- Research Article
- Open Access

# Joint Linear Filter Design in Multiuser Cooperative Nonregenerative MIMO Relay Systems

- Gen Li
^{1, 2}Email author, - Ying Wang
^{1, 2}, - Tong Wu
^{1, 2}and - Jing Huang
^{1, 2}

**2009**:670265

https://doi.org/10.1155/2009/670265

© Gen Li et al. 2009

**Received:**30 November 2008**Accepted:**1 July 2009**Published:**19 August 2009

## Abstract

This paper addresses the filter design issues for multiuser cooperative nonregenerative MIMO relay systems in both downlink and uplink scenarios. Based on the formulated signal model, the filter matrix optimization is first performed for direct path and relay path respectively, aiming to minimize the mean squared error (MSE). To be more specific, for the relay path, we derive the local optimal filter scheme at the base station and the relay station jointly in the downlink scenario along with a more practical suboptimal scheme, and then a closed-form joint local optimal solution in the uplink scenario is exploited. Furthermore, the optimal filter for the direct path is also presented by using the exiting results of conventional MIMO link. After that, several schemes are proposed for cooperative scenario to combine the signals from both paths. Numerical results show that the proposed schemes can reduce the bit error rate (BER) significantly.

## Keywords

- Mean Square Error
- Direct Path
- Minimum Mean Square Error
- Filter Design
- Uplink System

## 1. Introduction

Wireless relays are essential to provide reliable transmission, high-throughput, and broad coverage for next-generation wireless networks [1]. Deploying a relay between a source and a destination cannot only overcome shadowing due to obstacles but also reduce the required transmitted power from the source and hence interference to neighboring nodes. Relays can be regenerative [2] or nonregenerative [3]. The former employs decode-and-forward scheme and regenerates the original information from the source. The latter employs amplify-and-forward scheme, which only performs linear processing for the received signal and then transmits to the destinations. As a result of the above difference, a nonregenerative relay generally causes smaller delay than a regenerative relay.

MIMO techniques are well studied to promise significant improvements in terms of spectral efficiency and link reliability. In [4, 5], the capacity of point-to-point MIMO channel is investigated and extensive work on multi-user MIMO has been done for a decade [6]. Therefore, combined with the above two technologies, a novel system called MIMO-relay emerges to accommodate users with high data rate requests and extend the network coverage. Recently, there is a vigorous body of work on MIMO-relay systems [7–15]. For example, [7, 8] derives upper bounds and lower bounds for the capacity of MIMO-relay channels. In [9], the optimal design of non-regenerative MIMO relays is investigated. Assuming relays and receivers with multiple antennas, the optimal relay matrix that maximizes the capacity between the source and destination is developed when a direct link is not considered or is negligible. The same problem is studied in [10], and [11] extends the work to partial channel state information (CSI) scenario.

Despite significant research efforts and advances on MIMO relay systems, most of the aforementioned research is based on a point-to-point scenario with a single user equipped with multiple antennas. In practical systems, however, each relay will need to support multiple users. This motivates us to study multiuser MIMO-relay systems, where the relay forwards data to multiple users. The most different feature between the researches on the single-user (with multiantenna) and multiuser (with single antenna) system is that the signals of the multiple users cannot be cooperatively pretransformed (e.g., uplink of a cellular system) or posttransformed (e.g., downlink of a cellular system). While single-user MIMO-relay systems have been a primary focus of prior research, a few researchers begin to pay attention to multiuser scenario as well. In [16], the optimal design of nonregenerative relays for multiuser MIMO-relay systems based on sum rate is investigated. Assuming zero-forcing dirty paper coding at the base station and linear operations at the relay station , it proposes upper and lower bounds on the achievable sum rate, neglecting the direct links from the BS to the users.

In this paper, we consider the problem of joint linear optimization for both downlink and uplink in multiuser cooperative nonregenerative MIMO-relay systems based on MSE criterion, which is different from the sum rate criterion in [16]. The MSE criterion is motivated by robustness to channel estimation errors and a lower implementation complexity. Then our main contributions are as follows.

(i)We derive the optimal joint design of the BS and RS filter matrices that achieves the minimum mean squared error (MMSE) for both downlink and uplink of the multiuser MIMO relay systems at the absence of direct path.

(ii)We propose several schemes for the design of the BS and RS filter matrices based on MSE criterion in the presence of direct path, which is called cooperative scenario in this paper.

(iii)We compare different schemes for direct-path-only scenario, relay-path-only scenario and cooperative scenario, and the numerical results are provided to show the effectiveness joint filter design and cooperative combine operation.

The rest of this paper is organized as follows. Sections 2 and 3 formulate the system model and propose the joint filter design schemes for downlink and uplink of multiuser MIMO relay systems, respectively. Numerical results are given in Section 4. Finally, Section 5 concludes this paper.

Notations

Boldface capital letters and boldface small letters denote matrices and vectors, respectively. Superscripts , and stand for the conjugate, transpose, and complex conjugated transpose operation, respectively., while and represent inversion and pseudoinversion of matrices. Also, and denote the expectation and trace operation, respectively, and, finally, is the identity matrix.

## 2. Downlink Systems

### 2.1. System Model and Problem Formulation

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

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

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.

where and are diagonal matrices.

Proof.

where is the Lagrange multiplier.

can be obtained, since no other matrices satisfy the property. Note that is a permutation matrix.

which leads to the desired result (33).

Theorem 2.

Proof.

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

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.

## 3. Uplink Systems

### 3.1. System Model and Problem Formulation

Similar with that in downlink systems, the signal model for the direct path of proposed systems in uplink is

where is a zero-mean complex Gaussian noise vector received at the BS with covariance matrix . Also, denotes a zero-mean complex Gaussian vector whose covariance matrix is , which indicates uncorrelated data streams with equal power are transmitted. Note that is the total transmit power for all the . Here, is the filter matrix at the BS for the direct path.

Then the signal model for the relay path of the proposed multiuser nonregenerative MIMO-relay system in uplink is given by

where and are zero-mean complex Gaussian noise vectors received at the BS and RS with covariance matrices and , respectively. Also, is the filter matrix at the BS for the relay path. The afore-mentioned assumptions can be expressed as

Finally, we combine the signals from both the paths to get

that is,

Therefore the optimization problem based on MSE can be formulated as

where we assume that the RS uses the whole available average transmit power .

### 3.2. Filter Optimization for Direct Path

Based on the signal model (54), we first propose the optimization problem for the direct path as

whose optimal solution can be expressed as [18]

### 3.3. Filter Optimization for Relay Path

For the relay path, the MSE is given by

Then using (55) in (62), the optimization problem for the relay path is formulated as

where we assume that the RS uses the whole available average transmit power .

As discussed in downlink systems, the Lagrange function is constructed as

with the Lagrange multiplier and by setting its derivative, we have

Obviously, and are function of each other. Iterative algorithms can be applied to get the optimal solution. However, it is too complex to be practical. Thus, a close-form solution will be derived in the following. Before the derivation, we introduce a useful lemma first [19] as follows.

Lemma 1.

If and are both Hermitian, there exists a unitary such that and are both diagonal if an only if is Hermitian.

Next, let the SVD of and be . Here, we also assume for simplicity. Then two main theorems involving the optimal scheme in uplink with their proofs are presented as follows.

Theorem 3.

where and are diagonal matrices.

Proof.

Hence, substituting (71), (73), and (75) into the SVD of and , we can have the desired result (68), which decomposes the MIMO relay channel into parallel channels.

Theorem 4.

Proof.

Substituting (75) and (78) into (74), the desired results are obtained.

Therefore, the afore-mentioned theorems form the closed form local optimal solution for uplink of proposed systems, that is, the filter matrices and can be easily calculated according to Theorems 3 and 4.

### 3.4. Filter Design Schemes for Cooperative Scenario

Based on the results derived earlier, we propose two schemes to approach the optimal results.

Cooperative Scheme 1

*Linear MMSE receiver*in [18], that is,

Cooperative Scheme 2

In this scheme, the relay filer matrix is also given by the expression of as shown in Section 3.3. Besides, the BS detects the soft estimate of the data streams from the direct path and relay path using the filter matrix and respectively. Finally, the receiver performs MRC combination over the separate data stream and then decodes them.

## 4. Numerical Results

The bit error rates (BER) of the proposed schemes in the previous sections are evaluated by applying them to a -user MIMO-relay system with antennas at the BS and antennas at the RS. We obtain the BER plots of OJ-MMSE/RP (Section 2.3.1), SOJ-MMSE/RP (Section 2.3.2), MMSB/DP (Section 2.2), CS1-MMSE/RDP (Section 2.4) and CS2-MMSE/RDP (Section 2.4) in downlink systems, together with OJ-MMSE/RP (Section 3.3), MMSB/DP (Section 3.2), CS1-MMSE/RDP (Section 3.4), and CS2-MMSE/RDP (Section 3.4) in uplink systems. Note that RP and DP denote direct path and relay path, respectively, while RDP represents the cooperative scenario with both the paths. In addition, we also evaluate the following two schemes as a reference for downlink and uplink systems, respectively.

*Transmit Amplify-and-Forward MMSE for relay path of downlink systems (TAF-MMSE/RP).*This scheme only requires the relay to normalize the received signal to meet the power constraint and then forward the signal. In this case, the filter matrix at the relay is

where is given to meet the power constraint at the relay, and hence the BS filter matrix and the scalar are obtained by substituting (80) into (29).

*Receive Amplify-and-Forward MMSE for relay path of uplink systems (RAF-MMSE/RP).*In this scheme, the filter matrix at the relay is also , where is given to meet the power constraint at the relay and hence the uplink signal model becomes

which is similar with that in conventional MIMO systems by regarding
and
as equivalent channel matrix and noise vector. Then the received MMSE filter
can be obtained via the *Linear MMSE receiver* in [18].

### 4.1. BER versus SNR

### 4.2. BER versus the Number of Antennas per Node

### 4.3. BER versus the Relevant Path Loss of Direct Path

### 4.4. Complexity

Computational complexity of the proposed schemes in downlink systems.

### 4.5. Convergence of Iterative Algorithm

## 5. Conclusion

In this paper, the local optimal MSE-based joint (BS and RS) filters have been proposed for a multiuser cooperative nonregenerative MIMO-relay system. Both uplink and downlink are considered. It is clear that the cooperative system can be divided into two paths, that is, the direct path and the relay path. As the optimal filter for the direct path can be obtained by using the exiting results of conventional MIMO link, we focus on the optimization for the relay path first. To be more specific, we propose the joint local optimal filter scheme, which involves an iterative algorithm in downlink scenario. Thus a simpler suboptimal scheme is derived for practical use. Then, in uplink scenario a closed-form optimal solution is exploited based on matrix analysis theory. The proposed optimal scheme firstly transform the MIMO relay channel into parallel sub-channels and then the optimal power allocation among the sub-channels has been found to follow a water-filling pattern. Furthermore, based on the results for direct path and relay path, two schemes are proposed for downlink systems and uplink systems with different combination methods, respectively. Numerical results and analysis show that joint filter design and cooperative operation can offer significant performance gain in terms of BER.

## Declarations

### Acknowledgment

This work was supported in part by Ericsson Company, Beijing Science and Technology Committee under project no. 2007B053, National Natural Science Foundation of China (NSFC) under no. 60772112, National 973 Program under no. 2009CB320406, National 863 Program under no. 2009AA011802 and no. 2009AA01Z262.

## Authors’ Affiliations

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