Two-Hop MIMO Relay System

In this scenario, the channel condition in the direct link is poor such that the destination only receives the signal from the relay link. Here, we first consider the case that N = R = M = L = 4. Let the elements of each channel matrix be i.i.d. complex Gaussian random variables with zero mean and unity variance. Let SNRsr and SNRrddenote, respectively, the SNR per receive antenna of the source-to-relay and relay-to-destination links. Here, we set SNR_sr = 20 dB and vary SNRrd. Fig. 2.4 and Fig. 2.5 show the MSE and the BER comparisons, respectively for (a) an un-precoded system with ZF receiver, (b) an un-precoded system with MMSE receiver, (c) the optimal relay precoder with MMSE receiver [43], and (d) the linear source and relay precoded system. From those figures we can see that the linear source and relay precoded system outperforms not only the un-precoded system, but also the relay precoded system in [43].

This is because the linear source and relay precoded system incorporates the additional source precoder such that the performance can be enhanced even the direct link is not considered.

We also report the simulation result for cooperative beamforming, i.e., L = 1. As discussed in Theorem 2.1, our design for this case is optimal. We let N = R = M = 4 and SNR_sr = 5 dB. Fig. 2.6 shows the BER comparison for the antennas selection method in [45] and the linear source and relay precoded method. From the figure, we can see that the linear source and relay precoded method is superior to the antenna selection. This is expected since our design here is optimal.

§ 2.3.3 General MIMO Relay Channel

In this scenario, we consider a symmetric MIMO relay system, i.e., N = M = R = L = 4.

As the previous case, each element of the channel matrices is assumed to be i.i.d. complex Gaussian random variables with zero mean and same variance. We let SNR_sr, SNR_rd be the same as those defined in Section 2.3.2, and SNR_sd as the SNR per receive antenna for the source-to-destination link. Here, we set SNR_sr= 15 dB, SNR_rd= 10 dB and vary SNR_sd. Fig.

Table 2.1: Complexity of linear source and relay precoders (MMSE receiver).

Operation FLOPs

SVD, (2.14)-(2.16) (14MN²+ 8N³) + (14RN²+ 8N³) + (14MR²+ 8R³) B⁻¹, (2.24) 2MN²+ 2MN + 2N³ + 13/4N²+ N²

p_s,iand p_r,i, (2.29)-(2.30) (21LI_r+ 14LI_s)I_i

E, (2.33)˜ 14L + 10M + 4NL + 2L²N

SVD of ˜E, (2.34) 22L³

U_S, (2.35) 2L³

F_Sand F_R, (2.20)-(2.21) (2NL + 2NL²) + (2R²+ 2R³) N: number of transmit antennas

R: number of relay antennas M: number of receive antennas

L: number of transmitted symbol streams Ir: number of iteration for computing pr,i

I_s: number of iteration for computing p_s,i

Ii: number of iteration of the water-filling process

2.7 and Fig. 2.8 show the MSE and BER comparisons, respectively, for the linear source and relay precoded system and other systems described in Section 2.3.2. Note that the optimal relay precoder in [43] only considers the two-hop relay system. For fair comparison, we include the direct link at the destination when implementing the MMSE receiver. As expected, the linear source and relay precoded method outperforms all other systems.

HSR

#

Source: N antennas

MMSE

#

Relay: R antennas

#

Destination: M antennas

HSD

HRD

: First phase : Second phase

F

S

s

y

R

F

R

Nsubstreams

Figure 2.1: Linear source and relay precoded AF MIMO relay system with MMSE receiver.

0 5 10 15 20 10⁻²

10⁻¹ 10⁰

SNR (dB)

MSE

Un−precoded

Linear source and relay precoded

Figure 2.2: MSE performance comparison for un-precoded and linear source and relay precoded AF SISO-OFDM cooperative systems.

0 5 10 15 20 10⁻³

10⁻² 10⁻¹

SNR (dB)

BER

Un−precoded

Linear source and relay precoded

Figure 2.3: BER performance comparison for un-precoded and linear source and relay precoded AF SISO-OFDM cooperative systems.

0 5 10 15 20 25 10⁻¹

10⁰

SNR (dB)

MSE

Un−precoded (MMSE)

Linear relay precoded (MMSE)

Linear source and relay precoded (MMSE)

Figure 2.4: MSE performance comparison for existing un-precoded/precoded and linear source and relay precoded AF two-hop MIMO relay systems.

0 5 10 15 20 25 10⁻³

10⁻² 10⁻¹ 10⁰

SNR (dB)

BER

Un−precoded (ZF) Un−precoded (MMSE)

Linear relay precoded (MMSE)

Linear source and relay precoded (MMSE)

Figure 2.5: BER performance comparison for existing un-precoded/precoded and linear source and relay precoded AF two-hop MIMO relay systems.

−2 −1 0 1 2 3 4 5 6 10⁻⁴

10⁻³ 10⁻² 10⁻¹

SNR (dB)

BER

Antenna selection

Linear source and relay precoded

Figure 2.6: BER performance comparison for antenna selection [45] and linear source and relay precoded AF two-hop MIMO relay systems (L = 1 and N = R = M = 4).

0 2 4 6 8 10 12 14 16 10⁻¹

SNR (dB)

MSE

Un−precoded (ZF) Un−precoded (MMSE)

Linear relay precoded (MMSE)

Linear source and relay precoded (MMSE)

Figure 2.7: MSE performance comparison for existing un-precoded/precoded and linear source and relay precoded AF MIMO relay systems.

0 2 4 6 8 10 12 14 16 10⁻⁴

10⁻³ 10⁻² 10⁻¹

SNR (dB)

BER

Un−precoded (ZF) Un−precoded (MMSE)

Linear relay precoded (MMSE)

Linear source and relay precoded (MMSE)

Figure 2.8: BER performance comparison for existing un-precoded/precoded and linear source and relay precoded AF MIMO relay systems.

Chapter 3

Joint MMSE Transceiver Design with

Tomlinson-Harashima Source and Linear Relay Precoders

In this chapter, we address the problem of the MMSE transceiver design with a nonlinear THP.

In Section 3.1, we first formulate the precoded system model in which a THP cascaded with a linear precoder are used at the source, a linear precoder at the relay, and the MMSE receiver at the destination. As that in the previous section, we found that the MSE is a complicated function of the source and the relay precoders, and the corresponding optimization is difficult to conduct. In Section 3.2, we then propose a new method to overcome the problem. The main idea is to use the primal decomposition such that the two-precoder design problem can be translated into a single-relay precoder problem. With the proposed method, the optimization problem can be further expressed as a convex optimization problem, and the closed-form solution can be obtained by KKT conditions. Finally, we evaluate the performance of the proposed method in Section 3.3.

§ 3.1 System Model and Problem Formulation

§ 3.1.1 MMSE Receiver with Tomlinson-Harashima Source and Linear Relay Precoders

We consider the precoded three-node AF MIMO relay precoding system as shown in Fig. 3.1, where we include two precoders - a THP source precoder and a linear relay precoder F_R, and a linear MMSE receiver, G, is applied at the destination. Here, we also consider the general two-phase transmission protocol [41]- [45]. In the first two-phase, the source signal s ∈ C^{N ×1}is fed into the nonlinear THP in which a successive cancellation operation characterized by a backward squared matrix B and a modulo operation MOD_m(·). The source signals s = [s₁, · · · , s_N]^T are modulated by m-QAM where the real and image parts of s_kas the set {±1, · · · , ±(√

m − 1)}.

The feedback matrix B has a lower triangular structure and the diagonal elements are all zeros.

The modulo operation acts over the real and image parts of the inputs, respectively, is expressed as follows:

MOD_m(x) = x − 2√

mbx +√ m 2√

m c. (3.1)

It is clear that the transmitted signal x is bounded between −√

m and √

m. With B and the operation in (3.1), the elements of x can be recursively expressed as [16]

xk= sk− Xk−1

l=1

B(k, l)xl+ ek (3.2)

where x_k is the kth elements of vector x and B(k, l) is the (k, l) element of matrix B; e = [e₁, . . . , e_N]^T denotes the errors of the modulo operation (the difference of the input and the output). From (3.2), we can reformulate the transmitted signal x after THP with the following matrix form

x = C⁻¹v (3.3)

where C = B + I_N is a lower triangular with ones in its diagonal, and v = s + e. The THP precoded x is then passed through a unitary precoder matrix F_S and subsequently sent to the

relay and the destination simultaneously. The unitary precoder, as we will see, can greatly facilitate the joint precoders design and improve the BER performance.

In the second phase, the received signal at the relay is multiplied the relay precoder and then is transmitted to the destination. Therefore, the signal received at the destination in the two consecutive phases can be expressed as a vector form as

y_D :=

where H and n denote the equivalent channel matrix and the equivalent noise vector, respec-tively, as the same definition in (2.4). In (3.4), x ∈ C^{N ×1} is the THP precoded signal vector (3.3); y_D ∈ C^{2M ×1} is the received signal vector at the destination. Note that if v can be esti-mated at the destination, s can then be recovered by the modulo operation in (3.1). Thus, the optimum G ∈ C^{2M ×N} can be found by minimizing the MSE defined as

J = E©

kGy_D − vk²ª

. (3.5)

To solve the problem in (3.5), we assume that the precoded signal x_k’s are statistically indepen-dent and they have the zero-mean and the same variance. Let the variance of each element in s be denoted as σ_s². We then have E£

xx^H¤

= σ_s²I_N and E£ vv^H¤

= σ_s²CC^H. It is noted that the assumption is valid when the QAM size is large (m ≥ 16) [15], [16]. Then, the optimum solution of (3.5) can be obtained as [16]

G_opt = σ_s²CF^H_SH^H¡

σ_s²HF_SF^H_SH^H + R_n¢₋₁

, (3.6)

where G_opt is the optimum G; R_n = E[nn^H] is the covariance matrix of the equivalent noise vector n, as also defined in (2.6). Considering the noise components σ²_n,dand σ²_n,r in (3.6) and substituting (3.6) in (3.5), we can have the MSE matrix

E = C¡

and

is defined as the equivalent channel matrix after noise whitening. Note that the MSE is con-tributed by both the direct and relay links. By ignoring the direct link and adopting a single precoder at the relay, the problem is reduced those considered in [43] and [44]. Here, we incor-porate the THP as the source precoder and take the direct link into consideration. A significant performance enhancement can then be expected.

§ 3.1.2 Problem Formulation

From the MMSE criterion in (3.5)-(3.9), we now can formulate our joint design problem as:

C,FminS,FR

where the inequalities in (3.10) indicate the transmitted power constraints at source and relay (the maximal available power is P_S,T and P_R,T, respectively). Here, we force F_S = αU_S

in which α is a scalar and U_S is an unitary matrix. Taking a close look at (3.10), we can observe that the cost function and the power constraints are nonlinear functions of F_Sand F_R. Moreover, (3.10) is not a convex optimization problem. As a result, it is difficult to solve the problem, directly. In the next subsection, we propose a new approach to seek for a solution.

§ 3.2 Joint Source/Relay Precoders Design

We resort to the primal decomposition approach [51] translating (3.10) into a subproblem and a master problem. The subproblem is first optimized for the source precoder, and subsequently the master problem is optimized for the relay precoder. To proceed, we reformulate (3.10) as

C,FminS,FR

tr {E} = min

FR

C,FminS

tr {E}

s.t.

E = C³

σ_s⁻²I_N + α²F^H_SHe^HHFe _S´₋₁ C^H He^HH in (3.10),e

F_S = αU_S σ_s²tr©

F_SF^H_Sª

≤ P_S,T tr©

F_R¡

σ_n,r² I_R+ α²σ_s²H_SRH^H_SR¢ F^H_Rª

≤ P_R,T. (3.11)

In the subproblem, the relay precoder F_Ris assumed to be given. Then, the optimum C and F_S can first be derived as a function of F_R. Therefore, the joint precoders design is reduced to the master optimization problem in which the optimum relay precoder remains to be determined.

Since the unitary F_S = αU_S, the subproblem thus becomes optimizing α, U_Sand C, given

as

To find the solutions in (3.12), we first fixed U_S and C, finding optimum α, denoted α_opt. The solution can be easily obtain as

α_opt = s

P_S,T

Nσ²_s. (3.13)

This is because the cost function is a strictly decreasing function in α, we enlarge α with sat-isfying the source power constraint. In this manner, αopt can also maximize the SINR at the relay, reducing the noise enhancement at the relay node and thus minimizing the MSE value.

The subproblem thus becomes

The resultant relay power constraint tr n re-moved to the master problem since it is only the function of the relay precoder.

Fortunately, without considering the relay precoder, the problem in (3.14) has been ad-dressed in non-cooperative MIMO system [15], [16], and the optimum solutions can be ex-pressed as

is the Cholesky factorization of matrix that scales the elements on the diagonal of C to unity; V_He ∈ C^{N ×N} is the left singular matrices of eH; U⁰_S ∈ C^{N ×N} is an unitary matrix that needs to be further specified. Substituting (3.16) into (3.17), we have

LL^H = U^0H_S is diagonal matrix here. Applying GMD on eD, we can express eD as

De^1/2 = QRP^H, (3.19)

where Q, P are unitary matrix and R is upper triangular matrix with equal diagonal elements.

Letting U⁰_S = P and substituting (3.15), (3.16) in (3.8), we then have the resultant MSE as

J_min =

Now, the problem becomes the minimization of (3.20) in the master problem. From (3.15)-(3.19), we note that the original THP precoding does not include the unitary F_S [15]. Here the including unitary F_S has two main concerns: (i) The additional unitary precoder can facilitate the relay power constraint, as described in (3.14), in solving the optimization. (ii) By adequately designing U_S, we can make L(i, i) = L(j, j), ∀i 6= j in (3.20). In this manner, the minimum MSE can be expressed as (3.20) and, most importantly, as we will see, optimizations with (3.20) are more tractable for optimization.

Now, our residual problem is to solve the master problem. To proceed, let us first see the following equivalence:

proof: The result can be easily obtained since

in the master problem, we then reformulate the optimizations as

To solve (3.23), we use the Hardamard inequality, described in the following Lemma.

Lemma 3.1 [50]: Let M ∈ C^{N ×N} be a positive definite matrix, then

det(M) ≤ YN i=1

M(i, i), (3.24)

where M(i, i) denotes the ith diagonal element of M. The equality in (3.24) holds when M is a diagonal matrix. If we let M = eH^HH, it turns out that when M is diagonalized, the coste function in (3.23) is maximized. Unfortunately, from (3.10) we can see that eH^HH is a summa-e tion of two separated matrices and one of them dose not depend on F_R, and the diagonalization cannot be directly conducted. The following lemma suggests a feasible way to overcome the problem.

and A = _P^N

2 and det A are ignored since they are not functions of F_R. Equation (3.26) provides a feasible way to diagonalize the cost function. The optimization problem in (3.23) can now be reformulated as

maxFR

There exists certain structure for the relay precoder such that the diagonalization can be achieved.

Consider following SVD: respectively; We found that if the optimal F_Rhave the following structure, a full diagonalization of M⁰can be achieved:

F_R,opt = V_rdΣ_rU^0H_sr, (3.30)

where Σ_r is a diagonal matrix with ith diagonal element σ_r,i, i = 1, · · · , κ, yet to be deter-mined. Here, κ = min{N, R}. Let σ_rd,iand σ_sr,i⁰ be the ith diagonal element of Σ_rdand Σ⁰_sr,

respectively. Substituting (3.28), (3.29) and (3.30) into (3.27) and taking the ln operation to the cost function, we can then rewrite (3.27) as:

pr,imax,1≤i≤κ diagonal element of D⁰_sr. The cost function now is simplified to a function of scalar parameters.

Since the cost function and the inequalities are all concave for p_r,i ≥ 0 [51], (3.31) is a standard concave optimization problem. As a result, the optimal solutions p_r,i, i = 1, . . . , κ, can be solved by means of KKT conditions given by

p_r,i =

where µ is chosen to satisfy the power constraint in (3.31). We have also proposed a water-filling algorithm to solve (3.32). The detailed derivations of (3.32) and the water-filling algorithm are given in Appendix A.4 and A.5. Substituting (3.32) into (3.30), we can finally obtain the optimum relay precoder. With the relay precoder, eH in (3.9) can be obtained. Subsequently, the unitary source prefilter can be derived by substituting (3.19) into (3.16) and C can be obtained by (3.15). The computations of the THP source and linear relay precoders mainly involve SVD, GMD, and matrix inversion operations. The overall computational complexity, measured in terms of FLOPs, is summarized in Table 3.1.

Table 3.1: Computational Complexity of THP source and linear relay precoders (MMSE re-ceiver).

Step Operation FLOPs

1 H⁰_SR (3.27) O(N³+ RN²)

2 SVD HRD = UrdΣrdV^H_rd (3.28) O(MR²+ R³)

3 SVD H⁰SR = U⁰srΣ⁰srV^0H_sr (3.29) O(RN²+ N³)

4 Σr (3.32) O(κIr)

5 F_R (3.30) O(R³)

6 SVD eH O(MN²+ N³)

7 GMD eD^1/2 = QRP^H (3.19) O(N³)

8 L (3.18) O(N³)

9 C_opt (3.15) O(N³)

10 F_S,opt (3.16) O(N³)

I_ris denoted as the iteration number of the water-filling process in (3.32).

§ 3.3 Simulations

We consider an AF MIMO relay system with N=R=M=4. The elements of each channel matrix are assumed to be i.i.d. complex Gaussian random variables with zero-mean and unity variance.

Here, we let SNRsr=15 dB, SNRrd=15 dB and vary SNRsd=15. Also, we use 16-QAM for each transmitted symbols. Fig. 3.2 and Fig. 3.3 show the MSE and BER performances comparison, respectively, for (a) an un-precoded system with the MMSE receiver, (b) the optimum relay pre-coded system with MMSE receiver [43], (c) the linear source and relay prepre-coded with MMSE receiver in Chapter 2, and (d) the THP source and linear relay precoded with MMSE receiver in Chapter 3. Note that optimum relay precoder in [43] only considers the relay link. For better performance, we further include the direct link when implementing the MMSE receiver. As we can see, the proposed THP precoded system significantly outperforms other methods. Although two precoders are used in Chapter 2, the performance is limited. This is because both precoders are linear.

Table 3.2: Proposed water-filling algorithm solving (3.32)

µ_M = µ_M,0, µ_L = µ_L,0, δ_µ while δ_µ > ²

µ = ^µM₂^+µL ifP_κ

i=1

·r ai

³ µ + _a^bi_i

´

− ci

¸₊

di ≤ PR,T

µ_L= µ else

µ_M = µ end

µ⁰ = ^µM+µ₂ ^L, δ_µ = |µ⁰− µ|

end

HSR

#

Source: N antennas

# G

Relay: R antennas

Destination:

Mantennas

HSD

HRD

: First time slot : Second time slot

s

y

R

F

R

B x

F

S

THP source precoder

ˆx

MODm Decision

ˆs

#

MOD_m

Figure 3.1: THP source and linear relay precoded AF MIMO relay system with MMSE receiver.

0 5 10 15 20 10⁻²

10⁻¹

SNR (dB)

MSE

Un−precoded (MMSE)

Linear relay precoded (MMSE)

Linear source and linear relay precoded (MMSE) THP source and linear relay precoded (MMSE)

Figure 3.2: MSE performance comparison for existing precoded systems and THP source and linear relay precoded system with MMSE receiver.

0 5 10 15 20 10⁻⁶

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

SNR (dB)

BER

Un−precoded (MMSE)

Optimal relay precoded (MMSE)

Linear source and linear relay precoded (MMSE) THP source and linear relay precoded (MMSE)

Figure 3.3: BER performance comparison for existing precoded systems and THP source and linear relay precoded system with MMSE receiver.

Chapter 4

Joint QR-SIC Transceiver Design with Linear Source and Relay Precoders

We have addressed the precoded AF MIMO relay system with the linear MMSE receiver in the previous two chapters. In this chapter, we study the precoded system with a nonlinear re-ceiver. In general, nonlinear receivers require higher computational complexity. One exception is the QR-SIC receiver. It is known that the QR-SIC receiver has good performance while its computational complexity is low. Therefore, we consider the precoded system with the linear precoders at the source and the relay, and the QR-SIC receiver at the destination. In Section 4.1, we give the system model accommodating the QR-SIC receiver. In Section 4.2, we propose a GMD related method to derive the source and relay precoders. Finally, we report simulation results in Section 4.3 to evaluate the performance of the proposed method.

§ 4.1 System Model and Problem Formulation

§ 4.1.1 QR-SIC Receiver with Linear Source and Relay Precoders

Recall that the received signals with linear source and relay precoders are expressed as (2.4)

y_D = HF_Ss + n, (4.1)

Here, particularly, we assume that L = N ≤ M. This assumption can guarantee the existence of the solution of the proposed method (see Lemma 3.1). For the case of L < N , we can apply the antenna selection technique, to be discussed in Section 4.2.2. As previous mentioned, the equivalent channel matrix H does not include the source precoder.

By the same statistical assumptions in (2.6), we can also find that the equivalent noise vector is not white. To facilitate later analysis of the QR-SIC receiver, we first apply a whitening operation to the equivalent receive vector. Let W be a whitening matrix. Multiplying (4.1) with W, we can have I_2M. From (2.6) and (4.3), we can then obtain the whitening matrix as

W =

The equivalent channel matrix after the whitening process can then be written as

H = WH =˜

From (4.3), we can see that an AF MIMO relay system can be regarded as a MIMO system with the channel matrix shown in (4.5). However, note that the channel in (4.5) is a function of the relay precoder, and this is quite different from the scenario considered in conventional MIMO systems. Since F_R is unknown, existing design methods in MIMO systems cannot directly be applied.

It is well-known that nonlinear MIMO receivers perform better than linear receivers though their complexity may be higher. In this chapter, we consider a computationally efficient non-linear receiver, the QR-SIC receiver. In the receiver, the equivalent channel of the precoded system is first factorized by the QR method, i.e. ˜HF_S = QR, where Q is a 2M × 2M orthog-onal matrix, and R is a 2M × N upper triangular matrix. Equation (4.3) can then be rewritten as

= I2M. Note that the equivalent channel for QR factorization here includes the source precoder. Thus, signal detection of a QR-SIC receiver can then be conducted as:

where Dec[·] denotes the decision operation, ˆy_i the ith element of ˆy_D, and ˆs_j the estimation of the jth transmitted symbol.

§ 4.1.2 Problem Formulation

In the transceiver design, the most desirable criterion we want to use is the minimum BER.

However, for the QR-SIC receiver, the precoders which can minimize the BER is very difficult to design. Fortunately, for a MIMO system with CSI available at the transmitter, [11] and [12]

propose a well-known precoder design method, called GMD. In this approach, the precoded MIMO channel, is first QR factorized. Then, the precoding matrix is designed such that the diagonal elements of R in (4.6) are made equal and maximized. It has been proven that GMD can minimize the BLER, and maximize the lower bound of channel’s free distance [12]. As known, the free distance is the metric used in the MLD. This implies the GMD method can also improve the performance of the MLD.

So, in this section, we adopt the GMD design criterion to solve the precoding problem in AF MIMO relay systems. By this manner, we have the following advantages: (i) The BLER

So, in this section, we adopt the GMD design criterion to solve the precoding problem in AF MIMO relay systems. By this manner, we have the following advantages: (i) The BLER

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