Three-Way Channels with Multiple Unicast Sessions: Capacity Approximation via Network
arXiv:1510.09046v1 [cs.IT] 30 Oct 2015
Transformation Anas Chaaban, Henning Maier, Aydin Sezgin, and Rudolf Mathar
Abstract With the increase in the number of devices capable of communicating with other devices in their vicinity, new communication topologies arise. Consequently, it becomes important to investigate such topologies in terms of performance. In this paper, an elemental network of 3 nodes mutually communicating to each other is studied. This full-duplex multi-way network is a suitable model for 3-user device-to-device communications. The main goal of this paper is to characterize the capacity region of the underlying 3-way channel (3WC) in the Gaussian setting within a constant gap. To this end, the 3WC is first transformed into an equivalent star-channel. Then, this star-channel is decomposed into a set of ‘successive’ sub-channels. This decomposition leads to sub-channel allocation problem, and constitutes a general framework for studying the capacity of different types of Gaussian networks. Using backward decoding, interference neutralization, and known results on the capacity of the star-channel, an achievable rate region for the 3WC is obtained. Furthermore, an outer bound is derived using cut-set bounds and genie-aided bounds. It is shown that the outer bound and achievable rate region are within a constant gap. This paper highlights the importance of physical-layer network-coding schemes in this network. Index Terms Multi-way channel; full-duplex; successive sub-channels; capacity; constant gap.
I. I NTRODUCTION Future communication networks have to support much higher data-rates than today’s networks [3]. To enable this goal, several new ideas are presently intensively investigated. Major progress can be achieved by combinA. Chaaban is with the Division of Computer, Electrical, and Mathematical Sciences and Engineering (CEMSE) at King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia. Email:
[email protected]. H. Maier and R. Mathar are with the Institute for Theoretical Information Technology, RWTH Aachen University, 52056 Aachen, Germany. Email: {maier, mathar}@ti.rwth-aachen.de. A. Sezgin is with the Institute of Digital Communication Systems, Ruhr-Universit¨at Bochum (RUB), Universit¨atsstrasse 150, 44780 Bochum, Germany. Email:
[email protected]. The work of A. Sezgin was supported in part by the DFG, under grant SE 1697/5. Parts of the paper have been presented in Allerton 2014 [1] and at the ISIT 2014 [2].
ing two promising techniques for enhancing the capacity of future networks, namely, full-duplex and multi-way communication. [4] The two-way channel (TWC) [5], [6] is the simplest form of multi-way communication. The main idea of communication over a TWC is that both nodes communicate with each other in both directions. Such communication can be established in a half-duplex or full-duplex mode. Half-duplex is the common trend in today’s communication systems mainly due to its practical simplicity in comparison to full-duplex. Clearly, if the TWC operates in halfduplex mode, it can be modeled as two non-interacting point-to-point (P2P) channels, for which both theory and practice have matured throughout the decades following Shannon’s work in [7]. The draw-back of this mode of communication is that it requires two separate channels. For instance, assuming a communication bandwidth of B Hz, the channel has to be segregated into two channels, each with bandwidth B/2 Hz. This in turn reduces the data-rate per channel by a factor of two. A similar conclusion holds if we separate the users in time/code instead of frequency. Full-duplex operation can avoid such data-rate reduction. In a full-duplex TWC, communication in both directions takes place over the same channel with bandwidth B as in the previous example. Although the actual practical application of full-duplex communication has been a challenging problem until recently, it is now feasible to operate in full-duplex mode with acceptable effort [8]. Full-duplex communication has already reached a certain degree of maturity and even includes applications to MIMO [9]. The advantage of such operation is that the data rate is doubled [10]. This insight also follows from [6], which shows that the capacity of the full-duplex Gaussian TWC is not affected by self-interference. In other words, the capacity is equal to that of two parallel P2P channels, each of which operates over the whole resources available (time/frequency). From this point-of-view, full-duplex multi-way communication is regarded as a promising candidate for boosting the performance of future networks. One possible scenario where such full-duplex two-way operation can be exploited is a device-to-device communications (D2D) scenario [11]. D2D has been proposed by several researchers as a potential component of future networks [12], [13]. In a cellular network, D2D allows nearby users to communicate directly among each other. This communication occurs with limited involvement of the base-station (BS) in the so-called direct D2D communication with operator control (DC-OC), or with no involvement of the BS at all in the so-called direct D2D communication with device control (DC-DC) [14]. These modes of D2D can offload some traffic from the BS in dense areas, especially if the D2D pair and the BS operate in-band. Now consider a general scenario where a D2D pair and a BS share the same resources, and want to establish full message exchange.1 In its full generality, such a scenario can be modeled as a fully-connected network with 3 nodes communicating with each other. This network also models the scenario where 3 users establish device relaying with device control (DR-DC) communication or where 2 users and a BS establish device relaying with operator control (DR-OC) [14], [15]. The resulting 3-way channel (3WC) (Fig. 1(a)), denoted ∆-channel in [2], is the main focus of this work. 1 Information
exchange between the BS and the D2D nodes might encompass control signaling, or communication with a third party in another
cell for example.
As mentioned above, major practical improvements have been achieved in the design of full-duplex transceivers lately [8], [16]. As such, we assume perfect full-duplex operation, and focus on the multi-way communication aspect of the network. The 3WC can be thought of as an extension of the TWC to 3 users. It combines several aspects of wireless communication like multiple-access, broadcasting, and most importantly relaying. Each node in a 3WC is simultaneously a source, a destination, and a relay. These nodes do not only operate as traditional relays [17], but also as bi-directional relays [18]–[20]. Hence, finding the best transmission scheme over this channel is interesting from both theoretical and practical perspectives. Some special classes of the 3-way channel have been studied in [21] where a multi-cast setting was considered. That is, each user sends a common message to the other two users. The difference to our work is that we consider a multiple uni-cast setting, which takes into account that users exchange bidirectional independent streams with each of the other users. The main goal of this paper is the characterization of the capacity region of the 3WC with multiple uni-cast within a constant gap. For brevity, we will omit reference to multiple uni-cast in what follows, and refer to the channel simply as the 3WC. Instead of studying the 3WC directly, we rather resort to the detour of transforming the 3WC to a star-channel, or the so-called Y-channel [22], [23] first (Fig. 1(c)). Here, the Y-channel denotes a class of multi-way relay channels [24]–[27], in which multiple users exchange information via a relay node. The main difference between the 3WC and the Y-channel is the presence of a central node in the latter (i.e., the relay) which manages a centralized information exchange. Typically, studying a centralized network is easier than studying a distributed one. Then we derive the results on the capacity of the 3WC from results on the capacity of the transformed Y-channel. From this point-of-view, this 3WC–Y-channel transformation (in analogy to the ∆–Y transformation) serves as a useful tool for studying the 3WC via its Y-channel counterpart. It might also be beneficial for studying larger multi-way channels as well. On our quest for the approximate capacity of the 3WC, some intermediate results are obtained. The main contributions of the paper are summarized in the following paragraphs. A capacity outer bound: We use the cut-set bounds [28] to derive a set of upper bounds on the achievable rates in the 3WC. Some of the resulting bounds are then tightened by using a genie-aided MAC approach [29]. The resulting bounds are assembled to form an outer bound on the capacity region of the 3WC. This outer bound reveals an interesting resemblance to the approximate capacity region of the Y-channel given in [30]. In particular, the outer bound is identical (within a constant gap) to the approximate capacity region of a Y-channel with channel gains obtained as functions of the 3WC gains. This observation suggests investigating whether the transmission scheme applied for the Y-channel in [30] also extends to the 3WC. It turns out that this is indeed possible. We show this with the aid of a channel decomposition, which leads us to the next contribution of this work. Channel decomposition: We describe a decomposition of a Gaussian channel into a set of sub-channels, which we call the ‘successive channel decomposition’ (SCD). This decomposition is a generalization of the one described in [31] to a channel with an arbitrary topology. In the SCD, we split the power at a transmitter into a sequence of ‘power levels’, so that the ratios between successive power levels are equal. Each such power level constitutes a sub-channel, which can be used as a one-to-one, one-to-many, or a many-to-one channel, depending
on the channel topology. The receiver processes the sub-channels successively, starting with the sub-channel with highest power level, followed by the second highest, and so on. The idea of this decomposition appeared first, to the best of our knowledge, in [32], where it was described for the many-to-one interference channel (IC) and one-to-many IC. Later on, a similar decomposition was used for other networks, see [33], [34]. Our contribution here is to provide a systematic and topology-independent description of this decomposition, which can be applied in any network. Several channel decompositions were devised earlier for a similar purpose. For instance, the deterministic channel model, which we refer to as the binary channel decomposition (BCD) was devised in [35]. A Q-ary channel decomposition (QCD) was devised in [36]. Furthermore, a lower-triangular decomposition (LTD) was devised in [37]. This latter decomposition encompasses interference alignment into the constant gap characterization, and permits feasible communication schemes for a wider range of channels, where the BCD fails. Below is a short comparison between these decompositions and the SCD. The SCD is an exact decomposition, in contrast to the BCD which is only approximate. In particular, we avoid the assumptions on the Gaussian channel made in the BCD, namely, bounds on the magnitudes of channel inputs and noise variables. Similar assumptions appear in the QCD, which does not explicitly assume bounded noisemagnitude, but implicitly induces this from assuming that noise does not disturb sub-channels of higher powers. Another difference is that the BCD and the QCD allow random access of sub-channels at the receivers, contrary to the SCD and LTD, where sub-channels are accessed successively. Moreover, the SCD has interacting sub-channels contrary to the BCD and the QCD. The main difference with the LTD is that the sub-channels in the SCD are noisy whereas those are noiseless in the LTD. The consequence of these differences is that the SCD holds for any Gaussian channel under a mild condition: the signal-to-noise ratio (SNR) has to be lower bounded by a finite value which is a function of the decomposition parameters. The sub-channels of the SCD can support the same rate for decoding or computation [38]. Similar to the BCD, QCD, and LTD, this simplifies the problem of rate achievability over a network into a sub-channel allocation problem. The optimization of such an allocation can be tackled very efficiently by using linear programming methods. Furthermore, the resulting sub-channel allocation can be easily converted into an achievable rate by multiplying the number of allocated sub-channels by the achievable rate per sub-channel. Such a decomposition will be very useful in order to simplify the study of the 3WC and its Y-channel counterpart. Before applying the successive sub-channel decomposition approach to the 3WC and the Y-channel, we explain it for the elementary P2P, many-to-one, and one-to-many channels. Once defined, we use this decomposition to obtain a simple representation of an achievable rate region for the Y-channel as described next. Y-channel achievable rate region: The capacity region of the Y-channel has been characterized within a constant gap in [30], [39]. This region is achieved by using a combination of a cyclic communication scheme [40] and communication schemes that are known for the two-way relay channel [20], [41], [42] (namely bi-directional and uni-directional schemes). Recall that the goal for considering the Y-channel here is to examine whether a transmission scheme for the Y-channel can be re-employed in the 3WC. In particular, we would like to investigate
User 1
User 2
User 3
(a) The 3-way channel (3WC).
User 2
User ˜1
User 1
Relay
Relay
User 3
(b) Transformed 3WC (extended Y-
User 2
User 3 (c) Y-channel.
channel). Fig. 1.
The 3WC–Y-channel transformation transforms a 3WC into a Y-channel counterpart with an additional direct channel between two of
its users.
the optimality of the optimal Y-channel scheme (within a constant gap) for the 3WC. However, the description of the Y-channel scheme in [30] is not suitable for the 3WC due to some subtleties that we will demonstrate later in the paper. For this reason, we use the channel decomposition approach to express the Y-channel as a set of sub-channels. Then, we integrate the transmission scheme of [30] in the decomposed Y-channel, thus obtaining a description of the transmission scheme per sub-channel. The main difference between the scheme in [30] and the one in this paper is the following. The former treats the Y-channel as one sub-channel where a combination of bi-directional, cyclic, and uni-directional communication is applied. The latter treats the Y-channel as multiple sub-channels, where one and only one of the aforementioned schemes is applied over each sub-channel, leading to a simple description of the scheme. At this point, it remains to allocate the available sub-channels to the 3 users and the 3 possible schemes. An optimal allocation for this problem was given in [40] in the context of the linear-deterministic Y-channel. We use the allocation in [40] to derive an achievable rate region which is within a constant gap of the capacity region of the Y-channel. Due to its simpler description, the resulting scheme for the Y-channel is suitable for re-use in the 3WC. 3WC rate region: With all the required tools at hand, we are ready to extend the Y-channel scheme to a 3WC scheme by developing a 3WC–Y-channel transformation. The first step of this transformation is splitting the strongest user of the 3WC into two nodes: a relay and a virtual user (Fig. 1(b)). The resulting network is an extended Y-channel, with a direct connection between two of its users. Any transmission scheme for this extended Y-channel serves as a transmission scheme for the original 3WC. Decomposing this extended Y-channel into a set of sub-channels and applying the transmission scheme of the basic Y-channel (Fig. 1(c)) in the extended Y-channel leads to interfering users. This interference has to be resolved in order to obtain a feasible scheme for the 3WC. It turns out that the arising interference can be completely eliminated using two techniques: Backward decoding and interference neutralization. Backward decoding is used to eliminate interference that has already been decoded
by the user in a subsequent transmission block2 , while interference neutralization is used to eliminate interference that has not been decoded yet. Interference neutralization is achieved by pre-transmitting the interfering signal (one transmission block in advance) to the relay, which in turn sends a negative version of this signal to the interfered user, thus neutralizing interference. This modification leads to a generalized version of the basic Y-channel scheme, which applies to the basic Y-channel, and also to the extended Y-channel. Consequently, we obtain a transmission scheme for the 3WC. This scheme interestingly achieves the capacity region of the 3WC within a constant gap. Note that the resulting transmission scheme and capacity region approximation can be readily applied to the aforementioned D2D scenarios given in [14], for any mode of information exchange between the 3 nodes. The rest of the paper is organized as follows. Section II introduces the notation used throughout the paper in addition to the system models of the 3WC and the Y-channel. Section III presents the main result. Section IV provides an outer bound for the 3WC. Sections V and VI present a channel decomposition for the Gaussian channel, and an achievable rate region of the Y-channel, respectively. Section VII proves the achievability of the outer bound for the 3WC within a constant gap, thus proving the main result. Section VIII discusses special cases of the 3WC that have been studied earlier in the literature, namely cooperative multiple-access and broadcast channels. We conclude in Section IX. II. N OTATION
AND
S YSTEM M ODEL
A. Notation Throughout the paper, we denote a sequence of symbols (x(1), · · · , x(n)) by x(n) , we denote
1 2
log(1 + x) by
ˆ the function C(x), and we use C(x) to denote max{0, C(x − 1)} which is an approximation of C(x) for large x.
We write X ∼ N (µ, σ 2 ) to indicate that X is Gaussian distributed with mean µ and variance σ 2 , and we use i.i.d. to indicate that a random sequence has independent and identically distributed instances. B. System Model: 3-Way Channel The 3WC consists of 3 nodes (users) communicating with each other as shown in Figure 2. The input-output relations of this channel can be expressed as yi (t) = hj xk (t) + hk xj (t) + zi (t),
(1)
for distinct i, j, k ∈ {1, 2, 3}, where at time instant t, yi (t), xi (t) and zi (t) are the real-valued received signal, transmit signal, and noise at user i, respectively. The channel coefficient between users j and k is denoted by hi ∈ R and assumed to be constant during a transmission block. The noises at the receivers are independent Gaussian N (0, 1) i.i.d. over time. 2 It
users.
is worth to remark here that the basic Y-channel scheme allows some users to decode information which is not intended to these particular
(w21 , w31 )
h3
(w12 , w32 )
User 2
(w ˆ12 , w ˆ13 )
User 1
h2
h1
User 3
(w13 , w23 )
(w ˆ31 , w ˆ32 )
(w ˆ21 , w ˆ23 )
Fig. 2. The 3-way channel (3WC) showing the incoming and outgoing messages at each user, and the multiplicative coefficients of the physical channels between the users.
All the nodes are full-duplex, have a power constraint P each, and have global knowledge of channel coefficients3 . We assume channel reciprocity, i.e., the channel coefficient from user i to user j is equal to the coefficient from user j to user i, which is a valid assumption for wireless communicating nodes sharing the same resources in full-duplex mode. Without loss of generality we further assume that h23 ≥ h22 ≥ h21 .
(2)
User i wants to communicate messages wji and wki , uniformly distributed over the sets Wji and Wki , with rates
Rji and Rki to users j and k, respectively. At time instant t, user i applies an encoder Eit : Wji × Wki × Rt−1 → R to generate its transmit signal (t−1)
xi (t) = Eit (wji , wki , yi
),
(3)
i.e., adaptive encoding can be used in general. The encoding functions are revealed to all users before the start of the transmission. After n channel uses, user i uses a decoder Di : Wji × Wki × Rn → Wij × Wik to decode (n)
(wˆij , w ˆik ) = Di (wji , wki , yi ).
(4)
The overall process of encoding, transmission and decoding induces an error probability Pe,n (probability of w 6= ˆ where w = (w21 , w31 , w12 , w32 , w13 , w23 ) and w ˆ is defined similarly. A rate tuple R = (R21 , R31 , R12 , R32 , R13 , R23 ) w), is said to be achievable if the transmission at the given rates can be accomplished with Pe,n → 0 as n → ∞. The set of all achievable rate tuples is the capacity region of the 3WC and denoted C. Note that C depends on the signal-to-noise ratios (SNR) of the 3WC defined as Γi = h2i P , i ∈ {1, 2, 3}. For this reason, we will refer to the
3WC as a 3WC(Γ1 , Γ2 , Γ3 ) channel and its capacity as C(Γ1 , Γ2 , Γ3 ) to indicate this dependence. The capacity region C(Γ1 , Γ2 , Γ3 ) is the main focus of this work. We aim to find an approximation of this region which is highly precise at high SNRs Γ1 , Γ2 , Γ3 , or equivalently at high P . 3 The
nodes estimate their channels before transmission starts and share the estimated values among each other.
(w21 , w31 )
(w ˆ12 , w ˆ13 )
(w13 , w23 )
User 1
User 3 ˜1 h
(w ˆ31 , w ˆ32 )
˜3 h Relay ˜2 h User 2
(w ˆ21 , w ˆ23 ) Fig. 3.
(w12 , w32 )
The Y-channel showing the incoming and outgoing messages at each user, and the multiplicative coefficients of the physical channels
between the users and the relay.
To achieve this goal, we use the Y-channel [30] as an intermediate step. The system model of the Y-channel is thus introduced next. C. System Model: Y-channel The Y-channel is a 3-way relay channel where 3 users exchange information with each other in all directions via (and only via) a relay node as shown in Figure 3. The input-output relations of the Y-channel are expressed as4 ˜ 1 x˜1 (t) + h ˜ 2x ˜ 3x y˜r (t) = h ˜2 (t) + h ˜3 (t) + z˜r (t),
(5)
˜ix y˜i (t) = h ˜r (t) + z˜i (t),
(6)
i ∈ {1, 2, 3},
where at time instant t, y˜r (t), y˜i (t), x ˜r (t), x˜i (t), z˜r (t), and z˜i (t) are the real-valued received signal at the relay, the received signal at user i, the transmit signal of the relay, the transmit signal of user i, the noise at the relay, ˜ i ∈ R is the channel coefficient between user i and the relay. The and the noise at user i, respectively. The scalar h
˜ i is assumed to be reciprocal and to hold the same value throughout a transmission block. The noises coefficient h
at the receivers are independent Gaussian N (0, 1) i.i.d. over time. All the nodes are full-duplex, and have a power constraint P each. We further assume without loss of generality that ˜2 ≥ h ˜2 ≥ h ˜ 2. h 1 2 3
(7)
Note that the ordering of the squared coefficients is reversed to the one given in (2) for the 3WC. ˜ ji and R ˜ ki In analogy to the 3WC, user i wants to communicate messages wji ∈ Wji and wki ∈ Wki with rates R to users j and k, respectively. The encoders and the decoders at the three users are similar to those applied in the (t−1)
3WC. The encoding at the relay at time t is done using the function Ert : Rt−1 → R to generate x ˜r (t) = Ert (˜ yr 4 Throughout
the paper, we will use the tilde notation to discern the Y-channel variables from the 3WC variables.
).
˜ is the capacity region of the Y-channel, and is denoted CY . Similar to the 3WC, The set of achievable rate tuples R
e1 , Γ e2 , Γ e 3 ) channel and its capacity region by CY (Γ e1 , Γ e2, Γ e 3 ) where the SNR we will denote a Y-channel as a Y(Γ
ei = ˜ parameters are defined as Γ h2i P , i ∈ {1, 2, 3}.
III. S UMMARY
OF THE
M AIN R ESULTS
The main result of the paper is a capacity region approximation for the 3WC within a constant gap, as given in the following theorem. Theorem 1. The capacity region C(Γ1 , Γ2 , Γ3 ) of a 3WC(Γ1 , Γ2 , Γ3 ) channel is within a constant gap of the region defined by the following inequalities ˆ 2 ), R31 + R32 ≤ C(Γ
(8)
ˆ 2 ), R13 + R23 ≤ C(Γ
(9)
ˆ 3 ), R12 + R13 + R32 ≤ C(Γ
(10)
ˆ 3 ), R12 + R13 + R23 ≤ C(Γ
(11)
ˆ 3 Γ2 /Γ1 ), R21 + R23 + R13 ≤ C(Γ
(12)
ˆ 3 ), R21 + R23 + R31 ≤ C(Γ
(13)
ˆ 3 ), R31 + R32 + R21 ≤ C(Γ
(14)
ˆ 3 Γ2 /Γ1 ). R31 + R32 + R12 ≤ C(Γ
(15)
e1 , Γ e2 , Γ e 3 ) of a Y(Γ e1 , Γ e2 , Γ e 3 ) channel with Γ e 1 = Γ3 Γ2 /Γ1 , Γ e 2 = Γ3 , and Furthermore, the capacity region CY (Γ
e 3 = Γ2 is within a constant gap of the region defined by the same inequalities with Rij replaced with R ˜ ij . Γ
This approximation is obtained by exploiting an interesting correspondence between the 3WC and the Y-channel
(a 3WC–Y-channel (∆–Y) transformation). To perform this transformation, we treat user 1 of the 3WC as two nodes: a relay, and a ‘virtual’ user denoted user ˜1, so that the relay and user ˜1 are connected by a virtual channel. Thus, the 3WC resembles a Y-channel, but with one main difference - the existence of a direct physical channel between users 2 and 3 (Fig. 4(b)). We call the resulting Y-channel an extended Y-channel to discern it from the basic Y-channel. The additional link between users 2 and 3 in the extended Y-channel causes cross-talk (interference) between users 2 and 3, in both directions. It turns out that if the virtual channel between the relay and user ˜1 has an SNR of Γ3 Γ2 /Γ1 , then it is possible to modify the transmission scheme for the Y-channel given in [30], so that the impact of this cross-talk between users 2 and 3 is eliminated. This modification mainly involves introducing interference neutralization steps. Since this cross-talk can be eliminated, the remaining network is a Y(Γ3 Γ2 /Γ1 , Γ3 , Γ2 ) channel (Fig. 4(c)). The capacity of this Y-channel can be achieved within a constant gap using the scheme in [30]. Interestingly, the capacity of this transformed Y-channel is also within a constant gap of the capacity of the original 3WC, leading to the characterization in Theorem 1 above.
User 1
Γ3
Γ2
User ˜1
User ˜1
Γ3 Γ2 /Γ1
Γ3 Γ2 /Γ1
Relay
Relay
Γ3 User 2
Γ1
User 3
(a) A 3WC(Γ1 , Γ2 , Γ3 ) channel.
Fig. 4.
User 2
Γ2 Γ1
Γ3 User 3
Γ2
User 2
User 3
(b) The 3WC after splitting user 1 into
(c) The extended Y-channel becomes
two nodes, leading to an extended Y-
a basic Y-channel after applying inter-
channel.
ference neutralization.
Transforming the 3WC into a Y -channel with a similar capacity region (within a constant gap). The channels in these figures are
labeled by their SNRs.
Proving Theorem 1 involves two parts: (i) proving it for the 3WC and (ii) for the Y-channel. The second part of the proof can be immediately obtained from [30, Theorem 7]. Thus, it remains to prove the first part of Theorem 1, i.e., to characterize the capacity of the 3WC within a constant gap. The rest of the paper is devoted for proving this result, and discussing it both for the general case and also for some special cases. We start by presenting an outer bound on the capacity region C(Γ1 , Γ2 , Γ3 ). IV. C APACITY O UTER B OUND
FOR THE
3WC
Upper bounds on achievable rates in the 3WC can be derived using cut-set bounds and genie-aided bounds in general. We start by stating the cut-set bounds. A. Cut-set Bounds The cut-set bounds for the 3WC can be written as follows [28] (n)
n(Rji + Rki − εn ) ≤ I(Wji , Wki ; Yj
(n)
, Yk |Wij , Wkj , Wik , Wjk ) (n)
n(Rij + Rik − εn ) ≤ I(Wij , Wkj , Wik , Wjk ; Yi (n)
where εn → 0 as n → ∞, and Wij and Yi
(16)
|Wji , Wki ),
(17) (n)
are the random variables corresponding to wij and yi , respectively.
By using Gaussian inputs, the bounds above can be further upper bounded as follows Rji + Rki ≤ C(h2k P + h2j P ),
(18)
Rij + Rik ≤ C((|hk | + |hj |)2 P ).
(19)
Details can be found in Appendix A. These bounds can be relaxed by using relation (2) given by h23 ≥ h22 ≥ h21 to obtain ˆ Rji + Rki ≤ C(max{Γ k , Γj }) + 1,
(20)
ˆ Rij + Rik ≤ C(max{Γ k , Γj }) + 3/2.
(21)
The bounds above have to be evaluated using all possible distinct i, j, k ∈ {1, 2, 3} and result in six bounds. Tighter upper bounds on the achievable rates can be derived by using a genie-aided bounding approach as we shall see next. B. Genie-aided Bounds A genie-aided bound can be developed by noting that user 1 for instance can decode w32 given w23 , after some so-called ‘noise reduction’. To show how this is possible, we consider any transmission scheme under which each user is able to decode its desired messages reliably. User 1 can thus decode w12 and w13 . Now let us enhance (n)
user 1 by providing w23 and z¯3 (n)
generate y3
(n)
= z3
−
h1 (n) h3 z 1
as side information.5 Given this side information, user 1 can
as follows. First, w23 is combined with the decoded w13 in order to generate x3 (1) as given in (3),
i.e., x3 (1) = E31 (w13 , w23 ). Then, user 1 computes y3 (1) as follows y3 (1) =
h1 (y1 (1) − h2 x3 (1)) + h2 x1 (1) + z¯3 (1). h3
(22)
Consequently, knowing x3 (1) and z¯3 (1), user 1 can generate y3 (1). Notice that at this point, knowing w13 , w23 , and y3 (1) allows user 1 to generate x3 (2) as given in (3), i.e., x3 (2) = E31 (w13 , w23 , y3 (1)). This in turn allows user 1 (n)
to compute y3 (2) as in (22). Now user 1 can repeat this procedure until all y3 can use
(n) y3
have been generated. Finally, user 1 (n)
together with w13 and w23 to decode w32 as given in (4), i.e., using (w31 , w32 ) = D3 (w13 , w23 , y3 ).
Therefore, we can write for the enhanced user 1 (n) (n) n(R12 + R13 + R32 − εn ) ≤ I W12 , W13 , W32 ; Y1 , Z¯3 , W21 , W31 , W23 ,
(23)
using Fano’s inequality with εn → 0 as n → ∞, where Wij denotes the random variable corresponding to message (n)
wij , and similarly Y1
(n) and Z¯3 . After some manipulations, this bound can be recast as
ˆ 3 ) + 2. R12 + R13 + R32 ≤ C(Γ
(24)
A tighter version of this bound and its detailed derivation are given in Appendix B. Similarly we can write the following bounds ˆ 3) + 2 R12 + R13 + R23 ≤ C(Γ
(25)
ˆ 3) + 2 R21 + R23 + R31 ≤ C(Γ
(26)
ˆ 3 Γ2 /Γ1 ) + 2 R21 + R23 + R13 ≤ C(Γ
(27)
ˆ 3) + 2 R31 + R32 + R21 ≤ C(Γ
(28)
ˆ 3 Γ2 /Γ1 ) + 2, R31 + R32 + R12 ≤ C(Γ
(29)
as also shown in Appendix B. 5 Notice
(n)
z¯3
(n)
(n)
(n)
effectively reduces the noise at user 1, since now user 1 can calculate E[z1 |¯ z3 ] and subtract it from y1 .
By comparing the cut-set bounds (20) and (21) and the genie-aided bounds (24)-(29), we notice that, for sufficiently high P , four out of the six cut-set bounds are dominated by the genie-aided ones. The only two surviving cut-set bounds are ˆ 2) + 1 R13 + R23 ≤ C(Γ
(30)
ˆ 2 ) + 3/2. R31 + R32 ≤ C(Γ
(31)
This leads to the following Lemma. Lemma 1. The capacity region C(Γ1 , Γ2 , Γ3 ) of the Gaussian 3WC(Γ1 , Γ2 , Γ3 ) channel is outer bounded by C(Γ1 , Γ2 , Γ3 ) where
C(Γ1 , Γ2 , Γ3 ) = R ∈ R6+ | (24)-(31) are satisfied .
(32)
Lemma 1 proves the converse of Theorem 1 for the 3WC since the outer bound C(Γ1 , Γ2 , Γ3 ) is within a constant gap of the region given in Theorem 1. It remains to show that the region given in this theorem is also achievable within a constant gap. The achievability proof is based on the 3WC–Y-channel transformation that has been briefly discussed in Section III, and will be elaborately discussed in Section VII. The transformation allows reusing results from the Y-channel in the 3WC under consideration. An instrumental tool that facilitates reusing such results is the channel decomposition illustrated in the next section. V. C HANNEL D ECOMPOSITION In this section, we describe a decomposition of the Gaussian channel into a set of ‘successive’ sub-channels. Before proceeding, let us briefly review three available decompositions in order to highlight the main differences between the discussed decompositions later on. In [36], a decomposition for the Gaussian channel based on the linear-deterministic channel model of Avestimehr et al. [35] has been proposed. The main ingredient of the decomposition in [36] is a generalization of the lineardeterministic model in [35] from a binary to a Q-ary one. Both decompositions aim at representing the Gaussian channel as a set of non-interacting (non-interfering) sub-channels with increasing powers, i.e., parallel sub-channels. To achieve this representation, three particular assumptions are imposed on the Gaussian channel. One of the assumptions in the binary channel decomposition (BCD) of [35] is that the transmit signal x and noise √ signals z are of bounded magnitude. While the assumption of a bounded transmit signal |x| < P is legitimate since it satisfies the power constraint E[X 2 ] < P , the assumption of bounded noise z is not since Gaussian noise instances can be unbounded. It is further assumed in [35] that x and z are non-negative. Another simplification done in the BCD is replacing the real-valued addition of x and z by a modulo-2 component-wise addition of their binary representations. With this simplification, potential carry-over bits arising from the addition of binary values are ignored. Despite these assumptions, the BCD proved to be a powerful tool for approximating the capacity of wireless networks as demonstrated in [20], [43] and [25] for instance.
Some of the assumptions imposed in the BCD [35] have been levitated in the Q-ary channel decomposition (QCD) in [36]. For instance, the assumption of bounded noise has been removed in the QCD, while the assumption of a bounded transmit signals has been preserved. Furthermore, the real-valued addition has not been replaced by a modulo-Q addition. In this case, achieving the goal of parallel (non-interacting) sub-channels requires imposing further restrictions on the transmit signal. Consider for instance the addition of two single-digit Q-ary numbers a and b. If a + b ≥ Q, then this addition would produce a carry-over, and the sum has two digits. The implication of this behavior in a the QCD is that we get interacting sub-channels. That is, there might be cases where the transmit signal on one sub-channel affects the next sub-channel of higher power. This is not desired in a decomposition into a set of non-interacting sub-channels. To avoid such behavior, the QCD limits the transmit signals over a given sub-channel to the set {0, · · · , ⌊(Q − 1)/K⌋}, where K is the number of users. One further condition remains to be satisfied in order to guarantee that the sub-channels do not interact. Namely, the addition of noise should not produce any carry-overs. In [36], all sub-channels are treated as noiseless by arguing that the probability the noise changes the value of the received symbol over a sub-channel is very low. However, as shown in [36], while higher-power sub-channels might not be affected by noise, lower-power sub-channels might still be affected. This reduces the capacity of lower-power sub-channels; an effect which has not been taken into account in [36]. Nevertheless, the QCD is still a powerful tool for a generalized degrees-of-freedom characterization, being an accurate approximation of capacity at asymptotically high SNRs. A recently proposed method to refine the accuracy of the large approximation gap of the BCD and to include multi-user interference alignment is the lower-triangular deterministic channel model introduced by Niesen et al. in [37]. We will call this approach the lower-triangular decomposition (LTD). While in the BCD, the channel gains only depend on the most significant bits of the binary value of the channel gains, the LTD also depends on the diversity of the lower bits. The authors propose a lower-triangular binary Toeplitz matrix in order to incorporate lower bits involved in the real Gaussian channel gains. Moreover, both the BCD and the QCD entail several outage cases where the channel gain constellations6 cause the decoding procedures to fail, whereas the LTD still provides feasible decoding schemes for a much wider range of channel constellations. Due to the lower-triangular channel structure, there is interaction between signals sent over different sub-channels at the receiver. This interaction affects the lower levels in particular. Since the higher bit levels are less affected, their additive impact on lower levels is canceled using successive decoding. A main feature of the LTD is that it permits a constant-gap capacity approximation for multi-user channels with interference alignment. Note that the LTD is deterministic, i.e., the resulting sub-channels are noiseless.7 The channel decomposition we discuss here, henceforth denoted the successive channel decomposition (SCD), avoids all the aforementioned restrictive assumptions, and can be considered as a generalization of the decomposition 6 These 7 We
outage cases are also called singularities.
distinguish between sub-channel interaction and sub-channel noise. Sub-channel interaction means that signals over different sub-channels
interfere at the receiver. Sub-channel noise results from the additive noise at the receiver in the corresponding Gaussian channel.
Decomposition BCD [35] QCD [36]
Input alphabet F2 j
Fq , q =
Q−1 K
Addition k
Channel noise
Sub-channel Noise
Interaction
Access
mod2
Bounded
None
None
Random
Real
Gaussian
None
None
Random
LTD [37]
F2
mod2
Bounded
None
Interacting
Successive
SCD
R
Real
Gaussian
Gaussian
Interacting
Successive
TABLE I T HE MAIN SIMILARITIES / DIFFERENCES BETWEEN THE KNOWN CHANNEL DECOMPOSITIONS (BCD, QCD AND LTD) AND THE PROPOSED ONE
(SCD). N OTE THAT OUR DECOMPOSITION MATCHES ALL THE DISCUSSED PROPERTIES OF A G AUSSIAN CHANNEL . H ERE , K
DENOTES
THE NUMBER OF USERS IN A CHANNEL .
in [31] to an arbitrary topology. The main difference between the BCD and the QCD on one hand, and the SCD on the other hand, is that the latter does not aim for noiseless non-interacting sub-channels. The main difference between the LTD and the SCD is that the former yields a set of interacting noiseless binary sub-channels, while the latter yields a set of interacting noisy real sub-channels. Thus, the SCD is intuitively closer to the actual Gaussian channel. To cope with sub-channel interaction, a receiver is only allowed to access the sub-channels successively (hence the name SCD), using successive decoding (similar to the LTD). It starts with the sub-channel of the highest power level. While processing this sub-channel, the remaining sub-channels are treated as noise. After decoding the content of this sub-channel, its contribution is removed from the received signal, and the receiver proceeds with decoding the sub-channel of the second highest power level, and so on, until the lowest power level is decoded. Note that the SCD enables the application of both interference alignment [29] and interference neutralization [44] although the sub-channels are interacting and even noisy. The main features of these decompositions are summarized in Table I. Next, we describe the proposed SCD in more detail. A. Point-to-point Channels Consider a Gaussian point-to-point channel with input X satisfying a power constraint E[X 2 ] ≤ Γ, and output
Y = X + Z where Z is N (0, 1), i.i.d. over time.8 Thus, the SNR of this channel is also Γ. We would like to decompose this channel into a set of N sub-channels. To this end, we write Γ as follows Γ = 1+
N X
pℓ ,
(33)
ℓ=1
where pℓ = γ ℓ − γ ℓ−1 for some γ ∈ R with γ N = Γ. This decomposes Γ into N + 1 powers 1, p1 , p2 , · · · , pN . As we shall see later, γ is roughly the SNR of each of the N sub-channels. (n)
Let us construct the transmit signal x(n) (n being the length of a transmission block) as a sum of N signals xℓ , (n)
ℓ = 1, · · · , N where the power of xℓ
is pℓ , and its rate is Rℓ . Each of the constituent signals is a codeword from
a Gaussian code or a lattice code (both are capacity achieving for the Gaussian P2P channel [45]). Note that this 8 The
channel gain here (as well in the next two sub-sections) is normalized to 1 by absorbing it to the power constraint.
composite signal satisfies the power constraint since
PN
ℓ=1
(n)
(n)
pℓ = Γ − 1 < Γ. Now, if the receiver decodes the signals
successively, starting with xN and ending with x1 , each time treating the yet un-decoded signals as noise, the achievable rate of each signal can be written as Rℓ ≤ C
pℓ 1 + p1 + p2 + · · · + pℓ−1 ℓ γ − γ ℓ−1 =C γ ℓ−1 ℓ γ . = Cˆ γ ℓ−1
(n)
That is, the achievable rate of xℓ
(34) (35) (36)
is determined by the ratio of the cumulative powers of the first ℓ codewords
divided by that of the first ℓ − 1 codewords. We call these cumulative powers ‘power levels’. Note that the ratio of power levels is equal to γ for all ℓ = 1, · · · , N . Thus, Rℓ ≤ Cˆ (γ) =
1 ˆ C (Γ) , N
is achievable for all ℓ = 1, · · · , N . Therefore, to obtain the total achievable rate, we multiply (37) by N which yields R =
(37) PN
ℓ=1
ˆ Rℓ ≤ C(Γ). Note
that this rate is achievable at any finite Γ, although it is only meaningful if Γ > 1. This is a mild restriction when compared to the QCD [36] which holds at asymptotically high Γ. Note also that the rate R converges to the channel capacity as Γ increases. From this point of view, we interpret each power level as a sub-channel and we have N sub-channels available for communication (see Figure 5). Thus, if Γ is the SNR of the P2P channel, then γ is the SNR of each of the N sub-channels. Since these sub-channels are accessed successively starting with sub-channel N and ending with sub-channel 1, we call them ‘successive sub-channels’. This leads to the following definition. Definition 1 (Successive sub-channels). We call a set of sub-channels 1, · · · , N successive sub-channels if the receiver is only allowed to access sub-channel ℓ after it has decoded sub-channels ℓ + 1, · · · , N and subtracted their contribution from the received signal. Lemma 2 (SCD of a P2P channel). A Gaussian point-to-point channel with an SNR of Γ can be decomposed into ˆ as a set of N ∈ N \ {0} successive sub-channels each with an achievable rate of C(γ) where γ N = Γ. Next, we extend the SCD to multi-user channels. We start with many-to-one channels followed by one-to-many channels. B. Many-to-one Channels The main goal of this subsection is showing the main differences which arise when extending the SCD to manyto-one channels. To this end, we consider a 2×1 Gaussian channel (two transmitters and one receiver) for simplicity. The channel inputs are X1 and X2 with power constraints Γ1 and Γ2 ≤ Γ1 respectively, and the channel output is Y = X1 + X2 + Z where Z is N (0, 1), i.i.d. over time.
Power levels
xN xN −1
γ N −1 γ N −2 γ N −3
.. . x2 x1 0 Tx
xN
.. . x4
γ2 γ 1
x3
N sub-channels
N sub-channels
xN −2
γN = Γ
x2 x1 z Rx
Fig. 5. Decomposition of a point-to-point channel into a set of successive sub-channels. The vertical axis shows the power level of a sub-channel, which is the cumulative power of all signals on sub-channels below and including this sub-channel. The ratio between each two consecutive ˆ power levels is γ leading to a rate of C(γ). Note that the index of a sub-channel is the logγ of its power level.
We begin by ignoring the weaker second transmitter for now and we describe the channel as a set of N1 ∈ N\{0}
successive sub-channels as before, by setting γ N1 = Γ1 . The power of the ℓ-th signal x1,ℓ is pℓ = γ ℓ − γ ℓ−1 , ℓ = 1, · · · , N1 .
Now, we bring the (weaker) second transmitter back into the picture. We decompose the transmit signal of the second transmitter into N2 ∈ N \ {0} signals, such that their power levels align with the first N2 power levels of
x1 (see Figure 6). That is, we decompose x2 into x2,ℓ , ℓ = 1, · · · , N2 , with power pℓ = γ ℓ − γ ℓ−1 . The number
of sub-channels N2 is chosen as the largest integer satisfying γ N2 < Γ2 , i.e., log(Γ2 ) . N2 = log(γ)
In this 2 × 1 channel, the achievable rate of each sub-channel depends on two parameters. First, it depends on γ. Second, it depends on the manner that the users are allowed to access the sub-channels. The two possible access modes are explained next. 1) One user per sub-channel: As an example of the first mode, consider a 2-user multiple-access channel (MAC) with signal-to-noise ratios Γ1 and Γ2 ≤ Γ1 . Let us decompose it into N1 successive sub-channels as described above. User 1 can access sub-channels N2 + 1, · · · , N1 exclusively, while it shares the remaining sub-channels 1, · · · , N2 with user 2. Let us partition these N2 sub-channels into s ≤ N2 sub-channels for user 1, and N2 − s sub-channels for user 2. If we denote the total number of sub-channels allocated to users 1 and 2 as r1 and r2 ,
respectively, then we can write r1 ≤ N1 − N2 + s
(38)
r2 ≤ N2 − s
(39)
0 ≤ s ≤ N2 .
(40)
Using Fourier-Motzkin’s elimination [46, Appendix D], we can restate this as r2 ≤ N2
(41)
r1 + r2 ≤ N1 .
(42)
The achievable rate region can be calculated by multiplying the number of sub-channels by the achievable rate per sub-channel. To calculate the achievable rate per sub-channel, denote the set of sub-channels allocated to users 1 and 2 by I1 and I2 , respectively. Since we allow only one user per sub-channel exclusively, we have I1 ∩ I2 = ∅ and I1 ∪ I2 = {1, · · · , N1 }. Thus, the interference plus noise power that the receiver observes while decoding the signal on sub-channel ℓ is 1+
ℓ−1 X
i=1 i∈I1
pi +
ℓ−1 X
pi = 1 +
i=1 i∈I2
ℓ−1 X
pi = γ ℓ−1 .
i=1
Thus, the SNR while decoding the signal on the ℓ-th sub-channel is γ. This leads to an achievable rate per subˆ channel of C(γ). As a result, the region ˆ R2 ≤ N2 C(γ)
(43)
ˆ R1 + R2 ≤ N1 C(γ),
(44)
ˆ ˆ 1 ). Similarly, using the definition of N2 , we = C(Γ is achievable. Using the definition of N1 , we can write N1 C(γ) ˆ ˆ 2 ) where we have approximated ⌊x⌋ by x.9 Therefore, the achievable rate region by C(Γ can approximate N2 C(γ) is approximated as ˆ 2) R2 ≤ C(Γ
(45)
ˆ 1 ). R1 + R2 ≤ C(Γ
(46)
This approximation is within a constant gap of the capacity region of the MAC given by Ri ≤ C(Γi ),
i = 1, 2,
R1 + R2 ≤ C(Γ1 + Γ2 ).
(47) (48)
ˆ Thus, this case can be well approximated by N1 sub-channels, each with a rate of C(γ), where the sub-channels 1, · · · , N2 are shared between the two transmitters, while the remaining sub-channels are accessed only by transmitter 1. 9 This
approximation can be made precise by choosing N1 appropriately.
Power levels
x1,N1
x1,N1 −1
x1,N1 −1
γ N 1 = Γ1 γ N1 −1
x2,N2
.. .
.. .
x1,2
x2,2
x1,1
x2,1
0
0
z
Tx1
Tx2
Rx
.. .
γ N1 −2 = γ N2 = Γ2
x2,N2
x1,N1 −2
.. .
+
x1,2
x2,2
x1,1
x2,1
N2 sub-channels
N1 sub-channels
x1,N1 −2
N2 sub-channels
N1 sub-channels
Fig. 6.
x1,N1
γ N1 −3
γ2 γ 1
Decomposition of a 2 × 1 channel into a set of successive sub-channels. The receiver observes the sum of the signals plus noise.
2) Two users per sub-channel: Now we consider the second mode when two users are allowed to access a ˆ sub-channel simultaneously10. In this case, the rate per sub-channel is C(γ/2). To show this, we consider a similar decomposition of the channel as above. If both users are allowed to access some sub-channels in 1, · · · , N2 simultaneously, then the sets of allocated sub-channels intersect, i.e., I1 ∩ I2 6= ∅. Let I12 = I1 ∩ I2 denote the set of sub-channels which are shared by both users. Then, the interference plus noise power that the receiver experiences while decoding the signal on sub-channel ℓ is 1+
ℓ−1 X
i=1 i∈I1
pi +
ℓ−1 X
pi = 1 +
ℓ−1 X
pi +
ℓ−1 X
2pi +
i=1 i∈I1 \I12
i=1 i∈I2
≤1+
=1+
i=1 i∈I1 \I12 ℓ−1 X
2pi
ℓ−1 X
pi +
i=1 i∈I2 \I12 ℓ−1 X
i=1 i∈I2 \I12
ℓ−1 X
2pi
(49)
i=1 i∈I12
2pi +
ℓ−1 X
2pi
(50)
i=1 i∈I12
(51)
i=1
≤ 2γ ℓ−1 .
(52)
Here, we upper bounded the interference plus noise power by a worst case scenario where all users share all sub-channels in {1, · · · , ℓ − 1}, and where the noise power is doubled. This implies that the SNR, while decoding the signal on the ℓ-th sub-channel, is γ/2 in the worst case. This leads to an achievable rate per sub-channel of ˆ C(γ/2). This rate is also achievable for computation using lattice codes [38]. Namely, if the receiver wants to decode x1,ℓ + x2,ℓ from sub-channel ℓ = 1, · · · , N2 , then the rates of x1,ℓ and x2,ℓ denoted R1,ℓ and R2,ℓ should 10 This
access mode is of particular interest for problems of computation over multi-user channels [38] and for interference networks [47].
satisfy Cˆ
1 γ ℓ − γ ℓ−1 + 2 2γ ℓ−1
γ . = Cˆ 2
(53)
Therefore, the rates R1,ℓ , R2,ℓ ≤ Cˆ (γ/2)
(54)
are achievable for ℓ = 1, · · · , N2 . Indeed, transmitter 1 can decide not to send any signal on a sub-channel, which leads to R1,ℓ = 0 while R2,ℓ > 0, or vice versa. In general, the transmitters can decide which sub-channel to use and which to skip. We now combine the two modes above (i.e., a) one user per sub-channel and b) two users per sub-channel) into one composite mode, where the achievable rate per sub-channel is Cˆ (γ/κ), and where κ ∈ {1, 2} depends on the scenario. Namely, if κℓ is the number of allowed users transmitting on sub-channel ℓ, then κ = maxℓ κℓ . Generalizing this to K × 1 channels leads to the following lemma. Lemma 3 (SCD of many-to-one channels). A K × 1 Gaussian channel with SNR’s of Γ1 ≥ Γ2 ≥ · · · ≥ ΓK can be decomposed into a set of N1 ∈ N \ {0} successive sub-channels where user i has access to sub-channels k j i) for i = 2, · · · , K and γ N1 = Γ1 . The decoding/computation rate of each sub1, · · · , Ni , with Ni = log(Γ log(γ) ˆ channel is C(γ/κ) where κ ∈ {1, · · · , K} is the maximum over all sub-channels of the number of allowed users per sub-channel. Remark 1. This decomposition is only meaningful for Γ1 > κN1 . Otherwise, the rate per sub-channel Cˆ (γ/κ) is negative. ˆ Note that allowing multiple users per sub-channel leads to a reduction in the achievable rate by C(κ) bits per ˆ over all sub-channels. sub-channel in comparison to the case κ = 1. This is a total rate reduction of at most N1 C(κ) This reduction is however independent of Γ1 , · · · , ΓK , and thus yields a constant gap as a function of Γi . Lemma 3 simplifies the problem of rate achievability over a Gaussian many-to-one channel, by transforming it into a sub-channel allocation problem. The sub-channels are equivalent in capacity, and therefore, after sub-channel allocation, we can obtain the achievable rate by multiplying the number of allocated sub-channels by the achievable rate per sub-channel. An optimal sub-channel allocation is in general easier to find than an optimal achievable rate (within a constant gap). Furthermore, sub-channel allocation solutions obtained from the BCD can be directly reused in the Gaussian case with SCD. We will see an example in the next sections. Next, we discuss the decomposition for one-to-many channels. C. One-to-many Channels We consider a 1 × 2 Gaussian channel (with one transmitter and two receivers) for simplicity. The channel input
is X with a power constraint Γ, and the channel outputs are Yi = X + Zi , i = 1, 2, where Zi is N (0, σi2 ), i.i.d.
over time. We assume without loss of generality that σ12 = 1, and σ22 ≥ 1, and we denote the SNR of channel i as
Γi = Γ/σi2 .
Similar to Section V-B, we start by ignoring the weaker second receiver, and write the observed channel at receiver 1 as a set of N1 successive sub-channels by setting γ N1 = Γ1 . The power of the signal xℓ on the ℓ-th sub-channel is pℓ = γ ℓ − γ ℓ−1 , ℓ = 1, · · · , N1 . Receiver 2, however, receives some of the transmit signals at a power lower than noise due to its lower SNR. This increases the amount of interference when decoding higher-power sub-channels, say sub-channel q, since the accumulative power of interference and noise when decoding this sub-channel is γ q−1 − 1 + σ22 (instead of γ q−1 at receiver 1). Now, we want to bound the value of q so that all sub-channels q, · · · , N1 can supply a rate within
ˆ a constant of C(γ) at receiver 2. To this end, let q be the smallest integer such that γ q−1 > σ22 − 1.
(55)
For decoding a signal on sub-channel q at receiver 2, the following rate can be supported q γ − γ q−1 γ q − γ q−1 > C C γ q−1 − 1 + σ22 2γ q−1 q γ + γ q−1 ˆ =C 2γ q−1 γ > Cˆ . 2
(56) (57) (58)
Thus, sub-channel q can support a rate of Cˆ (γ/2) at receiver 2. It can be similarly shown that all sub-channels q + 1, · · · , N1 can support the same rate. These sub-channels are said to be ‘accessible’ by receiver 2. Now let us find q which satisfies the condition (55). From (55), we can write log(σ22 − 1) q= +1 . log(γ)
(59)
The goal now is to lower bound the number of sub-channels that are accessible by receiver 2. This can be found by upper bounding q as follows q
2N1 since otherwise, the rate per sub-channel Cˆ (γ/2) is negative. This channel decomposition will be used to transform results from the Gaussian Y-channel to the Gaussian 3WC. In the process, the transmission strategies discussed in the next sub-section will be required. D. Main Communication Strategies over the Sub-Channels We are interested in 3 strategies that can be used over these sub-channels, namely: decoding, computation, and neutralization. 1) Decoding: Decoding has been explained above for the point-to-point channel. In general, the rate that can be achieved when decoding the signal over sub-channel ℓ is ˆ Rℓ = C(γ/(κ + µ)),
(66)
where κ is the number of users per sub-channel, and µ = 0 if the channel has multiple receivers and µ = 1 otherwise. After decoding, the receiver reconstructs the signal observed on the ℓ-th sub-channel, and subtracts its contribution from the received signal. Thus, it can proceed with decoding the successive sub-channels. 2) Computation: Assume that users 1, · · · , κ′ , with κ′ ≤ κ, transmit over sub-channel ℓ of a many-to-one channel, and that a function of their transmit codewords has to be computed at the receiver. For this purpose, user i ∈ {1, · · · , κ′ } uses an n-dimensional nested-lattice codebook with fine lattice Λf , coarse lattice Λc , rate Ri , and power 1 (see [38] for more details on lattice codes). Then, they send the nested-lattice codewords λ1 , · · · , λκ′ after dithering the codewords using random dithers d1 , · · · , dκ′ , reducing the outcome modulo-Λc, and scaling it with the power of sub-channel ℓ. That is, the transmit signal of user i = 1, · · · , κ′ over sub-channel ℓ is (n)
xi,ℓ =
√ pℓ [(λi + di ) mod Λc ].
(67)
Over this sub-channel, the receiver observes ′
(n) yℓ
=
κ X
(n)
(n)
xi,ℓ + zℓ ,
(68)
i=1
(n)
is a length-n sequence that contains the interference plus noise signals (with power κγ ℓ−1 at most). P ′ κ The receiver intends to compute the cumulative sum of codewords i=1 λi mod Λc . As in [38], this is possible where zℓ
as long as the rates of λ1 , · · · , λκ′ satisfy
γ 1 , i = 1, · · · , κ′ . (69) + κ′ κ P ′ Pκ′ (n) κ mod Λc , the receiver can reconstruct i=1 xi,ℓ effectively. In [48], it was shown that after computing λ i i=1 Ri ≤ Rℓ = Cˆ
Thus, the receiver can subtract the contribution of these signals in order to proceed with decoding the remaining sub-channels. If the channel has multiple receivers, then the achievable rate becomes 1 γ γ ˆ ˆ Rℓ = C >C . + κ′ κ+1 κ+1
In general, we can write the computation rate as Rℓ = Cˆ (γ/(κ + µ)) ,
(70)
where µ = 0 if the channel has one receiver and µ = 1 otherwise. 3) Neutralization: Finally, neutralization over sub-channel ℓ can also be performed by using nested-lattice codes and computation [49]. For the sequel, we are interested in the following example. Consider a scenario where a receiver listens to multiple transmitters (e.g., a many-to-one channel) and is interested in λ1 . Moreover, one transmitter i has knowledge of λi = (λ1 − λ2 ) mod Λc and another transmitter j has λj = λ2 .11 These two 11 All
lattice codewords are from a code with unit power.
transmitters i and j can cooperate to send λ1 to the receiver as follows. First, a sub-channel ℓ is dedicated to these two transmitters. These transmitters send (n)
√
pℓ [(λi + di ) mod Λc ],
(71)
(n)
√
pℓ [(λj + dj ) mod Λc ],
(72)
xi,ℓ = xj,ℓ = respectively. The receiver obtains (n)
yℓ
(n)
(n)
(n)
= xi,ℓ + xj,ℓ + zℓ .
(73)
Now the receiver computes (λi + λj ) mod Λc . This is possible as long as the rates satisfy γ 1 , + Ri , Rj ≤ Rℓ = Cˆ 2 κ+µ
(74)
where κ is the maximum number of allowed users per sub-channel, and µ is zero if the channel has one receiver and one otherwise. Then the receiver recovers (λi + λj ) mod Λc = (λ1 ) mod Λc = λ1 , and the interference from λ2 is neutralized. Therefore, the rate ˆ Rℓ = C(γ/(κ + µ))
(75)
is also achievable for neutralization, where κ and µ are as defined in Lemmas 3 and 4. ˆ From this point on, we will use C(γ/(κ + µ)) as the achievable rate per sub-channel for decoding, computation, and neutralization purposes. In the next section, we use the proposed channel decomposition for the Y-channel, and obtain a simple expression of its achievable rate region. This decomposition will be a useful component for proving the achievability of Theorem 1. VI. A N ACHIEVABLE R ATE R EGION
FOR THE
Y- CHANNEL
e1 , Γ e2 , Γ e 3 ) channel as defined in Section II-C. The uplink over this Y-channel is a 3 × 1 channel Consider a Y(Γ
˜ 2P , Γ ˜ 2 P , and Γ e1 = h e2 = h e 3 = ˜h2 P . Using the channel decomposition introduced with signal-to-noise ratios Γ 1 2 j e k3 e 1 . Note that N e1 has e1 ∈ N \ {0}, and we define N ek = log(Γk ) for k = 2, 3 where γ Ne1 = Γ above, we choose N log(γ)
e2 , N e3 > 0. Thus, we can write the uplink phase of the Y-channel (from all 3 users to the to be chosen so that N
e1 successive sub-channels, where the first N e3 sub-channels are accessible by all 3 users, the relay) as a set of N
e3 + 1, · · · , N e2 are accessible by users 1 and 2, and the sub-channels N e2 + 1, · · · , N e1 are accessible sub-channels N by user 1 only.
e1 , Γ e 2 , and Γ e3 . In the downlink, we have a 1 × 3 channel from the relay to the 3 users, with signal-to-noise ratios Γ
e1 sub-channels, where sub-channels 1, · · · , N e1 − N e2 are accessible by user 1 We decompose this 3×1 channel into N e1 − N e2 +1, · · · , N e1 − N e3 are accessible by users 1 and 2, and sub-channels N e1 − N e3 +1, · · · , N e1 only, sub-channels N
are accessible by all three users. Note that the sequence of sub-channels that is accessible by user 1 only, then by users 1 and 2, and then by users 1, 2 and 3, is mirrored to the sequence of sub-channels previously discussed in the uplink phase.
We would like to find an achievable rate region for this Y-channel that is within a constant gap of the capacity e1 , Γ e2 , Γ e 3 ). Instead of tackling this problem directly, our approach is to partition the problem in two region CY (Γ
sub-problems by decomposing the channel into successive sub-channels. The first sub-problem is then: how to
distribute the sub-channels over the users? This problem corresponds to the sub-channel allocation. The second sub-problem is: how much rate is achieved for a given sub-channel allocation? The solution of the second problem is easy, since we can obtain the achievable rate from the sub-channel allocation by merely multiplying the number of allocated sub-channels by the achievable rate per sub-channel. Thus, with the aid of the channel decomposition above, the problem boils down to a sub-channel allocation problem over the Y-channel. A similar problem for the linear-deterministic Y-channel has been studied in [30]. We restate the result of [30] here as follows. Assume that user j ∈ {1, 2, 3} requires r˜ij sub-channels in order to send message wij to user i 6= j. What is an achievable sub-channel allocation? In other words, what conditions must the sub-channel allocation parameters r˜ij satisfy in order to have reliable information transfer? It turns out that as long as the required number of sub-channels satisfies e3 , r˜31 + r˜32 ≤ N
(76)
e2 , r˜12 + r˜13 + r˜32 ≤ N
(78)
e1 , r˜21 + r˜23 + r˜13 ≤ N
(80)
e2 , r˜31 + r˜32 + r˜21 ≤ N
(82)
e3 , r˜13 + r˜23 ≤ N
(77)
e2 , r˜12 + r˜13 + r˜23 ≤ N
(79)
e2 , r˜21 + r˜23 + r˜31 ≤ N
(81)
e1 , r˜31 + r˜32 + r˜12 ≤ N
(83)
this sub-channel allocation is achievable. The sub-channel allocation corresponding to the achievability of the above region includes •
Shared allocation: allocating two users per sub-channel for the purpose of physical-layer network-coding. The shared sub-channels are used to establish bi-directional and cyclic communication [40].
•
Exclusive allocation: allocating one user per sub-channel to establish uni-directional communication.
Let us briefly recapitulate what is meant by these 3 modes of communication, i.e., bi-directional, cyclic, and uni-directional communication. A. Bi-directional Communication In this mode, two users, say users 1 and 2, share a sub-channel in the uplink. User 1 sends the codeword λ21 and user 2 sends λ12 which are nested-lattice codewords. Now as explained above, the relay computes λr = (λ21 + λ12 ) mod Λc over this sub-channel. In the downlink, the relay sends this codeword λr to both users 1 and 2. User 1 decodes λr and calculates (λr − λ21 ) mod Λc = λ12 , and similarly, user 2 recovers λ21 . This allows
bi-directional exchange of information between users 1 and 2. A number of two messages is conveyed over only one sub-channel. An important requirement for this mode of communication is that both the uplink and downlink e2 in the uplink, sub-channels are accessible by both users. Here, users 1 and 2 should use a sub-channel in 1, · · · , N e1 − N e2 + 1, · · · , N e1 in the downlink. In general, for user and the relay should forward λr over a sub-channel in N pair (i, j), the relay decodes λr = (λji + λij ) mod Λc in the uplink, and forwards it in the downlink. Users i and
j both obtain their desired signals after decoding λr . B. Cyclic Communication In this mode, the three users share two sub-channels in the uplink and downlink to exchange information in a cyclic manner. Suppose that user 1 wants to communicate with user 2, user 2 with user 3, and user 3 with user 1, and let all users use nested-lattice codes. User 1 sends λ21 and user 2 sends λ32 over sub-channel ℓ. User 3 sends λ13 and user 2 sends λ32 (again) over sub-channel ℓ′ with ℓ′ 6= ℓ. The relay computes λr,1 = (λ21 + λ32 ) mod Λc and λr,2 = (λ32 + λ13 ) mod Λc over these sub-channels. In the downlink, the relay sends λr,1 to users 1 and 2 over sub-channel m, and sends λr,2 to users 2 and 3 (which is also heard by user 1) over sub-channel m′ with m′ 6= m. Then, user 1 decodes λr,1 and λr,2 and calculates (λr,2 − λr,1 − λ21 ) mod Λc = λ13 . User 2 decodes λr,1 and calculates (λr,1 − λ32 ) mod Λc = λ21 . User 3 decodes λr,2 and calculates (λr,2 − λ13 ) mod Λc = λ32 . This allows cyclic exchange of information between users 1, 2, and 3. A number of three messages is conveyed over only two sub-channels. An important requirement for this mode of communication is that one uplink sub-channel is accessible by user 3 and the other by user 2, and that one downlink sub-channel is heard by user 3 and the other by user 2. For cyclic communication with the opposite direction, i.e., 1 → 3 → 2 → 1, user 1 sends λ31 over
sub-channels ℓ and ℓ′ , user 2 sends λ12 over sub-channel ℓ, and user 3 sends λ23 over sub-channel ℓ′ . The relay forwards λr,1 = (λ31 + λ12 ) mod Λc and λr,2 = (λ31 + λ23 ) mod Λc over over sub-channels m (heard by users 1 and 2) and m′ 6= m (heard by users 2 and 3), respectively. C. Uni-directional Communication
This mode is a simple decode-and-forward mode, where a user sends a codeword to the relay, the relay decodes it and forwards it to the destination. In this case, a sub-channel is used exclusively by one user. For instance, user 1 who wants to send λ21 to user 2, sends this codeword to the relay, which decodes it. Then the relay sends the decoded λ21 to user 2 who decodes it. No other users are allowed to use the uplink and downlink sub-channels required for uni-directional communication. It is important that the allocated uplink channel is accessible by the source (here user 1), and the downlink channel is heard by the destination (here user 2). D. Achievable Rate Region In order to transform the achievable sub-channel allocation region (76)-(83) to an achievable rate region, we have to multiply by the achievable rate per sub-channel. Notice from the description above that a sub-channel is used by at most two users at a time in the uplink, with the relay acting as a destination. Therefore, for the uplink we
ˆ have κ = 2 and µ = 0 leading to an achievable rate per uplink sub-channel of C(γ/2). In the downlink, we have only one transmitter (the relay) with multiple destinations. Therefore, κ = 1 and µ = 1 leading to an achievable ˆ rate per downlink sub-channel of C(γ/2). This achievable rate per sub-channel is only meaningful when γ > 2, e 1 > 2Ne1 . i.e., Γ
e3 into the following Accordingly, we can transform the sub-channel allocation constraint in (76), i.e., r˜31 +˜ r32 ≤ N
rate constraint
e ˜ 31 + R ˜ 32 ≤ N3 log(γ/2) R 2 e3 e3 N N = log(γ) − 2 2 e3 N ˆ Γ e3 ) − ≈ C( , 2
(84) (85) (86)
e3 and the approximation ⌊x⌋ ≈ x. Note that this is already a good where in the last step, we used the definition of N
e1 . Using similar steps, the approximation at high SNR, and can be made more precise by appropriately choosing N
other constraints on the sub-channels can be transformed into rate constraints. We obtain the following achievable rate region. e1 , Γ e2 , Γ e3 , N e1 ) defined by the following set of inequalities Proposition 1. The rate region C Y (Γ ˜ 31 + R ˜ 32 ≤ C( ˆ Γ e3 ) − N e3 /2, R
˜ 13 + R ˜ 23 ≤ C( ˆ Γ e3 ) − N e3 /2, R
(87) (88)
˜ 12 + R ˜ 13 + R ˜ 32 ≤ C( ˆ Γ e2 ) − N e2 /2, R
(89)
˜ 12 + R ˜ 13 + R ˜ 23 ≤ C( ˆ Γ e2 ) − N e2 /2, R
(90)
˜ 21 + R ˜ 23 + R ˜ 13 ≤ C( ˆ Γ e1 ) − N e1 /2, R
(91)
˜ 21 + R ˜ 23 + R ˜ 31 ≤ C( ˆ Γ e2 ) − N e2 /2, R
(92)
˜ 31 + R ˜ 32 + R ˜ 21 ≤ C( ˆ Γ e2 ) − N e2 /2, R
(93)
˜ 31 + R ˜ 32 + R ˜ 12 ≤ C( ˆ Γ e1 ) − N e1 /2. R
(94)
e1 , Γ e2 , Γ e 3 ) channel with Γ e1 ≥ Γ e2 ≥ Γ e 3 , where N e1 , N e2 , N e3 ∈ N \ {0}, is an achievable rate region for the Y(Γ j e k ei = log(Γi ) and γ Ne1 = Γ e 1 > 2Ne1 . N log(γ) By decomposing the Y-channel into a set of successive sub-channels, we were able to derive an achievable rate
region by recasting the problem as a sub-channel allocation problem. This achievable rate region is within a constant gap of the outer bound given in [30, Theorem 5]. The gap can be shown to be less than
e1 +3 N 4
bits per dimension
e1 , which in turn has to satisfy two constraints. (or per stream). Note that this gap is controlled by our choice of N
e1 has to e1 has to be chosen so that all N ei > 0. Second, it has to be chosen so that Γ e 1 > 2Ne1 . Therefore, N First, N
satisfy e1 ) log(Γ e1 < log(Γ e 1 ). 2. Consequently, the achievable rate region above is valid under the conditions Γ e3 > 2 Such an N
e1 at this stage, it can be refined by ’sub-channel grouping’ as e 1 > 2Ne1 . Although the gap is a function of N and Γ we describe next.
E. Sub-channel Grouping Note that the gap depends on the number of sub-channels, or rather, on the number of decoding steps involved. This follows since each decoding step incurs a gap of
1 2
bit, which arises due to sub-channel sharing leading to
ˆ ˆ a sub-channel rate of C(γ/2) instead of C(γ). From this point-of-view, it is desired to minimize the number of decoding steps involved in the scheme. In order to achieve this goal, we first note that several sub-channels might be used similarly in the aforementioned scheme for the Y-channel. Let us count how many different usage cases we have in the given scheme. For bidirectional communication, we have 3 sub-channel usage cases corresponding to the user pairs (1, 2), (1, 3), and (2, 3). For cyclic communication, we have four such cases, two per each of the following cycles 1 → 2 → 3 → 1 and 1 → 3 → 2 → 1. For uni-directional communication, we have six cases, one per each direction i → j, i, j ∈ {1, 2, 3}, j 6= i. Thus, we have 13 different usage cases of sub-channels in total. Now consider two consecutive sub-channels ℓ and ℓ − 1. If the same transmitters use the same strategy over both sub-channels, then these sub-channels can be grouped into one sub-channel. This grouping reduces the gap incurred by decoding the two sub-channels from 2 ·
1 2
= 1 to
1 2
and is executed as follows.
The signals sent over sub-channels ℓ and ℓ − 1 have powers γ ℓ − γ ℓ−1 and γ ℓ−1 − γ ℓ−2 and rate constraints
˜ℓ = R ˜ ℓ−1 = C(γ/2) ˆ ˆ R = C(γ) − 1/2. Instead of sending two separate signals, the transmitter sends only one signal with a power of (γ ℓ − γ ℓ−1 ) + (γ ℓ−1 − γ ℓ−2 ) = γ ℓ − γ ℓ−2 ˜ While decoding this signal at the receiver, the signals at sub-channels 1, · · · , ℓ − 2 are treated as and a rate of R. noise. This leads to the following rate (we consider computation as a worst case) ℓ ℓ−2 ˜ ≤ Cˆ 1 + γ − γ R 2 2γ ℓ−2 ℓ γ = Cˆ 2γ ℓ−2 2 γ ˆ =C 2 1 = 2Cˆ (γ) − . 2
(95) (96) (97) (98)
˜ ℓ and R ˜ ℓ−1 plus 1 , thus reducing the gap The resulting rate constraint is equal to the sum of the rate constraints R 2 by 21 .
This signal grouping can be generalized to all signals and sub-channels. The number of signal groups will be equal to the number of sub-channels with different usage cases. For the Y-channel we have 13 such cases, and e1 as much as necessary to therefore, we have at most 13 sub-channel groups. With this in mind, we can increase N
ej ≈ sharpen the approximation N
ej ) log(Γ log(γ) ,
and then group the sub-channels into 13 groups. Since the bounds (87)-
(94) above bound combinations of 2 and 3 rates, and since each rate is split into 3 parts (bi-directional, cyclic, and uni-directional), each such rate constraint corresponds to decoding on either 6 or 9 sub-channel groups, respectively. This leads to a gap of at most
6 2
and
9 2
bits, respectively, leading to the following achievable rate region ˜ 31 + R ˜ 32 ≤ C( ˆ Γ e 3 ) − 3, R
(99)
˜ 13 + R ˜ 23 ≤ C( ˆ Γ e 3 ) − 3, R
(100)
˜ 12 + R ˜ 13 + R ˜ 32 ≤ C( ˆ Γ e 2 ) − 9/2, R
(101)
˜ 12 + R ˜ 13 + R ˜ 23 ≤ C( ˆ Γ e 2 ) − 9/2, R
(102)
˜ 21 + R ˜ 23 + R ˜ 13 ≤ C( ˆ Γ e 1 ) − 9/2, R
(103)
˜ 21 + R ˜ 23 + R ˜ 31 ≤ C( ˆ Γ e 2 ) − 9/2, R
(104)
˜ 31 + R ˜ 32 + R ˜ 21 ≤ C( ˆ Γ e 2 ) − 9/2, R
(105)
˜ 31 + R ˜ 32 + R ˜ 12 ≤ C( ˆ Γ e 1 ) − 9/2. R
(106)
˜ 2 P . By comparing this achievable rate region with the outer bound in [30, Theorem 5], we ei = h Recall that here Γ i
can conclude that the gap between this achievable rate region and the capacity region is no more than 2 bits per dimension, instead of
e1 +3 N 4
e1 . bits per dimension for large N
Using this channel decomposition approach leads to a simplified treatment of the problem resulting in a capacity characterization within a constant gap. A more involved study which avoids this decomposition (as in [30]) leads to a smaller gap at the expense of a more difficult analysis. Another advantage of this channel decomposition is that it constitutes a general framework for studying the capacity of different types of Gaussian networks. Next, we exploit this decomposition to derive an achievable rate region of the Gaussian 3WC. VII. A N ACHIEVABLE R ATE R EGION
FOR THE
3WC
Now we are ready to present an achievable rate region for the 3WC. For this goal, we benefit from the decomposition approach explained above, in addition to the achievable rate region of the Y-channel. First of all,
Extended Y-channel
User ˜1 h3 h2 h1
3WC
User 1 h3
User 2
Fig. 8.
Y-channel relay
h2 h1
User 3
The 3WC transformed to an extended Y-channel with user 1 acting as a relay connected to a virtual user ˜ 1, and with an additional
direct channel between users 2 and 3.
we make the following observation. The outer bound C(Γ1 , Γ2 , Γ3 ) given in Lemma 1 which can be written as ˆ 2) + 1 R13 + R23 ≤ C(Γ
(107)
ˆ 2 ) + 3/2 R31 + R32 ≤ C(Γ
(108)
ˆ 3) + 2 R12 + R13 + R32 ≤ C(Γ
(109)
ˆ 3) + 2 R12 + R13 + R23 ≤ C(Γ
(110)
ˆ 3) + 2 R21 + R23 + R31 ≤ C(Γ
(111)
ˆ 3 Γ2 /Γ1 ) + 2 R21 + R23 + R13 ≤ C(Γ
(112)
ˆ 3) + 2 R31 + R32 + R21 ≤ C(Γ
(113)
ˆ 3 Γ2 /Γ1 ) + 2, R31 + R32 + R12 ≤ C(Γ
(114)
bears some resemblance with the achievable rate region of the Y-channel given in Proposition 1. Namely, if we e1, Γ e 2 , and Γ e 3 in Proposition 1 by Γ3 Γ2 /Γ1 , Γ3 , and Γ2 , respectively, then the achievable rate region replace Γ defined by (87)-(94) is within a constant gap of the outer bound defined by (107)-(114). Thus, the outer bound
C(Γ1 , Γ2 , Γ3 ) of the capacity region of a 3WC(Γ1 , Γ2 , Γ3 ) channel with Γ1 ≤ Γ2 ≤ Γ3 (cf. (2)) is within a constant
e1 , Γ e2 , Γ e3 , N e1 ) (Proposition 1) of a Y-channel with gap of the achievable rate region C Y (Γ e 1 = Γ3 Γ2 /Γ1 ≥ Γ e 2 = Γ3 ≥ Γ e 3 = Γ2 ≥ Γ1 . Γ
This Y-channel is depicted in Figure 8. Notice that in this extended Y-channel, we include a physical channel between users 2 and 3 with coefficient h1 , and obtain a 3WC as a part of this extended Y-channel, where the relay represents user 1. In what follows, we will prove the following result. Theorem 2. The capacity region of the 3WC given by C(Γ1 , Γ2 , Γ3 ) is within a constant gap of the capacity region
e1 , Γ e2 , Γ e 3 ) where Γ e 1 = Γ3 Γ2 /Γ1 , Γ e 2 = Γ3 , and Γ e 3 = Γ2 . of a Y-channel CY (Γ
The proof of this theorem is based on an optimal scheme (within a constant gap) for the 3WC which is a slightly modified version of the one for the Y -channel. To obtain this scheme, we first choose the user observing the two strongest channel gains h2 and h3 of the 3WC to act as a relay. We ‘split’ this user into two nodes: a relay node, and a ‘virtual’ user denoted user ˜1. These two nodes are connected via a Gaussian channel with channel coefficient h3 h2 /h1 , and thus with an SNR of Γ3 Γ2 /Γ1 . This step leads to an extended Y-channel with an additional direct channel between users 2 and 3 as shown in Figure 8. Note that since user ˜1 and the relay (user 1) are in fact one and the same node, the channel arising between them due to this transformation has infinite capacity. That is, all the information available at user 1 is also available at user ˜1. For convenience, we assume that it has a finite capacity dictated by the channel gain h3 h2 /h1 . Any rate that is achievable with this channel gain h3 h2 /h1 is also achievable if the channel gain is infinite. Furthermore, any rate that is achievable in this extended Y-channel is also achievable in the original 3WC. Now what happens if we use the Y-channel scheme e1 , Γ e2, Γ e3, N e1 ) (Proposition 1) for this extended Y-channel? To answer this question, we start which achieves C Y (Γ with a simple case. A. Case h1 = 0 If h1 = 0, then the 3WC–Y-channel transformation above leads to an extended Y-channel with no direct channel between users 2 and 3, i.e., it leads to a basic12 Y-channel. This Y-channel has an achievable rate region given e1 ) (cf. (87)-(94)). Since this achievable rate region is within a constant gap of the outer by C Y (Γ3 Γ2 /Γ1 , Γ3 , Γ2 , N bound C, then it characterizes the capacity region of the 3WC within a constant gap.
Thus, for h1 = 0, each of the six messages of the users of the 3WC is split into a bi-directional, a cyclic, and a uni-directional message, and these messages are communicated between users ˜1, 2, and 3 via the relay as in the basic Y-channel (recall that user 1 plays the role of the relay in the given 3WC–Y-channel transformation). This transforms the transmission scheme of the Y-channel into a transmission scheme of the 3WC which achieves the capacity within a constant gap. Now let us see what happens if h1 6= 0. B. General Case h1 6= 0 If we apply the basic achievable scheme used for the Y-channel on the extended Y-channel with h1 6= 0, there will be cross talk between users 2 and 3. In the basic Y-channel scheme, this causes interference in the e1 sub-channels accessible by user ˜1, downlink. Now, consider a decomposition of the extended Y-channel into N j e k j k e2 = log(Γ2 ) of these sub-channels are shared with user 2, and N e3 = log(Γe3 ) of the latter N e2 subwhere N log(γ) log(γ) e
e1 . channels are also shared with user 3, with γ N1 = Γ
Due to the interference perceived over channel h1 , the channel from the relay and user 2 to user 3 is a 2 × 1
channel. Similarly, we have a 2 × 1 channel from the relay and user 3 to user 2. Since the interference over the 12 A
‘basic’ Y-channel is one with no direct channels between the users as defined in Section II-C.
xr,N ˜
1 −1
xr,N ˜
1 −2
.. .
xr,N˜1 x3,N˜3 .. . x3,2
.. . x3,N˜3 xr,ar
+
xr,1
x3,1
x3,a3
0
0
z
Tx Relay
Tx User 3
Rx User 2
N1
xr,2
˜2 sub-channels N N1 ˜3 sub-channels N
˜1 sub-channels N ˜2 sub-channels N ˜3 sub-channels N
xr,N˜1
Fig. 9.
In the extended Y-channel, the received signal at user 2 is a superposition of the relay signal, the transmit signal of user 3, and noise. e2 sub-channels from the relay, i.e., ar = N e1 − (N e2 − 1). On the other hand, N1 sub-channels at user 2 Due to noise, user 2 only observes N e are corrupted by the transmit signal of user 3, and thus a3 = N3 − (N1 − 1).
channel h1 has power h21 P , it will corrupt the lowest N1 =
j
log(Γ1 ) log(γ)
k
sub-channels at users 2 and 3 (see Fig. 9).
In accordance with the SNR values, the numbers of sub-channels are sorted as follows ˜1 = N3 N2 /N1 ≥ N ˜ 2 = N3 ≥ N ˜ 3 ≥ N2 ≥ N1 . N Fortunately, this interference can be nicely taken care of as we show next. The interference between users 2 and 3 over the channel h1 can be classified into three classes: (a) The interference received at user i ∈ {2, 3} is a dedicated signal from user j ∈ {2, 3}, j 6= i to user i, that is to be forwarded by the relay (user 1) to user i in the next time-instant. (b) The interference received at user 3 is a dedicated signal from user 2 to user ˜1. (c) The interference received at user 2 is a dedicated signal from user 3 to user ˜1. Next, we describe how to deal with this interference while maintaining the general structure of the basic Y-channel scheme. We start with interference of class (a). 1) Class (a) Interference: To compensate interference of class (a), we use backward decoding. The basic Y-channel transmission scheme is applied for B + 1 blocks of length n symbols each (codeword length), where the Y-channel users are active in the first B blocks, and the relay (user 1) is active in the last B blocks. Decoding at the users is postponed until the end of the last block B + 1. This is the only difference with the basic Y-channel scheme – the latter does not require backward decoding. In block B + 1, the users are silent, and therefore, there is no interference between users 2 and 3 in this transmission block. Hence, users 2 and 3 can decode their dedicated signals corresponding to block B +1 from the relay signal. The users then proceed to decode the signals of block B, where there is interference between users 2 and 3. However, this interference consists of desired signals (class (a)) that have been decoded by users 2 and 3 in the block B + 1. Thus, this interference can be canceled, rendering
the received signals of users 2 and 3 in block B free of class (a) interference. This cancellation is performed analogously for all blocks B − 1, B − 2, · · · , 1, and class (a) interference is resolved. 2) Class (b) Interference: To compensate interference of class (b), i.e., interference observed at user 3 which consists of signals from user 2 intended to user ˜1, we apply the following interference neutralization scheme. User 2 pre-transmits the interference signal one transmission block in advance as follows. Consider sub-channel (n)
(n)
ℓ ∈ {1, · · · , N1 } at user 3 in downlink block b, and assume that user 3 receives x ˜r,ℓ (b) from the relay and x˜2,ℓ (b) from user 2 over this sub-channel, i.e., (n)
(n)
(n)
(n)
y˜3,ℓ (b) = x˜r,ℓ (b) + x ˜2,ℓ (b) + z˜3,ℓ (b),
(115)
(n)
where z3,ℓ (b) includes the noise at user 3 and all the interference signals at levels below ℓ. Recall that all signals in the Y-channel scheme explained above are lattice coded. Let us write (n)
x ˜r,ℓ (b) = (λr,ℓ (b) + dr,ℓ (b)) mod Λc
(116)
(n)
x ˜2,ℓ (b) = (λ2,ℓ (b) + d2,ℓ (b)) mod Λc .
(117)
Here, λr,ℓ (b) can be either a combination of two bi-directional or cyclic signals, one of which is desired at user 3, or a uni-directional signal. Assume that this signal is decoded by the relay over sub-channel ℓ′ in the uplink in block b − 1 (due to causality). In general, we can write λr,ℓ (b) = (λ˜1,ℓ′ (b − 1) + λ2,ℓ′ (b − 1) + λ3,ℓ′ (b − 1)) mod Λc where λi,ℓ′ (b − 1) = 0 if user i does not occupy this sub-channel. To enable interference neutralization, the relay should send (λr,ℓ (b) − λ2,ℓ (b)) mod Λc instead of λr,ℓ in block b
(see Section V-D3). Thus, in uplink block b − 1, sub-channel ℓ′ should be altered by user 2 so that the relay decodes (λr,ℓ (b) − λ2,ℓ (b)) mod Λc instead of λr,ℓ (b). To this end, user 2 pre-transmits this interference λ2,ℓ (b) to the relay.
In particular, user 2 adds −λ2,ℓ (b) (modΛc ) to λ2,ℓ′ (b − 1) which it sends over sub-channel ℓ′ in uplink block (n)
b − 1. The transmit signal of user 2 becomes x2,ℓ′ (b − 1) = (λ2,ℓ′ (b − 1) − λ2,ℓ (b) + d2,ℓ′ (b − 1)) mod Λc instead (n)
of x2,ℓ′ (b − 1) = (λ2,ℓ′ (b − 1) + d2,ℓ′ (b − 1)) mod Λc . The relay decodes (λr,ℓ (b) − λ2,ℓ (b)) mod Λc in uplink block b − 1, and consequently, user 3 receives (n)
(n)
y˜3,ℓ (b) = ((λr,ℓ (b) − λ2,ℓ (b)) mod Λc + dr,ℓ (b)) mod Λc + (λ2,ℓ (b) + d2,ℓ (b)) mod Λc + z˜3,ℓ (b) (n)
= (λr,ℓ (b) − λ2,ℓ (b) + dr,ℓ (b)) mod Λc + (λ2,ℓ (b) + d2,ℓ (b)) mod Λc + z˜3,ℓ (b),
(118) (119)
where the last step follows since (a mod Λ+b) mod Λ = (a+b) mod Λ, as given in [38]. This achieves interference neutralization after user 3 computes the sum (λr,ℓ (b) − λ2,ℓ (b) + λ2,ℓ (b)) mod Λc = λr,ℓ (b). This pre-transmission solves the problem of class (b) interference under two conditions. First, the pre-transmission should not disturb any other user. Second, user 2 should have access to sub-channel ℓ′ from which λr,ℓ (b) is decoded by the relay in the uplink. The first condition can be easily checked. User ˜1 is not disturbed by this pre-transmission of λ2,ℓ (b) in block b − 1 since λ2,ℓ (b) is desired by user ˜ 1 (as per definition of class (b) interference), and hence can be removed by
backward decoding. That is, user ˜ 1 can cancel the impact of this pre-transmission on its received signal in downlink block b − 1 after it has decoded λ2,ℓ (b) in downlink block b. Moreover, the pre-transmission of λ2,ℓ (b) clearly does not disturb user 2 since λ2,ℓ (b) originates from user 2 itself. This pre-transmission is only received at user 3 if it is sent over the top-most N1 sub-channels at user 2 in the uplink. Here we distinguish between two cases: this pre-transmission is either received on sub-channel ℓ at user 3, or on some other sub-channel. In the latter case, this pre-transmission is combined with class (b) interference and neutralized as described above. The former case where this pre-transmission by user 2 on sub-channel ℓ′ in the uplink interferes with sub-channel ℓ at user 3 in the downlink, is not possible by construction of the Y-channel scheme. Namely, this case means that the relay signal on sub-channel ℓ in the downlink is desired by user 3 and contains a signal from user 2 to user ˜1 at the same time, which is not possible for the schemes in Sections VI-A to VI-C (cf. [30, Sec. VI.C.1]). The second condition needs further consideration. We need to make sure that user 2 has access to sub-channel e2 } where ℓ′ from which λr,ℓ is decoded by the relay in the uplink. That is, we need to insure that ℓ′ ∈ {1, · · · , N
e2 = N3 by the transformation above. The value of ℓ′ depends on the content of λr,ℓ . Recall that λr,ℓ is desired at N
user 3. By referring to Sections VI-A to VI-C, it can be checked that all signals desired by user 3 are accessible by user 2 in the uplink, except the uni-directional signal from user ˜1 to user 3. This signal, which we will refer to as e2 + 1, · · · , N e1 in the uplink since it originates from user ˜1 (Fig. 10(a)). In λu3˜1 , might be received on sub-channels N
this case, user 2 can not alter this signal for the purpose of interference neutralization. Therefore, we need to avoid e2 + 1, · · · , N e1 is N e1 − N e2 . this scenario. We first note that the number of sub-channels in the problematic range N
On the other hand, the number of sub-channels at user 3 which do not receive any interference from user 2 is e3 − N1 (Fig. 10(b)). By the 3WC–Y-channel transformation given in Theorem 2, we have N e1 − N e2 = N e3 − N1 . N
Therefore, we can avoid the aforementioned problematic scenario by exploiting this interesting equality. Namely, the relay of the extended Y-channel (user 1) forwards λu3˜1 over the non-interfered downlink sub-channels at user 3 e1 − N e3 + N1 + 1, · · · , N e1 }. By pursuing such an approach, the impact of class (b) interference by choosing ℓ ∈ {N
is completely eliminated.
3) Class (c) Interference: Class (c) interference is similar to class (b) interference. To compensate it, we apply interference neutralization again. That is, user 3 pre-transmits the signal causing class (c) interference one transmission block in advance. As in Sec. VII-B2, we have to guarantee two conditions. First that this pretransmission does not disturb the users, and second that user 3 has access to all sub-channels on which this pre-transmission should take place. As per definition of class (c) interference, this pre-transmission does not affect user ˜1 since the pre-transmitted signal itself is desired at user ˜ 1. Thus, user ˜1 can cancel it by backward decoding. Moreover, this pre-transmission does not affect user 3 since it originates from this same user. The pre-transmission disturbs user 2 only if it is received on the same sub-channel where this class (c) interference is received. However, this is not possible since there is no relay signal in the Y-channel scheme which is both desired at user 2, and contains a signal dedicated to user ˜1 from user 3 (cf. Sections VI-A to VI-C). Thus, interference neutralization by pre-transmission does not disturb the users of the Y-channel.
Sub-channel
e1 N
e1 N
˜ User 1 User 2
2
User 3
e3 N
ℓ
1
e1 − N e2 + 1 N 2 1
(a) Fig. 10.
User ˜ 1
e2 N
e1 − N e2 + N1 N e1 − N e3 + 1 N
User 2
ℓ′
e1 − N e3 + N1 N
User 3
Sub-channel
(b)
Sub-channels accessible by users ˜ 1, 2, and 3, in the uplink (a) and downlink (b) of an extended Y-channel. The shaded and dashed
sub-channels in (b) are the sub-channels at users 3 and 2 which receive interference from users 2 and 3, respectively.
It remains to show that user 3 has access to all sub-channels on which this pre-transmission should occur. User 3 e3 sub-channels in the uplink (Fig. 10(a)), whose signals are potentially forwarded to user 2 in the can access N e3 + 1, · · · , N e1 . Thus, downlink during the next transmission block. However, user 3 can not access sub-channels N
e3 + 1, · · · , N e1 } by the relay in if the interfered signal which is desired by user 2 is received on sub-channel ℓ ∈ {N
the uplink, then user 3 can not perform interference neutralization. However, this problem is similarly solved as in the class (b) interference. The number of these problematic sub-channels is equal to the number of sub-channels at e1 − N e3 = N e2 − N1 user 2 which do not receive any interference in the downlink (Fig. 10(b)). This follows since N
e 1 = Γ3 Γ2 /Γ1 (cf. Theorem 2). Thus, by sending all signals received on sub-channels {N e3 + 1, · · · , N e1 } due to Γ
in the uplink on the interference-free sub-channels of user 2 in the downlink, we can guarantee that interference neutralization is performed successfully. 4) Achievable Rate Region: We conclude that interference over the channel h1 between users 2 and 3 in the extended Y-channel can be successfully resolved. Thus, the transmission scheme of the basic Y-channel is used for the 3WC to achieve a similar rate region after applying two modifications: backward decoding and signal pretransmission for interference neutralization. Now let us express the achievable rate region of this scheme. For this, we need to find the achievable rate per sub-channel. First note that the uplink in the extended Y-channel resembles a 3 × 1 channel, where three users can have access the same sub-channel due to pre-transmissions. Thus, we have κ = 3 (instead of 2 in the basic Y-channel) and µ = 0. The downlink on the other hand resembles a 1 × 3 channel leading to µ = 1. Furthermore, due to interference between users 2 and 3 (through channel h1 ), some sub-channels at user 2 are accessed by both the relay and user 3, and some sub-channels at user 3 are accessed by both the relay and user 2. Thus, in the downlink we have κ = 2. Both the uplink and downlink have κ + µ = 3. Therefore, each ˆ sub-channel can support a rate of C(γ/3) in the uplink and the downlink. Consequently, we obtain the following
proposition. Proposition 2. The rate region C(Γ1 , Γ2 , Γ3 , N3 ) defined by the following set of inequalities ˆ ˆ 2 ) − N2 C(3), R31 + R32 ≤ C(Γ
(120)
ˆ ˆ 2 ) − N2 C(3), R13 + R23 ≤ C(Γ
(121)
ˆ ˆ 3 ) − N3 C(3), R12 + R13 + R32 ≤ C(Γ
(122)
ˆ ˆ 3 ) − N3 C(3), R12 + R13 + R23 ≤ C(Γ
(123)
ˆ 3 Γ2 /Γ1 ) − (N3 + N2 − N1 )C(3), ˆ R21 + R23 + R13 ≤ C(Γ
(124)
ˆ ˆ 3 ) − N3 C(3), R21 + R23 + R31 ≤ C(Γ
(125)
ˆ ˆ 3 ) − N3 C(3), R31 + R32 + R21 ≤ C(Γ
(126)
ˆ 3 Γ2 /Γ1 ) − (N3 + N2 − N1 )C(3). ˆ R31 + R32 + R12 ≤ C(Γ
(127)
is an achievable rate region for the 3WC(Γ1 , Γ2 , Γ3 ) channel with Γ3 ≥ Γ2 ≥ Γ1 , where N3 , N2 , N1 ∈ N \ {0}, j k i) Ni = log(Γ and γ N3 = Γ3 > 3N3 . log(γ) e1 , Γ e 2 , and Γ e 3 in C (Γ e1 , Γ e2 , Γ e3 , N e1 ) by Γ3 Γ2 /Γ1 , The statement of this proposition is obtained by replacing Γ Y
e1 = N3 + N2 − N1 , N e2 = N3 , and N e3 = N2 . The factor C(3) ˆ Γ3 , and Γ2 , which also leads to N arises from
ˆ ˆ ˆ ˆ ˆ replacing the sub-channel capacity by C(γ/3) = C(γ) − C(3) instead of C(γ/2) = C(γ) − 1/2. This replacement is necessary due to the presence of interference and the use of pre-transmissions to counter this interference. Similar to Proposition 1, N3 should satisfy the following conditions log(Γ3 ) log(Γ3 ) < N3 < , log(Γ1 ) log(3) and a valid N3 exists if Γ1 > 3. The gap between C(Γ1 , Γ2 , Γ3 , N3 ) and C(Γ1 , Γ2 , Γ3 ) is a constant that depends mainly on N3 . Similar to the Y-channel, this gap can be reduced to a universal constant by using the sub-channel grouping idea explained in Section VI-E. With Proposition 2, the proof of Theorem 1 is complete. VIII. S PECIAL C ASES In this section, we focus on some special cases of the 3WC, and show that our result agrees with existing results in literature for these special cases. We will discuss the multiple access channel (MAC) with cooperating/conferencing encoders and the broadcast channel (BC) with cooperating receivers. A. The MAC with Cooperation Cooperation between the users of a MAC is possible if the users can hear each others’ transmission. This in turn is enabled if the users can both transmit and receive. From this point of view, a 2-user MAC with cooperation can be modeled as a 3WC, with a specific message exchange scenario. In what follows, we discuss the scenario where
the users are connected by the stronger channel h3 . This is motivated by D2D communication in cellular networks, where users within close proximity can communicate with each other simultaneously while communicating with a base-station. A similar discussion can be pursued for other cases where the cooperation channel is weaker than one or both the channels to the base-station. 1) Conferencing: The MAC with cooperating transmitters was first studied by Willems in [50], in the so-called MAC with conferencing encoders. In this channel, users 1 and 2 want to communicate two independent messages W1 and W2 to a common receiver, respectively. The users are allowed to communicate with each other in a conferencing phase over h3 before they communicate with the receiver in the transmission phase. The conferencing takes place over two error-free channels with capacities C21 (from user 1 to 2) and C12 (from user 2 to 1). Let each of the two users have a power constraint P , and let the common receiver’s signal be given by y = h2 x1 + h1 x2 + z where z is a Gaussian noise with zero mean and unit variance. Thus, the SNR’s of the channels from users 1 and 2 to the receiver are Γ2 = h22 P and Γ1 = h21 P , respectively. The capacity region of this channel was given in [51] as CMAC,conf =
[
β1 ,β2 ∈[0,1]
RMAC,conf (β1 , β2 ),
where RMAC,conf (β1 , β2 ) is the set of rate pairs (R1 , R2 ) ∈ R2+ satisfying R1 ≤ C(β1 Γ2 ) + C21 ,
(128)
R2 ≤ C(β2 Γ1 ) + C12 ,
(129)
R1 + R2 ≤ C(β1 Γ2 + β2 Γ1 ) + C21 + C12 , q R1 + R2 ≤ C(Γ1 + Γ2 + 2 β¯1 β¯2 Γ1 Γ2 ),
(130) (131)
with β¯1 = 1 − β1 and β¯2 = 1 − β2 . Now assume that C12 = C21 = C(Γ3 ), with Γ3 = h23 P . Furthermore, assume that Γ3 ≥ Γ2 ≥ Γ1 . Under these assumptions, the MAC with conferencing encoders resembles a 3WC with R31 = R1 , R32 = R2 , and R12 = R13 = R21 = R23 = 0. In this case, it can be easily verified that the region CMAC,conf is within a constant gap of the region defined by ˆ 2 ). R1 + R2 ≤ C(Γ
(132)
But the latter region coincides with the statement of Theorem 1 for this special case. Thus, the statements of Theorem 1 and [51] agree within a constant gap, and our result characterizes the capacity region of the MAC with conferencing encoders within a constant gap. This leads to the following interesting consequence. Remark 3. If the conferencing channel has high capacity given by C12 = C21 = C(Γ3 ) with Γ3 ≥ Γ2 , Γ1 , then conferencing can be replaced by in-band cooperation with marginal impact on the capacity region. Here, by marginal impact we mean that the impact is bounded by a constant independent of the SNR’s. Thus, in this case, we do not need to spend any extra time for conferencing, since cooperation can be established
simultaneously while transmitting to the receiver. 2) In-band cooperation: The MAC with in-band cooperation (also known as the MAC with generalized feedback [50]) differs from the MAC with conferencing in that cooperation takes place simultaneously with transmission to the receiver. That is, the cooperation phase and the transmission phase are actually one and the same phase. Consider a Gaussian MAC with cooperation, where two users have a power constraint P , and where the received signals are given by y = h2 x1 + h1 x2 + z,
(133)
y1 = h3 x2 + z1 ,
(134)
y2 = h3 x2 + z2 ,
(135)
where z, z1 , and z2 are independent Gaussian noises with zero mean and unit variance. Thus, the SNR’s of the channels from users 1 and 2 to the receiver are Γ2 = h22 P and Γ1 = h21 P , respectively, and the SNR of the cooperation channel is Γ3 . An achievable rate region for this scenario is given by [52] RMAC,coop =
[
RMAC,coop (β 1 , β 2 ),
where the union is over βi = (βi1 , βi2 , βi3 ) satisfying βij ≥ 0 and R2+
P3
j=1
βij ≤ 1 for i = 1, 2 and j = 1, 2, 3, and
where RMAC,coop (β 1 , β 2 ) is the set of rate pairs (R1 , R2 ) ∈ satisfying βi1 Γ3 + C (βi2 Γj ) , i, j ∈ {1, 2}, i 6= j Ri ≤ C 1 + βi2 Γ3 p R1 + R2 ≤ C Γ2 + Γ1 + 2 β13 β23 Γ1 Γ2 2 X βi1 Γ3 C . R1 + R2 ≤ C (β12 Γ2 + β22 Γ1 ) + 1 + βi2 Γ3 i=1
(136) (137) (138)
For Γ3 ≥ Γ2 ≥ Γ1 , it can be easily shown that the above region is within a constant gap of the region ˆ 2 ). R1 + R2 ≤ C(Γ
(139)
The latter region coincides with the statement of Theorem 1 for this case as shown for the conferencing MAC above. Thus, the statements of Theorem 1 and the achievable rate region RMAC,coop obtained from [52] agree within a constant gap. Furthermore, our result provides an outer bound on the capacity region for this case, and thus shows that the region RMAC,coop characterizes the capacity region of the MAC with in-band cooperation within a constant gap. Note that this statement confirms Remark 3. B. The BC with Cooperation Similar to the MAC with cooperation, it is possible to establish cooperation in a broadcast channel (BC) if users hear each others’ transmission. A 2-user BC with cooperation is thus also a special case of a 3WC. Next, we consider a scenario where the receivers of the BC are connected by the stronger channel h3 , motivated by D2D communications. Other cases can be discussed similarly.
The input-output relationships of a BC with in-band cooperation can be written as y1 = h1 x3 + h3 x2 + z1 ,
(140)
y2 = h2 x3 + h3 x1 + z2 ,
(141)
where z1 and z2 are independent Gaussian noises with zero mean and unit variance. Each of the nodes has power P . Therefore, the SNR’s corresponding to channels h1 , h2 , and h3 are Γ1 = h21 P , Γ2 = h22 P , and Γ3 = h23 P , respectively. An achievable rate region for this channel was given in [53] as [ RBC,coop = RMAC,coop (β2 , β3 ), where the union is over β2 ∈ [0, 1], and β 3 = (β31 , β32 , β33 ) with β3j ≥ 0 and
RMAC,coop (β2 , β3 ) is the set of rate pairs (R1 , R2 ) ∈ R2+ satisfying β31 β2 Γ1 Γ3 R1 ≤ C β31 Γ2 + 1 + β2 Γ3 + β31 (Γ1 + Γ2 ) √ (β33 + β32 )Γ1 + Γ3 + 2 β32 Γ1 Γ3 R2 ≤ C 1 + β31 Γ1 β33 Γ2 R2 ≤ C . 1 + β31 Γ2 + β2 Γ3
P3
j=1
β3j ≤ 1, and where
(142) (143) (144)
The region RBC,coop is within a constant gap of the region ˆ 2 ), R1 + R2 ≤ C(Γ
(145)
which is the approximate capacity region of the 3WC with Γ3 ≥ Γ2 ≥ Γ1 , and with R31 = R1 , R32 = R2 , R12 = R13 = R21 = R23 = 0. In particular, by setting β3 = (1, 0, 0) we achieve R1 ≤ C(Γ2 ), and by setting β 3 = (0, 0, 1) and β2 = 0 we achieve R2 ≤ C(Γ2 ). Time-sharing between these two solutions is within a constant
ˆ 2 ). of R1 + R2 ≤ C(Γ
Therefore, also for the BC with cooperation, the statement of Theorem 1 and the achievable rate region RBC,coop obtained from [53] agree within a constant gap in this case (cooperation channel h3 ). Thus, RBC,coop is within a constant gap of the capacity region of the BC with in-band cooperation. Finally, we note that similar analysis can be applied for the relay channel and the two-way relay channel and variations thereof. The capacities of those channels are within a constant gap of C(Γ1 , Γ2 , Γ3 ) in Theorem 1. IX. C ONCLUSION We have studied the capacity region of the 3-way channel consisting of 3 users communicating in all directions, denoted 3WC. First, we derived a capacity region outer bound for the 3WC. A resemblance between the derived outer bound and the approximate capacity region of the Y-channel in [30] motivated us to seek a 3WC–Y-channel (∆–Y) transformation. It enabled the extension of a transmission strategy from the Y-channel to the 3WC. To achieve this goal, we used a 4-step approach. First, we proposed a channel decomposition approach which decomposes a Gaussian channel into a set of successive sub-channels. These successive sub-channels differ from parallel subchannels as follows: successive sub-channels interact (interfere) with each other, contrary to parallel sub-channels.
However, this interaction can be dealt with by using an appropriate power allocation for the sub-channels, and successive decoding, hence the designation successive sub-channels. Second, we used this decomposition approach to express the problem of finding an achievable rate region for the Y-channel as a sub-channel allocation problem. This leads to a simple representation of an achievable rate region, which is within a constant gap of the capacity of the Y-channel. Third, the 3WC is re-cast as an extended Y-channel using a 3WC–Y-channel transformation. Finally, the obtained transmission scheme for the Y-channel, based on the sub-channel allocation approach, is modified for the corresponding extended Y-channel, leading to a transmission scheme which achieves the capacity region of the 3WC within a constant gap. Interestingly, the channel decomposition approach we follow in this paper significantly simplifies the problem of rate region achievability for Gaussian channels. First, the problem is reformulated as a sub-channel allocation problem. Then, a sub-channel allocation is derived, either directly, or with the aid of results from the corresponding linear-deterministic model of the channel [35]. This leads to an achievable sub-channel allocation region, which can be transformed into an achievable rate region simply by multiplying by the achievable rate per sub-channel. It is worth to note that our channel decomposition does not impose any additional assumptions on a Gaussian channel. Furthermore, it is a valid decomposition under a mild conditions the SNR, namely, it is valid for high and even moderate SNR. The interesting relation between the 3WC and the Y-channel raises several interesting questions. Does this transformation extend to larger networks (of more than 3 users) with mesh and star topologies? Likewise, can rather sophisticated networks be broken down into simpler networks by using similar correspondences as the 3WC– Y-channel transformation? Answering these questions would greatly simplify studying the information-theoretic limits of larger networks. A PPENDIX A C UT- SET
BOUNDS FOR
G AUSSIAN I NPUTS
The first cut-set bound we consider is (16), i.e., (n)
n(Rji + Rki − εn ) ≤ I(Wji , Wki ; Yj (n)
= h(Yj
(n)
(n)
, Yk |Wij , Wkj , Wik , Wjk ) (n)
, Yk |Wij , Wkj , Wik , Wjk ) − h(Yj
(146) (n)
, Yk |W),
(147)
where εn → 0 as n → ∞ and W = (Wij , Wji , Wik , Wki , Wkj , Wjk ). The conditional joint differential entropy of (n)
Yj
(n)
and Yk
can be upper bounded as follows
(n)
h(Yj
= = = ≤ ≤
(n)
, Yk |Wij , Wkj , Wik , Wjk )
n X
t=1 n X
t=1 n X
t=1 n X
h(Yj (t), Yk (t)|Wij , Wkj , Wik , Wjk , Yjt−1 , Ykt−1 )
(148)
h(Yj (t), Yk (t)|Wij , Wkj , Wik , Wjk , Yjt−1 , Ykt−1 , Xjt , Xkt )
(149)
h(hk Xi (t) + Zj (t), hj Xi (t) + Zk (t)|Wij , Wkj , Wik , Wjk , Yjt−1 , Ykt−1 , Xjt , Xkt )
(150)
h(hk Xi (t) + Zj (t), hj Xi (t) + Zk (t))
(151)
t=1
n log((2πe)2 (1 + h2k P + h2j P )), 2
(152)
where in (148)-(152), we have used the chain rule, the encoding function (3), the equality that h(X|Y ) = h(X − Y |Y ), the fact that conditioning does not increase entropy, and property that the Gaussian distribution maximizes the differential entropy under a covariance constraint [28], respectively. (n)
The differential entropy term h(Yj
(n)
, Yk |W) can be bounded by the entropy of the noise random variables Zj
and Zk as follows (n) (n) h(Yj , Yk |W)
= ≥ = = = =
n X
t=1 n X
t=1 n X
t=1 n X
t=1 n X
t=1 n X
h(Yj (t), Yk (t)|W, Yjt−1 , Ykt−1 )
(153)
h(Yj (t), Yk (t)|W, Yjt−1 , Ykt−1 , Yit−1 )
(154)
h(Yj (t), Yk (t)|W, Yjt−1 , Ykt−1 , Yit−1 , Xjt , Xkt , Xit )
(155)
h(Zj (t), Zk (t)|W, Zjt−1 , Zkt−1 , Zit−1 , Xjt , Xkt , Xit )
(156)
h(Zj (t), Zk (t)|W, Zjt−1 , Zkt−1 , Zit−1 )
(157)
h(Zj (t), Zk (t))
(158)
t=1
=
n log((2πe)2 ), 2
(159)
where steps (153)-(159) follow by using the chain rule, the fact that conditioning does not increase entropy, the encoding function (3), the equality h(X|Y ) = h(X − Y |Y ), the fact that Xjt , Xkt , and Xit can be constructed from W, Zjt−1 , Zkt−1 , and Zit−1 (since hi , hj , and hk are known at all nodes), the independence of the noises and the
messages, and that Zj and Zk are N (0, 1). Combining (159) and (152), dividing by n, and letting n → ∞ yields the bound (18).
The second cut-set bound (17) is given by (n)
n(Rij + Rik − εn ) ≤ I(Wij , Wik ; Yi (n)
= h(Yi
|Wji , Wjk , Wki , Wkj ) (n)
|Wji , Wjk , Wki , Wkj ) − h(Yi
|W).
(160) (161)
Similar to above, by using the chain rule, the fact that conditioning does not increase entropy, and that the Gaussian distribution is a differential entropy maximizer, we get (n)
h(Yi
n log(2πe(1 + h2j P + h2k P + 2hj hk ρP )) 2 n ≤ log(2πe(1 + (|hj | + |hk |)2 P )), 2
|Wji , Wjk , Wki , Wkj ) ≤
(162) (163)
where ρ ∈ [−1, 1] is the correlation between the Gaussian Xj and Xk . Similarly, we can show that (n)
h(Yi
|W) ≥
1 log(2πe). 2
(164)
Combining (164) and (163), dividing by n, and letting n → ∞ yields the desired bound (19). A PPENDIX B G ENIE - AIDED B OUNDS We start with the bound (23), i.e., (n)
(n)
n(R12 + R13 + R32 − εn ) ≤ I(W12 , W13 , W32 ; Y1 , Z¯3 , W21 , W31 , W23 ), (n)
where we have denoted Z3
−
h1 (n) h3 Z 1
(165)
(n) by Z¯3 , which we can rewrite as
(n) (n) n(R12 + R13 + R32 − εn ) ≤ I(W12 , W13 , W32 ; Y1 , Z¯3 |W21 , W31 , W23 ),
(166)
due to the independence of the messages. By using the definition of mutual information, we get (n) (n) (n) (n) n(R12 + R13 + R32 − εn ) ≤ h(Y1 , Z¯3 |W21 , W31 , W23 ) − h(Y1 , Z¯3 |W).
(167)
(n) (n) Now we proceed by bounding each of the terms above separately. First we consider h(Y1 , Z¯3 |W21 , W31 , W23 )
which we can bound as follows (n) (n) (n) (n) h(Y1 , Z¯3 |W21 , W31 , W23 ) ≤ h(Y1 , Z¯3 )
= ≤ =
n X
t=1 n X
t=1 n X t=1
(168)
h(Y1 (t), Z¯3 (t)|Y1t−1 , Z¯3t−1 )
(169)
h(Y1 (t), Z¯3 (t))
(170)
(171)
h(Z¯3 (t)) + h(Y1 (t)|Z¯3 (t)) .
(n) (n) Next, we consider h(Y1 , Z¯3 |W) which we bound as follows (n) (n) h(Y1 , Z¯3 |W) =
≥ = = =
n X
h(Y1 (t), Z¯3 (t)|W, Y1t−1 , Z¯3t−1 )
(172)
h(Y1 (t), Z¯3 (t)|W, Y1t−1 , Z¯3t−1 , X1t , X2t , X3t )
(173)
h(Z1 (t), Z¯3 (t)|W, Z1t−1 , Z¯3t−1 , X1t , X2t , X3t )
(174)
h(Z1 (t), Z¯3 (t))
(175)
[h(Z1 (t)) + h(Z¯3 (t)|Z1 (t))]
(176)
t=1 n X
t=1 n X
t=1 n X
t=1 n X t=1
where that last but one step follows since the noise at time instant t is independent of all past noise samples, the messages, and the transmit signals up to time instant t (only the transmit signals at times t + 1, · · · , n can be dependent on the noise samples at time t (3)). By plugging (170) and (175) in (167) we obtain n(R12 + R13 + R32 − εn ) ≤
n X h(Y1 (t), Z¯3 (t)) − h(Z1 (t), Z¯3 (t)) .
(177)
t=1
This can be rewritten as follows n(R12 + R13 + R32 − εn ) ≤
n X t=1
h(Y1 (t)|Z¯3 (t)) − h(Z1 (t)|Z¯3 (t))
(178)
by using the chain rule. Using [54, Lemma 6], we can rewrite this bound as n(R12 + R13 + R32 − εn ) ≤
n X t=1
[h(h2 X3 (t) + h3 X2 (t) + V (t)) − h(V (t))] ,
(179)
h2
3 where V ∼ N (0, h2 +h 2 ). Now by using the Gaussian distribution for X2 and X3 , we can maximize this bound to 1
3
obtain n(R12 + R13 + R32 − εn ) ≤ nC By dividing by n, and letting n → ∞ we obtain R12 + R13 + R32 ≤ C
h21 + h23 (|h2 | + |h3 |)2 P h23
h21 + h23 (|h2 | + |h3 |)2 P h23
.
Next, we relax this bound by using h23 ≥ h22 ≥ h21 (cf. the ordering in (2)) as follows 2 h1 + h23 2 R12 + R13 + R32 ≤ C (|h2 | + |h3 |) P h23
.
(180)
(181)
(182)
≤ C(2(|h2 | + |h3 |)2 P )
(183)
≤ C(2(2|h3 |)2 P )
(184)
= C(8h23 P )
(185)
≤
1 log(h23 P ) + 2, 2
(186)
which is the desired upper bound in (24). The remaining bounds (25)-(29) can be derived similarly by giving each node the suitable side-information. The second bound (25) given by R12 + R13 + R23 ≤ C(h23 P ) + 2 (n)
h1 (n) to node 1 as side information. The third and fourth h2 Z 1 (n) (n) (n) (n) (W13 , Z3 − hh23 Z2 ) and (W31 , Z1 − hh12 Z2 ) to node 2 as side
can be derived by giving W32 and Z2 and (27) can be derived by giving
−
(187) bounds (26) information,
respectively. Finally, the side information that should be given to node 3 in order to obtain the bounds (28) and (n)
(29) are (W12 , Z2
−
h3 (n) h2 Z 3 )
(n)
and (W21 , Z1
−
h3 (n) h1 Z3 ),
respectively.
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