A New Achievable Rate and the Capacity of Some Classes of Multilevel Relay Network

A new achievable rate based on a partial decoding scheme is proposed for the multilevel relay network. A novel application of regular encoding and backward decoding is presented to implement the proposed rate. In our scheme, the relays are arranged in feed-forward structure from the source to the destination. Each relay in the network decodes only part of the transmitted message by the previous relay. The proposed scheme di ﬀ ers from general parity forwarding scheme in which each relay selects some relays in the network but decodes all messages of the selected relays. It is also shown that in some cases higher rates can be achieved by the proposed scheme than previously known by Xie and Kumar. For the classes of semideterministic and orthogonal relay networks, the proposed achievable rate is shown to be the exact capacity. The application of the deﬁned networks is very well understood in wireless networking scenarios.


INTRODUCTION
The relay Van der Meulen in [1], describes a single-user communication channel where a relay helps a sender-receiver pair in their communication. In [2], Cover and El Gamal proved a converse result for the relay channel, the socalled max-flow min-cut upper bound. Additionally, they established two coding approaches and three achievability results for the discrete-memoryless relay channel. They also presented the capacity of degraded, reversely degraded relay channel, and the relay channel with full feedback. In [3], partial decoding scheme or generalized block Markov encoding was defined as a special case of the proposed coding scheme by Cover and El Gamal [2,Theorem 7]. In this encoding scheme, the relay does not completely decode the transmitted message by the sender. Instead, the relay only decodes part of the message transmitted by the sender. Partial decoding scheme was used to establish the capacity of two classes of relay channels called semideterministic relay channel [3,4] and orthogonal relay channel [5].
The last few decades have seen tremendous growth in communication networks. The most popular examples are cellular voice, data networks, and satellite communication systems. These and other similar applications have motivated researches to extend Shannon's information theory to networks. In the case of relay networks, deterministic relay networks with no interference, first introduced by Aref [4], are named Aref networks in [6]. Aref determined the unicast capacity of such networks. The multicast capacity of Aref networks is also characterized in [6]. There also has been much interest in channels with orthogonal components, since in a practical wireless communication system, a node cannot transmit and receive at the same time or over the same frequency band. In [5], the capacity of a class of discrete-memoryless relay channels with orthogonal channels from the sender to the relay receiver and from the sender and relay to the sink is shown to be equal to the maxflow min-cut upper bound.
There also have been a lot of works that apply the proposed encoding schemes by Cover and El Gamal to the multiple relay networks [7][8][9][10][11][12][13][14]. In [7], authors generalize compress-and-forward strategy and also give an achievable rate when the relays use either decode-and-forward or compress-and-forward. Additionally, they add partial decoding to the later method when there are two relays. In their scheme, the first relay uses decode-and-forward, and the second relay uses compress-and-forward. Second, relay further partially decodes the signal from first relay before compressing its observation. They made the second relay output statistically independent of the first relay and the transmitter outputs. In [8], Gupta and Kumar applied irregular encoding/successive decoding to multirelay networks in a manner similar to [4]. In [9,10], Xie and Kumar developed regular encoding-/sliding-window decoding for multiple relays, and showed that their scheme achieves better rates than those of [4,8]. Regular encoding/backward decoding was similarly generalized [11]. The achievable rates of the two regular encoding strategies turn out to be the same. However, the delay of sliding-window decoding is much less than that of backward decoding. Regular encoding-/slidingwindow decoding is therefore currently the preferred variant of multihopping in the sense that it achieves the best rates in the simplest way. In [12,13], parity-forwarding protocol is introduced and a structured generalization of decode-andforward strategies for multiple-relay networks with feedforward structure based on such protocol is proposed. In their method, each relay chooses a selective set of previous nodes in the network and decodes all messages of those nodes. Parity forwarding was shown to improve previous decode-and-forward strategies, and it achieves the capacity of new forms of degraded multirelay networks. In [14], a generalization of partial decoding scheme was applied to multiple-relay networks and a new achievable rate was proposed. In this method, all relays in the network successively decode only part of the messages of the previous node before they arrive at the destination, In this way, using auxiliary random variables that indicate the message parts results the flexibility in defining some special classes of relay networks that the proposed rate obtain their exact capacities. For example, the capacity of feed-forward semideterministic and orthogonal relay networks that are obtained by the proposed method. To our knowledge, up to now, except the work done in [14], no other work was done for applying the partial decoding to the relay networks in which more than one relay partially decodes the message transmitted by the sender. In this paper, we generalize the results of [14] for Nrelay networks and prove some theorems.
The paper is organized as follows. Section 2 introduces modeling assumptions and notations. In Section 3, some theorems and corollaries about generalized block Markov encoding scheme or partial decoding method are reviewed. In Section 4, we introduce sequential partial decoding and drive a new achievable rate for relay networks based on this scheme. In Section 5, a class of semideterministic relay network is introduced and it is shown that the capacity of this network is obtained by the proposed method. In Section 6, we first give a review of orthogonal relay channel defined in [5], then we introduce orthogonal relay networks and obtain its capacity. Finally, some concluding remarks are provided in Section 7.

DEFINITIONS AND PRELIMINARIES
The discrete memoryless relay network shown in Figure 1 [4, Figure 2.1] is a model for the communication between a source X 0 and a sink Y 0 via N intermediate nodes called relays. The relays receive signals from the source and other nodes and then transmit their information to help the sink to resolve its uncertainty about the message. To specify the network, we define 2N + 2 finite sets: In this model, X 0 is the input to the network, Y 0 is the ultimate output, Y i is the ith relay output, and X i is the ith relay input.
An (M, n) code for the network consists of a set of integers W = {1, 2, . . . , M}, an encoding function x n 0 : W →X n 0 , a set of relay function { f i j } such that . , x in ), and a decoding function g : Y n 0 →W . For generality, all functions are allowed to be stochastic functions.
Let y j−1 i = (y i1 , y i2 , . . . , y i, j−1 ). The input x i j is allowed to depend only on the past received signals at the ith node, that is, (y i1 , . . . , y i, j−1 ). The network is memoryless in the sense that (y 0i , y 1i , . . . , y Ni ) depends on the past (x i 0 , x i 1 , . . . , x i N ) only through the present transmitted symbols (x 0i , x 1i , . . . , x Ni ). Therefore, the joint probability mass func- where p(w) is the probability distribution on the message w ∈ W . If the message w ∈ W is sent, let λ(w) Pr{g(Y n 0 ) / = W | W = w} denote the conditional probability of error. Define the average probability of error of the code, assuming a uniform distribution over the set of all messages w ∈ W , as P n e = (1/M) w λ(w). Let λ n max w∈W λ(w) be the maximal probability of error for the (M, n) code. The rate R of an (M, n) code is defined to be R = (1/n) log M bits/transmission. The rate R is said to be achievable by the network if, for any > 0, and for all n sufficiently large, there exists an (M, n) code with M ≥ 2 nR such that P n e < . The capacity C of the network is the supremum of the set of achievable rates.

GENERALIZED BLOCK MARKOV ENCODING
In [3], generalized block Markov encoding is defined as a special case of [2,Theorem 7]. In this encoding scheme, the relay does not completely decode the transmitted message by the sender. Instead the relay only decodes part of the message transmitted by the sender. A block Markov encoding timeframe is again used in this scheme such that the relay decodes part of the message transmitted in the previous block and cooperates with the sender to transmit the decoded part of the message to the sink in current block. The following theorem expresses the obtained rate via generalized block Markov encoding.
Theorem 1 (see [3]). For any relay network (X 0 × X 1 , p(y 0 , where the maximum is taken over all joint probability mass functions of the form If we choose the random variable U = X 0 , it satisfies the Markovity criterion and the result of block Markov coding directly follows as The above expression introduces the capacity of degraded relay channel as shown in [2]. Moreover, by substituting U = Y 1 in (3), the capacity of semideterministic relay channel in which y 1 is a deterministic function of x 0 and x 1 .

Corollary 1.
If y 1 is a deterministic function of x 0 and x 1 , then In the next section, we apply the concept of Theorem 1 to the relay networks with N relays and prove the main theorem of this paper.

SEQUENTIAL PARTIAL DECODING
In this section, we introduce sequential partial decoding method and drive a new achievable rate for N-relay networks. In sequential partial decoding, the message of the sender is divided into N parts. The first part is directly decoded by the sink, while the other parts are decoded by the first relay. With the same way, at each relay, one part of the message is directly decoded by the sink, while the other parts are decoded by the next relay. In the next blocks, the sender and the relays cooperate with each other to remove the uncertainty of the sink about the individual parts of the messages.
Sequential partial decoding scheme is useful in the cases that the relays are located in feed-forward structure from the sender to the sink with at most distance with each other in such a way that each node is able to decode some parts of the message of the previous node, while the sink is sensitive enough to be able to directly decode the remaining parts of the messages of the sender and the relays. The rate obtained by this method is expressed in the following theorem.

Theorem 2. For any relay network
where the supremum is over all joint probability mass functions such that form a Markov chain.
Proof. In this encoding scheme, the source message is split into (N + 1) parts, w N0 , w N−1,0 , . . . , w 00 . The first relay decodes messages w N0 , . . . , w 10 ; the second relay decodes w N0 , . . . , w 20 ; and so on the Nth relay decodes only w N0 . Each relay retransmits its decoded messages to the sink using the same codebook as the source, that is, regular encoding is used. Backward decoding is used at all nodes to decode messages, starting from the last block and going backward to the first block. We consider B blocks of transmission, each of n symbols. A sequence of B − N messages, will be sent over the channel in nB transmissions. In each nblock b = 1, 2, . . . , B, we will use the same set of codewords. We consider only the probability of error in each block as the total average probability of error can be upper-bounded by the sum of the decoding error probabilities at each step, under the assumption that no error propagation from the previous steps has occurred [15].

Random coding
The random codewords to be used in each block are generated as follows.
Label these u n l ( w km m∈{N,...,l},k∈{N Label these x n l−1 ( w km m∈{N,...,l−1},k∈{N,...,m} ), In the above random coding strategy, m and k denote the relay number and the message part number, respectively. The mth relay decodes {w km } k∈{N,...,m} and the (m − 1)th relay decodes {w k,m−1 } k∈{N,...,m−1} , where at the mth relay for each message part k ∈ {N , . . . , m}, the index w km represents the index w k,m−1 of the previous block. In this coding construction, N(N + 1) indices are used in total.
As an example, for two-relay network, the transmitter and the relay encoders send the following codewords: in block h = 1, the following codewords: in block h = B − 1, and the following codewords: in block h = B. Figure 2 shows the individual parts of the messages that should be decoded by the relays and the sink. It can be inferred from Figure 2 that  x n 0 (w 22 , w 21 , w 11 , w 20 , w 10 , w 00 ) Y 0 x n 2 (w 22 ) Figure 2: Schematic diagram of sequential partial decoding for two-relay network.
This way continues until first block such that all {w i0,h−N } h∈{B,...,N+1} are decoded by the sink.
(c) follows from chain rule for information and (9). For i = 0, by respect to the fact that U N+1 = 0, we have equations (36) and (37) along with R = min 0≤i≤N R ti result in (7). This completes the proof. Figure 3: A degraded chain network with additive noises N k , 1 ≤ k ≤ N, [13, Figure 5].

Remarks
(1) By putting U l = X l−1 for 1 ≤ l ≤ N in (7), that means omitting partial decoding and assuming that each relay decodes all messages of the previous relay, the following rate is the result: In [13], the above rate is obtained as a special case of parityforwarding method in which each relay selects the message of the previous relay, and it is stated that (38) is the capacity of degraded chain network as shown in Figure 3. This point can be regarded as a special example that two schemes coincide.
(2) By comparing (38) with the rate proposed by Xie and Kumar in [10, Theorem 3-1], as it is seen that in the cases which the following relation is true for 2 ≤ i ≤ N, (38) yields higher rates than (39).
(3) In our proposed rate (7), we offer more flexibility than parity forwarding scheme [13], by introducing auxiliary random variables that indicate partial parts of the messages. By this priority, we can achieve the capacity of some other forms of relay networks such as semideterministic and orthogonal relay networks as shown in the next section. However, our scheme is limited by the assumption that each relay only decodes part of the message transmitted by the previous relay.

A CLASS OF SEMIDETERMINISTIC RELAY NETWORKS
seminet A class of semideterministic multirelay networks In this section, we introduce a class of semideterministic relay network and show that the capacity of such network is obtained by using the proposed method, that is, it coincides with the max-flow min-cut upper bound. Consider the semideterministic relay networks with N relays as shown in

Theorem 3. For a class of semideterministic relay network
the capacity C is given by Proof. The achievability is proved by replacing U k = Y k for k = 1, . . . , N, in (7). The converse follows immediately from the max-flow min-cut theorem for general multiple-node networks stated in [

A CLASS OF ORTHOGONAL RELAY NETWORKS
In this section, we introduce a class of orthogonal relay networks that is a generalization of orthogonal relay channel [5]. First, we define orthogonal relay channel. A relay channel with orthogonal components is a relay channel where the channel from the transmitter to the relay is orthogonal to the channel from the sender and relay to the sink. In other words, transmission on direct channel from the sender to the sink does not affect the reception at the relay and also transmission at the channel from the sender to the relay does not affect the received signal at the sink. This channel is defined in as follows [5].

Definition 1.
A discrete-memoryless relay channel is said to have orthogonal components if the sender alphabet X 0 = X D × X R and the channel can be expressed as for all The class A relay channel is illustrated in Figure 5, where the channels in the same frequency band are shown by the lines with the same type. The capacity is given by the following theorem.
Theorem 4 (see [5,Theorem]). The capacity of the relay channel with orthogonal components is given by where the maximum is taken over all joint probability mass functions of the form Generalized block Markov coding is used for the proof of achievability part by assuming joint probability mass function of the form (46). The converse part of the theorem is proved for all joint probability mass function p(x 1 , x D , x R ) only based on the orthogonality assumption (48) or equivalently the following Markov chains: Now, we introduce a class of relay networks with orthogonal components where the channels reach at each node uses the same frequency band while the channels diverge from each node uses different frequency bands. By this assumption, the network with N relays (intermediate nodes) uses (N + 1) frequency bands. The network is defined as follows.

Definition 2.
A discrete-memoryless relay networks with N relays is said to have orthogonal components if the sender and the relays alphabet X k = X kR ×X kD , for k = 0, . . . , N −1, and the channel can be expressed as Figure 6: A class of orthogonal relay networks.
or equivalently, Orthogonal relay networks with N = 2 are illustrated in Figure 6, where the channels in the same frequency band are shown by the same line type. The channel from the sender to the first relay is shown by a dash-dot line. The channel between two relays is shown by a dash line. The channels converging at the sink are shown by dot lines. The capacity for the networks of Figure 6 is given by the following theorem.

Theorem 5.
For the network depicted in Figure 6, the capacity is given by Proof. The achievability is proved By replacing X k = (X kD , X kR ), for k = 1, . . . , N − 1 and U k = X k−1,R for k = 1, . . . , N, in (7) and assuming joint probability mass function of the form as follows: (53) where (a) and (b) follow from (49). (c) follows from the fact that according to (52), we have this along with (50) results in The converse follows immediately from the max-flow min-cut theorem for general multiple-node networks stated in [16,Theorem 15.10.1], where the node set is chosen to be {0}, {0, 1}, . . . , {0, 1, . . . , N} sequentially, and with the following equations: The maximization in (51) is over the choice of joint probability mass function {p(x N ) N−1 i=0 p(x iD , x iR )}. Without loss of generality, we can restrict the joint probability mass functions to be of the form (52).
This completes the proof of the theorem.

CONCLUSION
This paper presents a new achievable rate based on a partial decoding scheme for the multilevel relay network. A novel application of regular encoding and backward decoding is presented to implement the proposed rate. In the proposed scheme, the relays are arranged in feed-forward structure from the source to the destination. Each relay in the network decodes only part of the transmitted message by the previous relay. The priorities and differences between the proposed method with similar previously known methods such as general parity forwarding scheme and the proposed rate by Xie and Kumar are specified. For the classes of semideterministic and orthogonal relay networks, the proposed achievable rate is shown to be the exact capacity. One of the applications of the defined networks is in wireless networks which have nondeterministic or orthogonal channels.