Multiuser Cognitive Radio Networks:An Information Theoretic Perspective

# Multiuser Cognitive Radio Networks: An Information Theoretic Perspective

K. G. Nagananda, Parthajit Mohapatra, Chandra R. Murthy and Shalinee Kishore 1
11K. G. Nagananda and Shalinee Kishore are with the Department of Electrical and Computer Engineering, Lehigh University, Bethlehem, PA, USA (e-mail: {kgn209, skishore}@lehigh.edu). Parthajit Mohapatra and Chandra R. Murthy are with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India (e-mail: {partha, cmurthy}@ece.iisc.ernet.in)
###### Abstract

Achievable rate regions and outer bounds are derived for three-user interference channels where the transmitters cooperate in a unidirectional manner via a noncausal message-sharing mechanism. The three-user channel facilitates different ways of message-sharing between the primary and secondary (or cognitive) transmitters. Three natural extensions of unidirectional message-sharing from two users to three users are introduced: (i) Cumulative message sharing; (ii) primary-only message sharing; and (iii) cognitive-only message sharing. To emphasize the notion of interference management, channels are classified based on different rate-splitting strategies at the transmitters. Standard techniques, superposition coding and Gel’fand-Pinsker’s binning principle, are employed to derive an achievable rate region for each of the cognitive interference channels. Simulation results for the Gaussian channel case are presented; they enable visual comparison of the achievable rate regions for different message-sharing schemes along with the outer bounds. These results also provide useful insights into the effect of rate-splitting at the transmitters, which aids in better interference management at the receivers.

## I Introduction

Cognitive radios (CRs) [1] try to improve spectral efficiency by gathering and using knowledge of their Radio Frequency (RF) environment to adjust their transmission and reception parameters. An overview of the potential benefits offered by the CRs in physical layer research is provided in [2]. In [3], three main CR paradigms have been identified - underlay, overlay and interweave. In the underlay paradigm, CR users are allowed to operate only if their interference to noncognitive (or primary) users is below a certain threshold. While operating in the overlay paradigm, the CRs transmit their data simultaneously with the primary users but employ sophisticated techniques that maintain (or even improve) the performance of primary users. In the interweave paradigm, the CRs sense unused frequency bands called spectrum holes to communicate without disrupting primary transmissions. Of these, the information theoretic research has focused primarily on the overlay paradigm where CR transmitters cooperate using unidirectional message sharing in a noncausal manner. Here, the cognitive user gains access to messages and the corresponding codewords of the primary user before transmission. Although clairvoyant, such models are popular in establishing performance limits of cooperative multiuser channels. Then, the primary and cognitive users simultaneously transmit their messages, but the encoding is performed in such a way that the primary user’s achievable rates do not suffer. We present first a short survey of recent information theoretic work in this area, followed by a summary of our contributions.

Related work: Besides identifying the three CR network paradigms mentioned above, [3] explored some of the fundamental capacity limits and associated transmission strategies for CR wireless networks. In [4], [5], Devroye et al defined the two-user genie-aided CR channel and derived an achievable rate region by employing rate-splitting at both transmitters. The coding scheme comprised a combination of the scheme proposed by Han and Kobayashi for the interference channel [6], and one proposed by Gel’fand and Pinsker (GP) for coding over channels with random parameters [7]. In [8], Wu et al introduced terms like dumb and smart antennas to refer to primary and cognitive senders, respectively. They employed a combination of GP and superposition coding [9] techniques, without resorting to rate-splitting, to come up with an achievable rate region for the two-user CR channel. In [10], an achievable rate region for the two-user interference channel with degraded message sets was derived using a combination of superposition and GP coding techniques, where only the CR transmitter employs rate-splitting. In [11], Jovičić et al presented the Gaussian CR channel and derived capacity bounds/results for low and high interference regimes by employing dirty paper coding [12], and joint code design at the two transmitters and multiuser decoding at the primary receiver.

Other prominent information theoretic results in the area of CR networks are as follows. Capacity bounds for two-user interference channels with cognitive and partially cognitive transmitters were reported in [13] - [18]. In [19] - [23], information theoretic results for interference channels with common information were derived. The sum-capacity of the Gaussian MIMO cognitive radio network was presented in [24], where the results applied to the single-antenna CR channel as well. Capacity scaling laws for CR networks were presented in [25], while [26] considered achievable rates when the encoder non-causally knows different channel states. Multiple access channels with cooperation have been considered in [27] - [29]. Furthermore, the algebraic structure of random binning schemes of [7] and [12] have been studied in [30] - [33], paving the way for practical realization of channel codes for CR networks.

Our contribution: With increasing interest in CR technology, one is motivated to consider a network of CRs sharing the same channel with an incumbent primary user. In particular, how do the primary user and the network of CRs cooperate assuming the overlay network paradigm? In this paper, we consider the case of three-user CR interference channels, where two (or one) CRs and one (or two) primary user communicate with three respective receivers. We consider three message sharing mechanisms between the senders, which are extensions of the two-user unidirectional message sharing paradigm to the three-user case. We term these three approaches (i) cumulative message sharing (CuMS); (ii) primary-only message sharing (PrMS); and (iii) cognitive-only message sharing (CoMS). To deal with interference in this three-user channel, we use rate-splitting, which was first reported in [6], to enlarge the achievable rate region for the classical two-user interference channel. The main idea behind rate-splitting is to encode part of the message at a possibly low rate, so that the unintended receiver can decode the interference caused to it by performing simultaneous decoding. To this end, we define five cognitive channel models, two each for CuMS and PrMS, and one for CoMS, with different rate-splitting strategies. The types of message-sharing mechanisms and rate-splitting strategies will be made precise in the next section. We then employ the standard technique of combining GP’s binning principle [7] and superposition coding [9] to derive an achievable rate region for each of the five channels. As a result, we illustrate the generality of the techniques employed here, and provide useful insights into the rate regions and their characterization. Next, we specialize the achievable rate regions to the Gaussian channel; this enables comparisons of the different rate regions both analytically and through simulations. We also present simple corollaries that help enlarge the rate regions in the Gaussian case. Finally, we compare our achievable regions to some outer bounds, and thereby provide some insight into the optimality of the proposed coding scheme. Initial results of this work have appeared in [34] - [36].

The outline of the paper is as follows. In Section II, we introduce the discrete memoryless channel models for CuMS, PrMS and CoMS, and review the notation used in the paper. We also present the probability distribution functions characterizing these channels. In Section III, we present the achievability theorem for the channel models and work out the details of the proof for one of the channel models. In Section IV, we consider the Gaussian channel model and construct the framework for numerical evaluation. We also state corollaries that enlarge the rate regions in the Gaussian case and derive some outer bounds. Simulation results and related discussions are presented in Section V. We conclude the paper in Section VI. The achievable rate region equations for the five discrete memoryless channels considered in this paper, the proof of the achievability theorem for one channel model and proofs of corollaries are relegated to the Appendix.

## Ii Discrete Memoryless Channel Model and Preliminaries

The three-user discrete memoryless cognitive interference channel is described by . For ,

1. the senders and receivers are denoted by and , respectively;

2. finite sets and denote the channel input and output alphabets, respectively;

3. random variables and are the inputs and outputs of the channel respectively; and

4. denotes the finite set of conditional probabilities , when are transmitted and are obtained by the receivers.

The channels are assumed to be memoryless. In the classical three-user interference channel, the messages at the senders are given by ; being a finite set with elements. The messages are assumed to be independently generated.

### Ii-a Message-Sharing Mechanisms

We describe now the message-sharing mechanisms considered in this paper.

1. In the case of cumulative message-sharing (CuMS), sender has noncausal knowledge of the message and the corresponding codewords of the primary sender, . Sender has noncausal knowledge of the message of the primary transmitter as well as the message of , and their respective codewords. A schematic of CuMS is shown in Fig. 1.

2. In the case of primary-only message-sharing (PrMS), senders and have noncausal knowledge of the message and the corresponding codewords of the primary sender, . There is no message-sharing mechanism between and themselves. See Fig. 2 for a channel schematic.

3. In the case of cognitive-only message-sharing (CoMS), sender has noncausal knowledge of messages and , and the corresponding codewords of senders, and . There is no message-sharing mechanism between the and . A channel schematic for CoMS is shown in Fig. 3.

An code exists for these channels, if there exists the following encoding functions:

 f1: M1↦Xn1, f′1: M1↦Xn1, f′′1: M1↦Xn1 f2: M1×M2↦Xn2, f′2: M1×M2↦Xn2 f′′2: M2↦Xn2 f3: M1×M2×M3↦Xn3, f′3: M1×M3↦Xn3, f′′3: M1×M2×M3↦Xn3

and the following decoding functions, for :

 gk: Ynk↦Mk, g′k: Ynk↦Mk, g′′k: Ynk↦Mk,

such that the decoding error probability . is the average probability of decoding error computed using:

(or ) correspond to the encoders (or decoders) used by channels with CuMS, (or ) correspond to the encoders (or decoders) used by channels with PrMS and (or ) correspond to the encoders (or decoders) used by channels with CoMS.

We define two channels denoted , two channels denoted and one channel denoted ; . A non-negative rate triple is achievable for each of the channels, if for any there exists a code such that as . The capacity region for the channels is the closure of the set of all achievable rate triples . A subset of the capacity region gives an achievable rate region.

### Ii-B Rate-Splitting Strategies

In [6], it has been shown that achievable rate region for the classical two-user interference channel can be enlarged by rate-splitting. Specifically, each transmitter encodes part of the message at a possibly low rate and constructs its codewords using superposition coding. This results in the unintended or non-pairing receiver being able to decode and cancel out the low rate222In the literature, this is typically called the “public part” of the message. Its rate could be large if the cross-channel gains are large. sub-message from the interfering transmitter using simultaneous decoding, thereby enlarging the achievable rate region. This forms the motivation for employing rate-splitting, as an effective interference management mechanism. In the three-user scenario, however, many more rate-splitting strategies exist compared to the two-user case. For example, sender can perform rate-splitting in one of the following four ways: (i) it can encode a part of its message such that both unintended receivers, and , can decode the sub-message; (ii) encode a part of the message such that can decode it but not ; (iii) encode a part of the message such that can decode it but not ; and finally, (iv) encode in a manner such that the sub-message is not decodable at either or (i.e., decodable only at the ). In this paper, we consider the following rate-splitting strategies:

1. In and , the senders encode part of their respective messages at a rate such that it can be reliably decoded by all the receivers. The other part of the message will be encoded at a rate such that only the intended or pairing receiver can decode it.

2. In and , one part of the message is encoded such that only the intended receiver can decode it, while the other part is encoded at a rate such that it can only be decoded at the intended reciever and the receiver .

3. In , sender encodes one part of the message at a rate such that all receivers can decode it, while the other part is encoded at a rate such that it can only be decoded at its pairing receiver, . There is no rate-splitting at and .

Note that, regardless of the manner in which rate-splitting is performed, should always be able to reliably decode the codewords from , .

The notation for describing the achievable rates of these sub-messages and their respective description is tabulated in Table I. The decoding capabilities of receivers, resulting from rate-splitting at the transmitters, are summarized in Tables II, III and IV. We also introduce auxiliary random variables defined on finite sets and tabulate them in Table V. Depending on the rate-splitting strategy employed by the senders, only a subset of these sub-messages, their corresponding rates, and the corresponding auxiliary random variables will be used to derive an achievable rate region for each channel model. Note that we do not consider the practical aspects of the underlying physical realization of such models. Also, the capacity region for a general CR channel still remains an open problem.

### Ii-C Channel Modification

Rate-splitting necessitates modification of the channels , and . Here, we explicitly show the modification for one channel (); the modification for the other channel models is similar. Referring to the rate-splitting strategy for the channel , the messages at the three senders in the modified channel can be written as:

Sender 1: ,

Sender 2: , ,

Sender 3: , ,

with all messages being defined on sets with finite number of elements. Note that, there is no rate-splitting at sender , but for consistency in notation we write as .

We define an code for the modified channel as a set of codewords for , codewords for , and codewords for , such that the average probability of decoding error is less than . We call a tuple achievable if there exists a sequence of codes such that as . Here, corresponds to . The capacity region for the modified channel is the closure of the set of all achievable rate tuples . It can be shown that if the rate tuple is achievable for the modified channel, then the rate triple is achievable for the channel (see [6, Corollary 2.1]). In a similar fashion, the remaining channel models can be appropriately modified; the details are omitted to avoid repetition.

### Ii-D Probability Distributions

Here, we present the probability distribution functions which characterize the channels , , , and . Let denote the set of all joint probability distributions respectively, that factor as follows:

 p1CuMS(q,w0,w1,x1,u0,u2,x2,v0,v3,x3,y1,y2,y3)= p(q)p(w0,w1,x1|q)p(u0|w0,w1,q)p(u2|w0,w1,q)p(x2|u0,u2,w0,w1,q)p(v0|u0,u2,w0,w1,q) p(v3|u0,u2,w0,w1,q)p(x3|v0,v3,u0,u2,w0,w1,q)p(y1,y2,y3|x1,x2,x3), (1) p2CuMS(q,w,x1,u1,u2,x2,v1,v3,x3,y1,y2,y3)= p(q)p(w,x1|q)p(u1|w,q)p(u2|w,q)p(x2|u1,u2,w,q)p(v1|u1,u2,w,q)p(v3|u1,u2,w,q) p(x3|v1,v3,u1,u2,w,q)p(y1,y2,y3|x1,x2,x3). (2)

Let denote the set of all joint probability distributions respectively, that factor as follows:

 p1PrMS(q,w0,w1,x1,u0,u2,x2,v0,v3,x3,y1,y2,y3)= p(q)p(w0,w1,x1|q)p(u0|w0,w1,q)p(u2|w0,w1,q) p(x2|u0,u2,w0,w1,q)p(v0|w0,w1,q)p(v3|w0,w1,q)p(x3|v0,v3,w0,w1,q)p(y1,y2,y3|x1,x2,x3), (3) p2PrMS(q,w,x1,u1,u2,x2,v1,v3,x3,y1,y2,y3)= p(q)p(w,x1|q)p(u1|w,q)p(u2|w,q) p(x2|u1,u2,w,q)p(v1|w,q)p(v3|w,q)p(x3|v1,v3,w,q)p(y1,y2,y3|x1,x2,x3). (4)

Let denote the set of all joint probability distributions respectively, that factor as follows:

 pCoMS(q,w1,x1,u2,x2,v0,v3,x3,y1,y2,y3)=p(q)p(w1,x1|q)p(u2,x2|q)p(v0|w1,u2,q)p(v3|w1,u2,q) p(x3|v0,v3,u2,w1,q)p(y1,y2,y3|x1,x2,x3). (5)

The lower case letters ( etc.) are realizations of their corresponding random variables, and note that for notational simplicity, the same letter () is used to denote all the different probability distributions above. An achievable rate region for each channel is defined by a set of non-negative real numbers (referred to as rate tuples) that satisfy certain information-theoretic inequalities. An achievable rate region for each of the channels considered in this paper are given in Appendices A, B and C.

## Iii Achievability Theorem and Proof

###### Theorem III.1

Let denote the capacity region of the channel . Let

 RtCuMS=⋃ptCuMS% (.)∈PtCuMSRCuMS% (ptCuMS),RtPrMS=⋃ptPrMS(.)∈PtPrMSRPrMS(ptPrMS) and RCoMS=⋃pCoMS(.)∈PCoMSRCoMS(pCoMS).

In the above, denotes a set of achievable rates when the channel is characterized by the joint probability distribution function , and similar definitions apply for the other notations used. The region is an achievable rate region for the channel , i.e., or or

• We employ the standard technique of combining GP’s binning principle [7] and superposition coding [9] to prove the coding theorem and derive a set of achievable rates for each of the channel models. We show the proof for the channels and . The proof for the remaining three channels ( , and ) are along similar lines and are omitted.

Proof of achievability for the channel : The proof is presented in four parts, namely, codebook generation, encoding, decoding and analysis of probabilities of decoding errors at the three receivers. We start with the codebook generation scheme.

### Iii-a Codebook Generation

Let us fix . Generate a random time sharing codeword q, of length , according to the distribution . Generate independent codewords , according to . For every , generate one codeword according to .

For , generate independent codewords , according to . For every codeword triple , generate one codeword according to
. Uniformly distribute the codewords into bins indexed by such that each bin contains codewords.

For , generate independent codewords , according to . For every codeword quadruple , generate one codeword according to . Distribute codewords uniformly into bins indexed by such that each bin contains codewords. The indices are given by , and .

### Iii-B Encoding & Transmission

Let be a typical set. We will be using the notation to describe a typical set over many different random variables, but the definition will be clear from the context.

Let us suppose that the source message vector generated at the three senders is . At the encoders, the first component is treated as the message index and the last four components are treated as the bin indices. looks for a codeword in bin and a codeword in bin such that and , respectively. looks for a codeword in bin and a codeword in bin such that and , respectively. , and then transmit codewords , and , respectively, through channel uses. The transmissions are assumed to be synchronized.

### Iii-C Decoding

Recall that in , the primary receiver can decode the public parts of the non-pairing sender’s messages, while the secondary receivers can only decode the messages from their pairing transmitters. The three receivers accumulate an -length channel output sequence: at , at and at . Decoders 1, 2 and 3 look for all indices , and , respectively, such that , and . If in all the index triples found are the same, declares , for some and . If in all the index pairs found are indices of codewords from the same bin with index , and in all the index pairs found are indices of codewords from the same bin with index , then determines . Similarly, if in all the index pairs found are indices of codewords from the same bin with index , and in all the index pairs found are indices of codewords from the same bin with index , then determines . Otherwise, the receivers , and declare an error.

### Iii-D Analysis of the Probabilities of Error

In this section we derive upperbounds on the probabilities of error events which could happen during encoding and decoding processes. We assume that a source message vector is encoded and transmitted. We consider the analysis of the probability of encoding error at senders and , and the analysis of the probability of decoding error at each of the three receivers , , and separately.

First, let us define the following events:
,
,
,
,
,
,
.
complement of the event . Events will be used in the analysis of probability of encoding error while events will be used in the analysis of probability of decoding error.

#### Iii-D1 Probability of Error at the Encoder of S2

An error is made if the encoder cannot find a in the bin indexed by such that or it cannot find a in the bin indexed by such that . The probability of encoding error at can be bounded as

 Pe,S2≤P⎛⎜⎝⋂U1(l1)∈bin(k1)(W(j),U1(l1),q)∉A(n)ϵ⎞⎟⎠+P⎛⎜⎝⋂U2(l2)∈bin(k2)(W(j),U2(l2),q)∉A(n)ϵ⎞⎟⎠, ≤(1−P(Ejl1))2n(I(W;U1|Q)+4ϵ)+(1−P(Ejl2))2n(I(W;U2|Q)+4ϵ),

where is the probability of an event. Since q is predetermined, and w and are independent given q,

 P(Ejl1)=∑(w,u1,q)∈A(n)ϵP(W(j)=w|q)P(% U1(l1)=u1|q) ≥2n(H(W,U1|Q)−ϵ)2−n(H(W|Q)+ϵ)2−n(H(U1|Q)+ϵ)=2−n(I(W;U1|Q)+3ϵ).

Similarly, . Therefore,

 Pe,S2≤(1−2−n(I(W;U1|Q)+3ϵ))2n(I(W;U1|Q)+4ϵ)+(1−2−n(I(W;U2|Q)+3ϵ))2n(I(W;U2|Q)+4ϵ).

Now,

 (1−2−n(I(W;U1|Q)+3ϵ))2n(I(W;U1|Q)+4ϵ)=e2n(I(W;U1|Q)+4ϵ)ln(1−2−n(I(W;U1|Q)+3ϵ)) ≤e2n(I(W;U1|Q)+4ϵ)(−2−n(I(W;U1|Q)+3ϵ)) =e−2nϵ.

Clearly, as .

#### Iii-D2 Probability of Error at the Encoder of S3

An error is made if the encoder cannot find a in the bin indexed by such that or it cannot find a in the bin indexed by such that . The probability of encoding error at can be bounded as

 Pe,S3≤P⎛⎜⎝⋂V1(t1)∈bin(r1)(W(j),U1(l1),U2(l2),V1(t1),q)∉A(n)ϵ⎞⎟⎠ +P⎛⎜⎝⋂V3(t3)∈bin(r3)(W(j),U1(l1),U2(l2),V3(t3),q)∉A(n)ϵ⎞⎟⎠ ≤(1−P(Ejl1l2t1))2n(I(W,U1,U2;V1|Q)+4ϵ)+(1−P(Ejl1l2t3))2n(I(W,U1,U2;V3|Q)+4ϵ).

Since q is predetermined, we have,