MMSE Filter Design for Multi-source and Multi-destination MIMO Amplify-and-Forward Relay Systems

Abstract

Multiple antenna source, relay, and destination filter sets are developed that minimize the sum mean-squared error (MSE). Motivated by the equivalence of the transmit-centric sum MSE and the receive-centric sum MSE of multiple-source and multiple-destination amplify-and-forward relaying systems, we formulate the sum MSE minimization problem using the transmit-centric MSEs. Next, using Karush–Kuhn–Tucker conditions, analytical filters are derived and an alternating algorithm that calculates the optimum solutions is presented. The simulation results and analysis verify that, compared with the conventional method that provides gradient-based numerical solutions, the proposed scheme suffers a marginal loss in performance while exhibiting significantly improved implementation efficiencies.

Appendix

1.1 Derivation of the Source-Node Transmit Filter (15)

By setting the derivative of Lagrangian function (14) to zero, ${\partial {\mathcal {L}}}/{\partial {\tilde{\mathbf{A}}}_k^{*}}=0$, we can obtain

$$\begin{aligned} {\tilde{\mathbf{A}}}_k = \Big {(} \sum _{i}^{K} {\mathbf{H}}_k^{H} {\mathbf{B}}^{H} {\mathbf{G}}_i^{H} {\mathbf{D}}_i^{H} {\mathbf{D}}_i {\mathbf{G}}_i {\mathbf{B}} {\mathbf{H}}_{k} +\psi _k {\mathbf{I}}_{N_s} + \xi _k {\mathbf{H}}_k^{H} {\mathbf{B}}^{H} {\mathbf{B}} {\mathbf{H}}_{k} \Big {)}^{-1} {\mathbf{H}}_k^{H} {\mathbf{B}}^{H} {\mathbf{G}}_k^{H} {\mathbf{D}}_k^{H}, \end{aligned}$$

(23)

where $\psi _k=\lambda _k \alpha _k^2$ and $\xi _k=\lambda _r\alpha _k^2$ are the modified Lagrange multipliers for the power constraint of $\textsf {S}_k$ and that of $\textsf {R}$, respectively. From ${\partial {\mathcal {L}}}/{\partial \alpha _k}=0$ and ${\partial {\mathcal {L}}}/{\partial \lambda _r}=0$, we can obtain two equations as follows:

$$\begin{aligned} \psi _k = \frac{1}{P_k}\Big {(}\sigma _{n_r}^2 {\text{Tr}}\left( {\mathbf{D}}_k {\mathbf{G}}_k {\mathbf{B}} {\mathbf{B}}^{H} {\mathbf{G}}_k^{H} {\mathbf{D}}_k^{H} \right) + \sigma _{n_k}^2 {\text{Tr}}\left( {\mathbf{D}}_k {\mathbf{D}}_k ^H \right) - \xi _k\alpha _k^2 {\text{Tr}}\left( {\mathbf{B}} {\mathbf{H}}_{k} {\tilde{\mathbf{A}}}_{k} {\tilde{\mathbf{A}}}_{k}^{H} {\mathbf{H}}_k^{H} {\mathbf{B}}^{H} \right) \Big {)} \end{aligned}$$

(24)

and

$$\begin{aligned} \alpha _k^2 {\text{Tr}}\Big {(} {\mathbf{B}} {\mathbf{H}}_{k} {\tilde{\mathbf{A}}}_{k} {\tilde{\mathbf{A}}}_{k}^{H} {\mathbf{H}}_k^{H} {\mathbf{B}}^{H}\Big {)} =&P_r - {\text{Tr}} \Big {(} {\mathbf{B}} \left( \sum _{i\ne k}^{K} \alpha _i^2{\mathbf{H}}_i {\tilde{\mathbf{A}}}_i {\tilde{\mathbf{A}}}_i^{H} {\mathbf{H}}_i^{H} +\sigma _{n_r}^{2} {\mathbf{I}}_{N_s} \right) {\mathbf{B}}^H \Big {)} . \end{aligned}$$

(25)

By substituting (25) into (24) to eliminate the parameters of $\textsf {S}_{k}$, the modified Lagrange multiplier for the $\textsf {S}_k$ (24) can be rewritten as

$$\begin{aligned} \psi _k =&\,\frac{1}{P_k} \Big {(} \sigma _{n_r}^2 {\text{Tr}} ( {\mathbf{D}}_k {\mathbf{G}}_k {\mathbf{B}} {\mathbf{B}}^{H} {\mathbf{G}}_k^{H} {\mathbf{D}}_k^{H}) + \sigma _{n_k}^2 {\text{Tr}}({\mathbf{D}}_k {\mathbf{D}}_k^H ) \nonumber \\&- \xi _k \big {(} P_r - {\text{Tr}} \big {(} {\mathbf{B}} (\sum _{i\ne k}^{K} \alpha _i^2 {\mathbf{H}}_i {\tilde{\mathbf{A}}}_i {\tilde{\mathbf{A}}}_i^{H} {\mathbf{H}}_i^{H} + \sigma _{n_r}^{2} {\mathbf{I}}_{N_s} ) {\mathbf{B}}^H \big {)} \Big {)}. \end{aligned}$$

(26)

By substituting (26) in (23), the transmit filter of $\textsf {S}_k$ (15) can be obtained. Lastly, by incorporating $\alpha _k^2 = P_k/{\text{Tr}}({\tilde{\mathbf{A}}}_k {\tilde{\mathbf{A}}}_k^H )$ and (25), the composite power constraint of (16) that satisfies both $\textsf {S}_k$ and $\textsf {R}$ power constraints can be derived.

1.2 Derivation of the Relay Filter (21)

Setting the derivatives of Lagrangian function (20) to zero, we can find

$$\begin{aligned} {\tilde{\mathbf{B}}}= \Big {(}\sum _{k=1}^{K} {\mathbf{G}}_{k}^{H}{\mathbf{D}}_{k}^{H}{\mathbf{D}}_{k}{\mathbf{G}}_{k} + \mu \gamma ^2{\mathbf{I}}\Big {)}^{-1} \Big {(}\sum _{k=1}^{K} {\mathbf{G}}_{k}^H{\mathbf{D}}_{k}^H{\mathbf{A}}_{k}^H{\mathbf{H}}_{k}^H\Big {)} \Big {(}\sum _{k=1}^{K} {\mathbf{H}}_{k}{\mathbf{A}}_{k}{\mathbf{A}}_{k}^H{\mathbf{H}}_{k}^H +\sigma _{n_r}^2{\mathbf{I}}\Big {)}^{-1} . \end{aligned}$$

(27)

In addition, from ${\partial {\mathcal {L}}}/{\partial {{\gamma }}}=0$ and ${\partial {\mathcal {L}}}/{\partial {{\mu }}}=0$, i.e.,

$$\begin{aligned} {\partial {\mathcal {L}}}/{\partial {{\gamma }}}= -2\gamma ^{-3} \sum _{k=1}^{K} \sigma _{n_k}^2 {\text{Tr}} ({\mathbf{D}}_{k} {\mathbf{D}}_{k}^{H}) + 2\mu \gamma {\text{Tr}}\big ( {\tilde{\mathbf{B}}}( \sum _{k=1}^H {\mathbf{H}}_k{\mathbf{A}}_k {\mathbf{A}}_k^H {\mathbf{H}}_k^H + \sigma _{n_r}^2 {\mathbf{I}}_{N_r} {\tilde{\mathbf{B}}}^{H} \big ) =0 \end{aligned}$$

(28)

and

$$\begin{aligned} {\partial {\mathcal {L}}}/{\partial {{\mu }}}=\gamma ^2 {\text{Tr}}\big ( {\tilde{\mathbf{B}}} ( \sum _{k=1}^H {\mathbf{H}}_k{\mathbf{A}}_k {\mathbf{A}}_k^H {\mathbf{H}}_k^H + \sigma _{n_r}^2 {\mathbf{I}}_{N_r}{\tilde{\mathbf{B}}}^{H} \big ) - P_r = 0 . \end{aligned}$$

(29)

we can obtain

$$\begin{aligned} \mu \gamma ^2 =\frac{\sum _{k=1}^{K} \sigma _{n_k}^2 {\text{Tr}}({\mathbf{D}}_k{\mathbf{D}}_k^H)}{P_r}. \end{aligned}$$

(30)

By substituting (30) into (27), the relay filter ${\tilde{\mathbf{B}}}$ in (21) can be derived. Note that the condition (22) for the scaling parameter $\gamma$ can be found by using (29).

1.3 Proof of Algorithm 1

If we denote ${\mathbf{SMSE}} ^{\text {MOD}}(\lbrace {\mathbf{A}}_k^{(l)}\rbrace , {\mathbf{B}}^{(l)} ,\lbrace {\mathbf{D}}_k^{(l)}\rbrace )$ as the modified sum MSE function with the lth alternation transmit filter $\lbrace {\mathbf{A}}_k^{(l)}\rbrace$, relay filter ${\mathbf{B}}^{(l)}$, and receive filter $\lbrace {\mathbf{D}}_k^{(l)}\rbrace$, then

$$\begin{aligned} {\mathbf{SMSE}} ^{\text {MOD}}(\lbrace {\mathbf{A}}_k^{(l+1)}\rbrace , {\mathbf{B}}^{(l+1)} ,\lbrace {\mathbf{D}}_k^{(l+1)}\rbrace )&\le {\mathbf{SMSE}} ^{\text {MOD}}(\lbrace {\mathbf{A}}_k^{(l+1)}\rbrace , {\mathbf{B}}^{(l+1)} ,\lbrace {\mathbf{D}}_k^{(l)}\rbrace ) \\&\le {\mathbf{SMSE}} ^{\text {MOD}}(\lbrace {\mathbf{A}}_k^{(l+1)}\rbrace , {\mathbf{B}}^{(l)} ,\lbrace {\mathbf{D}}_k^{(l)}\rbrace ) \\&\le {\mathbf{SMSE}} ^{\text {MOD}}(\lbrace {\mathbf{A}}_k^{(l)}\rbrace , {\mathbf{B}}^{(l)} ,\lbrace {\mathbf{D}}_k^{(l)}\rbrace ) \end{aligned}$$

and the modified sum MSE is lower bound. Therefore Algorithm 1 is guaranteed to converge as $l\rightarrow \infty$.