Capacity of a Class of Linear Binary Field Multi-source Relay Networks
aa r X i v : . [ c s . I T ] F e b Capacity of a Class of Linear Binary FieldMulti-source Relay Networks
Sang-Woon Jeon,
Student Member , IEEE and Sae-Young Chung,
Senior Member , IEEE
Abstract
Characterizing the capacity region of multi-source wireless relay networks is one of the fundamental issues innetwork information theory. The problem is, however, quite challenging due to inter-user interference when thereexist multiple source–destination (S–D) pairs in the network. By focusing on a special class of networks, we showthat the capacity can be found. Namely, we study a layered linear binary field network with time-varying channels,which is a simplified model reflecting broadcast, interference, and fading natures of wireless communications. Weobserve that fading can play an important role in mitigating inter-user interference effectively for both single-hopand multi-hop networks. We propose new encoding and relaying schemes with randomized channel pairing, whichexploit such channel variations, and derive their achievable rates. By comparing them with the cut-set upper bound,the capacity region of single-hop networks and the sum capacity of multi-hop networks can be characterized forsome classes of channel distributions and network topologies. For these classes, we show that the capacity regionor sum capacity can be interpreted as the max-flow min-cut theorem.
I. I
NTRODUCTION
Capacity characterization of general wireless relay networks is a fundamental problem in network informationtheory. However, the capacity is not fully characterized even for the simplest network consisting of single source,single relay, and single destination [1]. In wireless environments, a transmit signal will be heard by multiple nodes,which we call the broadcast nature of wireless communications, and a receiver will receive the superpositionof simultaneously transmitted signals from multiple nodes, which we call the interference nature of wirelesscommunications. Furthermore wireless channels may be time-varying due to fading , and there is noise at eachreceiver. Considering all these makes the problem vary hard.Hence, one of the promising approaches is to study simplified relay networks, whose results can provide insightstowards exact or approximate capacity characterization for more general wireless relay networks. Let us first look atsome cases for which the capacity is known. For wireline relay networks, routing is enough to achieve the unicastcapacity [2]. On the other hand, routing alone cannot achieve the multicast capacity and network coding has beenshown to be optimal in this case [3]–[6]. For deterministic relay networks with no interference, the unicast capacityhas been characterized in [7] and the extension to the multicast case has been studied in [8]. The multicast capacityof erasure networks with no interference has been also characterized in [9]. When there is no broadcast, the unicastcapacity of erasure networks has been characterized in [10], which is the dual network studied in [9]. For all thesementioned networks, the unicast or multicast capacity can be interpreted as the max-flow min-cut theorem .Notice that although such orthogonal transmission or reception is possible in practice by using time, frequency,or code-division techniques, it is suboptimal in general. Therefore, simplification of wireless relay networks whilepreserving both broadcast and interference natures is crucially important to capture the essence of wireless com-munications. One of the simplest models that successfully reflect both broadcast and interference natures is a linearfinite field relay network [11]–[13], where a node transmits an element in the finite field and receives the sum oftransmit signals in the same finite field. Recently, the work in [13] has shown that the max-flow min-cut theoremalso holds for deterministic linear finite field relay networks. After the capacity characterization of linear finitefield relay networks, the approximate capacity of Gaussian relay networks has been characterized within a constantnumber of bits/s/Hz using the quantize-random-map-and-forward by the same authors [14].
S.-W. Jeon and S.-Y. Chung are with the Department of EE, KAIST, Daejeon, South Korea (e-mail: [email protected]; [email protected]).The material in this paper was presented in part at the Information Theory and Applications Workshop, University of California SanDiego, La Jolla, CA, February 2009, and at the IEEE International Symposium on Information Theory (ISIT), Seoul, Korea, June/July 2009. ( ) b s s s s s s Repeattransmission (1)1 H (2)1 H ( ) a s s s s s s s s s H H ( ) RepeatRepeatRepeat ( ) s ss s Fig. 1. Interference mitigation for the single-hop network (a) and for the two-hop network (b), where the solid lines and the dashed linesdenote the corresponding channels are ones and zeros, respectively.
In spite of the surging importance of multi-source relay networks, capacity characterization is much morechallenging if there exist multiple source–destination (S–D) pairs in a network. Even for linear finite field relaynetworks, the extension of the results in [13] to the multi-source does not seem to be straightforward. Noticethat the main difficulty arises from the fact that the transmission of other sessions acts as inter-user interference and, as a result, the cut-set upper bound is not tight in general. Due to these difficulties, the existing capacity orapproximate capacity results are limited in specific network topologies such as two-user interference channel [15],[16], many-to-one and one-to-many interference channel [17], two-way channel [18], [19], two-user two-hop relaynetwork [20], [21], and double Z-channel [22]. Therefore, one of the basic questions is whether we can characterizethe capacity or approximate capacity for more general network topologies or other classes of relay networks.In this paper, we study a layered multi-source linear binary field relay network with time-varying channels ,which captures three key characteristics of wireless environment, i.e., broadcast, interference, and fading. Note thata random coding strategy, which is still optimal in fading single-source networks [23], [24], does not work anymorefor our network model due to the inter-user interference. As mentioned before, a fundamental issue in multi-sourcenetworks is how to manage inter-user interference properly. We observe that fading can play an important role inmitigating such interference efficiently, which leads to the capacity characterization for certain classes of networks.More specifically, for single-hop networks, inter-user interference can be removed completely at each destinationby using two particular channel instances jointly. For multi-hop networks, by using a series of particular channelinstances over multiple hops, each destination can also decode its message without interference.As an example, consider the three-user linear binary field relay network in Fig. 1, where s k ∈ F denotes theinformation bit of the k -th source and the symbol in each node denotes the transmit signal of that node. Forsingle-hop networks, as shown in Fig. 1. (a), by transmitting the same bit twice at each source through H (1)1 and H (2)1 such that H (1)1 + H (2)1 = I , each destination can cancel interference by adding the two received signals, where H (1)1 and H (2)1 denote the two different channel instances of the first hop and I denotes the identity matrix. Relatedworks dealing with the inseparability of parallel interference channels can be found in [25]–[28] and the referencestherein. The idea of opportunistically pairing two channel instances, i.e., H (1)1 + H (2)1 = I , also appeared in [27],[28]. This can be considered as a different and simpler way of doing interference alignment [29], [30]. For two-hopnetworks, as shown Fig. 1. (b), we notice that each destination can receive the information bit without interferenceif H H = I , where H and H denote the channel instances of the first and second hop, respectively. In general,the interference-free communication is possible for M -hop networks, M ≥ , by opportunistically pairing the seriesof channel instances from H to H M such that H M H M − · · · H = I , where H m denotes the channel instance ofthe m -th hop.Based on these key observations, we propose encoding and relaying schemes which make such opportunisticpairing of channel instances possible. By comparing their achievable rate regions with the cut-set upper bound,we characterize the capacity region of single-hop networks and the sum capacity of multi-hop networks for someclasses of network topologies and channel distributions. x x ,1 K x , y x , y x ,1 ,2 , K K y x , y x , y x ,2 ,3 , K K y x
1, 1 1, , M M y x (cid:16)
2, 1 2, , M M y x (cid:16) , 1 , , M M
K M K M y x (cid:16) M y M y , M K M y (cid:14) first hop second hop -th hop M (1,1)-th node (2,1)-th node (K ,1)-th node (1, 2)-th node (2, 2)-th node (K , 2)-th node (1, 3)-th node (2, 3)-th node (K ,3)-th node (1, )-th node M (2, )-th node M M (K , )-th node M (1, 1)-th node M (cid:14) (2, 1)-th node M (cid:14) M+1 (K , 1)-th node M (cid:14) Fig. 2. Layered multi-source relay network.
This paper is organized as follows. In Section II, we define the network model and state the multi-source relayproblem and the notations used in the paper. In Section III, we derive the general cut-set upper bound, which willbe used to prove the converses in Section IV. In Section IV, new encoding and relaying schemes are proposedto mitigate inter-user interference, which characterizes the capacity region or sum capacity for certain classes ofnetworks. We conclude this paper in Section V and refer the proofs of the lemmas to Appendices I and II.II. S
YSTEM M ODEL
In this section, we first explain the underlying network model and then define the achievable rate region and thenotations used in the paper. Throughout the paper, A and a denote a matrix and a vector, respectively. The symbol A denotes a set and |A| denotes the cardinality of A . A. Linear Binary Field Relay Networks
We study a layered network in Fig. 2 that consists of M + 1 layers having K m nodes at the m -th layer, where m ∈ { , · · · , M + 1 } . Let us denote K max = max m { K m } and K min = min m { K m } . The ( k, m ) -th node refers tothe k -th node at the m -th layer. Then K = K = K M +1 is the number of S–D pairs and the ( k, -th node andthe ( k, M + 1) -th node are the source and the destination of the k -th S–D pair, respectively. Notice that if M = 1 ,the network becomes a K -user interference channel.Consider the m -th hop transmission. The ( i, m ) -th node and the ( j, m + 1) -th node become the i -th transmitter(Tx) and the j -th receiver (Rx) of the m -th hop, respectively, where i ∈ { , · · · , K m } and j ∈ { , · · · , K m +1 } . Let x i,m [ t ] ∈ F denote the transmit signal of the ( i, m ) -th node at time t and y j,m [ t ] ∈ F denote the received signalof the ( j, m + 1) -th node at time t . Let h j,i,m [ t ] ∈ F be the channel from the ( i, m ) -th node to the ( j, m + 1) -thnode at time t . The relation between the transmit and received signals is given by y j,m [ t ] = K m X i =1 h j,i,m [ t ] x i,m [ t ] , (1)where all operations are performed over F . We assume time-varying channels such that Pr( h j,i,m [ t ] = 1) = p j,i,m (2)and h j,i,m [ t ] are independent of each other for different i , j , m , and t . This assumption can be generalized to blockfading with coherence time of T symbols, where T ≫ such that there is enough time for CSI to be spread torelevant nodes. We assume T = 1 for notational simplicity since our result does not explicitly depend on T as longas it is big enough such that CSI is available at all relevant nodes. Let x m [ t ] and y m [ t ] be the K m × transmit signalvector and K m +1 × received signal vector of the m -th hop, respectively, where x m [ t ] = [ x ,m [ t ] , · · · , x K m ,m [ t ]] T , y m [ t ] = (cid:2) y ,m [ t ] , · · · , y K m +1 ,m [ t ] (cid:3) T . Then the transmission of the m -th hop can be represented as We focus on the binary field F in this paper, but some results can be directly extended to F q (see Remarks 1 and 2). y m [ t ] = H m [ t ] x m [ t ] , (3)where H m [ t ] is the K m +1 × K m channel matrix of the m -th hop having h j,i,m [ t ] as the ( j, i ) -th element. We assumethat both Txs and Rxs of the m -th hop causally know the global channel state information (CSI) up to the m -th hop.That is, at time t , the nodes in the m -th layer know { H [ t ] , · · · , H m [ t ] } t t =1 if m ≤ M and { H [ t ] , · · · , H M [ t ] } t t =1 if m = M + 1 .For a broad class of networks, if the channel dimension of a certain hop is smaller than those of the other hops,then the average channel rank of the hop is likely to be less than those of the other hops. The following definitionformally states this class of networks. Definition 1:
Let m = arg min m ∈{ , ··· ,M } E (rank( H m [1])) . A linear binary relay network is said to have a minimum-dimensional bottleneck-hop m if K m ≥ K m and K m +1 ≥ K m +1 or K m ≥ K m +1 and K m +1 ≥ K m for all m ∈ { , · · · , M } .In this paper, we will study the class of networks satisfying Definition 1. Notice that any networks with K m = K for all m ∈ { , · · · , M + 1 } or any one-hop or two-hop networks are included in this class of networks regardlessof channel distributions. B. Problem Statement
Based on the previous network model, we define a set of length- n block codes. Let W k be the message of the k -th source uniformly distributed over { , , · · · , nR k } , where R k is the rate of the k -th source. For simplicity,we assume nR k is an integer. Then a (cid:0) nR , · · · , nR K ; n (cid:1) code consists of the following encoding, relaying, anddecoding functions. • (Encoding)For k ∈ { , · · · , K } , the set of encoding functions of the k -th source is given by { f k, ,t } nt =1 : { , · · · , nR k } → F n such that x k, [ t ] = f k, ,t ( W k ) for t ∈ { , · · · , n } . (4) • (Relaying)For m ∈ { , · · · , M } and k ∈ { , · · · , K m } , the set of relaying functions of the ( k, m ) -th node is given by { f k,m,t } nt =1 : F n → F n such that x k,m [ t ] = f k,m,t ( y k,m − [1] , · · · , y k,m − [ t − for t ∈ { , · · · , n } . (5) • (Decoding)For k ∈ { , · · · , K } , the decoding function of the k -th destination is given by g k : F n → { , · · · , nR k } suchthat ˆ W k = g k ( y k,M [1] , · · · , y k,M [ n ]) . (6)If M = 1 , the sources transmit directly to the destinations without relays. The probability of error at the k -thdestination is given by P ( n ) e,k = Pr( ˆ W k = W k ) . A set of rates ( R , · · · , R K ) is said to be achievable if there existsa sequence of (2 nR , · · · , nR K ; n ) codes with P ( n ) e,k → as n → ∞ for all k ∈ { , · · · , K } . Then the achievablesum rate is simply given by R sum = P Kk =1 R k . The capacity region is the closure of all achievable ( R , · · · , R K ) and the sum capacity is the supremum of all achievable sum rates. C. Notations
In this subsection, we introduce the notations for directed graphs and define sets of channel instances and setsof nodes. Notice that E (rank( H m [ t ])) is the same for all t .
1) Notations for directed graphs:
The considered network can be represented as a directed graph G = ( V , E ) consisting of a vertex set V and a directed edge set E . Let v k,m denote the ( k, m ) -th node and V m = { v k,m } K m k =1 denote the set of nodes in the m -th layer. Then V is given by ∪ m ∈{ , ··· ,M +1 } V m . The sets of sources and destinationsare given by S = V and D = V M +1 , respectively.There exists a directed edge ( v i,m , v j,m +1 ) from v i,m to v j,m +1 if p j,i,m > . For V ′ ⊆ V and V ′′ ⊆ V , define E ( V ′ , V ′′ ) as the set of edges going from V ′ to V ′′ given by { ( v ′ , v ′′ ) | v ′ ∈ V ′ , v ′′ ∈ V ′′ , ( v ′ , v ′′ ) ∈ E} . We saynode v ′′ is reachable from node v ′ if there exists a series of edges from v ′ to v ′′ , where we assume v ′ is alwaysreachable from v ′ itself. We further define v ′′ is reachable under V ′ from v ′ if there exists a series of edges in E ( V ′ , V ′ ) from v ′ to v ′′ . We define cut Ω ⊆ V as a subset of nodes such that at least one source is in Ω and atleast one corresponding destination is in Ω c . We define the following sets related to Ω : K Ω = { k | v k, ∈ Ω , v k,M +1 ∈ Ω c , k ∈ { , · · · , K }} , D Ω = { v k,M +1 | k ∈ K Ω } , S Ω = { v k, | k ∈ K Ω } , Ω D = { v |E (Ω , { v } ) = φ, at least one of the destinations in D Ω is reachable under Ω c from v, v ∈ Ω c } , Ω ′ = { v | v ∈ Ω is reachable from at least one of the sources in S Ω } , Ω S = { v |E ( { v } , Ω D ) = φ, v ∈ Ω ′ } . (7)Let X V ′ [ t ] and Y V ′ [ t ] denote the sets of transmit and received signals of the nodes in V ′ at time t , respectively. Let H V ′ , V ′′ [ t ] be the |V ′′ | × |V ′ | channel matrix at time t from the nodes in V ′ to the nodes in V ′′ . Hence H V m , V m +1 [ t ] = H m [ t ] . For notational simplicity, we use H Ω [ t ] to denote H Ω S , Ω D [ t ] in this paper.
2) Sets of channel instances and nodes:
For ¯ V ′ ⊆ V ′ , ¯ V ′′ ⊆ V ′′ , and G ∈ F | ¯ V ′′ |×| ¯ V ′ | , we define the followingsets of channel instances. H V ′ , V ′′ (cid:0) G , ¯ V ′ , ¯ V ′′ (cid:1) = (cid:8) H V ′ , V ′′ [1] (cid:12)(cid:12) H ¯ V ′ , ¯ V ′′ [1] = G , H V ′ , V ′′ [1] ∈ F |V ′′ |×|V ′ | (cid:9) , H F V ′ , V ′′ (cid:0) G , ¯ V ′ , ¯ V ′′ (cid:1) = (cid:8) H V ′ , V ′′ [1] (cid:12)(cid:12) rank( H V ′ , V ′′ [1]) = rank( G ) , H ¯ V ′ , ¯ V ′′ [1] = G , H V ′ , V ′′ [1] ∈ F |V ′′ |×|V ′ | (cid:9) . (8)Note that H V ′ , V ′′ (cid:0) G , ¯ V ′ , ¯ V ′′ (cid:1) is the set of all H V ′ , V ′′ [1] ∈ F |V ′′ |×|V ′ | that contain G in H ¯ V ′ , ¯ V ′′ [1] . Similarly, H F V ′ , V ′′ (cid:0) G , ¯ V ′ , ¯ V ′′ (cid:1) is the set of all H V ′ , V ′′ [1] ∈ F |V ′′ |×|V ′ | that have the same rank as G and contain G in H ¯ V ′ , ¯ V ′′ [1] .We further define the following sets of nodes. For positive integers a ≤ |V ′ | and b ≤ |V ′′ | , V ( a, b, V ′ , V ′′ ) = (cid:8) ( ¯ V ′ , ¯ V ′′ ) (cid:12)(cid:12) | ¯ V ′ | = a, | ¯ V ′′ | = b, ( ¯ V ′ , ¯ V ′′ ) ⊆ ( V ′ , V ′′ ) (cid:9) (9)and for H ∈ F |V ′′ |×|V ′ | , V (cid:0) H , V ′ , V ′′ (cid:1) = (cid:8) ( ¯ V ′ , ¯ V ′′ ) (cid:12)(cid:12) rank( H ¯ V ′ , ¯ V ′′ [1]) = | ¯ V ′ | = | ¯ V ′′ | = rank( H ) where H V ′ , V ′′ [1] = H , ( ¯ V ′ , ¯ V ′′ ) ⊆ ( V ′ , V ′′ ) (cid:9) , (10)where V ( H , V ′ , V ′′ ) = φ if rank( H ) = 0 . The set V ( a, b, V ′ , V ′′ ) consists of all ( ¯ V ′ , ¯ V ′′ ) ⊆ ( V ′ , V ′′ ) such that thenumber of nodes in ¯ V ′ and the number of nodes in ¯ V ′′ are equal to a and b , respectively. The set V ( H , V ′ , V ′′ ) consists of all ( ¯ V ′ , ¯ V ′′ ) ⊆ ( V ′ , V ′′ ) such that H ¯ V ′ , ¯ V ′′ [1] is a full-rank matrix and has the same rank as H , where H V ′ , V ′′ [1] = H . III. U PPER B OUND
In this section, we derive a general cut-set upper bound, which will be used to show the converses in SectionIV. v v v v v v v v v v v v (cid:58) Fig. 3. Example of the cut-set upper bound, where the solid lines mean that the corresponding channels become ones with non-zeroprobabilities.
A. Cut-set Upper Bound
We show that any sequence of (2 nR , · · · , nR K ; n ) codes with P ( n ) e,k → for all k ∈ { , · · · , K } satisfies therate constraints in the following theorem. Theorem 1:
Suppose a linear binary field relay network. For a cut Ω , the set of achievable rates ( R , · · · , R K ) is upper bounded by X k ∈K Ω R k ≤ E (rank( H Ω [1])) . (11) Proof:
Let us define W K Ω = { W k (cid:12)(cid:12) k ∈ K Ω } . We further define a length- n sequence a n to denote { a [1] , · · · , a [ n ] } .Then n X k ∈K Ω R k = H ( W K Ω )= I ( W K Ω ; Y n D Ω , H n , · · · , H nM ) + H ( W K Ω |Y n D Ω , H n , · · · , H nM ) ( a ) ≤ I ( W K Ω ; Y n D Ω , H n , · · · , H nM ) + nǫ n ( b ) = I ( W K Ω ; Y n D Ω | H n , · · · , H nM ) + nǫ n ( c ) ≤ I ( W K Ω ; Y n Ω D | H n , · · · , H nM ) + nǫ n ( d ) ≤ H ( W K Ω |X n Ω \ Ω ′ , H n , · · · , H nM ) − H ( W K Ω |X n Ω \ Ω ′ , Y n Ω D , H n , · · · , H nM ) + nǫ n = I ( W K Ω ; Y n Ω D |X n Ω \ Ω ′ , H n , · · · , H nM ) + nǫ n ≤ H ( Y n Ω D |X n Ω \ Ω ′ , H n , · · · , H nM ) + nǫ n ( e ) = n X t =1 H ( Y Ω D [ t ] |X Ω \ Ω ′ [ t ] , H [ t ] , · · · , H M [ t ]) + nǫ n ( f ) ≤ n E (rank( H Ω [1])) + nǫ n , (12)where ǫ n > satisfies ǫ n → as n → ∞ . Notice that ( a ) holds from Fano’s inequality, ( b ) holds since the messagesare independent of channels, ( c ) holds since W K Ω − (cid:0) Y n Ω D , H n , · · · , H nM (cid:1) − Y n D Ω forms a Markov chain, ( d ) holdssince W K Ω is independent of X n Ω \ Ω ′ , H n , · · · , H nM and conditioning reduces entropy, ( e ) holds since channels arememoryless, and ( f ) holds with equality if X Ω S [ t ] is uniformly distributed over F | Ω S | . Therefore, we have (11),which completes the proof.Theorem 1 shows that the aggregate rate of the S–D pairs divided by a cut is upper bounded by the averagerank of the channel matrix constructed by the cut. Example 1 (Cut-set Upper Bound):
Consider the cut
Ω = { v , , v , , v , , v , , v , , v , , v , } in Fig. 3. Thenwe obtain D Ω = { v , , v , } , S Ω = { v , , v , } , Ω D = { v , , v , } , Ω S = { v , , v , } , and H Ω [1] is given by [[ h , , [1] , T , [0 , h , , [1]] T ] T . Therefore, R + R is upper bounded by E (rank( H Ω [1])) = p , , + p , , . B. Rate Bounds for Single-hop and Multi-hop Networks
In this subsection, we obtain useful rate upper bounds from Theorem 1, which will be used to show the conversesin Corollaries 1 and 2. Let us first consider single-hop networks, that is M = 1 . If we set Ω = { v k, } , then P i ∈K Ω R i = R k and H Ω [ t ] = h k,k, [ t ] . Thus, we obtain R k ≤ p k,k, (13)for all k ∈ { , · · · , K } . Let us now consider multi-hop networks, that is M ≥ . By setting Ω = ∪ i ∈{ , ··· ,m } V i ,we have P k ∈K Ω R k = R sum and H Ω [ t ] = H m [ t ] , where m ∈ { , · · · , M } . Hence, we obtain R sum ≤ min m ∈{ , ··· ,M } E (rank( H m [1])) (14)or equivalently R sum ≤ E (rank( H m [1])) . IV. A CHIEVABILITY
In this section, we propose transmission schemes and derive their achievable rate regions.
A. Achievability for M = 1 Consider a single-hop network, that is M = 1 . As mentioned in Introduction, each source can transmit onebit without interference by using two particular instances H (1)1 and H (2)1 jointly such that H (1)1 + H (2)1 = I . Theproposed encoding makes such pairing possible.
1) Proposed scheme:
Let us divide a block into two sub-blocks having length n/ for each sub-block. For H ∈ F K × K , define T b ( H ) as the set of time indices of the b -th sub-block whose channel instances are equal to H , where b ∈ { , } . We further define n ( H ) = c − nR min { Pr( H [1] = H ) , Pr( H [1] = H + I ) } , (15)where c = X H ∈ F K × K min { H [1] = Pr( H ) , Pr( H [1] = H + I ) } . (16)The detailed encoding is as follows. • (Encoding of the first sub-block)For all H ∈ F K × K , declare an error if |T ( H ) | < n ( H ) , otherwise each source transmits n ( H ) informationbits using the time indices in T ( H ) . • (Encoding of the second sub-block)For all H ∈ F K × K , declare an error if |T ( H ) | < n ( H ) , otherwise each source retransmits n ( H ) information bits that were transmitted during T ( H + I ) using the time indices in T ( H ) .Notice that, since each source transmits P H ∈ F K × K n ( H ) information bits during n channel uses, the trans-mission rates are given by R = · · · = R K = n P H ∈ F K × K n ( H ) = R . Let s k ( i ) denote the i -th informationbit of the k -th source, where i = { , · · · , nR } . Let t ( i ) and t ( i ) denote the time indices over which s k ( i ) wastransmitted. Then the detailed decoding is as follows. • (Decoding)For i ∈ { , · · · , nR } , the k -th destination sets ˆ s k ( i ) = y k, [ t ( i )] + y k, [ t ( i )] .
2) Achievable rate region:
We derive the achievable rate region of the proposed scheme. Let E b denote the eventsuch that |T b ( H ) | < n ( H ) for any H ∈ F K × K , where b ∈ { , } . The following lemma shows that there is noerror if ( E ∪ E ) c occurs. Lemma 1:
Suppose a linear binary field relay network with M = 1 . The probability of error is upper boundedby P ( n ) e,k ≤ Pr( E ) + Pr( E ) (17)for all k ∈ { , · · · , K } . Proof:
The proof is in Appendix I.
Then the remaining thing is to derive R that guarantees P ( n ) e,k → as n → ∞ . The following theorem characterizessuch R . Theorem 2:
Suppose a linear binary field relay network with M = 1 . Then R k = 12 X H ∈ F K × K min { Pr( H [1] = H ) , Pr( H [1] = H + I ) } (18)is achievable for all k ∈ { , · · · , K } . Proof:
Let us consider |T b ( H ) | . By the weak law of large numbers [31], there exists a sequence ǫ n → as n → ∞ such that the probability |T b ( H ) | ≥ n H [1] = H ) − δ n ) for all H (19)is greater than or equal to − ǫ n , where δ n → as n → ∞ . This indicates that Pr( E b ) ≤ ǫ n if n ( H ) ≤ n (Pr( H [1] = H ) − δ n ) for all H . Hence, from (15), if R ≤ c (Pr( H [1] = H ) − δ n )2 min { Pr( H [1] = H ) , Pr( H [1] = H + I ) } (20)for all H , then P ( n ) e,k ≤ ǫ n , where we use the result of Lemma 1. Thus we set R = c (1 − δ ∗ n ) , where δ ∗ n = δ n min H ′ ∈ F K × K { Pr( H [1]= H ′ ) } , which converges to zero as n → ∞ . In conclusion, (18) is achievable for all k ∈{ , · · · , K } , which completes the proof. Corollary 1:
Suppose a linear binary field relay network with M = 1 . If p k,k, = 1 / for all k ∈ { , · · · , K } ,the capacity region is given by all rate tuples ( R , · · · , R K ) satisfying R k ≤ (21)for all k ∈ { , · · · , K } . Proof:
Note that
Pr( H [1] = H ) = Pr( H [1] = H + I ) for all H if p k,k, = 1 / . Hence, from (18), R k = P H ∈ F K × K Pr( H [1] = H ) = is achievable for all k ∈ { , · · · , K } . Note that the achievable rateregion coincides with the upper bound in (13), which provides the capacity region. Therefore, Corollary 1 holds. Remark 1:
Corollary 1 can be directly extended to a general linear finite field relay network in which inputs,outputs, and channels are in F q and channels are i.i.d. uniformly distributed over F q . Specifically, the capacityregion is given by all rate tuples ( R , · · · , R K ) satisfying R k ≤ log q for all k ∈ { , · · · , K } .Corollary 1 shows that all S–D pairs can simultaneously achieve the capacity of the point-to-point channelassuming no interference if the direct channels are uniformly distributed. This result also shows that the max-flowmin-cut theorem holds for a certain class of channel distributions. Similar to the Gaussian interference channel inwhich / degrees of freedom is achievable for each S–D pair [29], each source can transmit data to its destinationwith a non-vanishing rate even as K tends to infinity. Example 2 ( – network): Consider the case where K = 2 and M = 1 with p j,i, = 1 / for all i and j . If weuse each channel instance separately, then R sum ≤ / is achievable. However, the proposed scheme achieves R sum ≤ . More specifically, R ≤ / and R ≤ / are achievable, which is the capacity region of this network. B. Achievability for M ≥ Consider a multi-hop network, that is M ≥ . As mentioned in Introduction, each source can transmit one bitto its destination without interference through particular instances from H to H M such that H M H M − · · · H = I . (22)Due to network topologies and channel distributions, however, some instances will be rank-deficient and it isimpossible to find a series of pairs satisfying (22) by using rank-deficient instances. Furthermore, a series of pairssatisfying (22) is not unique and the number of possible pairing increases exponentially as the number of nodesin a layer or the number of layers increases. Hence, we first reduce the size of effective channels by transmittingand receiving using subsets of nodes at each hop such that the average ranks are balanced between hops and theirinstances have full-rank. Then we randomize a series of pairs based on these effective channels.
1) Construction of effective channels:
Recall that the m -th hop becomes a bottleneck for the entire multi-hoptransmission, which can be verified from (14). Hence, we select V m, tx [ t ] ⊆ V m and V m, rx [ t ] ⊆ V m +1 randomlysuch that ( V m, tx [ t ] , V m, rx [ t ]) ∈ V ( K m , K m +1 , V m , V m +1 ) (23)with equal probabilities (or in V ( K m +1 , K m , V m , V m +1 ) ). Notice that this is possible since the considered networkhas a minimum-dimensional bottleneck-hop. Because the maximum number of bits transmitted at the m -th hop islimited by rank( H V m, tx [ t ] , V m, rx [ t ] [ t ]) , we further select ¯ V m, tx [ t ] ⊆ V m, tx [ t ] and ¯ V m, rx [ t ] ⊆ V m, rx [ t ] randomly suchthat ( ¯ V m, tx [ t ] , ¯ V m, rx [ t ]) ∈ V ( H V m, tx [ t ] , V m, rx [ t ] [ t ] , V m, tx [ t ] , V m, rx [ t ]) (24)with equal probabilities. For each time t , the nodes in ¯ V m, tx [ t ] transmit and the nodes in ¯ V m, rx [ t ] receive throughtheir effective channel H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] at the m -th hop. Then information bits can be transmitted using particulartime indices t , · · · , t M such that ¯ V , tx [ t ] = ¯ V M, rx [ t M ] , ¯ V m, tx [ t m ] = ¯ V m − , rx [ t m − ] for all m ∈ { , · · · , M } , and H ¯ V M, tx [ t M ] , ¯ V M, rx [ t M ] [ t M ] · · · H ¯ V , tx [ t ] , ¯ V , rx [ t ] [ t ] = I , (25)which guarantees interference-free reception at the destinations. It is possible to construct those pairs becauseeffective channels are always invertible . Let F i be the set of all full-rank matrices in F i × i , where i ∈ { , · · · , K min } .The following lemma shows useful probability distributions, which will be used to derive the achievable rate regionof the proposed scheme. Lemma 2:
Suppose a linear binary field relay network with M ≥ . If the network has a minimum-dimensionalbottleneck-hop and p j,i,m = p for all i , j , and m , then the following probabilities hold:1) For H ∈ F K m × K m (or F K m × K m ), Pr( H V m, tx [ t ] , V m, rx [ t ] [ t ] = H ) = p u (1 − p ) K m K m − u , (26)where u is the number of ones in H .2) For G ∈ F i , Pr( H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] = G )= X ( V ′ , V ′′ ) ∈ V ( i,i, V m , V m X H ∈H F V m , V m ( G , V ′ , V ′′ ) Pr( H V m, tx [ t ] , V m, rx [ t ] [ t ] = H ) |V ( H , V m , V m +1 ) | , (27)where Pr( H V m, tx [ t ] , V m, rx [ t ] [ t ] = H ) is given by (26). If p = 1 / , we have Pr( H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] = G ) = 2 − K m K m N K m ,K m ( i ) N i,i ( i ) , (28)where N a,b ( c ) is the number of channel matrices in F a × b having rank c .3) For G ∈ F i and ( V ′ m , V ′ m +1 ) ∈ V ( i, i, V m , V m +1 ) , Pr( H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] = G , ¯ V m, tx [ t ] = V ′ m , ¯ V m, rx [ t ] = V ′ m +1 )= Pr( H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] = G ) (cid:0) K m i (cid:1)(cid:0) K m +1 i (cid:1) , (29)where Pr( H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] = G ) is given by (27). Proof:
The proof is in Appendix II.Note that the probabilities in (26) to (29) are the same for all m and t . For notational simplicity, we use theshorthand notation P G ( G ) to denote Pr( H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] = G ) . That is, for G ∈ F i , P G ( G ) = X ( V ′ , V ′′ ) ∈ V ( i,i, V m , V m X H ∈H F V m , V m ( G , V ′ , V ′′ ) P H ( H ) |V ( H , V m , V m +1 ) | , (30)where P H ( H ) = p u (1 − p ) K m K m − u and u is the number of ones in H . We do not use the effective channels having all zeros, which give zero rate.
2) Proposed scheme:
Divide a block into B + M − sub-blocks having length n B for each sub-block, where n B = nB + M − . Since block encoding and relaying are applied over M hops, the number of effective sub-blocksis equal to B . Thus, the overall rate is given by BB + M − R k . As n → ∞ , the fractional rate loss − BB + M − willbe negligible because we can make both n B and B large enough. For simplicity, we omit the sub-block index indescribing the proposed scheme.We divide M hops into two parts, the first N hops and the rest of the M − N hops, where N ∈ { , · · · , M − } .Then, for G ∈ F i , define P α ( G ) = ζ − ( N − i X G , ··· , G N ∈F i , G N ··· G G N Y m =1 P G ( G m ) , (31) P β ( G ) = ζ − ( M − N − i X G N +1 , ··· , G M ∈F i , G M ··· G N +1= G M Y m = N +1 P G ( G m ) , (32)and n ( G ) = c − n B R min { P α ( G ) , P β ( G − ) } , (33)where ζ i = P G ′ ∈F i P G ( G ′ ) and c = K P K min j =1 j P G ′ ∈F j min { P α ( G ′ ) , P β ( G ′− ) } . We further define n α ( G , · · · , G N ) = c − n B Rζ − ( N − i N Y m =1 ( P G ( G m ) − ∆ α ( G , · · · , G N )) , (34) n β ( G N +1 , · · · , G M ) = c − n B Rζ − ( M − N − i M Y m = N +1 ( P G ( G m ) − ∆ β ( G N +1 , · · · , G M )) , (35)where G , · · · , G M ∈ F i . Here, ∆ α ( G , · · · , G N ) ≥ and ∆ β ( G N +1 , · · · , G M ) ≥ are set such that X G ′ , ··· , G ′ N ∈F i , G ′ N ··· G ′ G n α ( G ′ , · · · , G ′ N ) = X G ′ N +1 , ··· , G ′ M ∈F i , G ′ M ··· G ′ N +1= G − n β ( G ′ N +1 , · · · , G ′ M ) = n ( G ) (36)is satisfied for all G ∈ F i .For a given G ∈ F i , the proposed scheme transmits i × n α ( G , · · · , G N ) bits through a series of effective channels G to G N satisfying G N · · · G = G for all G , · · · , G N ∈ F i . Hence a total of i P G , ··· , G N ∈F i , G N ··· G G n α ( G , · · · , G N ) = i × n ( G ) bits are transmitted. Then these i × n ( G ) received bits are transmitted through G N +1 to G M satisfying G M · · · G N +1 = G − for all G N +1 , · · · , G M ∈ F i . More specifically, i × n β ( G N +1 , · · · , G M ) bits are transmittedthrough G N +1 to G M and, as a result, a total of i P G N +1 , ··· , G M ∈F i , G M ··· G N +1= G − n β ( G N +1 , · · · , G M ) = i × n ( G ) bits aretransmitted. Let n m ( G m ) = (P G , ··· , G m − , G m +1 , ··· , G N ∈F i n α ( G , · · · , G N ) for m ∈ { , · · · , N } , P G N +1 , ··· , G m − , G m +1 , ··· , G M ∈F i n β ( G N +1 , · · · , G M ) for m ∈ { N + 1 , · · · , M } , (37)where G m ∈ F i . Then i × n m ( G m ) is the total number of bits that are transmitted through G m at the m -th hop.Define T m ( G m , V ′ m , V ′ m +1 ) as the set of time indices of the sub-block at the m -th hop satisfying ¯ V tx ,m [ t ] = V ′ m , ¯ V rx ,m [ t ] = V ′ m +1 , and H V ′ m , V ′ m +1 [ t ] = G m , where G m ∈ F i and ( V ′ m , V ′ m +1 ) ∈ V ( i, i, V m , V m +1 ) . For all i ∈{ , · · · , K min } , the detailed encoding and relaying are as follows. • (Encoding)For all G ∈ F i and ( V ′ , V ′ ) ∈ V ( i, i, V , V ) , declare an error if |T ( G , V ′ , V ′ ) | < n ( G ) / (cid:0)(cid:0) K i (cid:1)(cid:0) K i (cid:1)(cid:1) , oth-erwise each source in V ′ transmits n ( G ) / (cid:0)(cid:0) K i (cid:1)(cid:0) K i (cid:1)(cid:1) information bits, which are supposed to be transmittedthrough G , using the time indices in T ( G , V ′ , V ′ ) to the nodes in V ′ . • (Relaying for m ∈ { , · · · , M } )For all G m ∈ F i and ( V ′ m , V ′ m +1 ) ∈ V ( i, i, V m , V m +1 ) , declare an error if |T m ( G m , V ′ m , V ′ m +1 ) | is less than n m ( G m ) / (cid:0)(cid:0) K m i (cid:1)(cid:0) K m +1 i (cid:1)(cid:1) , otherwise each node in V ′ m transmits n m ( G m ) / (cid:0)(cid:0) K m i (cid:1)(cid:0) K m +1 i (cid:1)(cid:1) received bits, which are supposed to be transmitted through G m , using the time indices in T m ( G m , V ′ m , V ′ m +1 ) to the nodes in V ′ m +1 . If m = M , the transmit bits are constructed by the received bits that originate from S ( V ′ M +1 ) , where S ( V ′ M +1 ) is the set of sources of V ′ M +1 .From the proposed scheme, the transmission rates are given by R = · · · = R K = 1 Kn B K min X i =1 i X G ∈F i n ( G )= 1 Kn B K min X i =1 i X G , ··· , G N ∈F i n α ( G , · · · , G N ) = R. (38)Let s k ( i ) denote the i -th information bit of the k -th source and t k,m ( i ) denote the time index of the received signaloriginating from s k ( i ) at the m -th hop, where i ∈ { , · · · , n B R } . That is, s k ( i ) is transmitted using the time indices t k, ( i ) to t k,M ( i ) during the multi-hop transmission. The detailed decoding of the k -th destination is as follows. • (Decoding)For i ∈ { , · · · , n B R } , the k -th destination sets ˆ s k ( i ) = y k,M [ t k,M ( i )] .
3) Achievable rate region:
We derive the achievable rate region of the proposed scheme. Let E m denote theevent such that |T m ( G m , V ′ m , V ′ m +1 ) | < n m ( G m ) (cid:0) K m i (cid:1)(cid:0) K m +1 i (cid:1) (39)for any G m ∈ F i , ( V ′ m , V ′ m +1 ) ∈ V ( i, i, V m , V m +1 ) , and i ∈ { , · · · , K min } . The following lemma shows that thereis no error if ( ∪ Mm =1 E m ) c occurs. Lemma 3:
Suppose a linear binary field relay network with M ≥ . If the network has a minimum-dimensionalbottleneck-hop and p j,i,m = p for all i , j , and m , then P ( n B ) e,k ≤ M X m =1 Pr( E m ) (40)for all k ∈ { , · · · , K } . Proof:
The proof is in Appendix I.The following theorem characterizes R that guarantees P ( n B ) e,k → as n B → ∞ . Theorem 3:
Suppose a linear binary field relay network with M ≥ . If the network has a minimum-dimensionalbottleneck-hop and p j,i,m = p for all i , j , and m , then R k = 1 K K min X i =1 i X G ∈F i min { P α ( G ) , P β ( G − ) } (41)is achievable for all k ∈ { , · · · , K } , where P α ( G ) and P β ( G ) are defined in (31) and (32), respectively. Proof:
Let us consider |T m ( G m , V ′ m , V ′ m +1 ) | . By the weak law of large numbers [31], there exists a sequence ǫ n B → as n B → ∞ such that the probability |T m ( G m , V ′ m , V ′ m +1 ) |≥ n B (Pr( H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] = G m , ¯ V m, tx [ t ] = V ′ m , ¯ V m, rx [ t ] = V ′ m +1 ) − δ n B )= n B (cid:18) P G ( G m ) / (cid:18)(cid:18) K m i (cid:19)(cid:18) K m +1 i (cid:19)(cid:19) − δ n B (cid:19) (42)for all G m ∈ F i , ( V ′ m , V ′ m +1 ) ∈ V ( i, i, V m , V m +1 ) , and i ∈ { , · · · , K min } is greater than or equal to − ǫ n B ,where δ n B → as n B → ∞ . Here the equality holds from the third property of Lemma 2. This indicates that Pr( E m ) ≤ ǫ n B if n m ( G m ) ≤ n B (cid:18) P G ( G m ) − (cid:18) K m i (cid:19)(cid:18) K m +1 i (cid:19) δ n B (cid:19) (43) for all G m ∈ F i and i ∈ { , · · · , K min } . For m ∈ { , · · · , N } , from (37), we also have n m ( G m ) ≤ X G , ··· , G m − , G m +1 , ··· , G N ∈F i c − n B Rζ − ( N − i N Y l =1 P G ( G l )= c − n B RP G ( G m ) , (44)where we use the fact that n α ( G , · · · , G N ) ≤ c − n B Rζ − ( N − i Q Nl =1 P G ( G l ) from (34). Similarly, from (35) and(37), n m ( G m ) ≤ c − n B RP G ( G m ) for m ∈ { N + 1 , · · · , M } . Then, the condition in (43) can be satisfied if R ≤ c (cid:0) P G ( G m ) − (cid:0) K m i (cid:1)(cid:0) K m +1 i (cid:1) δ n B (cid:1) P G ( G m ) (45)for all G m ∈ F i and i ∈ { , · · · , K min } . Hence, we set R = c (1 − δ ∗ n B ) , where δ ∗ n B = ( K max !) δ nB min i ∈{ , ··· ,K min } , G ′∈F i { P G ( G ′ ) } ,which converges to zero as n B → ∞ . Therefore, from Lemma 3, we have P ( n B ) e,k ≤ M ǫ n B , which converges tozero as n B → ∞ . In conclusion, R k = c = 1 K K min X i =1 i X G ∈F i min { P α ( G ) , P β ( G − ) } (46)is achievable for all k ∈ { , · · · , K } , which completes the proof.For M = 2 , the proposed scheme pairs G and G satisfying G = G − and the achievable rate in (41) isgiven by R k = 1 K K min X i =1 i X G ∈F i min (cid:8) P G ( G ) , P G ( G − ) (cid:9) . (47)Let us now consider the capacity achieving case. The following corollary shows that if p = 1 / , the sum capacityis given by the average rank of the channel matrix of the bottleneck-hop. Corollary 2:
Suppose a linear binary field relay network with M ≥ . If the network has a minimum-dimensionalbottleneck-hop and p j,i,m = 1 / for all i , j , and m , the sum capacity is given by C sum = 2 − K m K m X H ∈ F Km × Km rank( H ) . (48) Proof:
From (28), we have ζ i = X G ∈F i − K m K m N K m ,K m ( i ) N i,i ( i ) = 2 − K m K m N K m ,K m ( i ) (49)and P α ( G ) = ζ − ( N − i X G , ··· , G N ∈F i , G N ··· G G N Y m =1 − K m K m N K m ,K m ( i ) N i,i ( i )= 2 − K m K m N K m ,K m ( i )( N i,i ( i )) N X G , ··· , G N ∈F i G N ··· G G
1= 2 − K m K m N K m ,K m ( i ) N i,i ( i ) . (50)Similarly, P β ( G ) = 2 − K m K m N Km ,Km ( i ) N i,i ( i ) . Then, from (41), R k = 1 K − K m K m K min X i =1 iN K m ,K m ( i )= 1 K − K m K m X H ∈ F Km × Km rank( H ) (51) s s s s s s s s s s s s s s (cid:14) s s s s s s s s s s s s s s (cid:14) s s s s s s (cid:14) s s s s s (cid:14) s s s s s s s Fig. 4. Deterministic channel pairing between the first and the second hops. is achievable for all k ∈ { , · · · , K } . Hence the sum rate in (48) is achievable, which coincides with the sum rateupper bound in (14). In conclusion, Corollary 2 holds. Remark 2:
Corollary 2 can be directly extended to a general linear finite field relay network in which inputs,outputs, and channels are in F q and channels are i.i.d. uniformly distributed over F q . Then C sum = q − K m K m X H ∈ F Km × Km q rank( H ) log q. (52)Notice that Corollary 2 shows that the sum rate of E (rank( H m [1])) is achievable, which is the multi-inputmulti-output (MIMO) capacity of the bottleneck-hop . This result also shows that the max-flow min-cut theorem holds for a certain class of channel distributions and network topologies.For – – networks, we characterize the sum capacity for more general classes of channel distributions byapplying deterministic channel pairing. Theorem 4:
Suppose a linear binary field relay network with M = 2 and K = K = K = 2 .1) For a symmetric channel satisfying p , , = p , , = p , , = p , , and p , , = p , , = p , , = p , , or a Z channel satisfying p , , = p , , = 0 , p , , = p , , , p , , = p , , , and p , , = p , , , the sum capacityis given by C sum = E (rank( H [1])) . (53)2) For a Z channel satisfying p , , = p , , = 0 , p , , = p , , , p , , = p , , , and p , , = p , , , the sumcapacity is given by C sum = min { p , , , p , , } . (54) Proof:
We use deterministic channel pairing between the first and the second hops. The overall block encodingand relaying structure making such pairing possible is the same as in the previous scheme.Let us prove the first result. Fig. 4 illustrates the deterministic channel pairing between the first and the secondhops and related encoding and relaying. The solid lines and the dashed lines denote the corresponding channelsare ones and zeros, respectively. The symbols in the figure denote the transmit signals of the nodes and the nodeswith no symbol transmit zeros, where s k denotes the information bit of the k -th source. Let p (1) m to p (16) m denote possible instances of H m [ t ] as shown in Fig. 5, where m ∈ { , } . Then the achievable sum rate of the deterministicpairing in Fig. 4 is given by R sum = X i ∈{ , , , , , , } min { p ( i )1 , p ( i )2 } + min { p (3)1 , p (5)2 } + min { p (5)1 , p (3)2 } + 2 X i ∈{ , , , } min { p ( i )1 , p ( i )2 } + 2 min { p (8)1 , p (15)2 } + 2 min { p (15)1 , p (8)2 } . (55) (1) m p (2) m p (7) m p (8) m p (6) m p (5) m p (4) m p (3) m p (9) m p (10) m p (15) m p (16) m p (14) m p (13) m p (12) m p (11) m p Fig. 5. possible instances of H m [ t ] . C s u m [ b i t s / c hanne l u s e ] Fig. 6. Sum capacity when p j,i,m = p for all i , j , and m . Since the probabilities of each paired H [ t ] and H [ t ] are the same, from (55), we have R sum = X i ∈{ , , , , , , , , } p ( i )1 + 2 X i ∈{ , , , , , } p ( i )1 = E (rank( H [1])) . (56)Notice that the achievable sum rate coincides with the sum rate upper bound in (14).Now let us prove the second result. Unlike the previous case, the probabilities of some paired H [ t ] and H [ t ] in Fig. 4 are not the same. Let us denote p a = p , , = p , , , p b = p , , = p , , , and p c = p , , = p , , . For p a ≥ p c , from (55), R sum = p (2)1 + p (6)1 + p (9)2 + p (13)2 + 2 p (10)1 + 2 p (14)1 = (1 − p a )(1 − p b ) p c + (1 − p a ) p b p c + (1 − p a )(1 − p b ) p c + (1 − p a ) p b p c + 2 p a (1 − p b ) p c + 2 p a p b p c = 2 p c (57)is achievable. By setting Ω = { v , , v , , v , } , we have R sum ≤ E (rank( (cid:2) [ h , , [1] , T , [0 , h , , [1]] T ) (cid:3) T ) = 2 p c , (58)which coincides with (57). Similarly, for p a < p c , from (55), R sum = p (2)2 + p (6)2 + p (9)1 + p (13)1 + 2 p (10)1 + 2 p (14)1 = 2 p a (59)is achievable. From Ω = { v , } and Ω = { v , , v , , v , , v , , v , } , we have R ≤ p a and R ≤ p a , respectively.Then R sum ≤ p a , which coincides with (59). In conclusion, Theorem 4 holds. Example 3 ( – – and – – networks): Fig. 6 plots sum rates of two-hop networks with p j,i,m = p . For – – networks, the sum capacity is given by C sum = 4 pq +8 p q +8 p q + p , where q = 1 − p . Notice that the considered channel distribution is a special case of the symmetric channel in Theorem 4. Therefore, we can characterize thesum capacity for all p ∈ [0 , . For – – networks, we obtain C sum ≥ pq + 54 p q + 168 p q + 279 p q +216 p q + 72 | p q − p q | + 216 min { p q , p q } + 90 p q + 90 p q + 18 p q + p and C sum ≤ pq + 54 p q +168 p q + 279 p q + 324 p q + 198 p q + 90 p q + 18 p q + p . The lower and upper bounds are the same when p = , which coincides with the result of Corollary 2 (if p = 0 or the lower and upper bounds are trivially thesame). Example 4 (Networks with K = K = · · · = K M +1 ): Suppose a linear finite field relay network with K = K = · · · = K M +1 in which inputs, outputs, and channels are in F q and channels are i.i.d. uniformly distributed over F q .From Remarks 1 and 2, we have C sum = ( K log q if M = 1 , E (rank( H m [1])) log q if M ≥ . (60)For K = K = · · · = K M +1 = 2 and q = 2 , C sum is given by if M = 1 and / if M ≥ .V. C ONCLUSION
In this paper, we studied layered linear binary field relay networks with time-varying channels, which exhibitbroadcast, interference, and fading natures of wireless communications. Capacity characterization of such relaynetworks with multiple S–D pairs is quite challenging because the transmission of other session acts as inter-user interference. We observed that the fading can play an important role in mitigating interference that leads tothe capacity characterization for some classes of channel distributions and network topologies. For these classes,we showed that the capacity region of single-hop networks and the sum capacity of multi-hop networks can beinterpreted as the max-flow min-cut theorem. A
PPENDIX IU PPER B OUND ON THE P ROBABILITY OF E RROR
Proof of Lemma 1:
Let us assume that ( E ∪ E ) c occurs. Then, from the assumption, each source can transmit n ( H ) bits using the time indices in T ( H ) for all H . Since n ( H ) = n ( H + I ) , from the assumption, eachsource can retransmit all information bits that were transmitted during T ( H + I ) using the time indices in T ( H ) for all H . Lastly, there is no decoding error if ( E ∪ E ) c occurs since H [ t ( i )] + H [ t ( i )] = I , meaning ˆ s k ( i ) = s k ( i ) . In conclusion, from the union bound, we obtain P ( n ) e,k ≤ Pr( E ) + Pr( E ) , which completes theproof. (cid:4) Proof of Lemma 3:
Let us assume that ( ∪ Mm =1 E m ) c occurs. Then each source can transmit all information bitsto the nodes in the next layer. Consider the m -th hop transmission through G m ∈ F i , where m ∈ { , · · · , M − } .Each node in V ′ m receives n m ( G m ) / (cid:0)(cid:0) K m − i (cid:1)(cid:0) K m i (cid:1)(cid:1) bits from V ′ m − that should be transmitted through G m . Sincethere are (cid:0) K m − i (cid:1) candidates for V ′ m − , a total of n m ( G m ) / (cid:0) K m i (cid:1) bits should be transmitted through G m . From theassumption, each node in V ′ m is able to transmit n m ( G m ) / (cid:0)(cid:0) K m i (cid:1)(cid:0) K m +1 i (cid:1)(cid:1) bits to the nodes in V ′ m +1 using the timeindices in T m ( G m , V ′ m , V ′ m +1 ) . Since there are (cid:0) K m +1 i (cid:1) candidates for V ′ m +1 , each node in V ′ m can transmit a totalof n m ( G m ) / (cid:0) K m i (cid:1) bits through G m . Hence, each node in V ′ m can transmit all received bits. Consider the last hoptransmission. Similar to the previous hops, each node in V ′ M receives n M ( G M ) / (cid:0) K M i (cid:1) bits that should be transmittedthrough G M and, among them, n M ( G M ) / (cid:0)(cid:0) K i (cid:1)(cid:0) K M i (cid:1)(cid:1) bits are originated from S ( V ′ M +1 ) . From the assumption,each node in V ′ M is able to transmit n M ( G M ) / (cid:0)(cid:0) K M i (cid:1)(cid:0) K M +1 i (cid:1)(cid:1) bits to the nodes in V ′ M +1 using the time indicesin T M ( G M , V ′ M , V ′ M +1 ) . Hence, each node in V ′ M can transmit all received bits because n M ( G M ) / (cid:0)(cid:0) K i (cid:1)(cid:0) K M i (cid:1)(cid:1) is equal to n M ( G M ) / (cid:0)(cid:0) K M i (cid:1)(cid:0) K M +1 i (cid:1)(cid:1) , where we use the fact that K = K = K M +1 .Lastly, consider the estimated bit ˆ s k ( i ) at the k -th destination. Since the overall channel matrix from ¯ V tx , [ t k, ( i )] to ¯ V rx ,M [ t k,M ( i )] is given by H ¯ V tx ,M [ t k,M ( i )] , ¯ V rx ,M [ t k,M ( i )] [ t k,M ( i )] · · · H ¯ V tx , [ t k, ( i )] , ¯ V rx , [ t k, ( i )] [ t k, ( i )] = I , (61)we obtain ˆ s k ( i ) = s k ( i ) . Hence, there is no error if ( ∪ Mm =1 E m ) c occurs. In conclusion, from the union bound, weobtain P ( n B ) e,k ≤ P Mm =1 Pr( E m ) , which completes the proof. (cid:4) TABLE IN
OTATIONS USED IN A PPENDIX I. P (1) m ( H m ) Pr( H m [ t ] = H m ) P (2) m ( H ) Pr( H V m, tx [ t ] , V m, rx [ t ] [ t ] = H ) P (3) m ( V ′ , V ′′ (cid:12)(cid:12) H m ) Pr( V m, tx [ t ] = V ′ , V m, rx [ t ] = V ′′ (cid:12)(cid:12) H m [ t ] = H m ) P (4) m ( H (cid:12)(cid:12) H m , V ′ , V ′′ ) Pr( H V m, tx [ t ] , V m, rx [ t ] [ t ] = H (cid:12)(cid:12) H m [ t ] = H m , V m, tx [ t ] = V ′ , V m, rx [ t ] = V ′′ ) P (5) m ( V ′ , V ′′ ) Pr( V m, tx [ t ] = V ′ , V m, rx [ t ] = V ′′ ) P (6) m ( G ) Pr( H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] = G ) P (7) m ( V ′ , V ′′ (cid:12)(cid:12) H ) Pr( ¯ V m, tx [ t ] = V ′ , ¯ V m, rx [ t ] = V ′′ (cid:12)(cid:12) H V m, tx [ t ] , V m, rx [ t ] [ t ] = H ) P (8) m ( G (cid:12)(cid:12) H , V ′ , V ′′ ) Pr( H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] = G (cid:12)(cid:12) H V m, tx [ t ] , V m, rx [ t ] [ t ] = H , ¯ V m, tx [ t ] = V ′ , ¯ V m, rx [ t ] = V ′′ ) P (9) m ( G , V ′ , V ′′ ) Pr( H ¯ V m, tx [ t ] , ¯ V m, rx [ t ] [ t ] = G , ¯ V m, tx [ t ] = V ′ , ¯ V m, rx [ t ] = V ′′ ) A PPENDIX
IIP
ROBABILITY D ISTRIBUTIONS OF S UB - CHANNEL M ATRICES
In this appendix, we prove the probability distributions shown in Lemma 2. For notational simplicity, we willuse the shorthand notations in Table I.
Proof of Lemma 2.(1):
We assume that |V m, tx [ t ] | = K m and |V m, rx [ t ] | = K m +1 in the proof. But the sameresult holds for the case where |V m, tx [ t ] | = K m +1 and |V m, rx [ t ] | = K m . We have P (2) m ( H ) = X ( V ′ , V ′′ ) ∈ V ( Km ,Km , V m, V m +1) X H m ∈ F Km +1 × Km P (1) m ( H m ) P (3) m (cid:0) V ′ , V ′′ (cid:12)(cid:12) H m (cid:1) P (4) m (cid:0) H (cid:12)(cid:12) H m , V ′ , V ′′ (cid:1) ( a ) = X ( V ′ , V ′′ ) ∈ V ( Km ,Km , V m, V m +1) P (5) m (cid:0) V ′ , V ′′ (cid:1) X H m ∈ F Km +1 × Km P (1) m ( H m ) P (4) m (cid:0) H (cid:12)(cid:12) H m , V ′ , V ′′ (cid:1) ( b ) = X ( V ′ , V ′′ ) ∈ V ( Km ,Km , V m, V m +1) P (5) m (cid:0) V ′ , V ′′ (cid:1) X H m ∈H V m, V m +1 ( H , V ′ , V ′′ ) P (1) m ( H m ) ( c ) = p u (1 − p ) K m K m − u , (62)where ( a ) holds from the fact that P (3) m (cid:0) V ′ , V ′′ (cid:12)(cid:12) H m (cid:1) = P (5) m ( V ′ , V ′′ ) because V m, tx [ t ] and V m, rx [ t ] are chosenregardless of channel instances, ( b ) holds since P (4) m (cid:0) H (cid:12)(cid:12) H m , V ′ , V ′′ (cid:1) = ( if H m ∈ H V m , V m +1 ( H , V ′ , V ′′ )0 otherwise , (63)and ( c ) holds since P H m ∈H V m, V m +1 ( H , V ′ , V ′′ ) P (1) m ( H m ) = p u (1 − p ) K m K m − u . Therefore, Lemma 2.(1) holds. (cid:4) Proof of Lemma 2.(2):
We again assume that |V m, tx [ t ] | = K m and |V m, rx [ t ] | = K m +1 in the proof. We have P (6) m ( G ) = X ( V ′ , V ′′ ) ∈ V ( i,i, V m, tx[ t ] , V m, rx[ t ]) X H ∈ F Km × Km P (2) m ( H ) P (7) m ( V ′ , V ′′ | H ) P (8) m ( G | H , V ′ , V ′′ ) ( a ) = X ( V ′ , V ′′ ) ∈ V ( i,i, V m, tx[ t ] , V m, rx[ t ]) X H ∈H F V m, tx[ t ] , V m, rx[ t ] ( G , V ′ , V ′′ ) P (2) m ( H ) P (7) m ( V ′ , V ′′ | H ) ( b ) = X ( V ′ , V ′′ ) ∈ V ( i,i, V m, tx[ t ] , V m, rx[ t ]) X H ∈H F V m, tx[ t ] , V m, rx[ t ] ( G , V ′ , V ′′ ) P (2) m ( H ) |V ( H , V m, tx [ t ] , V m, rx [ t ]) | ( c ) = X ( V ′ , V ′′ ) ∈ V ( i,i, V m , V m X H ∈H F V m , V m ( G , V ′ , V ′′ ) P (2) m ( H ) |V ( H , V m , V m +1 ) | , (64)where ( a ) holds since P (8) m ( G | H , V ′ , V ′′ ) = ( if H ∈ H F V m, tx [ t ] , V m, rx [ t ] ( G , V ′ , V ′′ )0 otherwise , (65) ( b ) holds since P (7) m ( V ′ , V ′′ | H ) = |V ( H , V m, tx [ t ] , V m, rx [ t ]) | if H ∈ H F V m, tx [ t ] , V m, rx [ t ] ( G , V ′ , V ′′ ) , and ( c ) holds from thefacts that P (2) m ( H ) is the same for all m , which is the result of Lemma 2.(1), and |V ( H , V m, tx [ t ] , V m, rx [ t ]) | is thesame for all m .Now consider the case p j,i,m = 1 / . Since rank( H ¯ V m, tx [ t ] , ¯ V m, tx [ t ] [ t ]) = rank( H V m, tx [ t ] , V m, tx [ t ] [ t ]) , we obtain X G ′ ∈F i P (6) m ( G ′ ) = X H ∈ F Km × Km , rank( H )= i P (2) m ( H ) , (66)where P (2) m ( H ) = 2 − K m K m (67)and P (6) m ( G ′ ) = 2 − K m K m X ( V ′ , V ′′ ) ∈ V (rank( G ′ ) , rank( G ′ ) , V m , V m X H ∈H F V m , V m ( G ′ , V ′ , V ′′ ) |V ( H , V m , V m +1 ) | . (68)Here, (67) and (68) can be derived from Lemma 2.(1) and (64). Then we will prove the following two properties:1) P H ∈H F V m , V m ( G ′ , V ′ , V ′′ ) 1 |V ( H , V m , V m ) | is the same for all V ′ and V ′′ .2) P H ∈H F V m , V m ( G ′ , V ′ , V ′′ ) 1 |V ( H , V m , V m ) | is the same for all G ′ having the same rank.To prove the first property, consider two ( V ′ a , V ′′ a ) and ( V ′ b , V ′′ b ) . Then we can find a row permutation matrix E row and a column permutation matrix E col such that H F V m , V m ( G ′ , V ′ a , V ′′ a ) = { E row HE col (cid:12)(cid:12) H ∈ H F V m , V m ( G ′ , V ′ b , V ′′ b ) } . (69)Therefore, from the fact that |V ( H , V m , V m +1 ) | = |V ( E row HE col , V m , V m +1 ) | , the first property holds.Now consider the second property. We assume that V ′ = { v ,m , · · · , v i,m } and V ′′ = { v ,m +1 , · · · , v i,m +1 } for the proof, but the same property can be easily derived for arbitrary V ′ and V ′′ by using the first property. Fig.7 illustrates the construction of H F V m , V m ( G ′ , V ′ , V ′′ ) . We obtain i × ( K m − i ) matrix G = G ′ A , where A ∈ F i × ( K m − i )2 . Then ( K m +1 − i ) × K m matrix G is obtained by setting G = B [ G ′ , G ] , where B ∈ F ( K m − i ) × i .Therefore, we obtain H F V m , V m ( G ′ , V ′ , V ′′ ) = n (cid:2) [ G ′ , G ] T , [ G ] T (cid:3) T (cid:12)(cid:12) A ∈ F i × ( K m − i )2 , B ∈ F ( K m − i ) × i o . (70) ' G ' G G A [ ', ]
G B G G i i iK m ! iK m Fig. 7. Construction of H F V m , V m ( G ′ , V ′ , V ′′ ) , where A ∈ F i × ( K m − i )2 , and B ∈ F ( K m − i ) × i . Then, for given A and B , (cid:12)(cid:12) V (cid:0) (cid:2) [ G ′ , G ] T , [ G ] T (cid:3) T , V m , V m +1 (cid:1)(cid:12)(cid:12) is the same for all G ′ having the same rank.Therefore, the second property holds.From the above two properties, P H ∈H F V m , V m ( G ′ , V ′ , V ′′ ) 1 |V ( H , V m , V m ) | is the same for all V ′ , V ′′ , and G ′ having the same rank. We also know that |V (rank( G ′ ) , rank( G ′ ) , V m , V m +1 ) | is the same for all G ′ having thesame rank. As a result, P (6) m ( G ′ ) is the same for all G ′ having the same rank. Thus, from (66) and (67), we have P (6) m ( G ) X G ′ ∈F i − K m K m X H ∈ F Km × Km , rank( H )= i . (71)Since P G ′ ∈F i N i,i ( i ) and P H ∈ F Km × Km , rank( H )= i N K m ,K m ( i ) , we finally obtain P (6) m ( G ) = 2 − K m K m N K m ,K m ( i ) N i,i ( i ) . (72)In conclusion, Lemma 2.(2) holds. (cid:4) Proof of Lemma 2.(3):
From the definitions of P (6) m ( G ) and P (9) m ( G , V ′ , V ′′ ) , we obtain P (6) m ( G ) = X ( V ′ , V ′′ ) ∈V ( i,i, V m , V m +1 ) P (9) m ( G , V ′ , V ′′ )= (cid:18) K m i (cid:19)(cid:18) K m +1 i (cid:19) P (9) m ( G , V ′ m , V ′ m +1 ) , (73)where the second equality holds since |V ( i, i, V m , V m +1 ) | = (cid:0) K m i (cid:1)(cid:0) K m +1 i (cid:1) and P (9) m ( G , V ′ , V ′′ ) is the same for all V ′ and V ′′ . Thus, we have P (9) m ( G , V ′ m , V ′ m +1 ) = P (6) m ( G ) / (cid:18)(cid:18) K m i (cid:19)(cid:18) K m +1 i (cid:19)(cid:19) , (74)which completes the proof. (cid:4) R EFERENCES [1] T. M. Cover and A. El Gamal, “Capacity theorems for the relay channel,”
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