2.1. System Model and Problem Formulation
In this section, we focus on the downlink of the multiuser cooperative MIMO relay system as illustrated in Figure 1. Assuming half duplex relaying, the scenario under analysis consists of a base station
, a relay station
and
mobile station
transmitting through two orthogonal channels, for instance, two separate time slots as time division multiple access (TDMA). During the first slot, The BS deployed with
transmit antennas communicates with the fixed RS that has
antennas and the MSs,each of which has single antenna. A MIMO channel denoted by
is thus created between the BS and the RS while a MIMO broadcast channel (MIMO
) denoted by
is also established. The precoding strategy at the BS includes an encoding operation and a subsequent linear operation with a filter matrix
. The BS encodes
data streams that are targeted to the MSs and broadcasts it to the RS and the
. The RS processes the received signal with a filter matrix
, and then forwards the data streams to the MSs through a MIMO BC denoted by
in the second slot. Finally, in the cooperative scenario, each of the MSs combines the signals from the direct path
and the relay path
that are received in the first slot and the second slot, respectively. Note that all the matrices in this paper are assumed full rank for simplicity.
During the first slot, the signal model for the direct path of the proposed system in downlink is
where
and
is a zero-mean complex Gaussian noise vector received at the MSs with covariance matrix
. Also,
denotes a zero-mean complex Gaussian vector whose covariance matrix is
, which indicates that uncorrelated data streams are transmitted.
During the second slot, assuming
is the received signal at MS
and
, the signal model for the relay path of the proposed downlink system is
where
and
are zero-mean complex Gaussian noise vector received at the RS and MSs with covariance matrices
and
, respectively. In addition, we assume the signal and noise are uncorrelated as well. The assumptions with the afore-mentioned signal and noise can be summarized as
Then, the signal
and
are normalized as
where the scalar
and
can be interpreted as automatic gain control that are necessary to give reasonable expressions for the MSE in any real MIMO system.
Finally, we combine the signals from both the paths to get
Therefore, the optimization problem based on MSE can be formulated as
where we assume that BS and RS use the whole available average transmit power, that is,
and
, respectively. Since the transmitted signal from BS is Fs and the transmitted signal from the RS is
, by using the assumption (3) simultaneously, the power constraints can be obtained.
However, from the following explicit expression of the objective function, it can be seen that the problem (7) is too complex to be solved optimally:
Hence, we separate it to be several independent subproblems as the following sections produce.
2.2. Filter Optimization for Direct Path
Based on the signal model (1) and (4), we first propose the optimization problem for the direct path as
As the direct path is actually a conventional MIMO link, a closed form solution is found for the optimization in [17]
where we define
Thus the optimal result for problem (10) is obtained.
2.3. Filter Optimization for Relay Path
For the relay path, the MSE is given by
Then using (2) in (15), the optimization problem for the relay path is formulated as
Here, note that
2.3.1. Local Optimal Joint (OJ) MMSE Scheme
Aiming at the optimal solution of the problem (16), we can find necessary conditions for the transmit filter
, the relay filter
, and the weight
by constructing the Lagrange function
with the Lagrange multiplier
,
and setting its derivative to zero:
where we use
and
. By introducing
, the structure of the resulting relay filter follows from (21):
with
where the power constraint at the relay is used.
Applying (23) into (21), we get
which follows that
where
and the power constraint at the relay are used.
Hence, using (23) and (25) in (22), we obtain that
. Therefore, the filter matrix can be expressed as the function of the transmit matrix for the optimization in (16):
where we define
Similarly, the expression of the transmit filter matrix in terms of the relay filter matrix can be derived as
where we define
From the above results, it is obviously seen that
and
are functions of each other. Therefore, the solutions
and
for the problem (16) can be obtained via the following iterative procedures.
(1)Initialize the transmit filter matrix
, satisfying the transmit power constraint.
(2)Calculate the relay filter matrix
with the given
according to (26).
(3)Calculate the transmit filter matrix
with the new
according to (29).
(4)Go back to
2 until convergence to get
and
.
Although the MSE function in (15) is not jointly convex on both the transmit filter matrix and the relay filter matrix, it is convex over either of them. This guarantees that the proposed iterative algorithm could at least converge on a local minimum.
2.3.2. Suboptimal Joint (SOJ) MMSE Scheme
In this subsection, we present a simplified closed form solution to the suboptimal structure of
and
, in that the optimal scheme proposed above involves a complex iterative algorithm which is not quite practical in real systems.
First, we ignore the scalar
and the power constraint at the relay for simplicity, and the problem can be changed into
Let the singular value decomposition (SVD) of
be
. Here, for simplicity of the derivation, we assume
. Thus,
is a diagonal matrix of singular values while
and
are square and unitary matrices. Then, our main theories are described as follows.
Theorem 1.
The objective
of problem (32) can achieve its minimum when the BS filter and the relay filter are constructed as follows:
where
and
are diagonal matrices.
Proof.
The standard Lagrange multiplier technique, which is similar to that in the last section, is used to solve the optimization problem formulated in (32). By setting the derivative of the cost function to zero, we get
where
is the Lagrange multiplier.
Supposing
, the afore-mentioned two equations can be arranged as
which implies
is Hermitian.
Thus, combining (35) and (36) gives
which follows that
where
is a unitary matrix. Using (38) in (35), we have
Premultiply the equation by
and postmultiply by
to get
Let
,
and substituting the SVD of
and
in (40),we have
Since
is Hermitian from (40),
. Applying it in the afore-mentioned equation we get
which implies
must be diagonal. Hence,
can be obtained, since no other matrices satisfy the property. Note that
is a permutation matrix.
Similarly, we can also yield that
Substitute the SVD of
and
in (37) to get
Using uniqueness of SVD, we have
Without loss of generality, set the permutation matrix as
. Then, using (43), (44), and (46) in (15), the MSE expression becomes
Since the trace of matrix depends only on its singular values,
can be chosen to be any unitary matrix (e.g.,
) without affecting the MSE. Therefore, we have
which leads to the desired result (33).
Theorem 2.
The optimum MMSE power allocation policy can be expressed as
Proof.
Using (43) and (44) in (42), we have
which produces the desired water filling result (49). Besides, from (45), (50) can be easily obtained.
Therefore, the filter matrices
,
and the scalar
can be obtained via the following steps. First, calculate
and
according to Theorems 1 and 2. Then, let the relay filter matrix
, where
is chosen to meet the relay power constraint. In addition, the scalar
is set to be equal to
. Thus, the solutions of
,
, and
form a suboptimal scheme for the optimization problem (32), which is simpler than the local optimal scheme.
2.4. Filter Design Schemes for Cooperative Scenario
After the signal from the direct path and the relay path
and
are obtained, the optimization problem for combining them based on minimizing the MSE is formulated as
By applying the standard Lagrange multiplier technique, the optimal weighing parameters are written as
where we assume
and
.
As is known, we are unable to find the optimal solution for problem (7). Then based on the optimal results for the subproblems (10), (16), and (52), we propose two schemes to approach the optimal results.
Cooperative Scheme 1 (CS1)
In this scheme, we first present the transmit filter matrix from view of the direct path, that is,
. Then, based on (26), the relay filter
is fixed. Besides, the scalar
and
at the MSs can be easily obtained by using (13) and (27), respectively. Conditioned upon the results above, the covariance matrix
and
can be worked out to get the weight
and
. Therefore, the above solutions
form Cooperative scheme 1 for the downlink of proposed multiuser cooperative MIMO-relay systems.
Cooperative Scheme 2 (CS2)
Alternatively, this scheme takes the relay path into account primarily. Namely, the transmit filter matrix and the relay filter matrix follow the result deduced for the relay path, which is written as
and
. Then the scalar
and
can be calculated accordingly to normalize the received signal at the MSs. Similar with that in Cooperative scheme 1, the weight
and
are obtained. Thus the Cooperative scheme 2
is created.
In summary, all the schemes above are useful for different scenarios. As we know, there may be three kinds of users in relaying networks: direct users, pure relay users, and cooperative users. The direct users communicate with the BS directly and can use the filter design results in Section 2.2. For the pure relay users, they receive the data stream signal only from the relay path neglecting the direct link. These users can adopt the filter design results in Section 2.3. The cooperative users are those who combine the signal from the direct path in first time slot and the signal from the relay path in the second time slot. For these users, we propose two different filter design schemes in Section 2.4.