List of Figures. Acknowledgements. Abstract 1. 1 Introduction 2. 2 Preliminaries Superposition Coding Block Markov Encoding...

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2 Contents Contents i List of Figures iv Acknowledgements vi Abstract 1 1 Introduction 2 2 Preliminaries Superposition Coding Block Markov Encoding Two-Way channel Auxiliary random variables Markov Chains Main Results Notation Channel Models Full-duplex Channel Half-duplex Channel Definitions i

3 3.4 System Model Inner Bound Outer Bound Inner Bound Code generation Encoding Scheme Decoding Scheme Probability of Error Half-duplex Channel Channel Encoding scheme Decoding scheme Probability of Error Outer Bound Outer Bound for R Outer Bound for R 1 and R Outer Bound for R 1 + R Special cases Gaussian case Achievable Rate Region Symmetric Case Assymmetric Case Half-duplex MAC with feedback Reduction to Carleial rate region Reduction to Cover-Leung rate region ii

4 7 Conclusions 58 Bibliography 60 Index 64

5 List of Figures 1.1 Three user hidden node topology (a) Date in uplink and only feedback in the downlink and (b) No feedback (inner bound), genie-aided perfect feedback (outer bound) and MAC with finite feedback (two inner regions) with different amount of feedback power TDMA: Users orthogonally separated in time FDMA: Users orthogonally separated in frequency Superposition coding: Codeword occupies the whole bandwidth and time Markov chain formed by the channel blocks Shannon s two-way channel Half-duplex two-way channel Feedback based coding in multiuser two-way channels. For clarity of representation, the broadcast channel is p(y 1, y 2 x) = p(y 1 x)p(y 2 x) Half-duplex two-way network: two states in the system Markov chain model of the multiuser feedback scenario (a) Uplink and (b) Downlink for the half-duplex Gaussian channel Power distribution for Gaussian MAC-MC with half-duplex links Rate region without feedback, R nf iv

6 6.4 The 3-d rate region for the MAC + BC half-duplex two-way channel MAC sum-rate versus BC common rate with and without feedback. User powers are P 1 = P 2 = 50, and gray node power P = 50, 100, N = 1 and N 1 = N 2 = AWGN with pathloss: Near-far situation Percentage gain in the rates of the strong user (a) and weak user (b) Behaviour of Gaussian capacity at high and low SNRs Gaussian case with asymmetric power constraints No feedback (inner bound), genie-aided perfect feedback (outer bound) and MAC with finite feedback (two inner regions) with different amount of feedback power

7 Acknowledgements

8 ABSTRACT Bounds and Protocols for a Two-way Multiple Node Channel by Debashis Dash Feedback is known to enlarge the capacity region of multiuser channels. However, none of the analysis accounts for all the feedback resources. We show that feedback protocols lose the rate gains when analyzed with resource accounting. Also, in typical multiuser systems without feedback, uplink and downlink code designs are usually decoupled and independent of each other. By jointly designing the uplink and the downlink system we show that feedback is able to get back the rate gains partially, but only if the traffic is two-way. We find inner and outer bounds for a three node system and show that our scheme can be used for user cooperation in hidden node topologies. In near-far situations, the weak user gains appreciably from such cooperation. We also show that the well known inner bounds for multiuser channels with feedback can be derived as special cases of our generalized inner bound.

9 Chapter 1 Introduction The information theory of channels with feedback poses challenging problems which often have unexpected solutions G. Kramer, Ph.D. Thesis Feedback is known to improve the capacity of multiuser channels and in some cases reduce the computational complexity. It is interesting to notice that most of the practical feedback protocols are based on genie-aided theoretical analysis, which either have a noiseless feedback link or some other practically difficult assumption like full-duplex radios [6, 12, 10, 4, 32, 23]. Hence, it becomes necessary to put practical constraints and see if the existing feedback protocols still give us gains. We show in this thesis, that such a practical analysis does indeed give a somewhat unexpected solution. User 1 User 3 User 2 Figure 1.1: Three user hidden node topology We focus on a three-node network shown in Figure 1.1 and show that if feedback has to share resources with forward link in conventional multiple access, the receiver 2

10 CHAPTER 1. INTRODUCTION 3 (user 3) has to use an exponentially large power to offset the time loss. In essence, feedback is practically useless to implement in half-duplex networks, especially when there is no downlink data. To get an idea, consider Figure 1.2 which indicates that in order to do better than the well known MAC rate region, even if the downlink user employs enormous power it cannot even get close to the promised noiseless upper bound (more details in Chapter 6). But we claim that this is merely because we are not using the downlink channel optimally by reserving it solely for feedback. In this thesis, we show that the correct way to analyse practical feedback scheme is by using a two-way channel model. P P Data Data P f Feedback only (a) (b) Figure 1.2: (a) Date in uplink and only feedback in the downlink and (b) No feedback (inner bound), genie-aided perfect feedback (outer bound) and MAC with finite feedback (two inner regions) with different amount of feedback power. Two-way channels, first proposed and studied by Shannon in 1961 [29], is a natural model for many communication scenarios in which all parties may have data for each other. Shannon proposed both inner and outer bounds for the two-node two-way channel. Further, he also provided cases where the bounds match (deterministic channels like modulo-2 adding channel and non-deterministic channels like binary erasurenoiseless two-way channel and push-to-talk channel), and where they do not match (e.g. the well-studied example of binary multiplying channel [25, 26, 31, 21, 27, 28]).

11 CHAPTER 1. INTRODUCTION 4 Though comparatively less widely studied than the one-way channel model [30], both the inner and outer bounds were improved subsequently [15, 33, 16]. An interesting point to note is that while feedback is not needed to achieve the capacity of two-way Gaussian channel, we show that it is beneficial in multiple node networks like that in Figure 3.1. The gain can be attributed to cooperative beamforming by the two nodes in uplink communication. Perhaps due to limited attention given to two-way channels, its multi-node extension has received even less attention. Recent work on two-way networks can be attributed to [16, 19], where the authors study both inner and outer bounds for arbitrary network topologies with two-way networks as a special case. The elegant formulation using code trees [16] clearly demonstrates the complexity of incorporating feedback information in code construction for generalized networks. However, the inner and outer bounds derived in [16] are not computable in many cases of interest. Our key contribution is an inner bound giving a larger rate region for the multiuser two-way channel. The inner bound is derived for both full-duplex and half-duplex channels. Our proposed coding strategy is inspired by the two-way coding scheme of Han [15] and generalizes our recent results in [9]. In fact, we show that the achievable rate region presented in [8] is a special case of the new coding structure. The topology considered in this paper is inspired by two applications. First, it resembles the hidden node topology commonly considered in the design of contention resolution protocols, where the two senders do not have a channel between them. Due to the lack of a direct cooperative link and hence cooperative coding is not feasible. However, since the receivers are also senders, the feedback based on the received signals can be used to induce cooperation between uplink senders (Users 1 and 2). To some extent, the above conclusion was also independently derived for multiple access channels with noisy feedback [14]. Our results show that that conclusion holds for both full- and half-duplex networks, with and without downlink data (rate R 3 ).

12 CHAPTER 1. INTRODUCTION 5 Second, for cellular networks, the analysis shows that there is a sum-rate gain in jointly designing uplink and downlink communication. While all networks have decoupled uplink and downlink systems, a joint design can increase the system data rates. We note that the usual min-cut max-flow bound [5, Chapter 14] does not apply to two-way networks. An extension of the bound for full-duplex two-way networks was considered in [18]. An improved outer bound was found for the conventional AWGN multiple access channel (where the third link does not have any data of its own to send) in [14]. In this thesis, we derive the outer bound from first principles, which demonstrates the challenge of deriving such an outer bound. The two-way channel implicitly allows sending feedback about the signal received in one mode (say MAC) while transmitting in the second mode (say BC). Thus it leads to an infinite Markov chain (as shown in Figure 5.1). Our proof uses the techniques proposed in [33], where the authors studied the conventional two-node two-way network. As proposed by Shannon [29], the outer bound typically uses an arbitrary input distribution. On the other hand, the inner bounds are obtained with independent inputs. Following the line of reasoning of [29, 33], our proposed outer bounds too use non-independent inputs to reflect how the feedback can induce correlation in the inputs. In a typical cellular multiuser network, it is common to have data both in the uplink and downlink. For such scenarios, it is not clear if it is better to use disjoint design criteria for the uplink and downlink or design them together. While there has been a lot of work on individual design of multiple-access channel [20, 1, 12, 7, 23, 4, 32, 17] and broadcast channels [6, 13, 2, 11, 10, 22], there is no work on joint encoding of the two channels. But our analysis of feedback protocols using two-way channel models also answer this additional question that for half duplex channels having data in both the directions of the two-way link, it is better to design the uplink and downlink jointly.

13 CHAPTER 1. INTRODUCTION 6 The rest of the thesis is organized as follows. In Chapter 2, we shall discuss some basic principles which will be used very often in the later chapters. In Chapter 3 we shall discuss the channel model, assumptions, nomenclature and give the main results of the work. Chapter 4 contains the proof of the inner bound, where we derive special encoding and decoding functions to find the half-duplex special case of the inner bound. Chapter 5 contains the proof of the outer bound and Chapter 6 has some interesting gaussian half-duplex cases. Then we consider a pathloss model which simulates a near-far situation and show that such cooperation through feedback is more useful for the weak user. For the case when both the uplink transmitters are equidistant from the receiver, we consider cases when they have symmetric or asymmetric power constraints. In the last part of this chapter we show that some of the previous known results are special cases of our rate region in Chapter 4. We conclude our findings in Chapter 7.

14 Chapter 2 Preliminaries In this chapter we review some basic information theoretic ideas, which will be used throughout the thesis to derive our inner and outer bounds. We start with a common encoding method, known as superposition coding, used in multiuser communication systems. 2.1 Superposition Coding The simplest way to separate users (or messages) when the resources are constrained, is to do orthogonal allocation. Users (or messages) can be divided in time (TDMA) or in frequency (FDMA) or both. As shown in Figure 2.1, the messages W 1 and W 2 are independently mapped into channel input X 1 and X 2 respectively and they are sent over the channel in orthogonal time bands. Similar is the situation for FDMA as shown in Figure 2.2. Due to its orthogonal nature, the decoding scheme is very simple. However, this is not an optimal scheme if one can allow a more complicated multiuser decoding scheme. Superposition coding is known to perform better than time and frequency (orthogonal) division of resources. As shown in Figure 2.3, now the channel input X is a function of both W 1 and W 2, where the cloud centers denote the message from the first user and the around each cloud center the alphabet points 7

15 CHAPTER 2. PRELIMINARIES 8 code for the message from the second user, forming a larger code X, which now can take up the whole time bandwidth space on the channel. But due to its structure it is more complicated to decode it than any orthogonal scheme. W 2 X 2 (W 2 ) B f X 1 X 2 W 1 X 1 (W 1 ) t T Figure 2.1: TDMA: Users orthogonally separated in time W 2 X 2 (W 2 ) B X 2 f W 1 X 1 (W 1 ) X 1 t T Figure 2.2: FDMA: Users orthogonally separated in frequency Superposition coding was first used in [6] and was shown to be optimal and perform better than any orthogonal scheme [3]. The idea behind such a coding scheme is to use the channel simultaneously rather than dividing it between users or messages orthogonally. It was introduced in the context of the broadcast channel, so the authors also called it cooperative broadcasting. For Gaussian channels, the Gaussian codewords can just be added with a proper power scaling. For a more general

16 CHAPTER 2. PRELIMINARIES 9 channel, first typical sequences are selected for the input message layers and then a typical channel sequence is selected which is jointly typical with all the message layer sequences. Decoding of a superposition code is more complex than an equivalent orthogonal code. The best way is to first decode the message with the best SINR (considering all other messages as noise) and then subtract (for gaussian channels) the decoded message. Then the next message is decoded following the same procedure till the last message has been decoded. (W 1, W 2 ) X(W 1, W 2 ) (W 2 w 11 ) X(w 11, W 2 ) B W 1.. f X(W 1, W 2 ) (W 2 w 1k ) X(w 1k, W 2 ) t T Figure 2.3: Superposition coding: Codeword occupies the whole bandwidth and time 2.2 Block Markov Encoding The idea of Block Markov superposition codes was introduced in [7]. Instead of superimposing multiple user s data, the authors superimposed messages (independent and incremental) from different blocks and used noiseless feedback to attain a larger rate region for a multiple access channel. Superposition is used to encode messages (of more than one block) at different layers which can be decoded once enough information is acquired about that layer.

17 CHAPTER 2. PRELIMINARIES 10 N channel uses B 1 B 2 B 3... X 1 X 2 X 3... Figure 2.4: Markov chain formed by the channel blocks These layers give rise to different auxiliary random variables which increase the resolution at which we can manage the rate division between the different users depending on the noise variance at the receiver but it increases the complexity of the decoder. As the coding is done over blocks as a whole, as compared to some schemes which take adaptively modify each channel symbol, the blocks form a markov chain as shown in Figure 2.4. The block structure of the coding scheme enables to take off the part of the message that could be decoded after each block, which can reduce the amount of data to be sent as feedback. 2.3 Two-Way channel The two-way channel was first introduced by Shannon in 1961 [29]. A two-way channel model utilizes the previous channel outputs in addition to the message to form the channel input. As shown in Figure 2.5, the channel inputs also depend on the channel outputs for the same user in addition to the message. Such a channel model can be easily extended for feedback schemes. The two-way channel initially introduced by Shannon and those usually considered in the literature are assumed to be full-duplex, in the sense that the transmitters and recievers can send and recieve at the same time. In our system model, in order to put more practical constraints, we consider only half-duplex transmitters and recievers.

18 CHAPTER 2. PRELIMINARIES 11 W 1 ENC X 1 Y 2 DEC W 1 CHANNEL p(y 1,y 2 x 1,x 2 ) W 2 DEC Y 1 X 2 ENC W 2 Figure 2.5: Shannon s two-way channel We call our channel model a two-way model due to the fact that it utilizes a function of the channel output (which separates the feedback part of the message) and the message to form the channel input. X Y 1 2 W ENC 1 η DEC W 1 CHANNEL Y 1 p(y 2 x 1 ) X 2 X 1 1-η Y 2 CHANNEL W 2 DEC p(y 1 x 2 ) Y 1 X 2 ENC W 2 Figure 2.6: Half-duplex two-way channel 2.4 Auxiliary random variables The usual way of writing rate regions is done in terms of the channel input and output random variables. But superposition schemes typically, gives rise to random variables other than channel inputs and outputs due to the layered code structure. These additional random variables are called auxiliary variables. The auxiliary random variables help in encoding the messages into the layered code to form the channel inputs. One of the major concerns of any superposition code, to keep the rate region computable is the cardinality of such auxiliary variables [24].

19 CHAPTER 2. PRELIMINARIES Markov Chains Definition 2.1 Three random variables X, Y and Z form a markov chain (written as X Y Z) if the joint density function has the following form: p(x, y, z) = p(x)p(y x)p(z y) Some well known information theoretic properties of markov chains that are used to prove outer bounds are: Conditioning on functions of random variables H(A B, C) = H(A B, C,D) for D = f(b, C) Typical use: H(Y n W,Y n 1 ) = H(Y n W,Y n 1, X n ) for X n = f(w,y n 1 ). Conditioning reduces entropy H(A B) < H(A) Typical use: H(Y n W,X n, Y n 1 ) < H(Y n X n, Y n 1 ). Markov Chain property H(A B, C) = H(A B) for a markov chain C B A Typical use: H(Y n W,X 1n, X 2n, Y n 1 ) = H(Y n X 1n, X 2n, Y n 1 ). Memoryless channel property H(Y n X n, Y n 1 ) = H(Y n X n ) if p(y N x N ) = i p(y i x i ). Another example: H(Y n X n 1, X n 2, Y n 1 ) = H(Y n X 1n, X 2n ) Data processing theorem The most common way to write the data processing theorem is I(X; Y ) I(X; Z). While finding outer bounds, the following form: H(Z X) H(Z Y ) is also useful.

20 Chapter 3 Main Results Before mentioning our main results, we define our notation, the channel model we follow and the performance criteria. 3.1 Notation We define the following notation for representing all the random variables. Throughout this paper, random variables are represented by capitals, (e.g. U 1, X 1, W 1, etc.), their alphabet sets by the corresponding script symbols, (e.g. U 1, X 1, W 1, etc.), and any particular instance of those random variables by small case symbols, (e.g. u 1, x 1, w 1, etc.). In general, A b u,c will denote random variable A for the c th channel use of block b of user u. Boldface variables will be used to denote vectors. For example, X 1 = (X 1 1,X 2 1,...,X B 1 ) is the vector input to the channel for user 1 for B blocks. Each of the block inputs is a vector (n-tuple) itself (i.e. X i 1 = (X1,1, i X1,2, i...,x1,n)). i 3.2 Channel Models Now we describe the channel models considered in this work. We will consider two types of channels depending on certain assumptions on the transmitter-receiver ca- 13

21 CHAPTER 3. MAIN RESULTS 14 pabilities. If the transmitters and receivers can send and receive at the same time, they are said to be in a full-duplex mode. Usually, by operating in a full-duplex mode, we get a gain in the rate region. This gain is similar to what we had seen in superposition coding as compared to TDMA or FDMA in Chapter 2. On the other hand, if the transmitters and receivers can only send or receive and not do both at the same time, they are said to be in a half-duplex mode. The formal mathematical definitions of these channels are given below: Full-duplex Channel There will be three variables associated with each terminal, w i to represent the message, x i to represent the channel input and y i to represent the channel output at terminal i. Then the two-way network is described by the channel probability distribution p(y 1, y 2, y 3 x 1, x 2, x 3 ). In many cases of interest, the two-way channel can be decomposed such that p(y 1, y 2, y 3 x 1, x 2, x 3 ) = p(y 3 x 1, x 2, x 3 )p(y 1, y 2 x 1, x 2, x 3 ). In the case of Gaussian channels, the channel input-output relation can be described as follows: y 1 = x 1 + x 2 + x 3 + n 1, (3.1) y 1 = x 1 + x 2 + x 3 + n 2, (3.2) y 3 = x 1 + x 2 + x 3 + n 3, (3.3) where n 1, n 2 and n 3 are independent zero mean gaussians with variances N 1, N 2 and N 3 respectively. We further assume that each source operates under an average power constraint of P i, i = 1, 2, 3. Note that the inputs x 2 and x 1 do not appears in outputs

22 CHAPTER 3. MAIN RESULTS 15 y 1 and y 2, since we have assumed that there is no direct channel between Users 1 and Half-duplex Channel Since the users are assumed to be half-duplex, the network time-shares between two possible states: multiple access channel (MAC) mode for η fraction of time and broadcast channel (BC) mode for (1 η) fraction of time, where 0 η 1 denotes the time-sharing variable. In this case, the channel probability law can be written as p(y 3 x 1, x 2 ) p(y 1, y 2, y 3 x 1, x 2, x 3 ) = p(y 1, y 2 x 3 ) MAC BC. (3.4) In the case of Gaussian channels, the input-output relation is given by y 1 = x 3 + n 1, (3.5) y 2 = x 3 + n 2, (3.6) y 3 = x 1 + x 2 + n 3, (3.7) where n 1, n 2 and n 3 are independent zero mean gaussians with variances N 1, N 2 and N 3 respectively. We further assume that each source operates under an average power constraint of P i, i = 1, 2, 3 in their respective transmission modes. 3.3 Definitions We define the alphabet sets from which the input messages m 1, m 2, m 3 are chosen (m 1 M 1, m 2 M 2, m 3 M 3 ) as, M 1 = {1, 2,...,M 1 }, M 2 = {1, 2,...,M 2 }, M 3 = {1, 2,...,M 3 }. P e1 and P e2 are the error probabilities of decoding error at user 3 de-

23 CHAPTER 3. MAIN RESULTS 16 fined by, P e1 = P e2 = 1 M 1 M 2 M 3 1 M 1 M 2 M 3 M 1 M 2 M 3 m 1 =1 m 2 =1 m 3 =1 M 1 M 2 M 3 m 1 =1 m 2 =1 m 3 =1 Pr( ˆm 1 m 1 m 1, m 2, m 3 was sent) Pr( ˆm 2 m 2 m 1, m 2, m 3 was sent) Similarly, P 1 e3 and P 1 e3 are the error probabilities of decoding error at User 1 and User 2 (in decoding what User 3 sent to both of them) defined by P 1 e3 = P 2 e3 = 1 M 1 M 2 M 3 1 M 1 M 2 M 3 M 1 M 2 M 3 m 1 =1 m 2 =1 m 3 =1 M 1 M 2 M 3 m 1 =1 m 2 =1 m 3 =1 Pr( ˆm 1 3 m 1 m 1, m 2, m 3 was sent) Pr( ˆm 2 3 m 2 m 1, m 2, m 3 was sent) Definition 3.1 (Achievable Region) The rate tuple (R 1, R 2, R 3 ) is said to be achievable if there exists a (M 1, M 2, M 3, n) code with, R 1 log M 1 n, R 2 log M 2, R 3 log M 3 n n such that, P e1 0, P e2 0, P 1 e3 0 and P 2 e3 0, as n. Definition 3.2 (Capacity Region) The closure of all achievable rate tuples is called the capacity region of the two-way multiple access channel and denoted by C.

24 CHAPTER 3. MAIN RESULTS System Model Now we define our sytem model in greater detail. Consider the simplest multiuser case with three users as shown in Figure 3.1. Users have data for each other; in our case, User 1 and 2 have messages for User 3 and User 3 has a common message for Users 1 and 2. M 3 Y 1 M 1 X 1 ^ m M 1 i j Y 1 p(y 1 X) M 2 ^ m k l X 1 (i,j,i,k ) X (k,l,i,k ) 2 Y 2 p(y X 1,X 2 ) p(y 2 X) Y 3 X (m,i,k ) 3 X (m,i,k ) 3 i,k m ^^ ^^ (j, l, i, k ) (M ) 3 Figure 3.1: Feedback based coding in multiuser two-way channels. For clarity of representation, the broadcast channel is p(y 1, y 2 x) = p(y 1 x)p(y 2 x). Figure 3.1 shows the coding structure at each user. Since each user has data to send to other users, each user has an encoder to encode their transmissions and decode the received transmissions. In decoupled systems, the encoder and decoder are decoupled. However, in our proposed scheme, the input to decoder is also an input to encoder and vice versa, thereby implicitly allowing feedback from receivers to transmitters.

25 CHAPTER 3. MAIN RESULTS 18 The coding scheme can be understood to operate in three phases. In the first phase, both Users 1 and 2 split their messages, M 1 at rate R 1 and M 2 at rate R 2, into two parts. The rate pair (R 1, R 2 ) is outside the capacity region of multiple access region with no feedback [5]. Both Users 1 and 2 use superposition coding to encode their two-part messages. At the receiver of User 3, only the inner layer of superposition code can be decoded, and a portion of message from both Users 1 and 2 is left undecoded (similar to [4]). In the second phase, User 3 strips away the portion of information it has decoded from the received signal. In essence, it compresses the received message to only that portion of the message that could not be decoded. It then jointly encodes this codeword with the downlink message M 3 at rate R and transmits it in the broadcast channel (BC) mode. Since the users know the portion of their own message which could not be decoded, they use this side information to decode both M 3 and the undecoded message from the other user. Finally, in phase three, both Users 1 and 2 know a part of undecoded message sent by each other from previous transmissions. They superimpose that on top of their new messages leading to a three-layer superposition code. The receiver at User 3, uses the previous reception and the current reception to decode the undecodable portion from last instance and some new information (the inner two layers). The third layer is undecodable. Thus, the system goes through second and third phases for b consecutive blocks. Now we are ready for the two main results: Inner and Outer bounds for the half-duplex 3 user channel model with hidden node topology.

26 CHAPTER 3. MAIN RESULTS Inner Bound Theorem 3.1 All the rate points defined by (R 1, R 2, R 3 ) R which satisfy the following equations: R 1 I(Ũ1; Ũ2, X 3, Y 3, Ũ3, W 3 ) R 2 I(Ũ2; Ũ1, X 3, Y 3, Ũ3, W 3 ) R 3 min[(i(ũ3; X 1, Y 1, Ũ1, W 1 ), I(Ũ3; X 2, Y 2, Ũ2, W 2 )] R 1 + R 2 I(Ũ1, Ũ2; X 3, Y 3, Ũ3, W 3 ) are achievable (i.e. R C) where X 1, X 2, X 3 are the channel inputs, Y 1, Y 2, Y 3 are the channel outputs and U 1, U 2, U 3, Ũ1, Ũ2, Ũ3, W 1, W 2, W 3 are auxiliary random variables whose joint distribution is of the form, P(u 1, u 2, u 3, ũ 1, ũ 2, ũ 3, w 1, w 2, w 3, x 1, x 2, x 3, y 1, y 2, y 3 ) = P 1 (u 1 )P 2 (u 2 )P 3 (u 3 )P(ũ 1, ũ 2, ũ 3, w 1, w 2, w 3 ).P(x 1 u 1, ũ 1, w 1 )P(x 2 u 2, ũ 2, w 2 ).P(x 3 u 3, ũ 3, w 3 )P(y 1, y 2, y 3 x 1, x 2, x 3 ) We shall prove the inner bound and find the half-duplex special case in Chapter 4. The proof is based on typical decoding arguments and superpostion coding based on Carleial s inner bound for a full duplex MAC with feedback [4]. We shall further investigate the inner bound in Chapter 6 to find a lot of interesting special cases.

27 CHAPTER 3. MAIN RESULTS Outer Bound Theorem 3.2 The capacity region C of the three-node half-duplex two-way network is a subset of the following region C {(R 1, R 2, R 3 ) : R 1 I(X 1 ; Y 2 X 2, Z 2,W 3, Q) + I(X 1 ; Y 3 X 2, Z 2, Z 3, Q) R 2 I(X 2 ; Y 1 X 1, Z 1,W 3, Q) + I(X 2 ; Y 3 X 1, Z 1, Z 3, Q) R 3 min[i(x 3 ; Y 1 X 1, Z 1, Q), I(X 3 ; Y 2 X 2, Z 2, Q)] R 1 + R 2 I(X 1, X 2 ; Y 3 Z 3, Q)}, where X 1, X 2, X 3, Y 1, Y 2, Y 3, Z 1, Z 2, Z 3, Q are random variables whose joint distribution is of the form, p(q)p(z 1, z 2, z 3 q)p(x 1 z 1, q)p(x 2 z 2, q)p(x 3 z 3, q).p(y 3 x 1, x 2, q)p(y 1 x 3, q)p(y 2 x 3, q), where Q is the time sharing variable. We shall prove the outer bound in Chapter 5 and find some special cases like its half duplex extension and no feedback case. The proof of the outer bound is based on Han s outer bound for a two user two-way channel [15]. It is based on the markov structure of the channel input, output and auxiliary random variables.

28 Chapter 4 Inner Bound In this section we shall prove Theorem 3.1. In the last part of the chapter, we shall also look into the half-duplex special case. 4.1 Code generation Generate M 1 = M 1 independent identically distributed (i.i.d.) random vectors U 1 according to P 1 (u 1 ) = Π n i=1p 1 (u 1,i ). Then randomly assign each random vector to one of the messages from the input alphabet of User 1(M 1 ). Similarly, we generate M 2 = M 2 and M 3 = M 3 i.i.d. random vectors U 2 and U 3 according to P 2 (u 2 ) = Π n i=1p 2 (u 2,i ) and P 3 (u 3 ) = Π n i=1p 3 (u 3,i ) respectively. Then randomly assign each of these vectors to each member of the input alphabets of User 2 and 3 respectively. 4.2 Encoding Scheme Suppose at the b 1 th block the n-tuples x b 1 1,x b 1 2 and x b 1 3 had been sent by users 1, 2 and 3 and at the end of the b 1 th block, y b 1 1,y b 1 2 and y b 1 3 is received. In the 21

29 CHAPTER 4. INNER BOUND 22 b th block we define the feedback w b 1, w b 2 and w b 3 as: w b 1 = x b 1 1,y b 1 1 w2 b = x2 b 1,y2 b 1 w b 3 = x b 1 3,y b 1 3 and set the incremental part as: ũ b 1 = u b 1 1 (m b 1 1 ) ũ b 2 = u b 1 2 (m b 1 2 ) ũ b 3 = u b 1 3 (m b 1 3 ) The encoding functions use the channel output and the information about the previous blocks (1, 2,...,b 1) that it already has in addition to the new message for the present block to construct the channel inputs for block b. x b 1 = f 1 (u b 1(m b 1),ũ b 1,w b 1) x b 2 = f 2 (u b 2(m b 2),ũ b 2,w b 2) x b 3 = f 3 (u b 3(m b 3),ũ b 3,w b 3) 4.3 Decoding Scheme At the end of block b, x b 1 and y1 b are known to User 1, x b 2 and y2 b are known to User 2 and x b 3 and y3 b are known to User 3. The decoding procedure for the three users are as follows: User 1 decodes the message sent by User 3 during block b as ˆm b 3 K such that, (u b 3( ˆm b 3),x b 1,y1,ũ b b 1,w1) b T ǫ (Ũ3, X 1, Y 1, Ũ1, W 1 ) (4.1)

30 CHAPTER 4. INNER BOUND 23 Note that the information about y b 1 1 is hidden in W 1 and the incremental information about m b 3 is in y b 1. Similarly, User 2 decodes the message sent by User 3 during block b as ˆm b 3 K such that, (u b 3( ˆm b 3),x b 2,y b 2,ũ b 2,w b 2) T ǫ (Ũ3, X 2, Y 2, Ũ2, W 2 ) (4.2) User 3 decodes the messages sent by Users 1 and 2 during block b 1 as ˆm b 1 1 I and ˆm b 1 2 J such that, (u b 1 1 ( ˆm b 1 1 ),u b 1 2 ( ˆm b 1 2 ),x b 3,y b 3,ũ b 3,w b 3) T ǫ (Ũ1, Ũ2, X 3, Y 3, Ũ3, W 3 ) 4.4 Probability of Error Due to the symmetry in the random generation of the codes, any message sent by the users for a given block yields the same probability of error. So, without the loss of generality, we can assume that (m b 1 1, m b 1 2, m b 3) = (1, 1, 1) was sent. Consider decoding at User 1. For block b, define E k = ( ˆm b 3 = k m b 3 = 1) as (i.e. the event that User 1 decodes the message sent by User 3 during block b as k when User 3 sent the message m b 3 = 1). An error event happens if either the correct codeword is not found to be jointly typical with the channel output or an incorrect code word is found to be jointly typical. Therefore, P b e1 = Pr(Ē1 k 1 E k (1, 1, 1)was sent) P(Ē1) + k 1 P(E k ) where P is the conditional probability given that (1, 1, 1) was sent (as defined in Cover and Thomas pg 394). By using the properties of jointly typical sequences we

31 CHAPTER 4. INNER BOUND 24 can show that, P(Ē1) 0 and we can bound the second term as follows: P(E k ) = Pr((u b 1 3 (k),x b 1,y b 1,ũ b 1,w b 1) T ǫ ((Ũ3, X 1, Y 1, Ũ1, W 1 ) (1, 1, 1)was sent) = P(ũ 3 )P(x b 1,y1,ũ b b 1,w1) b (ũ b 3 (k),xb 1,yb 1,ũb 1,wb 1 ) Tǫ T ǫ 2 n(h(ũ3) ǫ) 2 n(h(x 1,Y 1,Ũ1,W 1 ) ǫ) 2 n(h(ũ3,x 1,Y 1,Ũ1,W 1 ) ǫ) 2 n(h(ũ3)) ǫ) 2 n(h(x 1,Y 1,Ũ1,W 1 ) ǫ) = 2 n(h(ũ3) ǫ) 2 n(h(ũ3 X 1,Y 1,Ũ1,W 1 ) 2ǫ) = 2 n(i(ũ3;x 1,Y 1,Ũ1,W 1 ) 3ǫ) Hence, it follows that: P b e1 P(Ē1) + k 1 2 n(i(ũ3;x 1,Y 1,Ũ1,W 1 ) 3ǫ) = P(Ē1) + (K 1)2 n(i(ũ3;x 1,Y 1,Ũ1,W 1 ) 3ǫ) P(Ē1) + 2 nr 3 2 n(i(ũ3;x 1,Y 1,Ũ1,W 1 ) 3ǫ) So, to make the probability of error to be arbitrarily small, R 3 I(Ũ3; X 1, Y 1, Ũ1, W 1 ) (4.3) Following exactly the same steps for User 2, it can be shown that R 3 I(Ũ3; X 2, Y 2, Ũ2, W 2 ) (4.4) Now consider the decoding at User 3. For block b, define E ij = (( ˆm b 1, ˆm b 1) = (i, j) (m b 1, m b 2) = (1, 1)) as (i.e. the event that User 3 decodes the message sent

32 CHAPTER 4. INNER BOUND 25 by User 1 and User 2 during block b as i and j respectively when (m b 1, m b 2) = (1, 1) was sent). An error event happens if either the correct codewords are not found to be jointly typical with the channel output or an incorrect code word is found to be jointly typical. Therefore, Pe3 b = Pr(Ē11 (i,j) (1,1) E k (1, 1, 1)was sent) P(Ē11) + P(E i1 ) + P(E 1j ) + P(E ij ) i 1,j=1 i=1,j 1 i 1,j 1 By using the properties of jointly typical sequences we can show that, P(Ē1) 0 and we can bound the second term as follows: P(E i1 ) = Pr((u b 1 1 (i),u b 1 2 (1),x b 3,y b 3,ũ b 3,w b 3) T ǫ ((Ũ1, Ũ2, X 3, Y 3, Ũ3, W 3 ) (1, 1, 1)was sent) = P(ũ 1 )P(ũ 2,x b 3,y3,ũ b b 3,w3) b (ũ b 1,ũb 2,xb 3,yb 3,ũb 3,wb 3 ) Tǫ T ǫ 2 n(h(ũ1) ǫ) 2 n(h(ũ2,x 3,Y 3,Ũ3,W 3 ) ǫ) 2 n(h(ũ1,ũ2,x 3,Y 3,Ũ3,W 3 ) ǫ) 2 n(h(ũ1)) ǫ) 2 n(h(ũ2,x 3,Y 3,Ũ3,W 3 ) ǫ) = 2 n(h(ũ1) ǫ) 2 n(h(ũ1 Ũ2,X 3,Y 3,Ũ3,W 3 ) 2ǫ) = 2 n(i(ũ1;ũ2,x 3,Y 3,Ũ3,W 3 ) 3ǫ) Hence, it follows that: R 1 I(Ũ1; Ũ2, X 3, Y 3, Ũ3, W 3 ) (4.5) Following the same steps we can show that to make the probability of error arbitrarily small, R 2 I(Ũ2; Ũ1, X 3, Y 3, Ũ3, W 3 ) (4.6) R 1 + R 2 I(Ũ1, Ũ2; X 3, Y 3, Ũ3, W 3 ) (4.7)

33 CHAPTER 4. INNER BOUND Half-duplex Channel In this case, the system is time duplexed between a multiple access (MAC) phase and a multicast (MC) phase 4.1. During the MAC phase, User 1 and User 2 send their codewords to User 3. We also refer to this phase as the uplink. This takes up η fraction of the total time. During the MC phase, User 3, sends a common codeword (message) to Users 1 and 2. This happens 1 η fraction of the total time. We call this phase MC to distinguish between the case when User 3 has independent messages for Users 1 and 2 (which we call a broadcast phase or BC phase). We also refer to this phase as the downlink. x 2 Node 2 Node 2 y 2 y Node 3 Node 3 x x 1 Node 1 Node 1 y 1 Figure 4.1: Half-duplex two-way network: two states in the system Channel Half-duplex two-way channel: ω(y 1, y 2, y 3 x 1, x 2, x 3 ) = ω MAC (y 3 x 1, x 2 )ω MC (y 1, y 2 x 3 )

34 CHAPTER 4. INNER BOUND 27 Test channel: ω (y 1, y 2, y 3 u 1, ũ 1, u 2, ũ 2, u 3, w 3 ) = ω MAC (y 3 x 1, x 2 )γ 1 (x 1 u 1, ũ 1, ũ 2 )γ 2 (x 2 u 2, ũ 1, ũ 2 ) x 1,x 2,x 3.ω BC (y 1, y 2 x 3 )γ 3 (x 3 u 3, w 3 ) where, p(u 1, u 2, u 3, ũ 1, ũ 2, w 3 ) = p(u 1 )p(u 2 )p(u 3 )p(ũ 1 u 1, y 1 )p(ũ 2 u 2, y 2 )p( w 3 y 3 ) (Note the independent new information and the dependent incremental and feedback information) Encoding scheme We shall denote the alphabets for the channel inputs and outputs in script letters. The coding scheme divides the input into blocks each of n channel uses. For each block, the channel input is formed, in general from three parts: New information independent of messages of any previous block (we shall denote this by u, and its alphabet by U ) Feedback information obtained from the previous channel output (we shall denote this by w, and its alphabet by W ) Incremental information about the previous channel input obtained using the previous feedback signal (we shall denote this by ũ, and its alphabet by U ) Let us define the input alphabets and the index sets first.

35 CHAPTER 4. INNER BOUND 28 Input alphabets: U 1 = U 10 U 11 (codewords of length n) U 2 = U 20 U 22 U 3 Index sets: U 10 I, U 11 J U 10 = I, U 11 = J (for the input alphabets) U 20 K, U 22 L U 20 = K, U 22 = L Incremental alphabets: Feedback alphabets: U 3 M Ũ 1 = U 10 Ũ 2 = U 20 W 3 U 3 = M So, in general, x b i = f b i (u i, ũ i, w i ), where (i = 1, 2 or 3). But note that due to our MAC-MC configuration, Users 1 and 2 donot send any feedback information and User 3 does not send any incremental information. Hence the encoding functions are as follows: User 1: x b 1 = f1(u b b 1(m b 1), ũ b 1) f1 b : U 1 n U1 n X1 n User 2: x b 2 = f2(u b b 2(m b 2), ũ b 2) f2 b : U 2 n U2 n X2 n User 3: x b 3 = f3(u b b 3(m), w 3) b f3 b : U3 n W 3 n X3 n At the beginning of block b, Users 1 and 2 form the incremental information ũ b 1andũ b 2 based on y b 1 1, ũ b 1 and u b 1 2, ũ b 2 respectively. It forms x 1 and x 2 as mentioned above and sends through the channel using it for η fraction of the time. User 3 receives y b 3 and forms x 3 as given above. The decoding part is explained in the next section. The incremental information and the feedback information donot assume any decoding. All the decoding is lumped into the encoding functions f 1, f 2 and f 3. (This gives rise to the problem of bounding the size of the incremental and feedback alphabets).

36 CHAPTER 4. INNER BOUND 29 Encoded msgs: u b 1 = (u b 10(i), u b 11(j)) u b 1 U 1, u b 10 U 10, u b 11 U 11 i I, j J u b 2 = (u b 20(k), u b 22(l)) u b 2 U 2, u b 20 U 20, u b 22 U 22 k K, l L u b 3 = u b 3(m) u b 3 U 3, m M Incremental msgs: ũ b 10(i ) = (y b 1 1, u b 1 10 (i )) u b 1 1 (i ) ũ b 20(k ) = (y b 1 2, u b 1 20 (k )) u b 1 2 (k ) Feedback msg: w b 3 = y b Decoding scheme Let us consider block b, (1 b B). It is divided into a MAC phase and a MC phase. During the MAC phase of the previous block, User 3 recieved y b 1 3, and decoded (i, j, k, l ). During block b, User 3 decodes (î, ĵ, ˆk,ˆl). Similarly, during the MC phase of the previous block, User 1 and User 2 have recieved y b 1 1, y b 1 2 and User 1 has decoded m, k and User 2 has decoded m, i. During the current block s MC phase, User 1 decodes ( ˆm, ˆk) and User 2 decodes ( ˆm, î) User 1: Find ˆk K and ˆm M such that (u b 3( ˆm), y1, b ũ b 10(i), ũ b 20(ˆk)) A n ǫ (U 3, Y 1, Ũ10, Ũ20) User 2: Find î I and ˆm M such that (u b 3( ˆm), y2, b ũ b 10(î), ũ b 20(k)) A n ǫ (U 3, Y 2, Ũ10, Ũ20) User 3: Find î I, ĵ J, ˆk K and ˆl L such that ũ b 10(î ), u b 11(ĵ), ũ b 20(ˆk ), u b 22(ˆl), w b 3, y b 3 A n ǫ (Ũ10, U 11, Ũ20, U 22, W 3, Y 3 ) ũ10 b 1 (i ), u10 b 1 (î ), u11 b 1 (j ), ũ20 b 1 (k ), u20 b 1 (ˆk ), u22 b 1 (l ), w 3 b 1, y3 b 1

37 CHAPTER 4. INNER BOUND 30 A n ǫ (Ũ10, U 10, U 11, Ũ20, U 20, U 22, W 3, Y 3 ) Probability of Error List Decoding Error During any block, User 3 makes a list of code indices (i, k) that are typical with the channel output. Let, 1, (u b 10(i), u b 20(k), y3) b A n ǫ (U 10, U 20, Y 3 ), ψ ik (y) = 0, otherwise. Size of the list can be estimated by taking the expected value of the number of sequences that are typical with the channel output. Define the size of the list made from the output signal y as S y and let the correct indices be i and k respectively. S y = i,k ψ ik (y) E S y =Eψ i k (y) + Eψ ik (y) i=i,k k + Eψ ik (y) + Eψ ik (y) i i,k=k i i,k k 1 + (2 nr 10 1)2 n[i(u 10;Y 3 Ũ10,U 20,U 11,Ũ20,U 22 ) ǫ] + (2 nr 20 1)2 n[i(u 20;Y 3 Ũ10,U 11,Ũ20,U 10,U 22 ) ǫ] + (2 nr 10 1)(2 nr 20 1)2 n[i(u 10,U 20 ;Y 3 Ũ10,U 11,Ũ20,U 22 ) ǫ] n(r 10 I(U 10 ;Y 3 Ũ10,U 20,U 11,Ũ20,U 22 )+ǫ) + 2 n(r 20 I(U 20 ;Y 3 Ũ10,U 11,Ũ20,U 10,U 22 )+ǫ) + 2 n(r 10+R 20 I(U 10,U 20 ;Y 3 Ũ10,U 11,Ũ20,U 22 )+ǫ)

38 CHAPTER 4. INNER BOUND 31 Decoding Errors Assume that the indices used to generate the channel inputs were (i = 1, i = 1, j = 1, k = 1, k = 1, l = 1, m = 1), during block b. User 1(E km ): (u b 3( ˆm), y1, b ũ b 10(i), ũ b 20(ˆk)) A n ǫ (U 3, Y 1, Ũ10, Ũ20) P e1 = (K 1)E (M 1)E (K 1)(M 1)E nr 20 E nr 3 E n(r 20+R 3 ) E nr 20 2 n[i(u 20;Y 1 U 10,U 3 ) ǫ] + 2 nr 3 2 n[i(u 3;Y 1 U 10,U 20 ) ǫ] + 2 n(r 20+R 3 ) 2 n[i(u 20,U 3 ;Y 1 U 10 ) ǫ] User 2 (E im ): (u b 3( ˆm), y2, b ũ b 10(î), ũ b 20(k)) A n ǫ (U 3, Y 2, Ũ10, Ũ20) P e2 = (I 1)E (M 1)E (I 1)(M 1)E nr 10 E nr 3 E n(r 10+R 3 ) E nr 10 2 n[i(u 10;Y 2 U 20,U 3 ) ǫ] + 2 nr 3 2 n[i(u 3;Y 2 U 10,U 20 ) ǫ] + 2 n(r 10+R 3 ) 2 n[i(u 10,U 3 ;Y 2 U 20 ) ǫ] User 3 (E ijkl F ik ): ũ b 10(î ), u b 11(ĵ), ũ b 20(ˆk ), u b 22(ˆl), w b 3, y b 3 A n ǫ (Ũ10, U 11, Ũ20, U 22, W 3, Y 3 ) ũ10 b 1 (i ), u10 b 1 (î ), u11 b 1 (j ), ũ20 b 1 (k ), u20 b 1 (ˆk ), u22 b 1 (l ), w 3 b 1, y3 b 1 A n ǫ (Ũ10, U 10, U 11, Ũ20, U 20, U 22, W 3, Y 3 ) P e3 =(I 1)E F (J 1)E (K 1)E F (L 1)E

39 CHAPTER 4. INNER BOUND 32 + (I 1)(J 1)E F (I 1)(K 1)E F (I 1)(L 1)E F (J 1)(K 1)E F (J 1)(L 1)E (K 1)(L 1)E F (I 1)(J 1)(K 1)E F (I 1)(J 1)(L 1)E F (I 1)(K 1)(L 1)E F (J 1)(K 1)(L 1)E F (I 1)(J 1)(K 1)(L 1)E1111F n[r 10 I(Ũ10;Y 3 Ũ20,U 11,U 22, W 3 ) I(U 10 ;Y 3 U 20,Ũ10,U 11,Ũ20,U 22, W 3 )+6ǫ] + 2 n[r 11 I(U 11 ;Y 3 Ũ10,Ũ20,U 22,Ũ10,U 11,Ũ20,U 22, W 3 )+3ǫ] + 2 n[r 20 I(Ũ20;Y 3 Ũ10,U 11,U 22, W 3 ) I(U 20 ;Y 3 U 10,Ũ10,U 11,Ũ20,U 22, W 3 )+6ǫ] + 2 n[r 22 I(U 22 ;Y 3 Ũ10,Ũ22,U 11, W 3 )+3ǫ] + 2 n[(r 10+R 11 ) I(Ũ10,U 11 ;Y 3 Ũ20,U 22, W 3 ) I(U 10 ;Y 3 U 20,Ũ10,U 11,Ũ20,U 22, W 3 )+6ǫ] + 2 n[(r 10+R 20 ) I(Ũ10,Ũ20;Y 3 U 11,U 22, W 3 ) I(U 10,U 20 ;Y 3 Ũ10,U 11,Ũ20,U 22, W 3 )+6ǫ] + 2 n[(r 10+R 22 ) I(Ũ10,U 22 ;Y 3 U 11,Ũ20, W 3 ) I(U 10 ;Y 3 U 20,Ũ10,U 11,Ũ20,U 22, W 3 )+6ǫ] + 2 n[(r 11+R 20 ) I(U 11,Ũ20;Y 3 Ũ10,U 22, W 3 ) I(U 20 ;Y 3 U 10,Ũ10,U 11,Ũ20,U 22, W 3 )+6ǫ] + 2 n[(r 11+R 22 ) I(U 11,U 22 ;Y 3 Ũ10,Ũ20, W 3 )+3ǫ] + 2 n[(r 20+R 22 ) I(Ũ20,U 22 ;Y 3 Ũ10,U 11, W 3 ) I(U 20 ;Y 3 U 10,Ũ10,U 11,Ũ20,U 22, W 3 )+6ǫ] + 2 n[(r 10+R 11 +R 20 ) I(Ũ10,U 11,Ũ20;Y 3 U 22, W 3 ) I(U 10,U 20 ;Y 3 Ũ10,U 11,Ũ20,U 22, W 3 )+6ǫ] + 2 n[(r 10+R 11 +R 22 ) I(Ũ10,U 11,U 22 ;Y 3 Ũ20, W 3 ) I(U 10 ;Y 3 U 20,Ũ10,U 11,Ũ20,U 22, W 3 )+6ǫ] + 2 n[(r 10+R 20 +R 22 ) I(Ũ10,Ũ20,U 22 ;Y 3 U 11, W 3 ) I(U 10,U 20 ;Y 3 Ũ10,U 11,Ũ20,U 22, W 3 )+6ǫ] + 2 n[(r 11+R 20 +R 22 ) I(U 11,Ũ20,U 22 ;Y 3 Ũ10, W 3 ) I(U 20 ;Y 3 U 10,Ũ10,U 11,Ũ20,U 22, W 3 )+6ǫ] + 2 n[(r 10+R 11 +R 20 +R 22 ) I(Ũ10,U 11,Ũ20,U 22 ;Y 3 W 3 ) I(U 10,U 20 ;Y 3 Ũ10,U 11,Ũ20,U 22, W 3 )+6ǫ] Hence, to make the probabilities of error arbitrarily small (as n ), we have to keep the rates in the rate region defined as: R 10 η min[i(u 10 ; Y 2 U 20, U 3 ), I(U 10 ; Y 3 Ũ10, U 20, U 11, Ũ20, U 22 )

40 CHAPTER 4. INNER BOUND 33, I(Ũ10; Y 3 Ũ20, U 11, U 22, W 3 ) + I(U 10 ; Y 3 U 20, Ũ10, U 11, Ũ20, U 22, W 3 )] R 11 ηi(u 11 ; Y 3 Ũ10, Ũ20, U 22, Ũ10, U 11, Ũ20, U 22, W 3 ) R 20 η min[i(u 20 ; Y 1 U 10, U 3 ), I(U 20 ; Y 3 Ũ10, U 11, Ũ20, U 10, U 22 ), I(Ũ20; Y 3 Ũ10, U 11, U 22, W 3 ) + I(U 20 ; Y 3 U 10, Ũ10, U 11, Ũ20, U 22, W 3 )] R 22 ηi(u 22 ; Y 3 Ũ10, Ũ22, U 11, Ũ10, U 11, Ũ20, U 22, W 3 ) R 10 + R 11 ηi(ũ10, U 11 ; Y 3 Ũ20, U 22, W 3 ) + I(U 10 ; Y 3 U 20, Ũ10, U 11, Ũ20, U 22, W 3 ) R 10 + R 20 η min[i(u 10, U 20 ; Y 3 Ũ10, U 11, Ũ20, U 22 ), I(Ũ10, Ũ20; Y 3 U 11, U 22, W 3 ) + I(U 10, U 20 ; Y 3 Ũ10, U 11, Ũ20, U 22, W 3 )] R 10 + R 22 ηi(ũ10, U 22 ; Y 3 U 11, Ũ20, W 3 ) + I(U 10 ; Y 3 U 20, Ũ10, U 11, Ũ20, U 22, W 3 ) R 11 + R 20 ηi(u 11, Ũ20; Y 3 Ũ10, U 22, W 3 ) + I(U 20 ; Y 3 U 10, Ũ10, U 11, Ũ20, U 22, W 3 ) R 11 + R 22 ηi(u 11, U 22 ; Y 3 Ũ10, Ũ20, W 3 ) R 20 + R 22 ηi(ũ20, U 22 ; Y 3 Ũ10, U 11, W 3 ) + I(U 20 ; Y 3 U 10, Ũ10, U 11, Ũ20, U 22, W 3 ) R 10 + R 11 + R 20 ηi(ũ10, U 11, Ũ20; Y 3 U 22, W 3 ) + I(U 10, U 20 ; Y 3 Ũ10, U 11, Ũ20, U 22, W 3 ) R 10 + R 11 + R 22 ηi(ũ10, U 11, U 22 ; Y 3 Ũ20, W 3 ) + I(U 10 ; Y 3 U 20, Ũ10, U 11, Ũ20, U 22, W 3 ) R 10 + R 20 + R 22 ηi(ũ10, Ũ20, U 22 ; Y 3 U 11, W 3 ) + I(U 10, U 20 ; Y 3 Ũ10, U 11, Ũ20, U 22, W 3 ) R 11 + R 20 + R 22 ηi(u 11, Ũ20, U 22 ; Y 3 Ũ10, W 3 ) + I(U 20 ; Y 3 U 10, Ũ10, U 11, Ũ20, U 22, W 3 ) R 10 + R 11 + R 20 + R 22 ηi(ũ10, U 11, Ũ20, U 22 ; Y 3 W 3 ) + I(U 10, U 20 ; Y 3 Ũ10, U 11, Ũ20, U 22, W 3 ) R 3 (1 η) min [I(U 3 ; Y 1 U 10, U 20 ), I(U 3 ; Y 2 U 10, U 20 )] R 20 + R 3 (1 η)i(u 20, U 3 ; Y 1 U 10 ) R 10 + R 3 (1 η)i(u 10, U 3 ; Y 2 U 20 )

41 Chapter 5 Outer Bound This chapter contains the proof of Theorem 3.2 and a dependence balance bound on the input distributions. We will denote channel inputs by X, channel outputs by Y and the messages by W with proper subscripts and superscripts as described below. The encoding will use B blocks of n channel uses each. So, we will introduce the following notation for all variables: First, A u,b will denote the b block of User u; note the block is of length n symbols. The vector of b consecutive blocks will be denoted by A b u = (A u,1,...,a u,b 1 ). Finally, the vector with all blocks b = 1,...,B will be denoted by A = (A u,1,...,a u,b ). Note that boldface variables denote vectors for B blocks each of length n drawn from the same distribution. For example, X 1 = (X 1,1, X 1,2,...,X 1,B ) is the vector input to the channel for User 1 for B blocks and (X 1,i = (X1,i, 1 X1,i, 2...,X1,i)) n is the channel input for User 1 for the i th block which is a vector (n-tuple) itself each independently drawn from the same distribution (X1,i k X 1 ). Finally, we introduce a random variable Z u,b = (X u,b 1, Y u,b 1 ). The model in Figure 3.1 can be mathematically defined by a Markov chain as shown in Figure 5.1. Note that X s are deterministic functions of W s and Y s (i.e. X 1,b = f 1 (W 1, Y 1,b 1 ), X 2,b = f 2 (W 2, Y 2,b 1 ) and X 3,b = g(w 3, Y 3,b ) for some deterministic encoding func- 34

42 CHAPTER 5. OUTER BOUND Y n 3,b 1 X n 3,b 1 W 3 Y 1,b 1 n W 1 X1,b n Y3,b n W 2 X n X2,b n W 3 3,b... Y n 2,b 1 Figure 5.1: Markov chain model of the multiuser feedback scenario. tions f 1, f 2, g). We prove each of the inequalities in the following subsections. To avoid notational overload, we will suppress conditioning on the time-sharing variable q in the proof, but the reader should modify the appropriate steps to account for time-sharing. 5.1 Outer Bound for R 3 We start by bounding R 3 first as follows. nbr 3 = H(W 3 W 1,W 2 ) = H(W 3 W 1,W 2,Y 1 ) + I(W 3 ;Y 1 W 1,W 2 ) (5.1) = H(W 3 W 1,W 2,Y 2 ) + I(W 3 ;Y 2 W 1,W 2 ). (5.2) The first term can be upper bounded by an arbitrarily small positive number using Fano s theorem (similarly for all other bounds given below). The mutual information (second term in equation 5.1) can be upper bounded as I(W 3 ;Y 1 W 1,W 2 ) = H(Y 1 W 1,W 2 ) H(Y 1 W 1,W 2,W 3 ) = (a) B b=1 B b=1 [H(Y 1,b W 1,W 2, Y b 1 1 ) H(Y 1,b W 1,W 2,W 3, Y b 1 1 )] [H(Y 1,b W 1,W 2, Y b 1 1, X b 1) H(Y 1,b W 1,W 2,W 3, X b 1, Y b 1 1, Y b 3 )]

43 CHAPTER 5. OUTER BOUND 36 (b) = (c) = (d) B b=1 [H(Y 1,b W 1,W 2, X b 1, Y b 1 1, Z b 1) H(Y 1,b W 1,W 2,W 3, X b 1, Y b 1 1, Y b 3, X b 3, Z b 1)] B [H(Y 1,b X 1,b, Z 1,b ) H(Y 1,b X 1,b, X 3,b, Z 1,b )] b=1 B I(Y 1,b ; X 3,b X 1,b, Z 1,b ) b=1 B b=1 n i=1 I(Y i 1,b; X i 3,b X i 1,b, Z i 1,b) nbi(x 3 ; Y 1 X 1, Z 1 ), where (a): Using X 1,b = f 1 (W 1, Y 1,b 1 ) in both the terms and the inequality is due to introducing Y b 3 in the second term. (b): Using Z 1,b = (X 1,b 1, Y 1,b 1 ) in both the terms and in the second term using the fact that X 3,b = g(w 3, Y 3,b ). (c): For the first term, we just remove a few variables and hence get an upper bound and for the second term, Y 1,b being the channel output is independent of all other terms given X 3,b and X 1,b as it uses feedback. (d): Using the DMC property for the n-channel uses in each block. Hence, R 3 I(X 3 ; Y 1 X 1, Z 1 ) and similarly from equation 5.2, R 3 I(X 3 ; Y 2 X 2, Z 2 ). So, R 3 min[i(x 3 ; Y 1 X 1, Z 1 ), I(X 3 ; Y 2 X 2, Z 2 )]. (5.3) 5.2 Outer Bound for R 1 and R 2 Note that nbr 1 = H(W 1 W 2,W 3 ) = H(W 1 W 2,W 3,Y 3 ) + I(W 1 ;Y 3 W 2,W 3 ).

44 CHAPTER 5. OUTER BOUND 37 The second term can be upper bounded as I(W 1 ;Y 3 W 2,W 3 ) I(W 1 ;Y 2,Y 3 W 2,W 3 ) = I(W 1 ;Y 2 W 2,W 3 ) + I(W 1 ;Y 3 W 2,W 3,Y 2 ). (5.4) The first term can be upper bounded very similar to the proof of Section 5.1, I(W 1 ;Y 2 W 2,W 3 ) = H(Y 2 W 2,W 3 ) H(Y 2 W 1,W 2,W 3 ) = (a) (b) = (c) = (d) B b=1 B b=1 B b=1 [H(Y 2,b W 2,W 3, Y b 1 2 ) H(Y 2,b W 1,W 2,W 3, Y b 1 2 )] [H(Y 2,b W 2,W 3, Y b 1 2, X b 2) H(Y 2,b W 1,W 2,W 3, X b 2, Y b 1 2, Y b 1 1 )] [H(Y 2,b W 2,W 3, X b 1 2, Y b 1 2, Z b 2) H(Y 2,b W 1,W 2,W 3, X b 1, Y b 1 1, Y b 1 2, X b 2, Z b 2)] B [H(Y 2,b X 2,b, Z 2,b,W 3 ) H(Y 2,b X 1,b, X 2,b, Z 2,b,W 3 )] b=1 B I(Y 2,b ; X 1,b X 2,b, Z 2,b,W 3 ) b=1 B b=1 n i=1 I(Y i 2,b; X i 1,b X i 2,b, Z i 2,b,W 3 ) nbi(x 1 ; Y 2 X 2, Z 2,W 3 ), where (a): Using X 2,b = f 2 (W 2, Y 2,b 1 ) in both the terms and the inequality is due to introducing Y b 1 1 in the second term. (b): Using Z 2,b = (X 2,b 1, Y 2,b 1 ) in both the terms and in the second term using the fact that X 3,b = g(w 3, Y 3,b ). (c): For the first term, we just remove a few variables and hence get an upper bound