Multihop Analog Network Coding: An AmplifyandForward Approach
Abstract
In this paper, we study the performance of an amplifyandforward (AF) based analog network coding (ANC) relay scheme in a multihop wireless network under individual power constraints. In the first part, a unicast scenario is considered. The problem of finding the maximum achievable rate is formulated as an optimization problem. Rather than solving this nonconcave maximization problem, we derive upper and lower bounds for the optimal rate. A cutset like upper bound is obtained in a closed form for a layered relay network. A pseudooptimal AF scheme is developed for a twohop parallel network, which is different from the conventional scheme with all amplification gains chosen as the maximum possible values. The conditions under which either the novel scheme or the conventional one achieves a rate within half a bit of the upper bound are found. Then we provide an AFbased multihop ANC scheme with the two schemes for a layered relay network. It is demonstrated that the lower bound of the optimal rate can asymptotically achieve the upper bound when the network is in the generalized highSNR regime. In the second part, the optimal rate region for a twohop multiple access channel (MAC) via AF relays is investigated. In a similar manner, we first derive an outer bound for it and then focus on designing low complexity AFbased ANC schemes for different scenarios. Several examples are given and the numerical results indicate that the achievable rate region of the ANC schemes can perform close to the outer bound.
I Introduction
Since the introduction of the amplifyandforward (AF) relay scheme, it has been studied in the context of cooperative communication [1][4]. It is an interesting technique from the practical standpoint because the complexity and cost of relaying, always an issue in designing cooperative networks, is minimal for AF relay networks. As the simplest coding scheme, AF is also used to estimate the network capacity of relay networks. Obviously, the achievable rate of AF scheme can be viewed as a lower bound to the network capacity. In addition to its simplicity, AF is known to be the optimal relay strategy in many interesting cases [5][7]. Gastpar and Vetterli [7] have shown that for a twohop network with AF relays, the cutset bound of the network capacity can be achieved in the limit of a large number of relays.
Network coding [9] is a novel and promising design paradigm for wired communication networks. As opposite to the conventional routing operation, network coding allows the intermediate relays processing the received packets to reduce the amount of transmissions and thus improves the total throughput of the network. Li et al. [8] have shown that linear network coding can achieve the multicast capacity [9] in a noiseless network. This result indicates that to send out a linear combination of the incoming packets at each node is sufficient to obtain the optimal capacity performance. Linear network coding has also been studied in [10] from an algebraic perspective. Each destination node effectively obtains source information multiplied by a transfer matrix consisting of global encoding kernels on the incoming edges of it, and can recover the original data provided that the matrix is invertible.
Applying the principle of network coding to wireless communication networks has recently received tremendous attention from the research community. AF relay scheme allows one to exploit the broadcast nature of the wireless medium and to introduce the concept of network coding into physical layer. Katti et al. [11] have studied an AFbased analog network coding (ANC) scheme. As opposite to the traditional approach of wireless communications, analog network coding fully use the interference rather than avoiding it. This technique significantly improves the network throughput in many scenarios. Since then, many works have been focused on the design of ANC relay schemes both for oneway and twoway relay channels [12][17]. In [15] Marić et al. have studied a multihop ANC scheme for a layered relay network under the individual power constraint. With each relay node amplifying the received signal to the maximum possible value, the achievable rate is shown to approach the network capacity in the highSNR regime. Later such result is extended to the generalized highSNR regime as defined in [16]. The results in [16] have shown the effectiveness of ANC in such generalized scenario. Recently, Agnihotri et al. [17] propose an iterative algorithm to obtain the optimal ANC scheme for a layered relay network with ”Equal Channel Gains between Adjacent Layers (ECGAL)” property in general SNR regime.
To employ ANC in a wireless network with AF relays, we are especially interested in deriving the optimal achievable rate. There have been many results for this problem [18][20]. For a twohop parallel relay network, Marić and Yates [18] have found the optimal AF relay scheme in closedform along with the maximum achievable rate under the sum power constraint. A more general result is obtained by Gomadam and Jafar [19]. A case when all the relay nodes introduce the correlated Gaussian noises is considered. They have found the optimal AF relay scheme for this scenario and investigated the influence of the correlation between noises on the endtoend performance. However, the optimal AF scheme for a general network where both the topology and the operation regime of it have no limitation, is still unknown, even in the case of a twohop parallel relay network under the individual power constraints. Agnihotri et al. [20] have provided a framework to compute the maximum achievable rate with AF schemes for a class of general wireless relay networks, which casts the problem as an optimization problem. The similar idea also appears in an independent work [16]. Unfortunately, the optimization problem in general case is hard to be solved.
To employ AFbased ANC scheme in a wireless network, there are still many important and interesting problems remaining to be solved. We list some of them as follows.

Problem 1: The cutset bound is usually used to justify the performance of different relay schemes. However, it is not a tight upper bound for the AF achievable rate in most cases because the relay nodes are only allowed to do linear operations. To derive a tight upper bound for the achievable rate of a network with AF relays is an issue of great importance since the optimal rate is always hard to be obtained.

Problem 2: Is it always optimal for relay nodes to amplify the received signals to the maximum possible values under individual power constraint? Intuitively, for different scenarios there may exist different AF schemes that have a better endtoend throughput performance than any others. Then how to characterize them?

Problem 3: A multiuser scenario is also an interesting topic [21]. For a twohop MAC with AF relays, how many AF schemes shall we use to obtain the entire achievable rate region? Obviously, it is infeasible from the practical standpoint for the relay nodes to store all the AF schemes. Hence, it is worth investigating the tradeoff between the performance and the complexity.
The main results of this paper give partial answers to the above questions and are summarized as follows:

In this paper, we pursue two related objectives. The first one is to investigate the optimal AFbased ANC scheme for a multihop unicast relay network. Assuming each relay node has a transmitting power constraint, we derive an upper bound of the optimal achievable ANC rate in a closed form for a layered relay network. The idea behind our method is similar to the technique used to derive the cutset bound. Thus we call it a cutset like upper bound. A novel AF scheme for a twohop parallel relay network is proposed, which is different from the conventional scheme with all relay nodes amplifying the received signal to the maximum possible value. We determine the different conditions under which either the novel scheme or the conventional one has a better performance to approach within bit to the upper bound and thus the optimal rate. Based on such observations, a mixed AFbased multihop ANC scheme is proposed. When the network is in the generalized highSNR regime [16], we demonstrate that the lower bound of the achievable rate can asymptotically achieve the upper bound. Furthermore, we point out that the result obtained in [15] can be viewed as a special case of the result obtained in our paper.

The second objective of this work is to extend the ANC scheme to the multiuser case. S. A. Jafar et al. [21] have shown the optimal rate region of the twohop parallel AF network under a sum power constraint. We find that to assume the relay nodes under individual power constraints makes the problem more complicated. It is observed that for a specific AF scheme, the network reduces to a conventional MAC. Each Gaussian capacity in the rate set of the MAC is equivalent to the capacity of a twohop parallel unicast network. With the similar idea, we first derive two outer bounds for the optimal rate region. The structures of them are fully characterized. Then a dynamic ANC scheme is proposed. According to different network settings, the number of the AF schemes stored in the relay nodes varies and with finitely many schemes a timesharing inner bound is obtained. The tradeoff between the storage space and the performance is decided for the practical purpose. Finally, we illustrate three ANC schemes. The numerical result shows that the inner bound of each ANC scheme is close to the respective outer bound in different scenarios.
Notation: Scalars are denoted by lowercase letters, e.g., , and boldface lowercase letters are used for vectors, e.g., , and boldface uppercase letters for matrices, e.g., . In addition, , , and denote the trace, determinant, transpose and inverse matrix of , respectively, and denotes a blockdiagonal square matrix with as the diagonal elements. denotes the th element of , and denotes the rank of . denotes the Euclidean norm of a vector . is the expectation operation. denotes the logarithm in the base 2 and denotes the natural logarithm. We use and to denote the convex hull and the closure of a set.
Ii Network Model
In the first part of this paper, we analyze a multihop Gaussian relay network with a single sourcedestination pair , which is represented by a directed graph depicted in Fig. 1. Assume each nonsource node introduces a Gaussian noise, all the relay nodes work in a fullduplex mode, and there is no circle path in the network. At instant , the channel output at node is a linear combination of all noisy signals transmitted from its upstream nodes and the Gaussian noise introduced by itself and can be expressed as
(1) 
where denotes the channel gain from node to node , represents neighboring nodes of node which have onehop path to node , is the transmitted signal at node , and is a sequence of independently and identically distributed (i.i.d.) Gaussian noises with zero mean and variance . All the channel gains are supposed to be fixed positive realvalued constants known through the network for the scope of the present paper.
A natural assumption is that there exists a transmitting power constraint at each node such that
(2) 
Each network node performs analog network coding via amplifyandforward relay scheme. Assume the relay nodes operate instantaneously as in [12], that is, the relay nodes amplify and forward their input signals without delay.
(3) 
where the amplification gain is chosen such that the power constraint (2) is satisfied. Actually the possible system instability resulting from this assumption is avoided by a ”buffering and subtracting” technique as observed in [20]. So, with this assumption, the time index then can be omitted in the sequel for the sake of brevity. Through a relaying path from node to node , the signal is multiplied by the amplification gains and the channel gains along this directed path. Since the multihop ANC scheme takes more of the network coding approach, the corresponding coding coefficients are defined as follows.
Definition 1 (Local Encoding Coefficient)
Let and be the input and output channels of node . We call
(4) 
the local encoding coefficient of pair , where, in particular, .
Definition 2 (Global Encoding Coefficient)
We call
(5) 
the global encoding coefficient from node to node , where represents the set of channels appearing in a relay path between the two nodes.
The global encoding coefficient defined here can be interpreted as the equivalent path gain between node and . The relationship between the local and global coefficients is given as follows.
(6) 
Iii Upper Bound to ANC Rate
Iiia Sufficient Condition of Power Constraint
We first derive the expression of the achievable rate for a unicast network with a specific ANC scheme. Then the problem of finding the maximum achievable rate is formulated as an optimization problem. In this subsection, we focus on capturing the feasible domain of each amplification gain according to the power constraint at each relay node.
According to the assumption that each nonsource node introduces a Gaussian noise, each of them can be viewed as a source with the noise as the input. Therefore, by the linear operation of AF relay scheme, the original network can be considered as a linear combination of several unicast subnetworks and the source of each subnetwork is either the source node or the nonsource node. Therefore, with the global encoding coefficient (5), the received signal at the destination is expressed as
(7) 
where represents the equivalent signal transmitted from the source node, and denotes the equivalent noise received at the destination node where the summation is over all relay nodes . By the assumption that all the Gaussian noises introduced at the nonsource nodes are independent, the sum of them is also drawn according to Gaussian distribution. Hence, from the RHS of the second equality above, the relay network employing the ANC relaying scheme can be considered as a pointtopoint Gaussian channel. As the wellknown result, the source node adopts the Gaussian codebook. The random variable (r.v.) used to generate the codewords is drawn according to , where denotes normal distribution with zero mean and variance . A sequence of codes containing codewords of length is proposed and it is shown that the error probability goes to zero as .
From (4), (5) and (7), both the equivalent signal and noise are related to the amplification gains chosen in the ANC scheme. For the equivalent Gaussian channel, we focus on the choice of the amplification gains to maximize the SNR value at the destination node, and thus the transmission rate . Certainly, this problem can be formulated as a standard optimization problem with the object function being the SNR received at the destination node which is shown as follows.
(8) 
which is subject to the power constraints at the relay nodes. Hence, to solve this optimization problem, it is necessary to convert the power constrains into the constraints with respect to the amplification gains first. However, for a multihop relay network, the selection of the amplification gain at one node also relies on the selections at its upstream nodes. Therefore, we find it hard to give an equivalent independent constraint for each amplification gain. To simplify the problem, we first provide such an independent constraint. Then we show it suffice for the power constraint. Fortunately, it will be shown that under such constraint, a good multihop ANC scheme can be proposed in a special SNR regime defined in the sequel. Several parameters are defined first as observed in [15].
Definition 3
When each node j transmits with given in (2), the power received at node k is upper bounded by
(9) 
and the reciprocal of is represented by
(10) 
Note that if the network is noiseless, then the received signal power at node approaches (9) when all the neighboring nodes of it transmit the received signals at the upper bounds of the power constraints (2). Then we extend the sufficient condition of the power constraint in [15] to a general network model described in section II by the following lemma.
Lemma 1
In a multihop Gaussian relay network, it is sufficient for each relay node to choose the amplification gain
(11) 
such that the power constraint (2) is satisfied.
Proof:
The proof is given in the appendix A. \qed
Note that by Definition 3, the value and its reciprocal are both constants determined by the network settings. Therefore, the constraints for the amplification gains proposed in the above lemma are independent of each other.
IiiB Upper Bound to ANC Rate
In this subsection, we first show a layered relay network depicted in Fig. 2. The source node is assumed to be at layer and the destination node at layer . The number of relay nodes at layer is denoted by . The transmitted signal vector at layer is denoted by and the received signal vector at layer can then be obtained as
(12) 
where and are the received signal vector and the noise vector at layer respectively, and denotes the channel matrix between these two layers with the element representing the channel gain from node at layer to node at layer . Especially, we use to denote the broadcast channel between the source node and the nodes in the first layer and to denote the multiple access channel between the nodes in the th layer to the destination node.
Specifically, the global encoding coefficients in such network are given as follows,
(13) 
(14) 
where represents the th columnvector in , and where denotes the node set of layer . From (7), (13) and (14), the signal received at the destination node is given by
(15) 
and the SNR function at the destination node (8) results to be
(16) 
To have an upper bound to the ANC rate, we recall the technique of obtaining the cutset bound of capacity of a noisy network. One may first assume all channels in the network are noiseless except at one cut and then have an upper bound. By taking the minimum upper bounds over all cuts, the cutset bound is derived. By applying the same idea to ANC in layered networks, an analysis of an ideal layered relay network is made first. We first fix a layer and assume the Gaussian noises are only introduced at nodes on this layer and other parts in the network are noiseless. By optimizing the SNR function received at the destination node of such network which is formulated by
(17) 
an upper bound of the optimal ANC rate is derived. Then by traversing , a collection of upper bounds is obtained. To choose the minimum of them, a better upper bound is derived. We draw the conclusion in the following theorem.
Theorem 1 (Upper Bound to ANC Rate)
An upper bound of the ANC rate of a Gaussian layered relay network with layers under the individual transmitting power constraints is given as follows.
where
Proof:
The proof is given in the appendix B. \qed
We refer to the optimal AF scheme at layer for the ideal network with only the nodes in layer introducing the noises as a ”pseudooptimal amplification scheme”. Note that the idea behind our upper bound is similar to the idea of cutset bound, we call this upper bound a cutset like upper bound. As we know, the cutset bound is an upper bound to the network capacity, where the relay nodes are allowed to do any operations, including decoding. So we intuitively feel that the cutset bound may not be tight for the optimal achievable rate of a network with AF relays in general case. To generalize the result of Theorem 1, we hope to find out a more general upper bound result for a nonlayered relay network with AF relays. By taking a cut set of the network, we use the notation to denote the nodes on the output sides of the channels in such cut set. If assume only the nodes ’s in introduce the Gaussian noises, the corresponding SNR function can be expressed as,
(18) 
Using the same argument as we derive Theorem 1, by optimizing , an upper bound of is derived. Then the upper bound of ANC rate in general network is given as
(19) 
However, we are unable to give a closedform of the optimal solution for in general case.
Remark 1
The upper bound of the ANC rate in [15] can be regarded as a corollary of Theorem . Let us revisit the result obtained in [15] in an alternative approach proposed in our paper. Since in the scenario considered in paper [15], all the relay nodes have sufficiently large transmitting powers except the ones at layer , the upper bound in Theorem shows that the minimum is taken when . The SNR function of an ideal network with , i.e., only the destination node introduces the Gaussian noise turns to be
(20) 
then the corresponding upper bound of the achievable ANC rate is
(21) 
Furthermore, from (20) we can see that the noise power received at the destination is independent of amplification gains in this case. Hence, the larger the transmitting powers are, the better the performance of the ANC scheme. With the power constraint, a practical ANC scheme is proposed as observed in [15].
(22) 
Remark 2
In [7], an upper bound to capacity of the relay network is computed using a weaker corollary of cutset theorem [25, Theorem 14.10.1] and the capacity formula for Gaussian vector channels with fixed transfer function [24]. Applying the similar idea used to verify the corollary, we can also compute the upper bound to capacity of the layered relay network, which can be formulated as an optimization problem as follows.
(23) 
subject to
(24) 
Let us relax this power constraint to
(25) 
Note that for given , (23) is formulated due to the assumption that the nodes in layer are the multiple transmitting antennas of the source node and the nodes in layer are the multiple receiving antennas of the destination node. Hence, the transmitting antennas may have any cooperation to encode the message and the multiple receiving antennas may have any cooperation to decode it. Consequently, the corresponding capacity is an upper bound to the network capacity. The solution of the problem defined by (23) and (24) cannot be larger than the solution of the problem defined by (23) and (25). For a specific , the latter can be evaluated by using the result in [24].
(26) 
where and the maximum is subject to . Note that the optimal solution is obtained by waterfilling. To investigate how far between the cutset bound and the upper bound derived in Theorem 1, several special cases are taken into account.
When the minimum in Theorem is obtained by taking , i.e., the first hop is on the bottleneck in the data transfer, we have the broadcast cutset bound as the upper bound for capacity.
(27) 
At this time, the upper bound obtained in Theorem 1 is
(28) 
which is equal to the former. Hence, in case when is achieved, then the network capacity can be achieved.
When the minimum in Theorem is obtained by taking , i.e., the last hop is on the bottleneck of the data transfer, we have the multiple access cutset bound as the upper bound for capacity.
(29) 
where . At this time, the upper bound obtained in Theorem 1 is
(30) 
By Schwarz inequality, . The equality holds when for some constant .
Iv Lower Bound to ANC Rate
We expect to obtain a good AFbased ANC scheme by solving an optimization problem in this section. The SNR function given in (16) is the object function, which is subject to the constraint given in Lemma 1. Note that Lemma 1 only provides a sufficient condition for the power constraint. Therefore, the optimal value of this optimization problem can only be served as a lower bound to the optimal ANC rate.
(31) 
As shown in [18], a similar optimization problem has a simple solution under the sum power constraint for a twohop relay network as depicted in Fig. 3. However, the optimization problem given above is essential a fractional programming [22] which is a hard problem. Even for a twohop network, the above problem is a fractional programming [23] where both the enumerator and denominator are convex functions. There is no such easy approach to solve the problem so far as we know.
Before investigating the lower bound to the optimal ANC rate for the layered relay network with arbitrary layers, we first examine three different AF schemes for two singlesource singledestination twohop parallel relay networks as shown in Fig. 4, and Fig. 6. Then we capture the characterizations of these schemes.

Scheme 1: As observed in most conventional amplifyandforward relaying schemes, all the relay nodes transmit the received signal at the upper bounds of the power constraints. Hence, the amplification gains are

Scheme 2: Motivated by the scheme used in layer when we obtain Theorem 1, we employ the pseudooptimal amplification scheme for this twohop network, that is
where .

Scheme 3: The destination node chooses one relay node to transmit the received noisy signal which is referred to as AF with selection. To be specific, only the relay node, with the assistant of which the destination receives the highest SNR value, amplifies and forwards its received noisy signal with the upper bound of the power constraint. Find
then choose
and .
For the network parameters given in Fig. 4, we compute the corresponding amplification gains and the SNR values received at the destination node for the above three schemes. Assume each Gaussian noise introduced by the nonsource node is i.i.d. with unit variance. Later, we will explain that the assumption does not lose the generality.

Scheme 1:
, and .

Scheme 2:
For ,
, , , and ,
and for ,
, , , and .

Scheme 3:
For ,
, , and ,
and for ,
, , and .
Then we compute the upper bound of the achievable rate for this network. By Theorem 1,
,
where and . The result shows that if , then is greater than , which means the bottleneck of data transfer is on the second hop, and if , then is no larger than , which means the bottleneck of data transfer is on the first hop.
In Fig. 5, the numerical result of the achievable rate for each scheme is given. From Fig. 5, we observe that when , the achievable rates of scheme 1 and 2 are almost the same, while when , the achievable rate of scheme 2 outperforms that of scheme 1. We are interested in how to select the AF scheme in different cases. Since in low SNR scenario, it is not expected to perform AF as the relaying scheme, to investigate the criterion, we consider the case when is sufficiently large, i.e., . The total noise received at the destination node has two portions, one is the propagation noise from relay nodes denoted by , which is referred to as the upper layer noise hereafter, and the other is the noise introduced by the destination node itself which is referred to as the lower layer noise later. We compute the propagation noise power both for scheme 1 and 2. In scheme 1,
and in scheme 2,
Since the lower layer noise has unit power, we observe that in both cases, the propagation noises dominate the total noise received at the destination node. Therefore, the AF scheme proposed in our paper is expected to be more effective in dealing with the upper layer noises. Note that the dominance only depends on the ratio of the variances of the upper layer and lower layer noises, hence assuming all the Gaussian noises have unit variance is without loss of generality.
For the network parameters given in Fig. 6, we compute the corresponding amplification gains with the above three schemes.

Scheme 1:
, , and .

Scheme 2:
For ,
, , , and ,
and for ,
, , , and .

Scheme 3:
For ,
, , and ,
and for ,
, , and .
Then we compute the upper bound of the achievable rate for this network. By Theorem 1,
,
where and . The result shows that if , then is less than , which means the bottleneck of data transfer is on the first hop, and if , then is no less than , which means the bottleneck of data transfer is on the second hop.
In Fig. 7, the numerical result of the achievable rate for each scheme is given. From Fig. 7, we observe that when , the achievable rates of scheme 1 and 2 are almost the same while when , the achievable rate of scheme 1 outperforms that of scheme 2. We are still interested in how to select the AF scheme in different cases. As explained previously, we consider the case when is sufficiently large, i.e., . We compute the propagation noise power both for scheme 1 and 2. In scheme 1,
and in scheme 2,
We observe that in both cases, the lower layer noise dominates the total noise. Therefore, from the numerical result, the scheme 1 is expected to be more effective in dealing with the lower layer noise.
Now let us summarize the results drawn from the above analysis. The first observation is that the more relay nodes assist the transmission the better performance we can obtain. The reason is that the source signals retransmitted by the relay nodes are all related; however, the noise signals are independent of each other. Therefore, from the aspect of cooperation communications, the more copies of the original signals received at the destination node the lager diversity gain will be obtained. Second, if the relay nodes always amplify and forward the received signal to the maximum possible value, then it may result in suboptimal endtoend throughput. This observation leads to an interesting problem that under what conditions the two schemes will have a better performance? From the above two examples, we conjecture that scheme 1 is more effective in managing lower layer noises and inversely, scheme 2 is more valid in dealing with upper layer noises. Finally, the first two AF schemes mentioned above can constitute a mixed ANC scheme which can be applied to a layered relay network with arbitrary layers. The relay nodes at each layer may select either of the conventional scheme (scheme 1) or the new scheme (scheme 2) proposed in our paper. However, in a layered relay network, the selection of the scheme for one layer also depends on the schemes applied by the other layers. Therefore, it is complicated to provide a localized criterion to select which scheme at each layer.
Let us consider the following AFbased ANC scheme motivated by the optimal solution in the ideal network analyzed in Theorem 1. With such ANC scheme, a lower bound to the optimal ANC rate is derived. To simplify the computation of the lower bound, we first define
(32) 
The amplification gain at node is chosen as
(33) 
and at node is chosen as some positive constant such that
(34) 
Note that the amplification vector selected at layer depends on these factors and moreover, no matter what feasible gains are selected will not affect the resulting upper bound. The amplification vector at layer is chosen as
(35) 
where and are defined in the proof of Theorem , and can be determined as
(36) 
Note that, in the above ANC scheme, both the conventional and the pseudooptimal amplification scheme are adopted. Although it is not expected to obtain the optimal ANC rate through a fixed scheme, the corresponding achievable rate of such ANC scheme can be served as a lower bound.
Theorem 2 (Lower Bound to ANC Rate)
A lower bound of the ANC rate of a Gaussian layered relay network with layers under the transmitting power constraints is as follows.
where is a bounded constant and .
Proof:
The proof is given in the appendix C. \qed
Note that is a bounded positive constant depending on the selection of amplification gains in nodes at layer to . Finally, we should emphasize that the lower bound obtained in the above theorem is a universal one since it is not required that should be a small positive value.
V Applications of AFbased ANC scheme
To the best of our knowledge, AF relay design for a layered relay network has been the focus of mush recent research. Thus by taking pseudooptimal amplification scheme at certain layers, we have an alternative multihop ANC scheme as opposite to the scheme with all relay nodes amplifying the received signals to the maximum [15]. A problem is whether pseudooptimal amplification scheme can make things better than conventional AF scheme. We turn to two particular classes of layered relay networks and compare the resulting lower bound of AF rates to our upper bound.
Va Application to TwoHop Network
We extend the results derived from the previous examples to a twohop parallel relay network with relay nodes. Note that scheme 2 can be viewed as a pseudooptimal amplification scheme for such network. Denote by and the two schemes and by and the upper layer noises corresponding to these two schemes respectively. Hence, we have
(37) 
and the lower layer noise introduced by the destination node itself has, as assumed, a unit power.
(38) 
It is easy to verify that is no less than .
Definition 4
We call the upper layer noise dominates the total noises received at the destination in a twohop relay network with AF relays when
Reversely, we call the lower layer noise dominates the total noises when
As we mentioned before, the individual power constraints make the problem of deriving the exact optimal AF rate complicated to be handled. However, we still obtain some interesting results on the performance of achievable AF rate shown as follows.
Theorem 3
For a twohop relay network with AF relays under individual power constraints, there exist AF schemes accodrding to different senarios when either the upper layer noise or the lower layer noise dominates the total noises recived at the destination node such that the gaps between the the corresponding achievable rates ’s and the optimal AF rate are at most bit.
Proof:
For a twohop relay network, we see that the achievable rate can be easily derived without employing the lower bound obtained in Theorem 2. For the first scenario, we compute the achievable rate with the pseudooptimal amplification scheme, i.e., .
(39) 
where (a) follows from the assumption that and (b) follows from Theorem 1 that
For the second scenario, we compute the achievable rate with the conventional scheme, i.e., .
(40) 
where (a) follows from the assumption that and (b) follows from Theorem 1 that
Then we complete the proof. \qed
Note that Theorem 3 rigorously interpret the intuition obtained from the examples in section IV. We are also interested in the case beyond these two scenarios, i.e., when . Are the performances of the two schemes in such general scenario also good enough? We draw the conclusion in the following theorem.
Theorem 4
For a twohop relay network with AF relays under individual power constraints, with either the pseudooptimal amplification scheme or the conventional one, the gaps between the corresponding achievable rates ’s and the optimal AF rate are upper bounded by a constant when neither the upper layer noise nor the lower layer noise dominates the total received noises at the destination.
Proof:
To verify the statement, we compare achievable rates corresponding to scheme and with two upper bounds and respectively. The achievable rate with the conventional scheme is as follows.
(41) 
where , (a) follows from the assumption , (b) follows from
(42) 
where the last inequality results from the generalized Rayleigh quotient [27], and (c) follows from the same argument as (b) in (40). Similarly, the achievable rate with the pseudooptimal amplification scheme is as follows.