1. Introduction
Wireless networks today are facing with challenges of high demands of reliable transmission and throughput while reducing signal interference. Relay-based cooperative networks are proposed to overcome these challenges and the essence here is to design relay strategies among sources and relay nodes. Compared with others like decode-forward (DF) and compress-forward (CF), amplified-forward (AF) has advantages of simple implementation and low relaying cost and is thus preferred in designing cooperative networks [
1,
2]. In fact, it has been shown optimal in some cases [
3].
One popular variant of AF is linear beamforming. It achieves high transmission rate by generating pencil beams to concentrate signals in a narrow direction towards intended receivers, and therefore significantly reduces interference from omni-directional antenna transmissions. For this reason, linear beamforming is widely applied in wireless relay networks as a promising relaying strategy [
4,
5]. The objective of linear beamforming design is to find the optimal beamforming vector that achieves maximum end-to-end data rate under power constraints, either individually or by sum-power [
6,
7,
8]. Depending on multiple antennas at relay, the objective becomes to find a beamforming matrix in multiple-input-multiple-output (MIMO) systems [
9,
10,
11]. In the latter case, it is actually to solve a matrix-monotonic optimization problem.
Another approach to avoid collision in wireless half-duplex mode is analog network coding (ANC). In [
12], Katti et al. proposes an analog network coding (ANC) relaying scheme, which allows relay nodes to receive the sum-signal from multiple sources in one time slot. When we have two separate sources, we can utilize ANC to combine the process of multiple signals and beamforming to address the capacity. Such a combination of ANC and AF [
13] has been shown to improve the performance.
In [
14], the authors make the first attempt to investigate the error rate and the power allocation of a MARN with ANC uninvolved with the system capacity. However, the capacity is more important in many cases. There have been several attempts to achieve high information rate by finding optimal beamforming in MARN. The pioneer work [
15] fully characterizes the complete achievable rate region of a dual-hop MARN where linear beamforming is employed under the sum power constraint. To obtain the optimal linear beamforming vector, the authors propose weighted sum-rate maximization approach. Unfortunately, when power is constrained individually rather than in a sum, such approach is hard to solve [
16] for its prohibitive computational complexity. A new and effective method is proposed in [
17] to find the optimal linear beamforming vector for the same MARN under an individual power constraint. Note that existing work assumes that the relays have only one antenna. To the best of our knowledge, the achievable rate region for a multi-antenna relay is still unknown.
In this paper, we investigate communications on MARN with two independent source nodes, one destination node and one multi-antenna relay. This network model captures various wireless systems like wireless monitoring system (WMS), in which multiple monitoring terminals transmit messages to a monitoring center through a multi-antenna relay station for remote monitoring under the sum-power constraint. Both ANC and beamforming are applied at the relay node.
Our main contribution of this paper is to propose an efficient scheme to characterize the achievable rate region and its corresponding linear beamforming matrix. Specifically, we give a families of corner point (CP) optimization formulations to determine the achievable rate region by solving a series of CP non-convex optimization problems. We then show how to transform them to semi-positive definite (SDP) convex problems. To lower the computational burden of solving SDP, we propose an optimal structure of linear beamforming matrix such that the number of independent variables in linear beamforming matrix is reduced to a small constant. Finally, we derive the mathematical close-forms of the maximum individual rates and the sum-rate. Our scheme has the following advantages:
High transmission rate: Beamforming can suppress the inference from antennas in relay for its targeted nature. ANC can fully utilize the inference from different users rather than avoid it to improve the transmission signal-to-noise ratio (SNR). Based on the simulation results, we find that our scheme can obtain a higher transmission rate at the same cost.
Low computational complexity: we propose an optimal structure of an AF matrix employing singular-value decomposition, and reduce the number of design variables in the relay beamforming matrix from to , . Then, the number of variable parameters in SDP will decrease significantly.
The rest of this paper is organized as follows. The network model and problem statement are presented in
Section 2. The method to design the ANC beamforming matrix is presented in
Section 3. The mathematical closed-forms of the maximum individual rates and the sum rate are derived in
Section 4. Then, the simulations are given in
Section 5, and all of the detailed proofs are arranged in appendixes.
Notation: Scalars are denoted by lower-case letters, e.g., x, and bold-face lower-case letters are used for column vectors, e.g., , and bold-face upper-case letters for matrices, e.g., and let . In addition, , , , , and denote the trace, determinant, conjugate, transpose, Hermitian transpose and inverse matrix, respectively. denotes a block-diagonal square matrix with as the diagonal elements, and . ⊗ denotes the Kronecker product. denotes the Euclidean norm. is the n identity matrix. is the expectation operation. denotes the logarithm in the base 2.
2. Network Model
Consider a wireless monitoring system in
Figure 1. In this paper, the direct links between the two wireless cameras and monitoring center are ignored since the monitor is far away the cameras. In addition, the destination receives the signals in maximal ratio combining (MRC). The system SNR is the sum of SNRs in all links. Apparently, the SNRs in these direct links are independent of the linear beamforming matrix. It does not affect the design of beamforming. This actual transmission model can be transformed mathematically as a MARN shown in
Figure 2.
The MARN consists of two single antenna sources,
and
, a helping
K antennas relay
R, and a single antenna destination
D. All of the channels are assumed flat-fading over a common narrow-band. The relay works in half-duplex mode and there is no direct link from
and
to
D. For a Gaussian multiple access channel, time division multiple access (TDMA), frequency division multiple access (FDMA), and code division multiple access (CDMA) are the general multiple access modes. In TDMA mode, the sources use non-overlapping time periods to complete the transmission. Thus, the receiver can separate signals according to different time periods. In FDMA mode, the sources transmit signals simultaneously in the mutually disjoint frequency bands. Then, the receiver can separate signals according to different frequency bands. In the CDMA mode, the sources transmit the signals using different codes simultaneously. Thus, the receiver decodes one by one to separate the signals. According to [
18], the multiple access channel can achieve a larger rate region in CDMA mode.
In this paper, the two sources will complete the communication in CDMA. Assume that the sources
and
independently generates the
and
codewords in the rates
and
with the block-length
n. Note, in this paper,
and
represent the information rates rather than the data rates or frame rates. The code symbols
and
.
and
denote the transmit powers of
and
. In practice,
and
transmit the codewords
and
with the block-length
n to the relay simultaneously. The destination performs decoding after receiving
with the block-length
n. The method of separating these two signals at the destination through decoding will be described later. In this case, the frame rates of the two sources are the same. Therefore, we can perform ANC at the relay without affecting decoding. Since the lengths of the two codewords from
and
are the same, we can consider this mode through discussing the single symbol transmission. During the first time slot, both
and
transmit simultaneously to
R, which uses a linear beamforming scheme to retransmit the received sum signal with noise to
D during the second time slot. It is also assumed that perfect synchronization are established among
and
prior to data transmission. The baseband signal received at
R in the first time slot is expressed as
where
is the signal vector received at
R;
and
represent the channel vectors from
to
R and from
to
R, respectively, which are assumed constant during the transmission.
is noise vector at the relay, and without loss of generality (w.l.o.g.),
. For ANC is exploited in the antennas of relay, the signals from
and
are summed in the antennas. Upon the
K sum-signals, the relay processes them employing a linear beamforming matrix
, and then retransmits it to
D during the second time slot. Mathematically, the signals vector retransmitted at
R can be concisely represented as
where
is the signal vector retransmitted at
R, and
is the beamforming matrix.
The relay has a power budget
. Thus, the signal vector retransmitted at
R should satisfy the following constraint:
We use
to denote the set of all beamforming matrices satisfying the relay power constraint. As a result, given a beamforming matrix
, the signal received at
D can be expressed as
where
is the noise at destination, and
denotes the channel vector from
R to
D. We assume that perfect channel state information (CSI) has been collected at
R prior to transmission.
From Equation (
4), the dual-hop MARN with linear beamforming can be considered as a conventional Gaussian multiple-access channel (MAC) on matrix
as follows:
where
and
are the information symbols of the two equivalent sources and
is the equivalent Gaussian noise drawn according to
. To distinguish them, we denote the former one as
. The capacity region of a Gaussian MAC
can be found in ([
18], Section 14.3). According to the capacity region of the Gaussian MAC, the achievable rate region of
is denoted by
given as follows
For notation brevity, we denote , , and the union of the achievable rate sets ’s by . By the time-sharing technique, the achievable rate region of an MARN is given by , where represents the convex hull of a set.
In information theory, the hypothesis that the decoder at the destination knows the codebooks of the source and the is practical. First, the decoder amplifies the all codewords with to obtain sequences with the length n not changing their corresponding messages. Similarly, the decoder can obtain new sequences employing . Second, the decoder selects the sequence combination with the smallest Euclidean distance from among all sequence combinations. Then, the decoder can determine the codeword combination transmitted by the sources and corresponding the sequence combination . In this paper, we do not discuss the specific decoding scheme in practice. We only cite the conclusion of Gaussian multiple access channel capacity region in information theory.
It should be pointed out that this achievable rate region is obtained in CDMA mode with the synchronization condition. Unfortunately, synchronization is very difficult to achieve even in the small distance. In addition, the paper [
19] pointed out that lack of synchronization can not reduce the capacity region for multiple access channel when the block lengths of the codes are long compared to the delay. Thus, in this paper, if the synchronization condition cannot be satisfied, as long as the packet length of the codes is long compared to the delay, the achievable rate region of the MARN remains unchanged.
From Equation (
6), we can determine the outer bound of the achievable rate region
by maximizing
,
and
under the condition
, respectively. We will obtain the maximum values
,
and
and the corresponding optimal beamforming matrices
,
and
. Note that they are not unique. Usually, there exists the situation as follows:
We will provide the relationship of the achievable rate regions in
Figure 3.
is in the blue area,
is in the red area, and
is in the violet area.
From
Figure 3, it is easy to obtain the outer bound of the rate region consisting of the polyline
, employing these three values
,
,
. However, according to the relationship of
,
,
, we can not determine the beamforming matrices corresponding to the dotted line subregions
and
in
Figure 3. Therefore, it is unknownwhether these two subregions are achievable. Additionally, using time-sharing, we can obtain the straight line
of the “upper corner” of regions
and
and
of the “low corner” of regions
and
. Obviously, the inner bound of the rate region with polyline
is achievable. Thus, these two subregions
and
between the inner bound and outer bound are unknown in
Figure 3. We will propose another new method to determine the achievable rate region and the corresponding beamforming matrices in
Section 3, and discuss whether this outer bound is tight in
Section 4.
4. Performance Analysis
To further investigate the performance of the proposed linear beamforming schemes, we study the maximum achievable individual and sum rates of . We first analytically obtain the maximum individual rates of and the corresponding linear beamforming scheme. Then, we derive the maximum sum rate of in closed form and the corresponding linear beamforming scheme can be determined by solving a system of linear equations.
We consider first the maximum individual rates. Mathematically, the problem can be formulated as follows:
According to Proposition 1 and the same argument above, it can be recast as
It is clear that the constraint holds with equality at the optimum, otherwise the optimal solution can always be scaled up yielding a larger objective value. Thus, substituting
in the objective function, it can be formulated as follows:
Using Lemma 1, we determine the maximum individual signal-to-noise ratios
,
, which is attained at
where
, which is chosen such that the constraint holds with equality. The maximum individual rates are easy to be determined
.
Next, we consider the maximum sum rate. Mathematically, the problem can be formulated as follows:
Similarly, it can be recast as
For the same reason as in the previous case, the constraint holds with equality at the optimum. Thus, substituting
in the objective function, it can be formulated as follows:
Using Lemma 1, we will determine the maximum sum rate by the following Theorem 2.
Theorem 2. The maximum sum signal-to-noise ratio is denoted , thenwhere , and . The corresponding maximum sum rate is easily given by
. The maximum is attained at
which can be determined by solving the following system of linear equations:
where
with an additional constraint that
. Through the mathematical analysis above, we obtain the two maximum individual rates
,
and the maximum sum rate
. These three values will determine the bound of the theoretical rate region easily, but we can not determine the beamforming matrices corresponding to all of the points on the bound of the rate region.
Next, we will discuss whether the outer bound determined by these three maximum rates
,
and
is tight. We know if this outer bound is tight, for the “upper corner” of the rare region
, there exists a beamforming matrix
, such that
all hold. According to the previous analysis, the condition achieving
is
where
. The condition achieving
is
where
. Obviously, (
40) and (
41) can not be established simultaneously. Therefore, this outer bound is not tight.
5. Numerical Results
In this section, we show some numerical results to quantify the achievable rate regions of a dual-hop MARN with linear beamforming, and analyze the transmission performance under different antenna numbers and relay schemes. All simulations are performed in MATLAB r2010a (MathWorks, Natick, MA, USA). We use CVX toolbox [
23] to solve the SDP problems. We assume that the power budgets of the two sources are
, and the relay is equipped with
K antennas having a relay power budget
. The channel coefficients are generated as independent complex Gaussian random variables with the distribution
. We randomly generate a set of channel vectors
and
as follows:
First, setting the channel vector
and
above and the number of antennas
K at relay. We show the achievable rate regions versus different relay power budgets
in
Figure 5. Meanwhile, according to the three rates
and
obtained employing the method in
Section 4, we determine an outer bound of the achievable rate region
. The simulation results show that the achievable rate regions obtained by using our scheme almost coincides with the corresponding outer bounds except the corners. It validates the conclusion obtained through analysis in
Section 4 that this outer bound is not tight. It is observed that the achievable rate region with
dBW is the interior of the achievable rate region with
dBW. The achievable rate region expands as the relay power budget
increases. It is in conformity with the reality that increasing the power budget of relay antennas always leads to better transmission performance.
In
Figure 6, we show the average sum-rates corresponding to the optimal beamforming matrices versus different relay power budget
when
over 5000 channel realization. Comparing these three curves, we find the average sum-rates increase with the increase of
K. It is in conformity with the reality that increasing the number of antennas
K can increase diversity gain to improve system transmission performance. In addition, the average sum-rates increase with the increase of the relay power budget
, which is consistent with the result in
Figure 5.
To further illustrate the system performance exploiting the scheme proposed in this paper, we show the average sum-rates versus different relay power budgets comparing with other several relay schemes that are usually applied in practice.
Direct relaying, where the relay beamforming matrix is in the form of and . It is a scheme to uniformly amplify all the signals of antennas at relay. In essence, it is not a beamforming scheme but a simple AF scheme.
Alternative relaying, which is a TDMA scheme. It requires four time slots to complete the communication between the two sources
and the destination. We assume that the communication between source
and destination is completed in the first and second time slots, while communication between source
and destination is completed in the other two time slots. The corresponding relay beamforming matrices are given in [
6].
Optimal diagonal beamforming, which is a beamforming scheme that the signals of the different antennas can not be superimposed. The corresponding relay beamforming matrices are diagonal matrices, which is an optimal form of the direct relaying scheme.
It is observed in
Figure 7 that the optimal linear beamforming scheme significantly outperforms the other three relay schems.