Scalable Capacity Bounding Models for Wireless ... - Semantic Scholar

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Scalable Capacity Bounding Models for Wireless Networks Jinfeng Du, Member, IEEE, Muriel M´edard, Fellow, IEEE, Ming Xiao, Senior

arXiv:1401.4189v1 [cs.IT] 16 Jan 2014

Member, IEEE, and Mikael Skoglund, Senior Member, IEEE

Abstract Based on the framework of network equivalence theory developed by Koetter et al., this paper presents scalable capacity upper and lower bounding models for wireless networks by construction of noiseless networks that can be used to calculate outer and inner bounds, respectively, for the original networks. A channel decoupling method is proposed to decompose wireless networks into point-to-point channels, and (potentially) coupled multiple-access channels (MACs) and broadcast channels (BCs). The upper bounding model, consisting of only point-to-point bit-pipes, is constructed by firstly extending the “one-shot” bounding models developed by Calmon et al. and then integrating them with network equivalence tools. The lower bounding model, consisting of both point-to-point and point-to-points bitpipes, is constructed based on a two-step update of the one-shot models to incorporate the broadcast nature of wireless transmission. The main advantages of the proposed methods are their simplicity and the fact that they can be extended easily to large networks with a complexity that grows linearly with the number of nodes. It is demonstrated that the gap between the resulting upper and lower bounds is usually not large, and they can approach the capacity in some setups.

Index Terms capacity, wireless networks, equivalence, channel emulation, channel decoupling

This work was presented in part at the IEEE International Symposium on Information Theory, Istanbul, Turkey, July 2013. Jinfeng Du is with Research Lab of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USA, and School of Electrical Engineering, Royal Institute of Technology, Stockholm, Sweden (Email: [email protected]). Muriel M´edard is with Research Lab of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USA (Email: [email protected]). Ming Xiao and Mikael Skoglund are with School of Electrical Engineering and the ACCESS Linnaeus Center, Royal Institute of Technology, Stockholm, Sweden (Email: {mingx, skoglund}@kth.se). January 20, 2014

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I. I NTRODUCTION A theory of network equivalence has been established in [1], [2] by Koetter et al. to characterize the capacity of a (large) memoryless noisy network: the original noisy network is first decomposed into many independent single-hop noisy channels; each of the single-hop noisy channels is then replaced by its corresponding upper (resp. lower) bounding model consisting of only noiseless bit-pipes; the capacity of the resulting noiseless network serves as an upper (resp. lower) bound for the capacity of the original noisy network. A noisy channel and a noiseless bit-pipe are said to be equivalent if the capacity region of any arbitrary network that contains the noisy channel remains unchanged after replacing the noisy channel by its noiseless counterpart. The equivalence of a point-to-point noisy channel and a noiseless point-to-point bitpipe has been established in [1] as long as the throughput of the latter equals the capacity of the former. For independent single-hop multi-terminal channels, such as the multiple-access channel (MAC), the broadcast channel (BC), and the interference channel (IC), operational frameworks for constructing upper and lower bounding models have been proposed in [2]. The constructive proofs presented in [1], [2] are based on a notion of channel emulation over a stack of N channel replicas, where the lower bounding models are established based on channel coding arguments, and the upper bounding models are constructed based on lossy source coding arguments. The bounding accuracy, in terms of both multiplicative and additive gaps between capacity upper and lower bounds, has been outlined in [2] for general noisy networks. Explicit upper and lower bounding models for MAC/BC/IC with two sources and/or two destinations have been constructed in [2]–[4]. For networks consisting of only point-to-point channels, MACs with two transmitters, and BCs with two receivers, the additive gap for Gaussian networks and the multiplicative gap for binary networks have been specified in [3]. The bounds obtained from network equivalence tools [2]–[4] can be tight in some setups, as shown in [4] for a multiple unicast network only consisting of noisy two-destination BCs, and in [5] for a frequency-division AWGN relay network in the wideband regime when the BC is physically degraded or when the source treats the stochastically degraded BC as physically degraded. A class of “one-shot” bounding models proposed in [6] by Calmon et al. introduces an auxiliary node for each BC/MAC such that the sum rate can be characterized by channel emulation over infinite number of channel uses as in [1], [2], whilst all the individual rates are characterized by emulating the transmission

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over each channel use (hence named “one-shot”). There are some other methods aiming at either emulating a noisy channel or characterizing the capacity of wireless networks. For point-to-point channels, the same upper bounding models established in [1] have also been developed in [7] for discrete memoryless channels with finitealphabet, and in [8] under the notion of strong coordination, where total variation (i.e., an additive gap) is used to measure the difference between the desired joint distribution and the empirical joint distribution of a pair of sequences (or a pair of symbols as in empirical coordination). The concept of channel emulation [1], [2], on the other hand, focuses on the set of jointly typical input-output pairs and the difference between the empirical joint distribution (averaged over ensembles of channel emulators) and the desired joint distribution is quantified by a multiplicative gap to ensure a small probability of error events1. As we focus on characterizing capacity (bounds) rather than reconstructing (exact) common randomness, we shall follow the channel emulation framework [1], [2] in construction of bounding models for BCs and MACs. A deterministic approach proposed in [9] can approximate the capacity of Gaussian networks within a constant gap in the high signal-to-noise ratio (SNR) regime, where amplify-and-forward (AF) has been proved to approach the capacity in multi-hop laying networks [10]. A layering approach with a global information flow routing technique proposed in [11] for wireless networks with non-coupled BCs and MACs can provide lower bounds that are within a multiplicative gap from the capacity upper bound. Since we are interested in an approach that can be used in all SNR regions and for both coupled2 and non-coupled networks, we do not follow the methods developed in [9], [11]. It is, however, non-trivial to apply the network equivalence tools [1]–[4] onto wireless networks owing to the broadcast nature of wireless transmission. On one hand, the bounding models proposed in [2] for MAC/BC with m transmitters/receivers contain up to (2m −1) bit-pipes3, leading to computational inefficiency when m is large (as in a wireless hot-spot which may contain potentially many users). On the other hand, the received signal at a terminal may contain several broadcasted signals, which creates dependence/interference among several transmitter1

Jointly typical pairs with larger decoding error probability than a threshold are expurgated from channel emulators.

2

The definition of coupled and non-coupled networks will be introduced in Sec. II-A.

3

For IC with m transmitter-receiver pairs, the number of bit-pipes is up to m(2m −1) as indicated in [2].

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receiver pairs. Although such dependence has been partially incorporated into ICs, the whole family of multi-hop channels (e.g., relay channels) have been excluded from consideration since the channel emulation techniques are developed for single-hop channels. Inspired by the idea of separate characterization of the sum and individual rates as in [6], we extend the one-shot bounding models for BCs/MACs to many-user scenarios and our new lower bounding models contain both point-to-point and point-to-points (hyper-arc) bit-pipes. We then propose a channel decoupling method that can decompose any memoryless wireless networks into independent point-to-point channels as well as (potentially coupled) BCs and MACs. The upper bounding model is constructed by integrating the one-shot models with the channel emulation techniques. The lower bounding model is obtained based on a two-step update of the one-shot models for coupled BCs and MACs by taking dependence between the networks into account. Throughout this paper, we assume that the distortion components (e.g., noise) are independent from the transmitted signals. This assumption can be relaxed in scenarios where the noise power is dependent on the power of input signals, and in such scenarios we take the smallest (resp. largest) noise power when constructing upper (resp. lower) bounding models. We further assume that the distortion components at receiving nodes within a coupled BC4 are mutually independent, and the scenario of coupled BC with correlated noise will be investigated in our future work. In this paper, we present a simple but efficient method that can construct both upper and lower bounding models for wireless networks with potentially many nodes, at a complexity that grows linearly with the number of nodes. The capacities of the noiseless bounding models serve as upper/lower bounds for the original wireless network whose capacity is otherwise difficult to characterize. Note that in this paper we focus on constructing noiseless bounding networks that can serve as a basis to compute capacity bounds, rather than finding the capacity of a noiseless network, which itself is a very difficult problem [12], [13]. We refer to [14]–[20] for various computational tools available to characterize (bounds on) the capacity of noiseless networks. The main advantage of our proposed bounding models are their simplicity and the fact that they can be easily extended to large networks. We demonstrate by examples that the gap between the upper and lower bounds is usually not large, and the resulting bounds can be tighter than benchmarks or even approaching the capacity in some setups. 4

Though we still allow noise correlation at receiving nodes within a non-coupled BC.

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The rest of this work is organized as follows. We first give a brief introduction of the network equivalence theory and the one-shot method in Sec. II, and then present our improvement on the bounding models for independent BCs and MACs in Sec. III. In Sec. IV we describe the network decoupling method for coupled networks and demonstrate how the upper and lower bounding models are updated by taking the coupled structure into account. We illustrate our bounding models in Sec. V by constructing upper and lower bounding models of coupled networks and conclude this work in Sec. VI. II. N ETWORK E QUIVALENCE T HEORY

AND THE

O NE -S HOT B OUNDING M ETHOD

A. Basic Definitions Definition 1 (Independent Channel): A point-to-point channel N i→j = (Xi , p(yj |xi ), Yj ) with input alphabet Xi , output alphabet Yj , and transition probability p(yj |xi ) within a memoryless Q Q network N T = ( n Xn , p(y|x), m Ym ), where T is the set of nodes, is said to be independent if the network transition probability p(y|x) can be partitioned as p(y|x) = p(y /j |x/i )p(yj |xi ),

(1)

where x/i denotes the vector of x without element xi , and similarly for y /j . To highlight the independence and simplify notation, we emphasize the notation for network N T as Y Y Ym ). N T = N i→j × ( Xn , p(y /j |x/i ), n6=i

(2)

m6=j

Similarly, a multi-terminal channel N S→D within N T is said to be independent, denoted by N T = N S→D × N S c →Dc ,

(3)

if the network transition probability can be partitioned as p(y|x) = p(y D |xS )p(y Dc |xS c ).

(4)

Here (S, S c ) and (D, D c) are two pairs of non-trivial complementary subsets of T . Definition 2 (Capacity Bounding Models): Given two independent channels C and N , C is said to upper bound N , or equivalently N lower bounds C, if the capacity (region) of N × W

is a subset of that for C × W for any network W. We denote their relationship by N ⊆ C. C and N are said to be equivalent if C ⊆ N ⊆ C. January 20, 2014

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For an independent noisy channel N , we shall construct channels C u and C l consisting of only noiseless bit-pipes, such that C u is the upper bounding model and C l is the lower bounding model for N , i.e., Cl ⊆ N ⊆ Cu.

(5)

Definition 3 (Coupled/Non-coupled Network): A network is said to be coupled if any of its point-to-point connections is part of a MAC and a BC simultaneously. Otherwise, the network is non-coupled. As expected, wireless networks are in general coupled owing to the broadcast nature of microwave propagation. Similarly, a MAC and a BC are said to be coupled if the transmitting node of the BC is also one of the source nodes of the MAC. B. Network Equivalence Theory for Independent Channels In [1], the equivalence between an independent point-to-point noisy channel N with capacity C and a noiseless point-to-point bit-pipe C of the same capacity has been established by showing that any code that runs on a network N × W can also operate on C × W with asymptotically vanishing error probability. The argument is based on a channel emulation technique over a stacked network where N parallel replicas of the network have been put together to run the code. As illustrated in Fig. 1, the proof can be divided into three steps. In Step I, a network and its N-fold stacked network (consisting of N parallel replicas of the network) are proved to share the same rate region by showing that any code that can run on the network can also run on its stacked network, and vice versa. Therefore we only need to show the equivalence between the stacked network for N and that for C, as illustrated in Fig. 1(a). In Step II, the proof of C ⊆ N employs a channel coding argument over the stack of N channel

replicas as illustrated in Fig. 1(b): A message W of 2N R bits is mapped by the channel encoder α(·) onto a codeword xN of length N, and then transmitted over the N-stack noisy channels, with one symbol on each replica, such that reliable transmission over the noisy stacked network can be realized with arbitrary small error probability as N goes to infinity for all R < C. In Step III, the proof of N ⊆ C is based on a lossy source coding argument as illustrated in

Fig. 1(c): The input sequence xN to the noisy stacked network is first quantized/compressed by a lossy source encoder β(·) into 2N R bits, represented by the message W , which is then transmitted January 20, 2014

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QN

i=1

p(y|x) X Y ⇔ XN N

p(yi |xi ) Y

N

{0, 1}N R

←→ W

W

⇔

{0, 1}R C

(a) The capacity regions of a network and its stacked network are identical QN p(y i |xi ) i=1 ˆ W W yN xN α(·) α−1 (·) (b) The channel coding argument to prove C ⊆ N for all R < C {0, 1}N R xN

β(·)

W

W

β −1 (·)

yN

(c) The lossy source coding argument to prove N ⊆ C for all R > C Fig. 1. A point-to-point noisy channel N = (X , p(y|x), Y) with capacity C = maxp(x) I(X; Y ) and a noiseless point-to-point bit-pipe C of rate R are said to be equivalent if R = C, where the equality comes from the continuity of the capacity region. ˆ ∈ {1, . . . , 2NR }. The input/output of their corresponding stacked networks are xN ∈ X N , y N ∈ Y N , and W, W

through the noiseless stacked network, and the reconstructed sequence y N is selected in such a way that it is jointly typical5 with the transmitted sequence xN , in contrast to the usual distortion measure. The existence of a good lossy source coding codebook for any R > C is proved by a random coding argument, i.e., by showing that the average error probability over the randomly chosen ensemble of codebooks is small. Finally, the equivalence between N of capacity C and C of throughput R can be established when R = C based on the continuity of the capacity region. Readers are referred to [1] for a rigorous and thorough treatment of the proof. The concept of capacity upper and lower bounding models developed in [1] has been extended to independent multi-terminal channels in [2] following similar arguments as illustrated in Fig. 1, and multiplicative and additive gaps between lower and upper bounding models for independent multi-terminal channels have been established. Illustrative upper and lower bounding models 5

As explained in [1], we only focus on jointly typical pairs (xN , y N ) whose associated decoding error probability (assuming

xN is transmitted through the original noisy channel) is smaller than a threshold. Other typical pairs that do not satisfy this condition are expurgated from the channel emulation codebooks.

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for MACs/BCs/ICs involving two sources and/or two destinations have been demonstrated [2]– [4]. Given a noisy network consisting of independent building blocks whose upper and lower bounding models are available, we can replace these building blocks with their corresponding upper (lower) bounding models and then characterize an outer (inner) bound for its capacity region based on the resulting noiseless network models. For wireless networks, however, it may be difficult in general to apply directly the channel emulation technique for construction of bounding models, as the coupled components may involve many transmitting/receiving nodes. For coupled single-hop networks which can be modeled as ICs, the bounding models are difficult to characterize even for the simplest 2 × 2 setup [2]. For coupled multi-hop channels, it is unclear how the channel emulation technique can be extended to incorporate the interaction among different transmitting-receiving pairs across different layers. Although one may apply the cut-set bound [21] to construct upper bounds for wireless networks, the resulting analysis may become quite involved, as illustrated in [22]–[25], for characterizing upper bounds for small size relay networks. Moreover, even if we manage to construct bounding models for a specific coupled network, we have to create new bounding models for each different network topology, which makes it unscalable for wireless networks that have diversified communication scenarios and topologies. C. One-shot Bounding Models Instead of exploiting emulation with channel or lossy source coding to construct bounding models as in [1], [2], a class of one-shot bounding tools have been proposed in [6] for independent MACs/BCs. As illustrated in Fig. 2, auxiliary operation nodes are introduced to facilitate separate characterization of the sum rate and individual rates. The channel emulation for sum rate is realized over each block of channel uses as in [1], [2], whilst the channel emulation for individual rates is realized in each instance corresponding to a channel use and hence referred to as “oneshot” approach. While the lower bounding models can be constructed based on achievable rate regions for MACs with independent source nodes and for BCs with non-cooperating destination nodes, the upper bounding models require special treatment, which will be outlined below. 1) Upper Bounding Models for Multiple-Access Channels: For MACs with m transmitters, each with transmitting alphabet Xi , i = 1, . . . , m, we can introduce an auxiliary operation node nI with an independent and orthogonal input channel li from the transmitter carrying Xi to emulate January 20, 2014

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X1 X2

l1 l2

Xm Fig. 2.

lm

ls

Y

nI

ls

X

l1 l2 nI l m

Y1 Y2 Ym

The one-shot bounding models for MACs/BCs with m transmitters/receivers. The white nodes indicated by nI are

auxiliary operation nodes to specify the rate constraints on the sum rate and on individual rates. All the channels are noiseless bitpipes and independent from others. The one-shot bounding models are fully characterized by the rate vector (Rls , Rl1 , . . . , Rlm ), where Rli is the rate of the noiseless bit-pipe li .

the individual rates, and one output channel ls to the destination node to emulate the sum rate. We then replace all the independent channels with noiseless bit-pipes and define a rate vector (Rls , Rl1 , . . . , Rlm ) to describe the rates of channels (ls , l1 , . . . , lm ). As the one-shot bounding model is fully characterized by this rate vector, we shall use it to represent the corresponding upper bounding model. One way to emulate the MAC channel is to let each input channel li carry exactly what the corresponding source node transmits, hence requiring a noiseless bit-pipe of rate Rli ≥ log(|Xi |), where |X | is the cardinality of the alphabet set X . The auxiliary node then combines all the inputs in such a way that the output signal at the destination node is exactly the same as in the original network. By the network equivalence theory for point-to-point channels [1], successful emulation of the original channel requires Rls ≥ RM AC ,

max

p(x1 ,...,xm )

I(X1 , . . . , Xm ; Y ).

(6)

Hence, we can construct the upper bounding model as C u,M AC,1 = (RM AC , log(|X1 |), . . . , log(|Xm |)).

(7)

Although C u,M AC,1 is tight on sum rate in the sense that there are some kind of networks where the sum rate constraint RM AC is tight, the constraints on individual rates are somewhat loose. Alternatively, one may tighten the individual rate constraints but relax the sum rate constraint. Denoting zi as the channel output of li , and letting the auxiliary node emulate the output at the destination node exactly via a predefined function g(·), i.e., y = g(z1 , z2 , . . . , zm ),

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the transition probability p(y|x1, . . . , xm ) and all its marginal distributions can then be exactly emulated as follows p(y|x1 , . . . , xm ) =

X

m Y

z1 ,...,zm : i=1 y=g(z1 ,...,zm )

p(zi |xi ).

(9)

We can therefore formulate an alternative upper bounding model as C u,M AC,2 = (log(|Y|), R1, . . . , Rm ),

(10)

Ri , max I(Xi ; Zi ).

(11)

where for i = 1, . . . , m, p(xi )

For Gaussian MACs, one way to construct the upper bounding model C u,M AC,2 is based on a noise partitioning approach, i.e., the additive noise at the destination is partitioned into independent parts and allocated to each of the individual channels. For a Gaussian MAC with two transmitters, as demonstrated in [6], the corresponding upper bounding model is     γ1  1 γ2 1 , log 1 + , C u,M AC,2 = log(|Y|), log 1 + 2 α 2 1−α

(12)

is the noise power partitioning parameter chosen to minimize the total input rates    1 γ1  1 γ2 Rs = log 1 + + log 1 + . 2 α 2 1−α

(13)

where γi is the effective link SNR6 at the receiver when only transmitter i is active, and α ∈ (0, 1)

For binary symmetric MAC channels with distortion parameter ǫ, the corresponding distortion ǫi for channel li should satisfy ǫ = ǫ1 (1 − ǫ2 ) + ǫ2 (1 − ǫ1 ).

(14)

2) Upper Bounding Models for Broadcast Channels with Two Receivers: Upper bounding models for BCs with two receivers have also been constructed in [6]. Similarly to the upper bounding models for MACs, there are also two different bounding models for BCs, i.e., C u,BC,1 = (RBC , log(|Y1 |), log(|Y2 |)), 6

(15)

The noise power is assumed to be 1 throughout this paper.

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where RBC , max I(X; Y1 , Y2),

(16)

C u,BC,2 = (log |X |, R1, R2 ),

(17)

Ri , max I(X; Yi ).

(18)

p(x)

and

where for i = 1, 2, p(x)

Note that C u,BC,2 is only valid when the noise at two receivers are independent, i.e., the transition probability can be factorized as p(y1 , y2 |x) = p(y1 |x)p(y2 |x). 3) Gap in One-shot Bounding Models: The gap between the upper and lower bounding models for Gaussian channels and for binary symmetric channels have been examined in [6], where a gap less than 1/2 bit per channel use has been established for MACs with two transmitters and BCs with two receivers. III. B OUNDING M ODELS

FOR

N ON - COUPLED N ETWORKS

For non-coupled networks, which can be decomposed into independent MACs/BCs and pointto-point channels, we first construct upper and lower bounding models for MACs/BCs, which can then be used to substitute their noisy counterparts in construction of noiseless bounding models for the original noisy network. To give a full description of all the rate constraints on any subset of transmitters/receivers, the upper and lower bounding models for independent MACs/BCs with m transmitters/receivers need to consist of (2m −1) rate constraints. However, such an approach is not scalable as m can be quite large in many practical scenarios. Instead, we introduce a rate vector of length up to (m+1) to specify our upper and lower bounding models. A. Upper Bounding Models for Independent MACs and BCs For independent MACs/BCs with m transmitters/receivers, our upper bounding models only contain constraints on each of the maximum allowed individual rate Ri , i = 1, . . . , m, and the

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total sum rate. All the constraints on subsets of individual rates, i.e., X R(S) , Ri , S ⊂ {1, . . . , m}, and |S| ≥ 2,

(19)

i∈S

are omitted to keep the structure illustrated in Fig. 2, which results in a looser but simpler upper bound. The benefits of keeping the simple structure are two fold: they can be easily and straightforwardly extended to BCs/MACs with m receivers/transmitters at low complexity; they facilitate our proposed channel decoupling method in a natural way for constructing the capacity upper bounds for coupled networks. For general BCs, we extend the one-shot upper bounding models developed in [6] to scenarios with m > 2 receivers and improve the bounding model C u,BC,1 on individual rates. For Gaussian

MACs, we improve the bounding models by generalizing the noise partitioning approach developed in [6]. 1) MACs with More than Two Transmitters: There are two one-shot upper bounding models for MACs with m transmitters: C u,M AC,1 = (RM AC , log(|X1 |), . . . , log(|Xm |)),

(20)

C u,M AC,2 = (log(|Y|), R1 , . . . , Rm ).

(21)

For Gaussian MAC with m transmitters, C u,M AC,1 is fully characterized by the sum rate constraint  !2  m X √ 1 γi  , RM AC = log 1 + (22) 2 i=1

and C u,M AC,2 can be determined by extending the noise partitioning approach [6]. Given effective link SNR γi , i = 1, ..., m, we construct a new upper bounding model parameterized by α ∈ (0, 1) as follows ′ C u,M AC,new (α) = (Rs (α), R1′ (α), . . . , Rm (α)),

(23)

where ! Pm √
2 γ + 1 − α 1 i i=1 , Rs (α) = log 1 + 2 α   γi 1 ′ , i = 1, . . . , m, Ri (α) = log 1 + 2 αi

January 20, 2014

(24) (25)

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with noise partitioning parameters αi > 0 and m X i=1

αi = 1 − α.

Note that for Gaussian MACs, C u,M AC,new (α) can include the previous two upper bounding models as special cases by allowing rate constraint of infinite (thus unbounded) capacity: setting α = 1 will lead to C u,M AC,1 and setting α = 0 will lead to C u,M AC,2 since the cardinality of a Gaussian signal is unbounded. One way to determine the noise partitioning parameters αi , i = 1, . . . , m is to solve the following optimization problem min

α1 ,...,αm

subject to

m X i=1

m X i=1

  γi , log 1 + αi αi = 1 − α,

(26)

αi > 0. This is a convex optimization problem whose solution can be explicitly found by Lagrangian methods [26] as follows (see Appendix A for details), p γi (γi + 4µ) − γi ∗ , i = 1, . . . , m, αi = 2 where µ satisfies

m

 1 X p γi (γi + 4µ) − γi = 1 − α. 2 i=1

(27)

Although solving this problem in closed-form is challenging, as the specific value depends both on α and the relative magnitude of all SNRs, its upper and lower bounds can be determined as shown by Lemma 1 below. Since the LHS of (27) is monotonously increasing with respect to µ, it is therefore simple to find µ numerically by evaluating (27) within the region defined by Lemma 1. Lemma 1: Given α ∈ [0, 1] and γi > 0, i = 1, ..., m, the µ defined by (27) is bounded by 1 − α (1 − α)2 1 1 − α (1 − α)2 1 P + + , ≤µ≤ m m m m2 mini γi i γi

(28)

where both equalities hold if and only if γ1 = . . . = γm . Proof: See Appendix B.

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X1

X2 Fig. 3.

R1′ (α) Rs (α)

Y

R2′ (α)

I(X1 ; U) X1 log(|X1 |)

Y I(X1 , X2 ; Y |U) log(|X2 |)

X2

Upper bounding models for Gaussian MAC with 2 transmitters: the general model C u,M AC,new (α) with α ∈ [0, 1]

(left) and the model developed in [2, Theorem 6] with p(u, y|x1 , x2 ) = p(u|x1 )p(y|x1 , x2 ) (right).

Remark 1: The parameterized bounding model C u,M AC,new (α) provides a tradeoff between the bounding accuracy on the sum rate and on each of individual rates. If we are interested in a tighter bound on any individual rate, say on Rli , we can partition the noise by setting α = 0 and αi = 1, which leads to a tighter constraint Ri =

1 2

log(1 + γi ) on rate Rli but unbounded

constraints on all other individual rates and the sum rate. If we are only interested in a tighter bound on the sum rate, setting α = 1 will give us a tight bound RM AC on the sum rate but unbounded constraints on all individual rates. From Lemma 2 below, we can see that the freedom of adjusting α ∈ [0, 1] in the optimized noise partition (26) can not improve the sum rate constraint RM AC . This is intuitive as RM AC is achievable when all the source nodes can fully cooperate. P ′ Lemma 2: Denote Rs′ (α) = m i=1 Ri (α). Given γ1 , . . . , γm > 0, for any α ∈ [0, 1], we have min{Rs (α), Rs′ (α)} ≥ RM AC ,

(29)

with equality when α = 1. Proof: See Appendix C. For Gaussian MAC with 2 transmitters, it is interesting to compare our model C u,M AC,new (α) to the upper bounding model developed in [2, Theorem 6], as illustrated in Fig. 3. On one hand, setting α=1 in C u,M AC,new (α) (i.e., C u,M AC,1) gives an equivalent upper bounding model as in [2, Theorem 6] by setting U = ∅, leading to the same sum rate constraint RM AC . On the other hand,

as shown by Lemma 3 below, setting α=0 in C u,M AC,new (α) (i.e., C u,M AC,2) leads to a tighter bound on R1 but looser constraints on R2 and on the sum rate compared to [2, Theorem 6]. Lemma 3: Compared to [2, Theorem 6], C u,M AC,new (α = 0) has a tighter constraint on R1 but looser constraints on R2 and on the sum rate. Proof: By partitioning the noise into independent parts Z1 and Z2 such that Y = X1 + January 20, 2014

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X2 + Z1 + Z2 , and then setting U = X1 + Z1 and V = X2 + Z2 , the rate region obtained from C u,M AC,new (α = 0) = (∞, R1′ , R2′ ) is    > R1′ = I(X1 ; U),   R1

R2 > R2′ = I(X2 ; V ),     R1 +R2 > R′ + R′ = I(X1 ; U) + I(X2 ; V ), 1 2

(30)

R2 > I(X1 , X2 ; Y |U),     R1 +R2 > I(X1 ; U) + I(X1 , X2 ; Y |U).

(31)

whereas the rate region given by [2, Theorem 6] is    > I(X1 ; U) + I(X1 , X2 ; Y |U),   R1

It is easy to see that C u,M AC,new (α = 0) has a tighter bound on R1 as I(X1 , X2 ; Y |U) ≥ 0. On the other hand, we have I(X1 , X2 ; Y |U) = H(Y |U) − H(Y |X1 , X2 , U) = H(Y − U|U) − H(Z2) = H(V |U) − H(V |X2 ) ≤ H(V ) − H(V |X2 ) = I(X2 ; V ),

(32)

where the equality holds if and only if X1 and X2 are independent. We can therefore conclude from (30) and (31) that C u,M AC,new (α = 0) has looser bounds on R2 and on the sum rate as X1 and X2 can be arbitrarily correlated. Furthermore, if we are only interested in a tight bound on sum rate, by Lemma 4 below, we can see that choosing a non-trivial auxiliary random variable U in [2, Theorem 6] can not improve the sum rate constraint either. Lemma 4: Given p(u, x1 , x2 , y) = p(u|x1)p(x1 , x2 )p(y|x1, x2 ), we have I(X1 ; U) + I(X1 , X2 ; Y |U) ≥ I(X1 , X2 ; Y ),

(33)

with equality if and only if I(X1 , X2 ; U|Y ) = 0. Proof: See Appendix D

January 20, 2014

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16

2) BCs with More than Two Receivers: The upper bounding model for broadcast channels with m receivers can be generalized straightforwardly from [6] as follows C u,BC,1 =(RBC , log(|Y1 |), . . . , log(|Ym |)),

(34)

C u,BC,2 =(log(|X |), R1, . . . , Rm ),

(35)

RBC , max I(X; Y1, . . . , Ym ),

(36)

where p(x)

Ri , max I(X; Yi ), i = 1, . . . , m.

(37)

p(x)

Note that C u,BC,1 is a valid upper bound for any channel transition function p(y1 , . . . , ym |x) whereas C u,BC,2 is only valid for BC with independent noise components at receivers, i.e., when the transition probability can be factorized as p(y1 , . . . , ym |x) =

m Y i=1

p(yi |x).

Below we construct step-by-step a new upper bounding model by combining the point-to-point channel emulation technique developed in [1] with the Covering Lemma and the Joint Typicality Lemma [27]. New bounding models for BC with m ≥ 2 receivers: Let [l1 , l2 , . . . , lm ] denote a permutation of the m receivers and [Y1 , Y2 , . . . , Ym ] be their corresponding channel outputs, whose dependence is characterized by the channel transition function p(y1 , . . . , ym|x). The new bounding model is represented by C u,BC,new = (Rs , Rl1 , Rl2 , . . . , Rlm ),

(38)

where Rs is the sum rate constraint from the transmitter to the auxiliary node nI , and Rlk is the rate constraint from nI to receiver lk . To simplify notation, for k=1, . . . , m, let Y[1:k] represent the sequence of k random variables {Y1 , Y2, . . . , Yk } and denote the sequence of k integers (w1 , w2 , . . . , wk ) by w[1:k] . As previously mentioned, the channel emulation is done over a stacked network which consists of N replicas of the original BC. (N )

Step I: Fix a channel input distribution pX (x). As defined in [1], let Aˆǫ (X) be a subset of (N ) (N ) the “classical” typical set Tǫ (X) [27] such that for any xN ∈ Aˆǫ (X) as the input to the BC,

N the probability that the corresponding output sequences {y1N , y2N , . . . , ym } are not jointly typical

January 20, 2014

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17

with xN is smaller than a predefined threshold. Furthermore, let pY[1:k] (y1 , . . . , yk ), k=1, . . . , m, be marginal distributions obtained from p(x, y1 , . . . , ym) = p(y1 , . . . , ym |x)pX (x), and define a series of conditional distributions as follows   0, p(yk+1|y1 , . . . , yk ) , pY[1:k+1] (y1 ,...,yk+1 )  , pY[1:k] (y1 ,...,yk )

if pY[1:k] (y1 , . . . , yk ) = 0, otherwise.

′

(39) ′

Step II: Generate independently at random 2N R1 sequences {y1N (w1 ) : w1 =1, . . . , 2N R1 }, Q (N ) each according to i pY1 (y1,i ). For any sequence xN ∈ Aˆǫ (X), according to the Covering

Lemma [27],

  ) N R′1 lim P r (xN , y1N (w1 )) ∈ Tǫ(N (XY ) for some w ∈ [1 : 2 ] = 1, 1 1 1

N →∞

(40)

if R1′ > I(X; Y1 ) + δ1 (ǫ1 ) for some ǫ1 > ǫ > 0 and δ1 (ǫ1 ) > 0 that goes to zero as ǫ1 → 0.

Following the channel emulation argument [1], we define a mapping function α1 (xN ) as   w , if ∃w s.t. (xN , y N (w )) ∈ Tǫ(N ) (XY ), 1 1 1 1 1 1 α1 (xN ) = (41)  1, otherwise.

If there is more than one sequence that is jointly typical with xN , then α1 (xN ) chooses one of them uniformly at random. ′

Step III: For each sequence y1N (w1 ), generate independently 2N R2 sequences {y2N (w1 , w2 ) : Q ′ w2 =1, . . . , 2N R2 }, each according to i p(y2,i |y1,i (w1 )), where p(y2 |y1 ) is defined in (39). Given

w1 = α1 (xN ) which implies


) P r (xN , y1N (w1 )) ∈ Tǫ(N (XY1 ) → 1 as N → ∞, 1

(42)

′

and according to the Joint Typicality Lemma [27], for all w2 ∈ [1 : 2N R2 ],
) P r (xN , y1N (w1 ), y2N (w1 , w2 )) ∈ Tǫ(N (XY Y ) ≥ 2−N (I(X;Y2 |Y1 )+δ2 (ǫ2 )) , 1 2 2

(43)

for ǫ2 > ǫ1 and some δ2 (ǫ2 ) > 0 that goes to zeros as ǫ2 → 0. Then we have   ′ ′ ) P r ∃w2 ∈ [1 : 2N R2 ] s.t. (xN , y1N (w1 ), y2N (w1 , w2 )) ∈ Tǫ(N (XY Y ) ≥ 2N (R2 −I(X;Y2 |Y1 )−δ2 (ǫ2 )) , 1 2 2

(44)

which goes to 1 as N → ∞ if R2′ > I(X; Y2|Y1 ) + δ2 (ǫ2 ). We define a mapping function α2 (xN , w1 ) as follows

  w, 2 α2 (xN , w1 ) =  1,

January 20, 2014

(N )

if ∃w2 s.t. (xN , y1N (w1 ), y2N (w1 , w2 )) ∈ Tǫ2 (XY1 Y2 ),

(45)

otherwise. DRAFT

18

If there is more than one candidate that satisfies the joint typicality condition, then α2 (·) chooses one of them uniformly at random. N Step IV: For k = 3, . . . , m, we treat the set of sequences {y1N (w1 ), . . . , yk−1 (wk−1 )} together

as one unit and repeat Step III, which generates the corresponding sequences {ykN (w[1:k−1], wk ) : ′

wk =1, . . . , 2N Rk }, the mapping function αk (xN , w[1:k−1]), and the rate constraint Rk′ > I(X; Yk |Y[1:k−1]) + δk (ǫk ),

(46)

where ǫk > ǫk−1 , and δk (ǫk ) > 0 that goes to zeros as ǫk → 0. Step V: Define a channel emulation codebook ′

{ykN (w1 , . . . , wk ) : k=1, . . . , m, wk =1, . . . , 2Rk },

(47)

and the associated encoding function α(xN )=[α1 (·), . . . , αm (·)] and the decoding function αk−1 (w[1:k]) for receiver lk , k=1, . . . , m. For any input xN , α(xN ) generates a sequence P ′ (w1 , w2 , . . . , wm ) of N m i=1 Ri bits that are transmitted from the transmitter of the BC to the P auxiliary node nI , which then forwards (w1 , . . . , wk ) (of N ki=1 Ri′ bits) to receiver lk . At receiver lk , the decoding function αk−1 (w[1:k] ) selects a sequence from the codebook {ykN } based

on the received information bits, i.e., αk−1 (w1 , . . . , wk ) = ykN (w1 , . . . , wk ). Note that the rate constraints in (46) should be satisfied for k=1, . . . , m, and for all pX (x). Let N → ∞ and ǫm → 07 , we can specify the rate constraints in the upper bounding model (38) as follows Rlk =

k X i=1

and

Ri′ = max I(X; Y1, . . . , Yk ), k=1, . . . , m, p(x)

Rs = Rlm = max I(X; Y1, . . . , Ym ). p(x)

(48)

(49)

The second equality in (48) comes from the fact that I(X; Y1, . . . , Yk ) = I(X; Y1) + I(X; Y2 |Y1) + . . . + I(X; Yk |Y1, . . . , Yk−1),

(50)

and the first equality in (49) comes from our design of the emulator as specified in Step V. 7

Since 0 < ǫ < ǫ1 < . . . < ǫm , letting ǫm → 0 implies that all of them go to zero.

January 20, 2014

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19

Compared to the upper bounding model C u,BC,1 specified in (34), the new model C u,BC,new maintains the tight sum rate constraint RBC as specified in (36) and meanwhile improves all the individual rate constraints. Remark 2: There are in total m! different permutations of l1 , . . . , lm , each leading to a different upper bounding model following our construction method. For each of these upper bounding models, the sum rate constraint and one of the individual rate constraints are tight. Depending on the needs, we can select a specific permutation to design the upper bounding model. Remark 3: For BC with m = 2 receivers, the proposed upper bounding model has two different layouts C u,BC,a = (RBC , R1 , RBC ) and C u,BC,b = (RBC , RBC , R2 ), where the latter turns out to be equivalent to the upper bounding model developed in [2, Theorem 5]. This is not surprising as the channel emulation codebook {y1N , y2N } used in our construction is generated in the same way as in [2, Theorem 5]: superposition encoding. Note that the proof in [2, Theorem 5], restricted for BC with m = 2 receivers, provides explicit error analysis. In contrast, our construction is valid for general cases but only claims that the error probability can be made arbitrarily small with the help of the Covering Lemma and the Joint Typicality Lemma. As a result, a discretization procedure [27, Chp. 3.4.1] is necessary when extending our results from finite-alphabet channels to more general (e.g., Gaussian) channels8 . B. Lower Bounding Models for Independent MACs and BCs Our lower bounding models for MACs/BCs are constructed directly based on some operating points within the achievable rate region assuming no transmitter/receiver cooperation. However, the structure of the resulting lower bounding models are quite different for MACs and BCs. The main difference between MACs and BCs is the encoding process. When there is no transmitter/receiver cooperation, distributed encoding is performed in MACs while centralized encoding is done in BCs. As a consequence, in MAC setups, only one rate constraint is needed for each point-to-point bit-pipe to fully describe any operation point within the rate region. In BCs, each of the private messages dedicated for one specific receiver may also be decoded by other receivers. Such “overheard” messages (the common messages) should be reflected in the 8

As described in [27, Chp. 3.4.1], the transition function of the continuous-alphabet channel should be “well-behaved” to

facilitate the discretization and quantization procedure. We shall not mention this explicitly in the subsequent sections when applying our results to Gaussian channels. January 20, 2014

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20

TABLE I A LL POSSIBLE RATE CONSTRAINTS IN THE LOWER BOUNDING MODEL FOR BROADCAST CHANNELS WITH m RECEIVERS .

Rate Ri , i =

1

2

3

···

2m − 1

i : {0, 1}m

0 · · · 001

0 · · · 010

0 · · · 011

···

1 · · · 111

{Dn }, n =

1

2

2, 1

···

m, · · · , 3, 2, 1

rate region, which requires the usage of point-to-points bit-pipe (i.e., hyper-arc) in the lower bounding model. 1) MACs with More than Two Transmitters: The lower bounding models can be constructed by choosing an operating point in the capacity region of the MAC assuming independent sources. We choose the point that can be achieved by using independent codebooks at transmitters and successive interference cancellation decoding at the receiver, from the strongest received signal to the weakest. For Gaussian MAC with m transmitters, each with received SNR γi , i = 1, . . . , m, the following sum rate is achievable Rl,s =

1 log 1 + 2

m X i=1

γi

!

.

(51)

2) BCs with More than Two Receivers: For BCs, each of the private messages dedicated for one specific receiver may also be decoded by other receivers, and such overheard messages can be useful when the BCs are part of a larger network. To model such message overhearing, we need to introduce point-to-points bit-pipes (i.e., hyper-arcs) to represent multicast rate constraints. For BCs with m receivers {Dn , n = 1, . . . , m}, there are in total (2m −1) subsets of receivers, each corresponding to a unique rate constraint. As illustrated in Table I, we denote Ri as the rate constraint corresponding to successful decoding at receivers indicated by the locations of ‘1’ in the length-m binary expression of the index i. For example, R3 is the constraint for the multicast rate to receivers D2 and D1 , and R2m −1 is the constraint for multicast rate to all receivers. Depending on the channel quality, we represent the lower bounding mode by a vector9 R which contains one sum rate constraint (denoted by R0 ) and up to m constraints10 9

For each Ri in R we also need to store its index i to specify the receiving subset.

10

For statistically degraded m-receiver BCs (e.g., Gaussian BCs), m constraints are sufficient by creating a physically degraded

channel via proper coding schemes, and the rate loss will vanish in low SNR regime [5]. For non-degraded channels, we only focus on the first m most significant non-zero constraints. January 20, 2014

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21

from Table I. We illustrate this by an example of Gaussian BCs with m receivers. Example: Gaussian BCs with m Receivers Let γi be the effective link SNR at receiver Di , i = 1, . . . , m. Without loss of generality, assuming γ1 ≤ γ2 ≤ ... ≤ γm , we divided the total information into m distinct messages {Wi , i = 1, . . . , m}. By superposition coding of Wi with P power allocation parameters βi ∈ [0, 1], m i=1 βi =1, at the transmitter, and successive interference

cancellation at each receiver11 , successful decoding of Wi can be realized at a set of receivers {Dn , n = i, . . . , m} with multicast/unicast rate R2m −2i−1 =

1 βi γi P log 1 + 2 1 + γi m j=i+1 βj

!

.

(52)

For example, successful decoding of W1 can be realized at all receivers with a multicast rate of R2m −1 , and successful decoding of Wm can only be realized at receiver Dm with a unicast rate of R2m−1 . The resulting rate vector is therefore R = [R0 , R2m −2i−1 : i = 1, . . . , m], where the sum-rate constraint R0 is m m X X 1 log R2m −2i−1 = R0 = 2 i=1 i=1

! P 1 + γi m j=i βj P 1 + γi m j=i+1 βj ! P m β 1 + γi m 1 1X j Pj=i = log(1 + γ1 ) + . log 2 2 i=2 1 + γi−1 m j=i βj

The last equality comes from the fact that m X

(53)

(54)

(55)

βi = 1.

i=1

Since γi−1 ≤ γi , the function 1+γi 1+γi−1

1+xγi 1+xγi−1

is monotonically increasing on x ∈ [0, 1], with its maximum

achieved when x = 1. It is simple to show that R0 ≤

1 log(1 + γm ), 2

(56)

where the equality is achieved when βm = 1 (i.e, βi = 0 for all i 6= m). Remark 4: Note that power allocation at the transmitter of a BC allows elimination of weakest receivers. For example, by setting β1 = β2 = 0 in (52) the weakest two receivers D1 and D2 will have nothing to decode and hence be removed from the set of destinations. 11

Alternatively, one can encode W1 to Wm successively by dirty paper coding [28] and use a maximum likelihood decoder

at each receiver. January 20, 2014

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22

C. Gaps between Upper and Lower Bounding Models for Gaussian MACs and BCs For Gaussian MACs with m transmitters, the sum rate is upper bounded by RM AC given by (22), and lower bounded by Rl,s given by (51). The gap between the upper and the lower bounds on sum rate, measured in bits per channel use, is therefore bounded by P √ 2  γi ) 1+( m 1 i=1 P ∆M AC = RM AC − Rl,s = log m 2 1 + i=1 γi Pm   1 + m i=1 γi 1 1 Pm ≤ log < log(m), 2 1 + i=1 γi 2

(57)

where the first inequality comes from Jensen’s inequality based on the convexity of the function f (x) = x2 . Hence, for Gaussian MACs with transmitters in isolation, feedback and transmitter cooperation can increase the sum capacity by at most

1 2

log(m) bits per channel use.

For Gaussian BCs with m receivers, the sum rate is lower bounded by R0 given by (56) and upper bounded by RBC

! m X 1 γi . = log 1 + 2 i=1

(58)

Note that RBC can be achieved only when full cooperation among all receivers is possible. The gap between the upper and the lower bounds on the sum rate is therefore P     1+ m 1 1 1 + mγm 1 i=1 γi ≤ log < log(m), ∆BC = RBC − R0 = log 2 1 + γm 2 1 + γm 2

(59)

where the first inequality comes from the assumption γi ≤ γm for all i. Hence, for m-receiver Gaussian BCs with all receivers in isolation, feedback and receiver cooperation can increase the sum capacity by at most

1 2

log(m) bits per channel use.

The gap between upper and lower bounding models becomes considerably smaller at low SNR or when the SNR for each link diverges. For example, with γ1 = 1, γ2 = 2, γ3 = 100 (e.g, 0, 3, 20dB, respectively), the gaps (measured in bits per channel use) are ! √ 2 1 + (1 + 2 + 10) 1 ≈ 0.29, ∆M AC = log 2 1 + 1 + 2 + 100   1 1 + 1 + 2 + 100 ∆BC = log ≈ 0.02, 2 1 + 100 which are much smaller than

January 20, 2014

1 2

log(3) ≈ 0.79.

DRAFT

23

IV. B OUNDING M ODELS

FOR

C OUPLED N ETWORKS

Capacity bounding models developed in [2]–[4], [6] and extensions presented in Sec. III are all designed for networks with non-coupled MACs/BCs. In wireless networks, however, a signal dedicated for one receiver may also be overheard by its neighbors, owing to the broadcast nature of wireless transmission. A transmit signal can be designed for multiple destinations (as in BC) and the received signal may consist of signals from several source nodes (as in MAC) and thus interfere with each other. Although such dependence among coupled BCs and MACs has been partially treated in [2] by grouping coupled transmitter-receiver pairs together as ICs, whose bounding models require up to m(2m −1) bit-pipes for IC with m transmitter-receiver pairs, the whole family of multi-hop channels (e.g., relay channels) remains untreated. Inspired by the idea of separate sum and individual rate constraints [6], we choose to incorporate dependent transmitter-receiver pairs into coupled BCs and MACs. With the help of a new channel decoupling method, we can decompose any memoryless networks into independent point-to-point channels, MACs, BCs, and coupled MACs/BCs. When a noisy connection between two nodes is part of both a BC and a MAC, the bounding models for corresponding BC and MAC will be updated by taking their dependence into account. A. Channel Decoupling Given a memoryless noisy network, we first identify all the BCs and MACs. For each BC with m receivers, we introduce an auxiliary node with one input channel connected to the broadcasting transmitter to describe the constraint on sum rate, and m output channels each connected to a receiver to reflect constraints on individual rates. For each MAC with n transmitting sources, we introduce an auxiliary node with n input channels each connected to a transmitter to describe individual rate constraints, and one output channel connected to the receiver for modelling the sum rate. If a node is both the receiver of a MAC and the transmitter of a BC, we introduce two auxiliary nodes, one for the receiving functionality and one for the transmitting functionality, interconnected by a directed bit-pipe of infinite capacity from the reception part to the transmission part. Assuming the self-interference can be perfectly cancelled, there will be no backward connection from the transmission part to the reception part. If a wireless connection between two nodes is part of both a BC and a MAC (i.e., the BC and the MAC are coupled), we can always identify an independent multiple-input multiple-output January 20, 2014

DRAFT

24

X1

Y1

X1

Xi

Yj ⇒ Xi

Xn

Ym

Xn

d1 ls dp

nI

Y1

l1 lj di

ds nJ lq

Yj Ym

The noisy connection Xi → Yj is fully described by the smallest independent multiple-input multiple-output subQ Qm network N = ( n l=1 Xl , p(y|x), k=1 Yk ) that contains it. After channel decoupling of N and then applying the one-shot Fig. 4.

bounding models to the BC with input Xi and the MAC with output Yj , two rate constraints will be introduced on the same noisy link Xi → Yj : the bit-pipe lj is due to the BC and the bit-pipe di comes from the MAC. Both lj and di will be replaced by a single bit-pipe from auxiliary node nI to auxiliary node nJ when constructing the noiseless bounding networks.

sub-network that contains the coupled BC/MAC. Such independent sub-network should be the smallest in size in the sense that it does not contain any independent channels, i.e., its transition function cannot be partitioned into non-trivial product format as defined in (4). As illustrated in Q Q the left part of Fig. 4, an independent sub-network N = ( nl=1 Xl , p(y|x), m k=1 Yk ) containing

n transmitting nodes and m receiving nodes, is fully described by a transition probability p(y|x) , p(y1 , . . . , yj , . . . , ym |x1 , . . . , xi , . . . , xn ).

Note that p(y|x) does not necessarily indicate a layered structure of the sub-network, rather it is only an abstraction of the input-output dependence of the network (cf. the relay channel with p(y, yr |x, xr )). For i=1, . . . , n and j=1, . . . , m, the noisy connection Xi → Yj is part of both the BC with transmitting signal Xi and the MAC with received signal Yj , which will be decoupled in the sense that the resulting BC/MAC can be fully (hence independently) described by its transaction function derived from p(y|x) as described below in Sec. IV-A1. When constructing bounding models, there will be two individual rate constraints on the bit-pipe corresponding to the connection Xi → Yj , one from the decoupled BC with input Xi and one from the decoupled MAC with output Yj . Since both of the two constraints are introduced to characterize the same individual rate, they should be simultaneously respected when constructing the noiseless bounding models. We shall demonstrate it in Sec. IV-B for the lower bounding models and in Sec. IV-C for the upper bounding models.

January 20, 2014

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25

1) Transaction Functions for Decoupled BCs/MACs: In order to derive the one-shot bounding models for the decoupled BCs/MACs, we need to construct transaction functions to describe the decoupled BCs and MACs based on the transaction function of the sub-network that contains them. As mentioned in Sec. I, in this paper we assume that the distortion components in a noisy coupled network are mutually independent, i.e., the transition probability can be partitioned as p(y|x) =

m Y j=1

p(yj |x),

where p(yj |x) ,

X

p(y|x),

(60)

y /j

is the marginal distribution for Yj given X. After introducing auxiliary nodes, the decoupled MAC with received signal Yj can be fully described by the marginal distribution p(yj |x), which preserves the possibility of source cooperation (allowing all possible p(x) as in N ). We can

therefore construct its upper and lower bounding models based on p(yj |x) by following the techniques developed in Sec. III-A1 and in Sec. III-B1, respectively. Note that we still have a valid upper bound by considering individually each of the decoupled MACs, I(X; Y ) = h(Y ) − h(Y |X) = h(Y ) − ≤

m X j=1

h(Yj ) − h(Yj |X) =

m X

m X j=1

h(Yj |X)

I(X; Yj ),

(61)

(62)

j=1

where the inequality is due to the correlation among Y . The decoupled BC with transmitting signal Xi , however, cannot be fully described by its marginal distribution p(y|xi ) ,

X x/i

p(y|x)p(x/i |xi ),

(63)

since p(y|xi ) is determined not only by the channel transition function p(y|x) itself, but also by all the possible distributions of channel inputs X /i . Therefore p(y|xi ) only provides a description of the average behavior of the correlation among different channel outputs but erases both the explicit dependence of Y on a specific channel input and the interaction among different channel

January 20, 2014

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26

inputs X. To preserve the structure of the original coupled network, we define a transition function for the decoupled BC as p˜L (˜ y |xi ) , p(y|xi , x/i = ∅),

(64)

˜ where x/i = ∅ represents the scenario that there is no input signal12 except Xi = xi , and y denotes the corresponding output signal. The transition function defined in (64) will be used in construction of lower bounding models for the decoupled BC following the techniques developed in Sec. III-B2. Note that the definition (64) accommodates the explicit dependence of channel outputs on the input signal Xi = xi but erases source cooperation. It therefore enables efficient and simple characterization of the individual and sum rate constraints (optimized over p(xi )) which are otherwise difficult to obtain (optimized over p(x)). Furthermore, it preserves possible noise correlation in Y which will be useful in future work. To construct upper bounding models for decoupled BCs, we introduce a group of axillary variables Z = {Zi,j |i=1, . . . , n, j=1, . . . , m} and a predefined function g(z) such that p(y|x) =

n X Y

z:y=g(z) i=1

p(z i |xi ),

(65)

where z i = [zi,1 , . . . , zi,m ] is the corresponding “output” vector of the input signal Xi = xi . The corresponding upper bounding models for the decoupled BC with transition probability p(z i |xi ) can be therefore constructed following the techniques developed in Sec. III-A2. Given Y = g(Z) and the fact that Xi − Xj − Z j forms a Markov chain for all i 6= j, we have I(X; Y ) ≤ I(X; Z 1 , . . . , Z n )

(66)

= h(Z 1 , . . . , Z n ) − h(Z 1 , . . . , Z n |X) X = h(Z 1 , . . . , Z n ) − h(Z i |Xi )

(67)

≤

(69)

(68)

i

=

X i

X

h(Z i ) −

X i

h(Z i |Xi )

I(Xi ; Z i ),

(70)

i

where the equality in (68) comes from (65). Hence focusing on each individual decoupled BC still gives us a valid upper bound on the original coupled network. 12

With additive noise this implies that we force x/i = 0 even if 0 6∈ Xj , j 6= i.

January 20, 2014

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27

Therefore, the decoupled MAC with receiving signal Yj is fully described by (60) and the decoupled BC with transmitting signal Xi is described by (64) when constructing lower bounding models and by (65) when constructing upper bounding models. 2) Channel Decoupling via Noise Partition for Gaussian Networks: For Gaussian coupled networks, the auxiliary variables z = {zi,j |i=1, . . . , n, j=1, . . . , m} can be determined by the noise partition approach as in Sec. III-A1. Let H = [hi,j ]n×m be the matrix of channel coefficients such that y = H T x + w = [h1 , . . . , hn ]x + w,

(71)

where hi is the column vector of channel coefficients from source node i to all receiving nodes, x is the transmitting vector with average power constraint E[|xi |2 ] ≤ Pi , and w is the vector of noise with unit variance. We can therefore partition the noise components into independent terms such that z i = hi xi + w i ,

(72)

where w i = [wi,1 , . . . , wi,m ]T is the vector of partitioned noise components w.r.t. the input signal xi and the variance of wi,j is denoted by αi,j > 0. Define γi,j = Pi ∗ |hi,j |2 , the noise partition parameters αi,j (and hence zi,j ) can be determined by the following optimization problem ! n m X X γi,j , min log 1 + αi,j α i,j i=1 j=1 n X (73) subject to αi,j = 1, ∀j ∈ {1, . . . , m}, i=1

αi,j > 0.

Note that (73) is a convex optimization problem whose solution can be explicitly found by Lagrangian methods [26] as shown in Appendix E. B. Lower Bounding Models The lower bounding models presented in Sec. III-B, see also [2]–[4], [6], are designed for non-coupled networks assuming isolated source/destination nodes, without taking into account the possibility of signal transmission/receiption by neighboring nodes. When a noisy connection between two nodes is part of both a BC and a MAC, the bounding models for the BC and for

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28

the MAC have to be updated. We demonstrate how the update should be done step by step as follows. Step I: Network Decomposition Apply the channel decoupling method proposed in Sec. IV-A to decompose the coupled network into independent point-to-point channels, independent BCs and MACs, and decoupled pairs of BCs and MACs. Step II: Apply Lower Bounding Models for Point-to-Point Channels, BCs, and Independent MACs We replace each point-to-point channel with a bit-pipe whose throughput equals its capacity [1]. For each BC and each non-coupled MAC, i.e., a MAC where none of its input signals is part of the output of a BC, we replace them with the corresponding lower bounding models as described in Sec. III-B2 and in Sec. III-B1, respectively. Step III: Construct Lower Bounding Models for Decoupled MACs If (some of) the input signals to a MAC are the output signals from BCs, part of the received signals at the MAC receiver cannot be decoded and therefore behaves as interference. The original lower bounding models for a non-coupled MAC, which assumes that all input signals can be decoded, need to be updated based on the sum power of the interfering signals. This can be calculated by taking into account the signal structure of each input source node. We illustrate this procedure by a coupled Gaussian MAC as follows. Example: Gaussian MAC with m Transmitters Q Consider a Gaussian MAC N = ( m i=1 Xi , p(y|x), Y), where Xi (i = 1, . . . , m) is the input

signal generating with SNR γi (incorporating the transmitted signal power and the corresponding channel gain). If Xi can only be observed by the receiver in channel N , all the components of Xi can be fully decoded by the receiver. If Xi is the transmitted signal from a broadcast source node, it may contain components that are not intended to be decoded by the receiver owing to rate and power allocation at the broadcast node, as described in Sec. III-B2 for constructing the lower bounding model for Gaussian BCs. The remaining component of Xi cannot be decoded by the receiver and therefore behaves as interference during the decoding process. We denote the power of the interfering component by Γi , and the exact value can be obtained from the power allocation parameters chosen by the corresponding BC that transmits Xi . We have Γi = 0 if all messages contained in Xi are intended for successful decoding, and Γi = γi if nothing is to be January 20, 2014

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29

decoded. After careful examination of the structure of all the input signals, we can calculate the total power of interfering components contained in Y as follows PI =

m X

Γi ,

(74)

i=1

out of which

PI,i = PI − Γi =

X

Γj

(75)

j6=i

is the amount of interference power introduced by input signals other than Xi . We call PI,i the “extrinsic interference” of Xi . We can now construct the lower bounding model for the MAC N based on the effective SNR, which is defined as γˆi =

γ i − Γi , i = 1, ..., m. 1 + PI

(76)

Step IV: Rate Adjustment for Decoupled BCs Let a noisy connection X-Yi , i = 1, . . . , m, be part both of a m-receiver BC transmitting X and a MAC with received signal Yi , and denote the corresponding link SNR by γi . After Step III, we can obtain by (75) the extrinsic interference power PI,i , caused by input signals other than X in the decoupled MAC with output Yi . Without loss of generality, assuming γ1 ≤ . . . ≤ γm , the rate constraint R2m −2i−1 defined in (52) which corresponds to the multicast rate constraint to a subset of receivers associated with {Yk : k = i, i + 1, . . . , m}, should be adjusted by taking into account the extrinsic interference power {PI,k : k = i, i + 1, . . . , m}. The corresponding

new rate constraint, denoted by R2′ m −2i−1 , is therefore defined as R2′ m −2i−1

γk βi 1 P log 1 + = min k∈{i,i+1,...,m} 2 1 + PI,k + γk m j=i+1 βj

!

.

(77)

The sum-rate constraint should be adjusted accordingly, i.e., R0′ =

m X

R2′ m −2i−1 .

(78)

i=1

Note that the minimum operation in (77) comes from the fact that given γi ≤ . . . ≤ γm we cannot guarantee γi γm ≤ ... ≤ 1 + PI,i 1 + PI,m January 20, 2014

(79)

DRAFT

30

due to the effect of the extrinsic interference caused by decoupled MACs. Here we simply keep the structure of the original lower bounding model unchanged without claiming its optimality. The resulting lower bounding network consists of only noiseless bit-pipes, but it may contain hyper-arcs (point-to-points bit-pipe channels) that carry the same data from one point to multiple points if the original noisy network has broadcast channels. The problem of finding the optimal scheme to manage the data flows over such noiseless networks is in general open. However, there exist many heuristic (and thus suboptimal in general) methods, see [29] for example, for constructing a valid inner bound. C. Upper Bounding Models Given a memoryless coupled noisy network with independent noise, we first apply the channel decoupling method proposed in Sec. IV-A to decompose the coupled network into independent point-to-point channels, independent BCs/MACs, and decoupled pairs of BCs and MACs. For each independent point-to-point connection, we replace it with a bit-pipe of rate equals its capacity. For each MAC/BC, independent or decoupled, we replace it with the corresponding one-shot bit-pipe models developed in Sec. III-A1 and Sec. III-A2. Then, for each pair of decoupled BC and MAC, there will be two rate constraints on the same bit-pipe that connects them. We take the maximum of the two rate constraints as the upper bounding rate constraint for that bit-pipe. According to the max-flow min-cut theorem, the maximum throughput from source to sink can be no larger than the value of the minimum cut in between. For each transmission task (unicast or multiple cast), we identify all the cuts in the resulting upper bounding network (which contains only noiseless point-to-point connections) and calculate the flows across each cut. The resulting capacity region is therefore an outer bound13 for the upper bounding network, and hence also an outer bound for the original coupled noisy network. 13

As shown in [14], the max-flow min-cut theorem is tight on some noiseless networks, which include noiseless networks

associated with single-source multiple-unicast transmission, single-source (two-level) multicast transmission, and multi-source multicast transmission. Therefore the bound we obtained by the max-flow min-cut theorem might be the capacity region for the corresponding upper bounding network.

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31

V. I LLUSTRATIVE E XAMPLES In this section we illustrate our capacity bounding models by several coupled noisy networks and compare the capacity inner and outer bounds obtained based on our bounding models with some benchmarks. Given a coupled noisy network N , we first apply the channel decoupling method proposed in Sec. IV-A to decompose the coupled network into independent point-to-point channels, independent BCs/MACs, and decoupled pairs of BCs and MACs. We then construct a lower bounding model C l,i following the procedure described in Sec. IV-B and an upper bounding model C u,i as in Sec. IV-C. As there are more than one upper (resp. lower) bounding model for every BC/MAC, each of such combinations will result in a noiseless upper bounding model C u,i (resp. lower bounding model C l,i ), whose capacity region serves as a capacity outer (resp. inner) bound for the original noisy network N . We then take the intersection of all the capacity outer bounds, one for each C u,i , to obtain the final (and tighter) outer bound, \ E(N ) ⊆ E(C u,i ),

(80)

i

where E(·) denotes the capacity region of the corresponding network. For the inner bound, we

compute the achievable rate region for each lower bounding model C l,i and then take the convex

hull of all the achievable rate regions to create the final (and tighter) inner bound (with abuse of notation), [ i

E(C l,i ) ⊆ E(N ).

(81)

A. The Smallest Coupled Network: the Relay Channel We first look at the smallest (in size) coupled network – the classical 3-node Gaussian relay channel – as illustrated in Fig. 5(a), which can be modelled as √

Y =

√

γsdX +

Yr =

√

γsr X + Zr ,

γrd Xr + Z,

where γsd, γsr , and γrd are effective link SNRs, Z and Zr are independent Gaussian noise with zero mean and unit variance, and X and Xr are transmitting signals subject to unit average power constraint. Note that the connection between the source node S and the destination node D is part of both the BC starting from S and the MAC ending at D, and therefore need to January 20, 2014

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32

S

γsd γsr

γrd

D

S

R (a) Gaussian relay channel S ′ R1 R4 D ′ D S R6 R0 R3 R5 R

(c) lower bounding model for γsd>γsr

Rs

S ′max{Rl1 , Rd1 }D ′ Rd Rl2 Rd2 R

D

(b) upper bounding model S ′ R′ R4′ D ′ 3 S ′ R6′ D R0 ′ R5′ R2 R

(d) lower bounding model for γsd ∆P > 0 are parameters such that γ2n > · · · > γ21 > γ11 > · · · > γ1n .

(93)

The transmission of S1 is aided by S2 via a Q-ary symmetric channel (Xs → Ys ) such that for all m, k ∈ {0, 1, . . . , Q − 1}

  1 − ξ, P r(Ys = m|Xs = k) =  ξ Q−1

if m = k, if m 6= k.

(94)

We can first decompose (via channel decoupling) the original network into a point-to-point channel, two BCs originating from S1 and S2 , and n MACs ending at each destination node, and then replace them by corresponding upper and lower bounding models. An illustration of the January 20, 2014

DRAFT

37 6 Cut−set bound Equv. upper bound Equv. lower bound Rate of S−R MAC Rate of S−R coop

Sum rate Rs [bits]

5

4

3

2

1

0 −10

Fig. 9.

−5

0 5 10 15 20 average value of channel gain, P [dB]

25

30

Bounds on the sum rate for the scenario with 10 destinations, δP /P = −3dB, and the Q-ary symmetric channel

(Xs → Ys ) with Q = 8 and ξ = 0.1.

resulting lower bounding model for n = 3 is presented in Fig. 8, where the rate constraints of poin-to-point(s) bit-pipes are determined following the process as presented in Sec. IV-B. From Fig. 8 we can see that multicast of W2 can only be achieved via the hyper-arc of rate R7 , while multicast of W1 can be achieved either via the hyper-arc of rate R7′ , or via the collaboration with S2 at rates min{R3′ , R4 } and/or min{R1′ , R6 }. The collaboration from S2 is subject to the rate constraint C12 which is the capacity of the Q-array symmetric channel from S1 to S2 . The bounds on sum rate obtained from upper and lower bounding networks with respect to varying channel quality have been illustrated in Fig. 9 for a scenario with n = 10 destinations. The Q-ary symmetric channel (Xs → Ys ) has parameters Q = 8 and ξ = 0.1, which results in a capacity of C12 = 2.85 bits per channel use [bpcu]. We also plot three benchmarks as references: the rate achieved by transmitting identical signals from S1 and S2 (denoted by “S1S2 coop”), the rate achieved by transmitting independent signals from S1 and S2 (denoted by

“S1-S2 MAC”), and the the cut-set upper bound following the method developed in [22]16 . Our 16

In [22] the cut-set bound is obtained by starting from multi-letter expressions and then combining various inequalities,

average power constraints, and some “properly” chosen auxiliary random variables. Here “properly” is to highlight the fact that it is a kind of art to decide when and where to introduce auxiliary random variables to quantify the possible correlation among transmitting signals, since only proper choice leads to nice upper bounds. See [23]–[25] for extensions of the method developed in [22].

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38

upper bound obtained from noiseless bounding networks is very good17 and it even approaches the capacity (meeting the lower bound provided by S1 -S2 cooperation) in low to medium SNR regions. Our lower bounding models discard the possibility of source cooperation and therefore suffers some performance degradation (less than 0.4 bits18 from the capacity). In high SNR region, it outperforms the two benchmarks since our lower bounding models can make use of the overhead messages to increase the multicast rate of W1 : extra bits of W1 can be transmitted via the collaboration with S2 at rate ∆R1 = min{R3′ , R4 } + min{R1′ , R6 },

(95)

if such operation is permitted by the link from S1 to S2 , i.e., when R4 + R6 < C12 . VI. S UMMARY In this work we have presented capacity upper and lower bounding models for wireless networks, where the upper bounding models consist of only point-to-point bit-pipes while the lower bounding models also contains point-to-points bit-pipes (hyper-arcs). We have extended the bounding models for two-user BCs/MACs to many-user scenarios and established a constant additive gap between upper and lower bounding models. For networks with coupled links, we have proposed a channel decoupling method which can decompose the coupled network into independent point-to-point channels, independent BCs/MACs, and decoupled pairs of BCs and MACs. We have proposed strategies to construct step-by-step upper and lower bounding models for the originally coupled networks. We have demonstrated by examples that the gap between the resulting upper and lower bounds is usually not large, and the upper/lower bound can approach capacity in some setups. The proposed methods for constructing upper and lower bounding models, simple and computationally efficient, can be easily extended to large networks. They therefore, combined with methods calculating the capacity of noiseless networks, provide additional powerful tools for characterizing the capacity region of general wireless networks. 17

In the sense that the gap from the cut-set bound is negligible.

18

As P and n increase, the gap from cut-set bound converges to a constant that depends only on ∆P /P .

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39

A PPENDIX A O PTIMAL N OISE PARTITIONING

FOR

G AUSSIAN MAC S

Following the Lagrangian method [26], the optimal noise partitioning for the optimization problem (26) can be obtained by taking partial derivatives of its Lagrangian X X L= log(1 + γi /αi ) + µ−1 ( αi − (1 − α)), for some µ > 0, i

(96)

i

with respect to αi , i = 1, . . . , m, and setting them to zero. Denoting αi∗ the optimal noise power for αi , we have (αi∗ )2 + γi αi∗ − γi µ = 0,

(97)

which leads to (omitting the negative root as αi∗ > 0) p γi (γi + 4µ) − γi . (98) αi∗ = 2 P The exact value of µ is determined by the condition m i=1 αi = 1 − α, which yields (27). A PPENDIX B P ROOF

OF

L EMMA 1

The upper bound is obtained by contradiction. Assuming µ >

1−α m

2

+ (1−α) m2

1 , mini γi

(27) is evaluated as follows s      m 2 1 X γj (1 − α)2 1−α LHS > − 1 − γj  + γj + 2 2 j=1 m m2 mini γi s  2  m X 1 1−α  ≥ − γj  γj + 2 2 j=1 m

the LHS of

(99)

= 1 − α,

which contradicts to the equality constraint stated in (27). Therefore we have 1 − α (1 − α)2 1 µ≤ , + m m2 mini γi

(100)

where the equality holds if and only if γj = mini γi for all j=1, . . . , m, i.e., when γ1 = . . . = γm .

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The lower bound is obtained as follows. From (27) we have !2 X Xp γi (γi + 4µ))2 2(1 − α) + γi =( i

(101)

i

≤

X

!

γi (4mµ +

i

= 4mµ

X

X

γi )

(102)

i

γi + (

i

X

γ i )2 ,

(103)

i

where the inequality in (102) is due to Cauchy-Schwarz inequality with equality holds if and only if γ1 = . . . = γm . By expanding the LHS and removing common items at both sides, we can easily obtain the lower bound. A PPENDIX C P ROOF

OF

L EMMA 2

From (24) it is straightforward to observe that Rs (α) is a monotonously decreasing function with respect to α, with its minimum Rs (α < 1) > Rs (α = 1) = RM AC , Therefore we only need to prove Rs′ (α) > RM AC .



1 log 1 + 2

m X √ i=1

!2  γi  .

(104)

From (25) and (27)it is easy to see that Ri′ (α) (and therefore also Rs′ (α)) is a monotonously increasing function with respect to α ∈ [0, 1), which leads to Rs′ (α > 0) > Rs′ (α = 0).

(105)

On the other hand, we have Rs′ (α

= 0) =

m X 1 i=1



γi log 1 + 2 αi m

X γi 1 > log 1 + 2 αi i=1



(106)

!

(107)

m X γi 1 min ≥ log 1 + P xi >0; xi =1 2 x i=1 i   !2 m X 1 √ = log 1 + γi  2 i=1

= RM AC , January 20, 2014

!

(108)

(109) (110) DRAFT

41

where (109) holds by the optimal solution x∗i =

√

γi /(

Rs′ (α) > RM AC , and hence prove the lemma.

P √ j

γj ). Therefore we can conclude that

A PPENDIX D P ROOF

OF

L EMMA 4

Since U − X1 − X2 forms a Markov chain, which leading to I(X2 ; U|X1 ) = 0, we have I(X1 , X2 ; Y |U) = I(X1 , X2 ; Y ) + I(X1 , X2 ; U|Y ) − I(X1 , X2 ; U)

(111)

= I(X1 , X2 ; Y ) + I(X1 , X2 ; U|Y ) − I(X1 ; U) − I(X2 ; U|X1 )

(112)

= I(X1 , X2 ; Y ) + I(X1 , X2 ; U|Y ) − I(X1 ; U)

(113)

≥ I(X1 , X2 ; Y ) − I(X1 ; U),

(114)

with equality if and only if I(X1 X2 ; U|Y ) = 0. A PPENDIX E N OISE PARTITION

FOR

G AUSSIAN C HANNEL D ECOUPLING

Let L=

n X i=1

m X γi,j log 1 + α j=1 i,j

!

+

m X j=1

λj

n X i=1

!

αi,j − 1 ,

(115)

be the Lagrangian, by taking partial derivative of L w.r.t. αi,j and setting them to zero, we get ! m X γi,j γi,k . (116) = λj 1 + 2 αi,j αi,k k=1

By introducing auxiliary variables

µi = 1 +

January 20, 2014

m X γi,j , ∀i ∈ {1, . . . , n}, α i,j j=1

(117)

DRAFT

42

we can derive from (116) the following equations √ γi,j αi,j = p √ , λj µ i γi,j √ p √ = γi,j λj µi , αi,j n √ X p γi,j λj = √ , µi i=1 µi = 1 +

√

µi

m X √ j=1

γi,j

p

(118) (119) (120)

λj ,

(121)

v  !2 u m m u X X p p 1 √ √ √ µi = t γi,j λj + 4 + γi,j λj  , 2 j=1 j=1

where (120) comes from (118) and the fact that

Pn

i=1

(122)

αi,j = 1, (121) is obtained by substituting

(119) into (117), and (122) is the unique feasible solution to (121). Therefore the equivalent SNRs

γi,j αi,j

for decoupled BCs are uniquely determined by (119) where the optimal value of

µi and λj can be easily obtained by iterating (120) and (122). The convergence to the global optimum is guaranteed by observing the fact that λj is a monotonically decreasing function of {µi : ∀i} via (120) and µi is a monotonically increasing function of {λj : ∀j}, via (122). R EFERENCES [1] R. Koetter, M. Effros, and M. M´edard, “A theory of network equivalence–part I: point-to-point channels,” IEEE Transactions on Information Theory, vol. 57, pp. 972–995, Feb. 2011. [2] R. Koetter, M. Effros, and M. M´edard, “A theory of network equivalence, part II,” 2010, arXiv:1007.1033. [3] M. Effros, “On capacity outer bounds for a simple family of wireless networks,” in Proceedings Information Theory and Applications Workshop (ITA) Feb. 2010. [4] M. Effros, “Capacity Bounds for Networks of Broadcast Channels,” in Proceedings IEEE International Symposium on Information Theory (ISIT), Jun. 2010. [5] N. Fawaz and M. M´edard, “A Converse for the Wideband Relay Channel with Physically Degraded Broadcast,” in Proceedings IEEE Information Theory Workshop (ITW), Oct. 2011. [6] F. P. Calmon, M. M´edard, and M. Effros, “Equivalent models for multi-terminal channels,” in Proceedings IEEE Information Theory Workshop (ITW), Oct. 2011. [7] C. Bennett, P. Shor, J. Smolin, and A. Thapliyal, “Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem,” IEEE Transactions on Information Theory, vol. 48, pp. 2637–2655, Oct. 2002. [8] P. W. Cuff, H. H. Permuter, and T. M. Cover, “Coordination Capacity,” IEEE Transactions on Information Theory, vol. 56, pp. 4181–4206, Sep. 2010.

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