# Robust Signaling for Bursty Interference

This paper studies a bursty interference channel, where the presence/absence of interference is modeled by a block-i.i.d. Bernoulli process that stays constant for a duration of T symbols (referred to as coherence block) and then changes independently to a new state. We consider both a quasi-static setup, where the interference state remains constant during the whole transmission of the codeword, and an ergodic setup, where a codeword spans several coherence blocks. For the quasi-static setup, we study the largest rate of a coding strategy that provides reliable communication at a basic rate and allows an increased (opportunistic) rate when there is no interference. For the ergodic setup, we study the largest achievable rate. We study how non-causal knowledge of the interference state, referred to as channel-state information (CSI), affects the achievable rates. We derive converse and achievability bounds for (i) local CSI at the receiver-side only; (ii) local CSI at the transmitter- and receiver-side, and (iii) global CSI at all nodes. Our bounds allow us to identify when interference burstiness is beneficial and in which scenarios global CSI outperforms local CSI. The joint treatment of the quasi-static and ergodic setup further allows for a thorough comparison of these two setups.

## Authors

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## I Introduction

Interference is a key limiting factor for the efficient use of the spectrum in modern wireless networks. It is therefore not surprising that the interference channel (IC) has been studied extensively in the past; see, e.g, [1, Ch. 6] and references therein. Most of the information-theoretic work developed for the IC assumes that interference is always present. However, certain physical phenomena, such as shadowing, can make the presence of interference intermittent or bursty. Interference can also be bursty due to the bursty nature of data traffic, distributed medium access control mechanisms, and decentralized networking protocols. For this reason, there has been an increasing interest in understanding and exploring the effects of burstiness of interference.

Seminal works in this area were performed by Khude et al. in [2, 3]. They tried to harness the burstiness of the interference by taking advantage of the time instants when the interference is not present to send opportunistic data. Specifically, [2, 3] considered a channel model where the interference state stays constant during the transmission of the entire codeword, which corresponds to a quasi-static channel. Motivated by the idea of degraded message sets by Körner and Marton [4], Khude et al. studied the largest rate of a coding strategy that provides reliable communication at a basic rate and allows an increased (opportunistic) rate when there is no interference. The idea of opportunism was also used by Diggavi and Tse [5] for the quasi-static flat fading channel and, recently, by Yi and Sun [6] for the -user IC with states.

Wang et al. [7] modeled the presence of interference using an independent and identically distributed (i.i.d.) Bernoulli process that indicates whether interference is present or not, which corresponds to an ergodic channel. Wang et. al. mainly studied the effect of causal feedback under this model, but also presented converse bounds for the non-feedback case. Mishra et al. considered the generalization of this model to multicarrier systems, modeled as parallel two-user bursty ICs, for the feedback [8] and non-feedback case [9].

The bursty IC is related to the binary fading IC, for which the four channel coefficients are in the binary field

according to some Bernoulli distribution.

111Note, however, that neither of the two models is a special case of the other. While a zero channel coefficient of the cross link corresponds to intermittence of interference, the bursty IC allows for non-binary signals. Conversely, in contrast to the binary fading IC, the direct links in the bursty IC cannot be zero, since only the interference can be intermittent. Vahid et al. studied the capacity region of the binary fading IC when the transmitters have access to the past channel coefficients [10, 11, 12, 13].

The focus of the works by Khude et al. [2] and Wang et al. [7] was on the linear deterministic model (LDM), which was first introduced by Avestimehr [14], but falls within the class of more general deterministic channels whose capacity was obtained by El Gamal and Costa in [15]. The LDM maps the Gaussian IC to a channel whose outputs are deterministic functions of their inputs. Bresler and Tse demonstrated in [16]

that the generalized degrees of freedom (first-order capacity approximation) of the two user Gaussian IC coincides with the normalized capacity of the corresponding deterministic channel. The LDM thus offers insights on the Gaussian IC.

### I-a Contributions

In this work, we consider the LDM of a bursty IC. We study how interference burstiness and the knowledge of the interference states (throughout referred to as channel-state information (CSI)) affects the capacity of this channel. We point out that this CSI is different from the one sometimes considered in the analysis of ICs (see, e.g., [17]), where CSI refers to knowledge of the channel coefficients. (In this regard, we assume that all transmitters and receivers have access to the channel coefficients.) For the sake of compactness, we focus on non-causal CSI and leave other CSI scenarios, such as causal or delayed CSI, for future work.

We consider the following cases: (i) only the receivers know the corresponding interference state (local CSIR); (ii) transmitters and receivers know their corresponding interference states (local CSIRT); and (iii) both transmitters and receivers know all interference states (global CSIRT). For each CSI level we consider both i) the quasi-static channel and ii) the ergodic channel. Specifically, in the quasi-static channel the interference is present or absent during the whole message transmission and we harness the realizations when the channel experiences better conditions (no presence of interference) to send extra messages. In the ergodic channel the presence/absence of interference is modeled as a Bernoulli random variable which determines the interference state. The interference state stays constant for a certain coherence time

and then changes independently to a new state. This model includes the i.i.d. model by Wang et al. as a special case, but also allows for scenarios where the interference state changes more slowly.222Note, however, that when the receivers know the interference state (as we shall assume in this work), then the capacity of this model becomes independent of and coincides with that of the i.i.d. model. All these analyses are performed for the two cases where the states of each of the interfering links are independent, and where states of the interfering links are fully correlated.

Our analysis shows that, for both the quasi-static and ergodic channels, for all interference regions except the very strong interference region, global CSIRT outperforms local CSIR/CSIRT. This result does not depend on the correlation between the states of the interfering links. For local CSIR/CSIRT and the quasi-static scenario, the burstiness of the channel is of benefit only in the very weak and weak interference regions. For the ergodic case and local CSIR, interference burstiness is only of clear benefit if the interference is either weak or very weak, or if it is present at most half of the time. This is in contrast to local CSIRT, where interference burstiness is beneficial in all interference regions.

Specific contributions of our paper include:

• A joint treatment of the quasi-static and the ergodic model: Previous literature on the bursty IC considers either the quasi-static model or the ergodic model. Furthermore, due to space constraints, the proofs of some of the existing results were either omitted or contain little details. In contrast, our paper discusses both models, allowing for a thorough comparison between the two.

• Novel achievability and converse bounds: For the ergodic model, the achievability bounds for local CSIRT, and the achievability and converse bounds for global CSIRT, are novel. In particular, novel achievability strategies are proposed that exploit certain synchronization between the users. To keep the paper self-contained, we further present the proof of the achievability bound for local CSIR that has appeared in the literature without proof.

• Novel converse proofs for the quasi-static model: In contrast to existing converse bounds, which are based on Fano’s inequality, our proofs of the converse bounds for the rates of the worst-case and opportunistic messages are based on an information density approach (more precise, they are based on the Verdú-Han lemma). This approach does not only allow for rigorous yet clear proofs, but it would also enable a more refined analysis of the probabilities that worst-case and opportunistic messages can be decoded correctly.

• A thorough comparison of the sum capacity of various scenarios: Inter alia, the obtained results are used to study the advantage of featuring different levels of CSI, the impact of the burstiness of the interference, and the effect of the correlation between the channel states of both users.

The rest of this paper is organized as follows. Section II introduces the system model, where we define the bursty IC quasi-static setup, the ergodic setup, and briefly summarize previous results on the non-bursty IC. In Sections IIIV we present our results for local CSIR, local CSIRT and global CSIRT, respectively. Section VI studies the impact on featuring different CSI levels. Section VII analyzes in which scenarios exploiting burstiness of interference is beneficial. Section VIII concludes the paper with a summary of the results. Most proofs of the presented results are deferred to the appendix.

### I-B Notation

To differentiate between scalars, vectors, and matrices we use different fonts: scalar random variables and their realizations are denoted by upper and lower case letters, respectively, e.g.,

, ; vectors are denoted using bold face, e.g., , ; random matrices are denoted via a special font, e.g., ; and for deterministic matrices we shall use yet another font, e.g., . For sets we use the calligraphic font, e.g., . We denote sequences such as by . We define as .

We use to denote the binary Galois field and to denote the modulo 2 addition. Let the down-shift matrix , a matrix of dimension , be defined as

 Su=[0Tu×(q−u)0Iu0u×(q−u)]q×q

with the all-zero vector and

the identity matrix.

Similarly, we define the matrix of dimension that selects the lowest components of a vector of dimension :

 Ld=[00Td×(q−d)0d×(q−d)Id]q×q.

We shall denote by the entropy of a binary random variable with probability mass function (), i.e.,

 Hb(p)≜−plogp−(1−p)log(1−p). (1)

Similarly, we denote by the entropy where and are two independent binary random variables with probability mass functions and , respectively:

 Hsum(p,q)≜Hb(p(1−q)+(1−p)q). (2)

For this function it holds that . Finally, denotes the indicator function, i.e., is if the statement is true and if it is wrong.

## Ii System Model

Our analysis is based on the LDM, introduced by Avestimehr et. al. [14] for some relay network. This model is, on the one hand, simple to analyze and, on the other hand, captures the essential structure of the Gaussian channel in the high signal-to-noise ratio regime.

We consider a bursty IC where i) the interference state remains constant during the whole transmission of the codeword of length (quasi-static setup) or ii) the interference state remains constant for a duration of consecutive symbols and then changes independently to a new state (ergodic setup). For one coherence block, the two user bursty IC is depicted in Figure 1, where and are the channel gains of the direct and cross links, respectively. We assume that and are known to both the transmitter and receiver and remain constant during the whole transmission of the codeword. For simplicity, we shall assume that and are equal for both users. Nevertheless, most of our results generalize to the asymmetric case. More precisely, all converse and achievability bounds generalize to the asymmetric case, while the direct generalization of the proposed achievability schemes may be loose in some asymmetric regions.

For the -th block, the input-output relation of the channel is given by

 Y1,k = SndX1,k⊕B1,kSncX2,k, (3) Y2,k = SndX2,k⊕B2,kSncX1,k. (4)

Let . In (3) and (4), and , . The interference states , , , are sequences of i.i.d. Bernoulli random variables with activation probability .

Regarding the sequences and , we consider two cases: i) and are independent of each other and ii) and are fully correlated sequences, i.e., . For both cases we assume that the sequences are independent of the messages and .

We shall define the normalized interference level as , based on which we can divide the interference into the following regions (a similar division was used by Jafar and Vishwanath [18]):

• very weak interference (VWI) for ,

• weak interference (WI) for ,

• moderate interference (MI) for ,

• strong interference (SI) for ,

• very strong interference (VSI) for .

### Ii-a Quasi-static channel

The channel defined in (3) and (4) may experience a slowly-varying change on the interference state. In this case, the duration of each of the transmitted codewords of length , is smaller than the coherence time of the channel and the interference state stays constant over the duration of each codeword, i.e., , . In the wireless communications literature such a channel is usually referred to as a quasi-static channel [19, Sec. 5.4.1]. In this scenario, the rate pair of achievable rates is dominated by the worst case, which corresponds to the presence of interference at both receivers. However, in absence of interference, it is possible to communicate at a higher date rate, so planning a system for the worst case may be too pessimistic. Assuming that the receivers have access to the interference states, the transmitters could send opportunistic messages that are decoded only if the interference is absent, in addition to the regular messages that are decoded irrespective of the interference state. We make the notion of opportunistic messages and rates precise in the subsequent paragraphs.

Let indicate the level of CSI available at the transmitter-side in coherence block , and let indicate the level of CSI at the receiver-side in coherence block :

1. local CSIR: ,

2. local CSIRT: ,

3. global CSIRT: .

We define the set of opportunistic messages according to the level of CSI at the receiver as , where denotes the set of possible interference states . Specifically,

1. for local CSIR: ,

2. for local CSIRT: ,

3. for global CSIRT: .

Then, we define an opportunistic code as follows.

###### Definition 1 (Opportunistic code for the bursty IC)

An opportunistic code for the bursty IC is defined as:

1. two independent messages and uniformly distributed over the message sets ;

2. two independent sets of opportunistic messages and uniformly distributed over the message sets , ,

3. two encoders:

4. two decoders: .

Here and denote the decoded message and the decoded opportunistic message, respectively. We set , (for local CSIR/CSIRT) and (for global CSIRT).

To better distinguish the rates from the opportunistic rates , , we shall refer to as worst-case rates, because the corresponding messages can be decoded even if the channel is in its worst state (see also Definition 2).

###### Definition 2 (Achievable opportunistic rates)

A rate tuple is achievable if there exists a sequence of codes such that

 (5)

and

 Pr{(^W1,Δ^W1(V1))≠(W1,ΔW1(V1))|V1=v1}→0asN→∞, v1∈V1, (6)
 Pr{(^W2,Δ^W2(V2))≠(W2,ΔW2(V2))|V2=v2}→0asN→∞, v2∈V2. (7)

The capacity region is the closure of the set of achievable rate tuples [1, Sec. 6.1]. We define the worst-case sum-rate as and the opportunistic sum-rate as . The worst-case sum-capacity is the supremum of all achievable worst-case sum-rates, the opportunistic sum-capacity is the supremum of all opportunistic sum-rates, and the total sum-capacity is defined as . Note that the opportunistic sum-capacity depends on the worst-case sum-rate.

### Ii-B Ergodic channel

In this setup, we shall restrict ourselves to codes whose blocklength is an integer multiple of the coherence time . A codeword of length thus spans independent channel realizations.

###### Definition 3 (Code for the bursty IC)

A code for the bursty IC is defined as:

1. two independent messages and uniformly distributed over the message sets

2. two encoders:

3. two decoders:

Here denotes the decoded message, and and indicate the level of CSI at the transmitter- and receiver-side, respectively, which are defined as for the quasi-static channel in Section II-A.

###### Definition 4 (Ergodic achievable rates)

A rate pair is achievable for a fixed if there exists a sequence of codes (parametrized by ) such that

The capacity region is the closure of the set of achievable rate pairs. We define the sum rate as , the sum capacity is the supremum of all achievable sum rates.

### Ii-C The sum-capacities of the non-bursty and the quasi-static bursty IC

When the activation probability is , we recover in both the ergodic and quasi-static scenarios the deterministic IC. For a general deterministic IC the capacity region was obtained in [15, Th. 1] and then by Bresler and Tse in [16] for a specific deterministic IC. For completeness, we present the sum capacity region for the deterministic non-bursty IC in the following theorem.

###### Theorem 1

The sum-capacity region of the 2-user deterministic IC is equal to the set of all sum rates satisfying

 R≤2nd (9) R≤(nd−nc)++max(nd,nc) (10) R≤2max{(nd−nc)+,nc}. (11)
###### Proof:

The proof is given in [15, Sec. II]. For the achievability bounds, El Gamal and Costa [15, Th. 1] use the Han-Kobayashi scheme [20] for a general IC. Bresler and Tse [16, Sec. 4] use a specific Han-Kobayashi strategy for the special case of the LDM. Jafar and Vishwanath [18] present an alternative achievability scheme for the -user IC, which particularized for the 2-user IC will be referenced in this work.

We can achieve the sum rates (9)-(11) over the quasi-static channel by treating the bursty IC as a non-bursty IC. The following theorem demonstrates that this is the largest achievable sum-rate irrespective of the availability of CSI and the correlation between and .

###### Theorem 2 (Sum capacity for the quasi-static bursty IC)

For , the worst-case sum capacity of the bursty IC is equal to the largest sum rate satisfying

• For ,

 R ≤ 2nd. (12)
• For

 R ≤ (nd−nc)++max(nd,nc) (13) R ≤ 2max{(nd−nc)+,nc}. (14)
###### Proof:

The converse bounds are proved in Appendix A-1. Achievability follows directly from Theorem 1 by treating the bursty IC as a non-bursty IC.

Theorem 2 shows that the sum capacity does not depend on the level of CSI available at the transmitter- and receiver-side. However, this is not the case for the opportunistic rates as we will see in the next sections.

###### Remark 1

In principle, one could reduce the worst-case rates in order to increase the opportunistic rates. However, it turns out that such a strategy is not beneficial in terms of total rates , . In other words, setup , (for local CSIR/CSIRT) and (for global CSIRT), as we have done in Definition 2, incurs no loss in total rate. Furthermore, in most cases it is preferable to maximize the worst-case rate, since it can be guaranteed irrespective of the interference state.

## Iii Local CSIR

For the quasi-static and ergodic setups, described in Sections II-A and II-B, respectively, we derive converse and achievabillity bounds for the independent and fully correlated scenarios when the interference state is only available at the receiver-side.

### Iii-a Quasi-static channel

#### Iii-A1 Independent case

We present converse and achievability bounds for local CSIR when and are independent. The converse bounds are derived for local CSIRT, hence they also apply to this case. Since converse and achievability bounds coincide, this implies that local CSI at the transmitter is not beneficial in the quasi-static setup.

###### Theorem 3 (Opportunistic sum-capacity for local CSIR/CSIRT)

Assume that and are independent of each other. For , the opportunistic sum-capacity region is the set of rate pairs satisfying (12)–(14) and

 R+ΔR(0) ≤ 2nd (15) 2R+ΔR(0) ≤ 2(nd−nc)++2max(nd,nc). (16)
###### Proof:

The converse bounds are proved in Appendix A-2 and the achievability bounds are proved in Appendix A-3.

As discussed in Remark 1, one could reduce the worst-case sum-rate and increase the opportunistic rates . For instance, when the interference is absent, one can achieve by transmitting in all sub-channels uncoded bits. However, in the case of one-shot transmission333With one-shot transmission we refer to the case where we transmit one codeword of length over the quasi-static channel. This is in contrast to the case discussed, e.g., in Section III-C, where we are interested in transmitting many codewords, each over channel uses of independent quasi-static channels. this is not desirable, since the worst-case sum-rate is the only rate that can be guaranteed irrespective of the interference state. Thus, one is typically interested in the opportunistic sum-capacity when is maximized. For this case, the results of Theorem 3 are summarized in Table I for the VWI, WI, MI and SI regions.

Observe that converse and achievability bounds coincide. Further observe that opportunistic messages can only be transmitted reliably for VWI or WI. In the other interference regions, the opportunistic sum-capacity is zero.

#### Iii-A2 Fully correlated case

Assume now that the sequences and are fully correlated (). For local CSIR, the correlation between and has no influence on the opportunistic sum-capacity region. Indeed, in this case the channel inputs are independent of and the opportunistic sum-capacity region of the quasi-static bursty IC depends on only via the marginal distributions of , . Hence, it follows that Theorem 3 as well as Table I apply also to the fully correlated case and local CSIR scenario. For completeness, a proof of the converse part is given in Appendix A-6. The achievability part is proved in Appendix A-7.

### Iii-B Ergodic channel

#### Iii-B1 Independent case

For the case where the sequences and are independent of each other, we have the following theorems.

###### Theorem 4 (Converse bounds for local CSIR)

Assume that and are independent of each other. The sum rate for the bursty IC is upper-bounded by

 R ≤21−p1+pnd+2p1+p[(nd−nc)++max(nd,nc)] (17)

and

 R≤{2(1−2p)nd+2p[(nd−nc)++max(nd,nc)] p≤12,2(1−p)[(nd−nc)++max(nd,nc)]+2(2p−1)[max{(nd−nc)+,nc}] p>12. (18)
###### Proof:

Bound (17) coincides with [7, Eq. (3)]. Specifically, [7, Eq. (3)] derives (17) for the considered channel model with and feedback. The proof for this bound under local CSIRT (without feedback) is given in Appendix B-1. Bound (18) coincides with [21, Lemma A.1]. Specifically, [21, Lemma A.1] derives (18) for the model considered with . The proof of [21, Lemma A.1] directly generalizes to arbitrary .

###### Theorem 5 (Achievability bounds for local CSIR)

Assume that and are independent of each other. The following sum rate is achievable over the bursty IC:

 (19)
###### Proof:

The achievability scheme for VWI for all values of , and for WI and MI when , is described in Appendix B-2. The achievability scheme for WI and is described in Appendix B-2. The scheme for SI and is summarized in Appendix B-2. For MI and SI when , the achievability bound in the theorem corresponds to the one of the non-bursty IC [18]. This also implies that in this sub-region we do not exploit the burstiness of the IC.

###### Remark 2

Wang et al. claim in [21, Lemma A.1] that the converse bound (18) is tight for without providing an achievability bound. Instead, they refer to Khude et al. [2] for the inner bound which, alas, does not apply to the ergodic setup. For completeness, we include the achievability schemes for the ergodic setup and in Appendix B-2.

Table II summarizes the results of Theorems 4 and 5. The gray zones represent the regions where the converse and achievability bounds do not match. In Table II, we define

 CLMI ≜ min{2[2(nd−nc)+p(3nc−2nd)],2[1−p1+pnd+p1+p(2nd−nc)]} (20) CLSI ≜ min{2pnc,2[1−p1+pnd+p1+pnc]} (21)

where “L” stands for “local CSIR”.

#### Iii-B2 Fully correlated case

For local CSIR, the correlation between and has no influence on the capacity region. Indeed, in this case the channel inputs are independent of and decoder has only access to and , . Furthermore, vanishes as if, and only if, , , vanishes as . Since depends only on , the capacity region of the bursty IC depends on only via the marginal distributions of and . Hence, Theorems 4 and 5 as well as Table II apply also to the case where . This is consistent with the observation by Sato [22] that “the capacity region is the same for all two-user channels that have the same marginal probabilities.”

### Iii-C Quasi-static vs. ergodic setup

In general, the sum capacities of the quasi-static and ergodic channels cannot be compared, because in the former case we have a set of sum capacities (worst case and opportunistic), whereas in the latter case only one is defined. To allow for a comparison, we introduce for the quasi-static channel the average sum-capacity as

 ¯C≜sup(R,ΔR(0)){R+(1−p)ΔR(0)} (22)

where the suprema is over all pairs that satisfy Theorems 2 and 3

. Intuitively, the average rate corresponds to the case where we send many messages over independent quasi-static fading channels. By the law of large numbers, a fraction of

transmissions will be affected by interference, the remaining transmissions will be interference-free. Table III summarizes the average sum-capacity for the different interference regions.

By comparing Tables II and III, we can observe that for and all intereference regions, and for and VWI/WI, the average sum-capacity in the quasi-static setup coincides with the sum capacity in the ergodic setup. For , and MI/SI (where converse and achievability bounds do not coincide), the average sum-capacities in the quasi-static setup coincide with the achievability bounds of the ergodic setup.

## Iv Local CSIRT

For the quasi-static and ergodic setups, we present converse and achievabillity bounds when transmitters and receivers have access to their corresponding interference states. We shall only consider the independent case here, because when local CSIRT coincides with global CSIRT, which will be discussed in Section V.

### Iv-a Quasi-static channel

For the quasi-static channel, the converse and achievability bounds were already presented in Theorem 3 in Section III-A1. Indeed, the converse bounds were derived for local CSIRT, whereas the achievability bounds in that theorem were derived for local CSIR. Since these bounds coincide for all interference regions and all probabilities of it follows that, for the quasi-static channel, availability of local CSI at the transmitter in addition to local CSI at the receiver is not beneficial. The converse and achievability bounds are then given in Theorem 3.

### Iv-B Ergodic channel

As mentioned in the previous section, the converse bound (17) presented in Theorem 4 was derived for local CSIRT, so it applies to the case at hand. We next present achievability bounds for this setup that improve upon those for CSIR. The intuition behind the corresponding achievability schemes will be explained with the following toy example.

Example: Let us assume that , and suppose that at time the transmitters send the bits . If there is no interference, then receiver receives . If there is interference, then receiver receives . Consequently, the channel flips if , and it flips if . It follows that each transmitter-receiver pair experiences a binary symmetric channel (BSC) with a given crossover probability that depends on and on the probabilities that are one. Specifically, let

 PX1|B1(X1=1|B1=0)≜p1 (23) PX1|B1(X1=1|B1=1)≜p2 (24) PX2|B2(X2=1|B1=0)≜q1 (25) PX2|B2(X2=1|B1=1)≜q2 (26)

and define and , which are the crossover probabilities of the BSCs experienced by receivers and , respectively, when they are affected by interference. By drawing for each user two codebooks (one for and one for ) i.i.d. at random according to the probabilities , , , and , and by following a random-coding argument, it can be shown that this scheme achieves the sum rate

 R = (1−p)[Hb(p1)+Hb(q1)]+p[Hsum(p2,q3)−Hb(q3)]+p[Hsum(q2,p3)−Hb(p3)]. (27)

This expression holds for any set of parameters , and the largest sum-rate achieved by this scheme is obtained by maximizing over .

In the following, we present the achievable sum-rates that can be obtained by generalizing the above achievability scheme to general and . The achievability schemes that achieve these rates are presented in Appendix E. The largest achievable sum-rates can then be obtained by numerically maximizing over the parameters (which depend on the interference region).

1. For the VWI region, we achieve the sum rate

 R=2(nd−pnc). (28)
2. For the WI region, we can achieve for any

 R1 = (nd−nc)+(1−p)[(nd−nc)+(2nc−nd)Hb(p1)]+p(2nc−nd)(1−Hb(q3)) (29) R2 = (nd−nc)+(1−p)[(nd−nc)+(2nc−nd)Hb(q1)]+p(2nc−nd)(1−Hb(p3)) (30)

where and .

3. To present the achievable rates for MI, we need to divide the region into the following four subregions:

1. For , we can achieve for any and

 R1 = (nd−nc) (31) +(1−p)[(3nc−2nd2)(Hb(η1)+Hb(^p1)+Hb(p1))+(4nd−5nc2)Hb(~p1)+(nd−nc)] +p[(3nc−2nd2)(1+H% sum(p2,~γ)−Hb(~γ)+Hsum(~p2,q3)−Hb(q3)−Hb(^q3)) +(4nd−5nc2)(1−Hb(~q3))]

where , , , and , and

 R2 = (nd−nc) (32) +(1−p)[(3nc−2nd2)(Hb(γ1)+Hb(^q1)+Hb(q1))+(4nd−5nc2)Hb(~q1)+(nd−nc)] +p[(3nc−2nd2)(1+H% sum(q2,~η)−Hb(~η)+Hsum(~q2,p3)−Hb(p3)−Hb(^p3)) +(4nd−5nc2)(1−Hb(~p3))]

where , , , and .

###### Remark 3

After combining (31) and (32), and appear only through the functions and , respectively. Hence, and can be optimized separately from the remaining terms.

2. For , we can achieve for any and

 R1 = (nd−nc) (33) +(1−p)[(3nc−2nd2)(Hb(p1)+Hb(η1)+Hb(^p1))+(4nd−5nc2)Hb(~p1)+(nd−nc)] +p[(3nc−2nd2)(H% sum(p2,~γ)−Hb(~γ)+1−Hb(^q3)) +(4nd−5nc2)(H% sum(~p2,q3)−Hb(q3)+1−Hb(~q3))]

where , , , and , and

 R2 = (nd−nc) (34) +(1−p)[(3nc−2nd2)(Hb(q1)+Hb(γ1)+Hb(^q1))+(4nd−5nc2)Hb(~q1)+(nd−nc)] +p[(3nc−2nd2)(H% sum(q2,~η)−Hb(~η)+1−Hb(^p3)) +(4nd−5nc2)(H% sum(~q2,p3)−Hb(p3)+1−Hb(~p3))]

where ,