Capacity bounds for multiple access-cognitive interference channel

Motivated by the uplink scenario in cellular cognitive radio, this study considers a communication network in which a point-to-point channel with a cognitive transmitter and a Multiple Access Channel (MAC) with common information share the same medium and interfere with each other. A Multiple Access-Cognitive Interference Channel (MA-CIFC) is proposed with three transmitters and two receivers, and its capacity region in different interference regimes is investigated. First, the inner bounds on the capacity region for the general discrete memoryless case are derived. Next, an outer bound on the capacity region for full parameter regime is provided. Using the derived inner and outer bounds, the capacity region for a class of degraded MA-CIFC is characterized. Two sets of strong interference conditions are also derived under which the capacity regions are established. Then, an investigation of the Gaussian case is presented, and the capacity regions are derived in the weak and strong interference regimes. Some numerical examples are also provided.

Interference avoidance techniques have traditionally been used in wireless networks wherein multiple source-destination pairs share the same medium. However, the broadcasting nature of wireless networks may enable cooperation among entities, which ensures higher rates with more reliable communication. On the other hand, due to the increasing number of wireless systems, spectrum resources have become scarce and expensive. The exponentially growing demand for wireless services along with the rapid advancements in wireless technology has lead to cognitive radio technology which aims to overcome the spectrum inefficiency problem by developing communication systems that have the capability to sense the environment and adapt to it [1].

In overlay cognitive networks, the cognitive user can transmit simultaneously with the non-cognitive users and compensate for the interference by cooperation in sending, i.e., relaying, the non-cognitive users' messages [1]. From an information theoretic point of view, Cognitive Interference Channel (CIFC) was first introduced in [2] to model an overlay cognitive radio and refers to a two-user Interference Channel (IFC) in which the cognitive user (secondary user) has the ability to obtain the message being transmitted by the other user (primary user), either in a non-causal or in a causal manner. An achievable rate region for the non-causal CIFC was derived in [2], by combining the Gel'fand-Pinsker (GP) binning [3] with a well-known simultaneous superposition coding scheme (rate splitting) applied to IFC [4]. For the non-causal CIFC, where the cognitive user has non-causal full or partial knowledge of the primary user's transmitted message several achievable rate regions and capacity results in some special cases have been established [5–14]. More recently a three-user cognitive radio network with one primary user and two cognitive users is studied in [15, 16], where an achievable rate region is derived for this setup based on rate splitting and GP binning.

In the interference avoidance-based systems, i.e., when the communication medium is interference-free, uplink transmission is modeled with a Multiple Access Channel (MAC) whose capacity region has been fully characterized for independent transmitters [17, 18] as well as for the transmitters with common information [19]. Recently, taking the effects of interference into account in the uplink scenario, a MAC and an IFC have been merged into one setup by adding one more transmit-receive pair to the communication medium of a two-user MAC [20, 21], where the channel inputs at the transmitters are independent and there is no cognition or cooperation.

In this paper, we introduce Multiple Access-Cognitive Interference Channel (MA-CIFC) by providing the transmitter of the point-to-point channel with cognition capabilities in the uplink with interference model. Moreover, transmitters of MAC have common information that enables cooperation among them. As shown in Figure 1, the proposed channe consists of three transmitters and two receivers: two-user MAC with common information as the primary network and a point-to-point channel with a cognitive transmitter that knows the message being sent by all of the transmitters in a non-causa manner. A physical example of this channel is the coexistence of cognitive users with the licensed primary users in a cellular or satellite uplink transmission, where the cognitive radios by their abilities exploit side information about the environment to maintain or improve the communication of primary users while also achieving some spectrum resources for their own communication. In this scenario, the primary non-cognitive users can be oblivious to the or aware of the cognitive users [1]. When the non-cognitive user is oblivious to the cognitive user's presence, its receiver's decoding process is independent of the interference caused by the cognitive user's transmission. In fact, the primary receiver treats interference as noise. However, in the aware cognitive user's scenario, the decoding process at the primary receiver can be adapted to improve its own rate. For example, the primary receiver can decode the cognitive user's message and cancel the interference when the interfering signal is strong enough. If the multi-antenna capability is available at the primary receiver, it can also reduce or increase the interfering signal by beam-steering, which results in the occurrence of the weak or strong interference regimes [1].

Graphic representation for MA-CIFC.

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To analyze the capacity region of MA-CIFC, we first derive three inner bounds on the capacity region (achievable rate regions). The first two bounds assume an oblivious primary receiver, which does not decode the cognitive user's message but treats it as noise. Two different coding schemes are proposed based on the superposition coding, the GP binning and the method of [6] in defining auxiliary Random Variables (RVs). Later, we show that these strategies are optimal for a degraded MA-CIFC and also in the Gaussian weak interference regime. In the third achievability scheme, we consider an aware primary receiver and obtain an inner bound on the capacity region based on using superposition coding in the encoding part and allowing both receivers to decode all messages with simultaneous joint decoding in the decoding part. This strategy is capacity-achieving in the strong interference regime. Next, we provide a general outer bound on the capacity region and derive conditions under which the first achievability scheme achieves capacity for the degraded MA-CIFC. We continue the capacity results by the derivation of two sets of strong interference conditions, under which the third inner bound achieves capacity. Further, we compare these two sets of conditions and identify the weaker set. We also extend the strong interference results to a network with k primary users.

Moreover, we consider the Gaussian case and find capacity results for the Gaussian MA-CIFC in both the weak and strong interference regimes. We use the second derived inner bound to show that the capacity-achieving scheme in weak interference consists of Dirty Paper Coding (DPC) [22] at the cognitive transmitter and treating interference as noise at both receivers. We also provide some numerical examples.

The rest of the paper is organized as follows. Section 2 introduces MA-CIFC model and the notations. Three inner bounds and an outer bound on the capacity region are derived in Section 3 and Section 4, respectively, for the discrete memoryless MA-CIFC. Section 5 presents the capacity results for the discrete memoryless MA-CIFC in three special cases. In Section 6, the Gaussian MA-CIFC is investigated. Finally, Section 7 concludes the paper.

Throughout the paper, upper case letters (e.g. X) are used to denote RVs and lower case letters (e.g. x) show their realizations. The probability mass function (p.m.f) of a RV X with alphabet set \mathcal{X} is denoted by p _X (x), where subscript X is occasionally omitted. A_{\epsilon }^n\left(X,Y\right) specifies the set of ϵ-strongly, jointly typical sequences of length n. The notation X_i^j indicates a sequence of RVs (X _i , X_i+1, ..., X _j ), where X^jis used instead of X_1^j, for brevity. \mathcal{N}\left(0,{\sigma }^2\right) denotes a zero mean normal distribution with variance σ².

Consider the MA-CIFC in Figure 2, which is denoted by \left({\mathcal{X}}_1\times {\mathcal{X}}_2\times {\mathcal{X}}_3,p\left(y_1^n,y_3^n|x_1^n,x_2^n,x_3^n\right),{\mathcal{Y}}_1\times {\mathcal{Y}}_3\right), where X_1\in {\mathcal{X}}_1,X_2\in {\mathcal{X}}_2 and X_3\in {\mathcal{X}}_3 are channel inputs at Transmitter 1 (Tx1), Transmitter 2 (Tx2) and Transmitter 3 (Tx3), respectively; Y_1\in {\mathcal{Y}}_1 and Y_3\in {\mathcal{Y}}_3 are channel outputs at Receiver 1 (Rx1) and Receiver 3 (Rx3), respectively; and p\left(y_1^n,y_3^n|x_1^n,x_2^n,x_3^n\right) is the channel transition probability distribution. In n channel uses, each Txj desires to send a message pair (m₀, m _j ) to Rx1 where j ∈ {1,2}, and Tx3 desires to send a message m₃ to Rx3.

Multiple Access-Cognitive Interference Channel (MA-CIFC).

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Definition 1: A \left(2^{n{R}_0},2^{n{R}_1},2^{n{R}_2},2^{n{R}_3},n\right) code for MA-CIFC consists of (i) four independent message sets {\mathcal{M}}_j=\left\{1,...,2^{n{R}_j}\right\}, where j ∈ {0, 1, 2, 3}; (ii) two encoding functions at the primary transmitters, f_1:{\mathcal{M}}_0\times {\mathcal{M}}_1\mapsto {\mathcal{X}}_1^n at Tx1 and f_2:{\mathcal{M}}_0\times {\mathcal{M}}_2\mapsto {\mathcal{X}}_2^n at Tx2; (iii) an encoding function at the cognitive transmitter, f_3:{\mathcal{M}}_0\times {\mathcal{M}}_1\times {\mathcal{M}}_2\times {\mathcal{M}}_3\mapsto {\mathcal{X}}_3^n; and (iv) two decoding functions, g_1:{\mathcal{Y}}_1^n\mapsto {\mathcal{M}}_0\times {\mathcal{M}}_1\times {\mathcal{M}}_2 at Rx1 and g_3:{\mathcal{Y}}_3^n\mapsto {\mathcal{M}}_3 at Rx3. We assume that the channel is memoryless. Thus, the channel transition probability distribution is given by

The probability of error for this code is defined as

Definition 2: A rate quadruple (R₀, R₁, R₂, R₃) is achievable if there exists a sequence of \left(2^{n{R}_0},2^{n{R}_1},2^{n{R}_2},2^{n{R}_s},n\right) codes with P _e → 0 as n → ∞. The capacity region \mathcal{C}, is the closure of the set of all achievable rates.

Now, we derive three achievable rate regions for the general setup. Theorems 1 and 2 assume an oblivious primary receiver (Rx1), which does not decode the cognitive user's message (m₃) and treats it as noise. The decoding procedure at the cognitive receiver (Rx3) differs in these schemes. In Theorem 1, the cognitive receiver (Rx3) decodes the primary messages (m₀, m₁, m₂), and all the transmitters use superposition coding. However, in Theorem 2, the cognitive receiver (Rx3) also treats the interference from the primary messages (m₀, m₁, m₂) as noise, while the cognitive transmitter (Tx3) uses GP binning to precode its message for interference cancelation at Rx3. We also utilize the method of [6] in defining auxiliary RVs, which helps us to achieve the outer bound in special cases. In fact, we achieve the outer bound of Theorem 4 using the region of Theorem 1 for a class of degraded MA-CIFC in Section 5. The region of Theorem 2 is used in Section 6 to derive the capacity region in the weak interference regime. In the scheme of Theorem 3, we consider an aware primary receiver (Rx1) which decodes the cognitive user's message (m₃). The cognitive receiver (Rx3) also decodes the primary messages (m₀, m₁, m₂). Therefore, this region is obtained based on using superposition coding in the encoding part and by allowing both receivers to decode all messages with simultaneous joint decoding in the decoding part. In Section 5, we show that this strategy is capacity-achieving in the strong interference regime. Proofs are provided in "Appendix A".

Theorem 1: The union of rate regions given by

is achievable for MA-CIFC, where the union is over all p.m.fs that factor as

Theorem 2: The union of rate regions given by (3)-(6) and

is achievable for MA-CIFC, where the union is over all p.m.fs that factor as

Theorem 3: The union of rate regions given by

is achievable for MA-CIFC, where the union is over all p.m.fs that factor as

Remark 1: We utilize the region of Theorem 1 in Section 5 to achieve capacity results for a class of degraded MA-CIFC, and the region of Theorem 2 to derive the results for the Gaussian case in Section 6. The region of Theorem 3 is also used to characterize the capacity region under strong interference conditions in Section 5.

Here, we derive a general outer bound on the capacity region of MA-CIFC which is used to obtain the capacity region for a class of the degraded MA-CIFC in Section 5 and also to find capacity results for the Gaussian MA-CIFC in the weak interference regime in Section 6. Let {\mathcal{R}}_o^1 denote the union of all rate quadruples (R₀, R₁, R₂, R₃) satisfying (3)-(6) and

where the union is over all p.m.fs that factor as (11).

Theorem 4: The capacity region of MA-CIFC satisfies

Proof: Consider a \left(2^{n{R}_0},2^{n{R}_1},2^{n{R}_2},2^{n{R}_3},n\right) code with the average error probability of P_e^n\to 0. Define the following RVs for i = 1, ..., n:

Considering the encoding functions f₁ and f₂, defined in Definition 1, and the above definitions for auxiliary RVs, we remark that (X_1,i, U _i ) → T _i → (X_2,i, V _i ) forms a Markov chain. Thus, these choices of auxiliary RVs satisfy the p.m.f (11) of Theorem 4. Now using Fano's inequality [23], we derive the bounds in Theorem 4. For the first bound, we have:

where (a) follows since messages are independent and (b) holds due to Fano's inequality and the fact that conditioning does not increase entropy. Hence,

where (a) is due to the encoding functions f₁, f₂ and f₃, defined in Definition 1, and the non-negativity of mutual information, (b) is obtained from the chain rule, (c) follows from the memoryless property of the channel and the fact that conditioning does not increase entropy, and (d) is obtained from (20)-(22).

Now, applying Fano's inequality and the independence of the messages, we can bound R₁ as:

where (a) follows from the chain rule and the encoding functions f₁ and f₂, and (b) from (20)-(22). Similarly, we can show that

Next, based on similar arguments, we bound R₁ + R₂ as

The last sum-rate bound can be derived as follows:

where (a) follows since conditioning does not increase entropy. Using the standard time-sharing argument for (24)-(28) completes the proof.

In this section, we characterize the capacity region of MA-CIFC under specific conditions. First, we consider a class of degraded MA-CIFC and derive conditions under which the inner bound in Theorem 1 achieves the outer bound of Theorem 4. Next, we investigate the strong interference regime by deriving two sets of strong interference conditions under which the region of Theorem 3 achieves capacity. We also compare these two sets of conditions and identify the weaker set. Finally, we extend the strong interference results to a network with k primary users.

A. Degraded MA-CIFC

Now, we characterize the capacity region for a class of MA-CIFC with a degraded primary receiver. We define MA-CIFC with a degraded primary receiver as a MA-CIFC where Y₁ and X₃ are independent given Y₃, X₁, X₂. More precisely, the following Markov chain holds:

or equivalently, X₃ → (X₁, X₂, Y₃) → Y₁ forms a Markov chain. This means that the primary receiver (Rx1) observes a degraded or noisier version of the cognitive user's signal (Tx3) compared with the cognitive receiver (Rx3).

Assume that the following conditions are satisfied for MA-CIFC over all p.m.fs that factor as (11):

Under these conditions, the cognitive receiver (Rx3) can decode the messages of the primary users with no rate penalty. If MA-CIFC with a degraded primary receiver satisfies conditions (30)-(33), the region of Theorem 1 coincides with {\mathcal{R}}_o^1 and achieves capacity, as stated in the following theorem.

Theorem 5: The capacity region of MA-CIFC with a degraded primary receiver, defined in (29), satisfying (30)-(33) is given by the union of rate regions satisfying (2)-(6) over all joint p.m.fs (11).

Remark 2: The messages of the primary users (m₀, m₁, m₂) can be decoded at Rx3 under conditions (30)-(33). Therefore, Rx3-Tx3 achieves the rate in (2). Moreover, we can see that due to the degradedness condition in (29), treating interference as noise at the primary receiver (Rx1) achieves capacity. We show in Section 6 that, in the Gaussian case the capacity is achieved by using the region of Theorem 2 based on DPC (or GP binning), where the cognitive receiver (Rx3) does not decode the primary messages and conditions (30)-(33) are not necessary.

Proof: Achievability: The proof follows from the region of Theorem 1. Using the condition in (30), the sum of the bounds in (2) and (3) makes the bound in (7) redundant. Similarly, conditions (31)-(33), along with the bound in (2), make the bounds in (8)-(10) redundant and the region reduces to (2)-(6).

Converse: To prove the converse part, we evaluate {\mathcal{R}}_o^1 of Theorem 4 with the degradedness condition in (29). It is noted that the p.m.f of Theorem 5 is the same as the one for {\mathcal{R}}_o^1. Moreover, the bounds in (3)-(6) are equal for both regions. Hence, it is only necessary to show the bound in (2). Considering (19), we obtain:

where (a) is obtained by applying the degradedness condition in (29). This completes the proof.

B. Strong interference regime

Now, we derive two sets of strong interference conditions under which the region of Theorem 3 achieves capacity. First, assume that the following set of strong interference conditions, referred to as Set1, holds for all p.m.fs that factor as (18):

In fact, under these conditions, interfering signals at the receivers are strong enough that all messages can be decoded by both receivers. Condition (34) implies that the cognitive user's message (m₃) can be decoded at Rx1, while conditions (35)-(37) guarantee the decoding of the primary messages (m₀, m₁, m₂) along with m₃ at Rx3 in a MAC fashion.

Theorem 6: The capacity region of MA-CIFC satisfying (34)-(37) is given by:

Remark 3: The message of the cognitive user (m₃) can be decoded at Rx1, under condition (34) and (m₀, m₁, m₂) can be decoded at Rx3 under conditions (35)-(37). Hence, the bound in (38) gives the capacity of a point-to-point channel with message m₃ with side-information X₁, X₂ at the receiver. Moreover, (38)-(41) with condition (34) give the capacity region for a three-user MAC with common information where R₁ and R₂ are the common rates, R₃ is the private rate for Tx3, and the private rates for Tx1 and Tx2 are zero.

Remark 4: If we omit Tx2, i.e., X_2=\varnothing, and Tx2 has no message to transmit, i.e., R₂ = 0, the model reduces to a CIFC, and {\mathcal{C}}_1^{\mathsf{\text{str}}} coincides with the capacity region of the strong interference channel with unidirectional cooperation (or CIFC), which was characterized in [8, Theorem 5]. It is noted that in this case, the common message can be ignored, i.e., T=\varnothing and R₀ = 0.

Proof: Achievability: Considering (35)-(37), the proof follows from Theorem 3.

Converse: Consider a \left(2^{n{R}_0},2^{n{R}_1},2^{n{R}_2},2^{n{R}_3},n\right) code with an average error probability of P_e^n\to 0. Define the following RV for i = 1, ..., n:

It is noted that due to the encoding functions f₁, f₂ and f₃, defined in Definition 1, the independence of messages, and the above definitions for Tⁿ, RVs satisfy the p.m.f (18) of Theorem 6. First, we provide a useful lemma which we need in the proof of the converse part.

Lemma 1: If (34) holds for all distributions that factor as (18), then

Proof: The proof relies on the results in [24, Proposition 1] and [25, Lemma]. By redefining X₂ = X₃, Y₂ = Y₃, X₁ = (X₁, X₂, T) in [8, Lemma 5], the proof follows.

Now, using Fano's inequality [23], we derive the bounds in Theorem 6. Using (23) provides:

where (a) is due to (42) and the encoding functions f₁, f₂ and f₃, defined in Definition 1, (b) follows from two facts; conditioning does not increase entropy and \left(M_1,M_2,M_3\right)\to \left(X_1^n,X_2^n,X_3^n\right)\to Y_3^n forms a Markov chain, (c) is obtained from the chain rule, and (d) follows from the memoryless property of the channel and the fact that conditioning does not increase entropy.

Now, applying Fano's inequality and the independence of the messages, we can bound R₁ + R₃ as

where (a) follows from encoding functions f₁, f₂ and f₃, (b) follows from (42) and the fact that M_3\to \left(X_1^n,X_2^n,X_3^n\right)\to Y_3^n forms a Markov chain, (c) is obtained from (43), (d) follows from the chain rule, and (e) follows from the memoryless property of the channel and the fact that conditioning does not increase entropy.

Applying similar steps, we can show that,

Finally, the sum-rate bound can be obtained as

where (a) follows from steps (a)-(c) in (45), (b) is due to the fact that \left(M_1,M_2\right)\to \left(X_1^n,X_2^n,X_3^n\right)\to Y_1^n forms a Markov chain, and (c) follows from the memoryless property of the channel and the fact that conditioning does not increase entropy. Using a standard time-sharing argument for (44)-(47) completes the proof.

Next, we derive the second set of strong interference conditions, called Set2, under which the region of Theorem 3 is the capacity region. For all p.m.fs that factor as (18), Set2 includes (34) and the following conditions:

Remark 5: Similar to the condition Set1, under these conditions interfering signals at the receivers are strong enough that all messages can be decoded by both receivers. The first condition in (34) is equal in the two sets under which the cognitive user's message (m₃) can be decoded at Rx1. However, conditions (48)-(50) imply that the primary messages (m₀, m₁, m₂) can be decoded at Rx3 in a MAC fashion, while in Set1, they can be decoded along with m₃.

Theorem 7: The capacity region of MA-CIFC, satisfying (34) and (48)-(50), referred to as {\mathcal{C}}_2^{\mathsf{\text{str}}}, is given by the union of rate regions satisfying (14)-(17) over all p.m.fs that factor as (18).

Proof: See "Appendix B".

Remark 6: Similar to Remark 4, by omitting \mathsf{\text{Tx}}2\left(T=X_2=\varnothing ,R_0=R_2=0\right), the model reduces to a CIFC. Moreover, {\mathcal{C}}_2 and Set2 reduce to the capacity region and strong interference conditions which have been derived in [13] for non-causal CIFC.

Remark 7 (Comparison of two sets of conditions): In the strong interference conditions of Set1, the first condition in (34) is used in the converse part, while (35)-(37) are used to reduce the inner bound to {\mathcal{C}}_1^{\mathsf{\text{str}}}. However, all the conditions of Set2 are utilized to prove the converse part. Now, we compare the conditions in these two sets. We can write (35) as

Considering (34), it can be seen that I_diff ≥ 0. Hence, condition (35) implies condition (48), but not vice versa. Similar conclusions can be drawn for other conditions of these two sets. Therefore, Set1 implies Set2, and the conditions of Set2 are weaker compared to those of Set1.

C. Multiple access-cognitive interference network (MA-CIFN)

Now, we extend the result of Theorem 6 to a network with k + 1 transmitters and two receivers; a k-user MAC as a primary network and a point-to-point channel with a cognitive transmitter. We call it Multiple Access-Cognitive Interference Network (MA-CIFN). Consider MA-CIFN in Figure 3, denoted by \left({\mathcal{X}}_1\times {\mathcal{X}}_2\times \cdot \cdot \cdot \times {\mathcal{X}}_k\times {\mathcal{X}}_{k+1},p\left(y_1^n,y_{k+1}^n|x_1^n,x_2^n,\cdot \cdot \cdot x_k^n,x_{k+1}^n\right),{\mathcal{Y}}_1\times {\mathcal{Y}}_{k+1}\right), where X_j\in {\mathcal{X}}_j is the channel input at Transmitter j (Txj), for j\in \left\{1,...,k+1\right\};Y_1\in {\mathcal{Y}}_1 and Y_{k+1}\in {\mathcal{Y}}_{k+1} are channel outputs at the primary and cognitive receivers, respectively, and p\left(y_1^n,y_{k+1}^n|n_1^n,x_2^n,...,x_k^n,x_{k+1}^n\right) is the channel transition probability distribution. In n channel uses, each Txj desires to send a message pair m _j to the primary receiver where j ∈ {1, ..., k}, and Txk + 1 desires to send a message m_k+1to the cognitive receiver. We ignore the common information for brevity. Definitions 1 and 2 can be simply extended to the MA-CIFN. Therefore, we state the result on the capacity region under strong interference conditions.

Graphic representation for the MA-CIFN.

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Corollary 1: The capacity region of the MA-CIFN, satisfying

for all S ⊆ [1: k] and for every p(x₁)p(x₂)...p(x _k ) p(x_{k+ 1}|x₁,x₂,...,x _k )p(y₁,y_k+1|x₁,x₂,...,x _k ,x_k+1), is given by