simultaneously, i.e., we keep the ratio or be unchanged as input power increases.

The following single-letter characterization of the capacity region of OIMAC readily follows from the capacity region of discrete memoryless multiple access channel and the discretization procedure [References, Sec. 3.4] (cf. [14]).

Lemma 1 (Capacity region of OIMAC): The capacity region of the OIMAC (3) is the convex closure of , where is the set of rate pairs satisfying

 R1 ≤I(X1;Y|X2), (7) R2 ≤I(X2;Y|X1), (8) R1+R2 ≤I(X1,X2;Y), (9)

for a fixed product distribution satisfying given input constraint.

However, evaluating the above capacity region is difficult since the inputs have continuous amplitude. Even for the single-user optical intensity channel, no analytic expression for the capacity is known. To characterize the capacity region, we will provide outer and inner bounds.

For an additive multiple access channel as (3), two simple but useful facts are given as follows. The first is

 I(Xi;Y|X~i)=I(Xi;Xi+Z), (10)

due to

 I(Xi;Y|X~i) =h(Y|X~i)−h(Y|X1,X2) =h(Xi+Z)−h(Z) =I(Xi;Xi+Z). (11)

The second is

 I(X1,X2;Y)=I(X1+X2;Y), (12)

due to

 I(X1,X2;Y) =h(Y)−h(Y|X1,X2) =h(Y)−h(Z) =h(Y)−h(Y|X1+X2) =I(X1+X2;Y). (13)

These facts help us utilize single-user capacity results in our study on the capacity region of OIMAC.

## Iii Average Power Constrained OIMAC

### Iii-a Known Single-User Capacity Results

For an OIMAC with average power constraint as (4), we utilize results of single-user optical intensity channel and additive exponential noise channel [17, 18] to derive capacity bounds. We introduce these results as follows.

Lemma 2 [8, 9, 10]: The capacity of the Gaussian optical intensity channel (2) with an average power constraint as is upper bounded by

 CAP-OIC≤12log(e2π(SNR+2)2), (14)

where . The capacity is lower bounded by

 CAP-OIC ≥IE(SNR)≜I(XE;XE+Z) (15) ≥12log(1+e2πSNR2), (16)

where is an exponential random variable with mean . The capacity is also lower bounded by

 CAP-OIC≥IG(SNR)≜maxℓ>0I(XG;XG+Z), (17)

where is a geometric random variable with mean

and probability mass function (PMF)

 (18)

The asymptotic capacity is given by

 CAP-OIC≐12log(e2πSNR2). (19)

Lemma 3[17]: The capacity of an additive exponential noise (AEN) channel

 Y=X+Z,X≥0,E[X]≤E%s, (20)

where is an exponential random variable with mean , is given by

 CAEN=log(1+EsEn). (21)

The probability density function (PDF) of the capacity-achieving input distribution is

 pX(x)=EnEs+E% nδ(x)+Es(Es+En)2exp(−xEs+En),x≥0, (22)

and the corresponding output distribution is an exponential distribution with mean .

### Iii-B Bounds on Capacity Region of Average Power Constrained OIMAC

Our main results for the average power constrained OIMAC are given in the following two propositions.

Proposition 1 (Outer bound): The capacity region of the OIMAC (3) with a per-user average power constraint as (4) is outer bounded by

 ¯¯¯¯Ci =12log(e2π(SNRi+2)2),i=1,2, (23) ¯¯¯¯Csum =12log(e2π(SNR1+SNR2+2)2). (24)
###### Proof:

From (7), the rate of user must satisfy . Combining this with the fact (10) and the single-user capacity upper bound (14), we obtain (23). From (9), the sum rate must satisfy . Combining this with (12), by noting that must satisfy an average power constraint , and applying the single-user upper bound (14), we obtain (24).

Proposition 2 (Inner bound): The capacity region of the OIMAC (3) with a per-user average power constraint as (4) is inner bounded by a polytope with the following five rate pairs as corner points:

 (R1,R2)={ (0,0),(IG(SNR1),0), (IE(SNR1),IE(SNR1+SNR2)−IE(SNR1)), (IE(SNR1+SNR2)−IE(SNR2),IE(SNR2)), (0,IG(SNR2))}. (25)

A closed-form inner bound weaker than (III-B) is given by

 C––i =12log(1+e2πSNR2i),i=1,2, (26) C––sum =12log(1+e2π(SNR1+SNR2)2). (27)
###### Proof:

The achievability of the second and last rate pairs in (III-B) follows directly from Lemma 2. To prove the achievability of the third and fourth rate pairs, we employ an input distribution . Let be an exponential distribution with mean , and let be as (22) in which we set and . According to Lemma 3, the sum random variable is exponentially distributed with mean . By combining (7), (8) with (10) we obtain that a rate is achievable for user , and by combining (9) with (12) we obtain that a sum rate is achievable. So user can achieve . Therefore the third and fourth rate pairs in (III-B) are both achievable. All other rate pairs in the inner bound can be achieved by time sharing [16].

According to (15) and (16), the region determined by (26) and (27) is a subset of the convex closure of (III-B) (noting that [10]). Therefore it is also an inner bound.

Combining Proposition 1 and the inner bound (26) (27) in Proposition 2 we immediately obtain the following corollary.

Corollary 1 (Asymptotic capacity region):222This result can also be obtained by combining Lemma 1 and the asymptotic optimality of exponential distribution in average power constrained optical intensity channels (this optimality is guaranteed by the convergence property of differential entropy [19] as SNR grows without bound). However, finite-SNR bounds cannot be obtained in this way. The high-SNR asymptotic capacity region of average power constrained OIMAC is given by

 Ci ≐12log(e2πSNR2i),i=1,2, (28) Csum ≐12log(e2π(SNR1+SNR2)2). (29)

Remark 1 (Rate of the second user): The asymptotic capacity region determined by (28) and (29) is a pentagon, as shown in Fig. 1, where the RHS of (28) is denoted as . Combining (28) and (29), we note that when user asymptotically achieves the rate given in the RHS of (28), the rate of the second user satisfies

 R~i≐˙R~i=log(1+SNR~iSNRi). (30)

This can be interpreted as follows. From Proposition 2, when user employs an exponential input distribution and achieves the rate , which is lower bounded by the RHS of (26), the other user , employing an input distribution like (22), can achieve , which is lower bounded by333By (26) and (27) we can show that when user achieves in (26), user can achieve , which exceeds in (31). However, the rate pair is not necessarily achievable because may exceed .

 R––~i=12log1+e2π(SNR1+SNR2)2e2π(SNRi+2)2. (31)

This lower bound can be obtained by 1) combining (14), (15), and (16) to obtain upper and lower bounds of , and 2) applying both bounds as

 R~i≥I–E(SNR1+SNR2)−¯¯¯IE(SNRi). (32)

Similarly, an upper bound on can be obtained as

 ¯¯¯¯R~i =¯¯¯IE(SNR1+SNR2)−I–E(SNRi) =12loge2π(SNR1+SNR2+2)21+e2πSNRi2. (33)

Comparing (31) and (III-B) we obtain (30). Therefore, a corner point of the pentagonal asymptotic capacity region given in Corollary 1, i.e., the rate pair

 (˙Ci,˙R~i)=(12log(e2πSNR2i),log(1+SNR~iSNRi)), (34)

is achieved.444Note that although the noise is Gaussian, to approach this rate pair, user may use a maximum energy decoder (see [18]) other than a nearest neighbor decoder (typical decoder in channels with Gaussian noise). This is because for an AEN channel, a maximum energy decoder (which is a maximum likelihood decoder in the AEN channel) achieves capacity. This result is proved in [18] by using the generalized mutual information (GMI).

Fig. 2 shows our capacity bounds for average power constrained OIMAC by two examples. At high SNR, the closed-form inner bound in Proposition 2 is very tight. At moderate SNR, the closed-form inner bound (26), (27) becomes looser, but the inner bound given by (22), which can be evaluated numerically, is still fairly tight.

### Iii-C Extension to K-User OIMAC

Consider a -user OIMAC as (6). Let , , (i.e., is allowed), , and . Let denote the complement of . By directly extending Lemma 1, we obtain that the capacity region of the -user OIMAC is the convex closure of the rate tuples satisfying

 (35)

for some product distribution satisfying given input power constraint.

Denote the maximum achievable for all feasible input distributions as (it is the single-user capacity for user when , and the sum capacity when ). The following results on the capacity region of the -user OIMAC can be obtained following the same approach in our study on the two-user case. For brevity we only give outlines of proofs.

Proposition 3 (Outer bound): The capacity region of the -user OIMAC (6) with a per-user average power constraint as is outer bounded by

 ¯¯¯¯CJ=12log⎛⎝e2π(∑k∈JSNRk+2)2⎞⎠,∀J⊆K. (36)

Outline of Proof: The bound can be derived from (35) by noting that

 E[∑k∈JXk]≤∑k∈JEk (37)

must be satisfied .

Proposition 4 (Inner bound): Let be a permutation on and be the order of in . For a given , let be the set of rate tuples satisfying (38).555When does not exist (i.e., when ), we let . The capacity region of the -user OIMAC (6) with a per-user average power constraint as is inner bounded by

 RK=Conv(⋃J⊆KVJ). (39)

A closed-form inner bound slightly weaker than (38) is given by

 (40)
###### Proof:

An outline of the proof is given in Appendix A.

The following asymptotic capacity region can be obtained by noting that the gap between the upper and lower bounds on vanishes in the high-SNR limit.

Corollary 2 (Asymptotic capacity region): The asymptotic capacity region of the -user OIMAC (6) with a per-user average power constraint as is determined by

 CJ≐12log⎛⎝e2π(∑k∈JSNRk)2⎞⎠,∀J⊆K. (41)

Note that each of (40) and (41) includes equations. The rate regions determined by (38), (40) and the high SNR rate region determined by (41) are all convex polytopes in the -dimensional space.

Remark 2: According to Proposition 4, a sum rate

 max(R1,...RK)∈RK∑k∈KRk=IE(∑k∈KSNRk) (42)

is achievable. We can also provide an inner bound on the capacity region by

 RJ≤IE(∑k∈JSNRk),∀J⊆K, (43)

which is tighter than (40). At first glance, the inner bound (43) is equivalent to the inner bound determined by (38) except when . However, this is true only at high SNR. When SNR is sufficiently low, the rate may exceed . In this case the inner bound (43) is strictly smaller than that determined by (38), even if we replace in (38) by .

To evaluate the gap between our outer and inner bounds, consider a symmetric OIMAC with average power constraint satisfying for example. In this case the gap on the sum capacity is

 Δsum(SNR) ≜¯¯¯¯Csum−max(R1,...RK)∈RK∑k∈KRk =12log(e2π(K⋅SNR+2)2)−IE(K⋅SNR). (44)

This gap is equal to the gap between (14) and (15) when their is scaled by (cf. [References, Fig. 2]). For fixed SNR, as increases, our bounds on the sum capacity becomes tighter; see Fig. 3.

### Iii-D Discussions on K-User OIMAC

In contrast to the Gaussian MAC, to achieve the capacity region of average power constrained OIMAC, different input distributions must be employed. Consider the asymptotic capacity region determined by (41). At high SNR, the boundary of this region includes a face on which the sum rate is maximized (max-sum-rate face). A corner point of this face is given by

 R1 ≐12log(e2πSNR1), (45) Rk ≐log(1+SNRk∑j1,k∈K, (46)

which is asymptotically achieved by employing the input distribution described in the proof of Proposition 4. That input distribution and the achieved rate for different users, however, is highly asymmetric. Take, for example, an OIMAC with a symmetric per-user average power constraint (i.e., ). In this case, as SNR increases, the rate in (45) grows without bound, and the corresponding input distribution is an exponential distribution with mean . But according to (46),

 Rk≈log(1+1k−1),k>1, (47)

and the corresponding input distribution of the th user has a singleton at zero satisfying . If our target is maximizing the sum rate with equal rates for all users (called symmetric capacity in [16, 14]), then time sharing or rate splitting must be used, while in Gaussian MAC a single Gaussian input distribution suffices [16]. The symmetric capacity can also be achieved using time-division multiple access (TDMA) with power (intensity) control [16], which has lower detection complexity than transmitting simultaneously. However, the optimality of TDMA in terms of sum capacity does not hold if there exists a per-user peak power constraint (some examples on this fact can be found in [14]).

According to our results, to achieve the sum capacity at high SNR, the input distribution for each user must be carefully chosen based on the input power constraints of all users. A natural question is that if the users still follow single-user transmission strategy (i.e., employing some near-optimal input distributions for the single-user OIMAC), then how large is the loss on the sum rate? We give an example to shed some insight on this. Consider the sum rates achieved by two types of input distributions as follows.

• Type I: the inputs of all users obey the asymptotically optimal distribution given in the proof of Proposion 4.

• Type II: the input of each user obeys an exponential distribution with maximum allowed average intensity (asymptotically optimal at high SNR in the single-user case).

For simplicity let us focus on the symmetric case with . For Type I, the sum of the inputs (we denote it by ) is exponentially distributed with mean , while for Type II, the sum of the inputs obeys an Erlang distribution with PDF [20]

 (48)

Using the fact

 I (S;SNR⋅S+Z) =h (SNR⋅S+Z)−h(SNR⋅S+Z|S) =h (SNR⋅S+Z)−h(Z) ≐h (SNR⋅S), (49)

the high-SNR gap between the sum capacity (asymptotically achieved by Type I input) and the sum rate achieved by Type II input can be evaluated by the gap between the differential entropies of and . The differential entropy of is , and the differential entropy of is [21]

 h(SII)=log(eK(K−1)!E)+(1−K)ψ(K), (50)

where is the digamma function which satisfies

 ψ(K)=K∑k=1k−1−γ, (51)

and is Euler’s constant:

 γ≜limn→∞[n∑k=1k−1−lnn]≈0.5772. (52)

Then we obtain

 Δ(SNR) ≜[Csum(SNR)−I(SII;SII+Z)] ≐(K−1)ψ(K)−log(eK−1(K−1)!K−1). (53)

In Fig. 5, the values of (III-D), denoted as , are provided. It is shown that the gap increases linearly as increases exponentially (in fact ). Therefore, at high SNR, the performance loss of Type II input is more important for relatively small number of users. In Fig. 5, by numerically evaluating the input-output mutual information, the sum rates achieved by Type I and Type II inputs are provided for different numbers of users and finite SNR values. It is shown that the performance loss of Type II input is more severe when SNR is lower.

## Iv Peak Power Constrained OIMAC

### Iv-a Known Single-User Capacity Results

For an OIMAC with peak power constraint as (5), we utilize results of single-user optical intensity channels and certain kinds of peak power constrained channels to derive capacity bounds. Here we review these results as follows.

The following lemma comes from upper bounds on the capacity of peak power constrained additive white Gaussian noise (AWGN) channels given in [22, 23]. Here we have translated the result to optical intensity channels by noting that an optical intensity channel with peak power constraint is equivalent to an AWGN channel with peak power constraint when .

Lemma 4: The capacity of the Gaussian optical intensity channel (2) with a peak power constraint as is upper bounded by the McKellips bound [22] as

 CPP-OIC≤¯¯¯¯CM(PNR) ≜min{log(1+√12πePNR),12log(1+14PNR2)}. (54)

When satisfies , the capacity is also upper bounded by [23]

 CPP-OIC≤¯¯¯¯CTKB(PNR) =H2(12−Q(PNR))+(12−Q(PNR))log√12πePNR, (55)

where is the binary entropy function, is the Q funtion. The capacity is lower bounded by [9]

 CPP-OIC ≥IU(PNR)≜I(XU;XU+Z) ≥12log(1+12πePNR2), (56)

where is a uniformly distributed random variable with support . The asymptotic capacity is given by

 CPP-OIC≐12log(12πePNR2). (57)

Lemma 5 [References, Problem 7.5, pp. 556]:666This result has also been noted in German literature in 1960’s; see [25] and references therein. Consider an additive noise channel with input constraint , and uniformly distributed over . The capacity of this channel is

 C(a)=log(n+1)−(n−a)logn+1n, (58)

where , and the capacity-achieving input distribution is

 fX(x)=n−1∑m=0n−mn(n+1)(δ(a−2m)+δ(−a+2m)). (59)

When is an integer, we have and , which is a discrete uniform distribution.

### Iv-B Bounds on Capacity Region of Peak Power Constrained OIMAC

Our main results for the peak power constrained OIMAC are given in the following two propositions.

Proposition 5 (Outer bound): The capacity region of the OIMAC (3) with a per-user peak power constraint as (5) is outer bounded by

 ¯¯¯¯Ci =¯¯¯¯C(PNRi),i=1,2, (60) ¯¯¯¯Csum =¯¯¯¯C(PNR1+PNR2). (61)

where

 ¯¯¯¯C(PNR)≜ ⎧⎨⎩min{¯¯¯¯CM(PNR),¯¯¯¯CTKB(PNR)},PNR≤PNR∗¯¯¯¯CM(PNR),PNR>PNR∗, (62)

where and are given in (IV-A) and (IV-A), respectively, and is the unique solution of 777The solution is approximately .

 12−Q(PNR)=PNRPNR+√2πe. (63)
###### Proof:

This outer bound can be obtained following the same approach of the proof of Proposition 1. Specifically, the bound is obtained by applying the single-user capacity upper bound in Lemma 4 directly; The bound is obtained by , noting that , and applying the single-user capacity upper bound in Lemma 4.

Using techniques pioneered in [26], the capacity of the Gaussian optical intensity channel (2) with a peak power constraint can be numerically evaluated accurately with respect to PNR. So the outer bound in Proposition 5 can be refined by replacing in (60) and (61) by the numerical result of . The gain of this refinement is limited, however, since the upper bound in Lemma 4 is already very tight for moderate to high PNR. But the numerical result of is indeed helpful for tightening the inner bound.

Proposition 6 (Inner bound): The capacity of the OIMAC (3) with a per-user peak power constraint as (5) is inner bounded by a polytope with the following five rate pairs as corner points:

 (R1,R2)={ (0,0),(CPP-OIC(PNR1),0), (IU(PNR1),IU+(a2)−IU(PNR1)), (IU+(a1)−IU(PNR2),IU(PNR2)), (0,CPP-OIC(PNR2))}, (64)

where is the capacity of a single-user OIC with peak power constraint; for , ,

 IU+(ai)=I(XU+XD;XU+XD+Z), (65)

where , , and has a PDF as

 fX(x)=ni−1∑m=0ni−mni(ni+1)(δ(Ai−mA~i)+δ(mA~i)), (66)

where . A closed-form inner bound weaker than (IV-B) is given by

 Ri≤C––i=12log(1+12πePNR2i),i=1,2, (67) (C––1+C––2−C––12)R1+(C––1+C––2−C––21)R2 ≤C––1C––21+C––2C––12−C––12C––21, (68)

where for ,

 C––~ii=12log(1+(nini+1)2(ni−ai)(ni+1)22πePNR2~i). (69)
###### Proof:

The achievability of the second and last rate pairs in (IV-B) follows directly from Lemma 4. To prove the achievability of the third and fourth rate pairs, we employ an input distribution , where is the PDF of a uniform input distribution over , and is as (66). By combining (7), (8) with (10), we obtain that a rate for user is achievable; simultaneously, by combining (9) with (12), we obtain that a sum rate is achievable, which is exactly . So user can achieve , and therefore the third and fourth rate pairs in (IV-B) are both achievable. All other rate pairs in the convex closure of (IV-B) can be achieved using time sharing [16].

To establish the closed-form lower bound (67) (IV-B), we first prove that the rate pair is achievable. We employ the input distributions and (where we set and ) for user 1 and 2, respectively. Then the achieved sum rate and single-user rate are exactly and . The achievability of can be obtained directly by the single-user capacity lower bound (IV-A). Thus, to show the achievability of , we only need a proof of

 IU+(a2)≥C––12. (70)

First, the LHS of (70) can be lower bounded as

 IU+(a2) =I(XU1+XD2;X