Linear Pre-coding in MIMO–CDMA Relay Networks

Abstract

Our aim in this paper is to define a novel beamforming approach in wireless multiple-input-multiple-output (MIMO) code-division-multiple-access (CDMA) relay networks, which involves communication between multiple source-destination pairs. It is assumed that full channel state information of source-relay and relay-destination channels are available. Our design consists of a two-step amplify-and-forward protocol. The first step includes signal transmission from the sources to the MIMO relay, and the second step contains transmitting a version of the linear precoded signal to the destinations. Beamforming is investigated only in MIMO-relay node to reduce end user’s hardware complexity and save the computational power. Accordingly, the optimization problem is defined to find the MIMO relay beamforming coefficients that minimize total relay transmit power by keeping the signal-to-interference-plus-noise ratio (SINR) of all destinations above a certain threshold value. It is shown that such optimization problem is a non-convex quadratically constrained quadratic programming, which is NP-hard in general. However, by relaxing this problem to a semi-definite programming problem, the problem can be solved efficiently. Simulation results verify the performance gain implied by MIMO–CDMA relay system compared to the non-CDMA coded system.

Appendices

Appendix A: Derivation of (15)

The pre-coded signals retransmitted by MIMO-relay is given by

$$\begin{aligned} {\varvec{\Omega }}( t )=\sum _{l=1}^d {{\varvec{\upxi }}_l } u_l (t) =\sum _{l=1}^d {\mathbf{W}_l {\varvec{\Gamma }}} u_l (t) \end{aligned}$$

(39)

where ${\varvec{\Omega }}(t)\in {\mathbb {C}}^{R\times 1}$, hence the total MIMO-relay transmit power can be calculated as

$$\begin{aligned} P_T&= E\left( {\left\langle {{\varvec{\Omega }}(t), {\varvec{\Omega }} (t)} \right\rangle } \right) =E\left( {\left. {\left( {\sum _{l=1}^d {\mathbf{W}_l u_l (T_0 -t){\varvec{\Gamma }}} } \right) ^{H}*\left( {\sum _{n=1}^d {\mathbf{W}_n u_n (t){\varvec{\Gamma }}}} \right) } \right| _{t=T_0 } } \right) \nonumber \\&= E\left( {\left. {{\varvec{\Gamma }}^{H}\sum _{l=1}^d {\mathbf{W}_l ^{H}u_l (T_0 -t)} *\sum _{n=1}^d {\mathbf{W}_n u_n (t){\varvec{\Gamma }}} } \right| _{t=T_0 } } \right) \nonumber \\&= E\left( {{\varvec{\Gamma }} ^{H}\sum _{l=1}^d {\sum _{n=1}^d {\mathbf{W}_l ^{H}} \left. {\underbrace{u_l (T_0 -t)*u_n (t)}} \right| _{t=T_0 } \mathbf{W}_n {\varvec{\Gamma }}} }\right) \end{aligned}$$

(40)

where the inner product for vectors $\mathbf{x}(t),\mathbf{y}(t)$ is defined as

$$\begin{aligned} \left\langle \mathbf{x}(t),\mathbf{y}(t) \right\rangle \triangleq {\mathop {\int \limits _{-\infty }}^{\infty }} \mathbf{x}^{H}(t) \mathbf{y}(t) dt=\mathbf{x}^{H} ( {{T_0} - t} )*\mathbf{{y}}(t) |_{t = {T_0}} \end{aligned}$$

(41)

By using the correlation of the code-words defined in (5), the total relay transmit power can be written as

$$\begin{aligned} P_T =E\left( {{\varvec{\Gamma }}^{H}\mathbf{Q} {\varvec{\Gamma }}} \right) ,\quad \mathbf{Q}\triangleq \sum _{l=1}^d {\sum _{j=1}^d {\mathbf{W}_l ^{H}\rho _{l,j} \mathbf{W}_j } } \end{aligned}$$

(42)

For simplicity, we represent $\mathrm{Q}$ in (42) into the following quadratic form

$$\begin{aligned} \mathbf{Q}=\left[ {{\begin{array}{l} {\mathbf{W}_\mathbf{1} } \\ {\mathbf{W}_\mathbf{2} } \\ \vdots \\ {\mathbf{W}_\mathbf{d} } \\ \end{array} }} \right] ^{H}\left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\rho _{1,1} \mathbf{I}_{R\times R} }&{} {\rho _{1,2} \mathbf{I}_{R\times R} }&{} \cdots &{} {\rho _{1,d} \mathbf{I}_{R\times R} } \\ {\rho _{2,1} \mathbf{I}_{R\times R} }&{} &{} &{} \\ \vdots &{} &{} \ddots &{} {\rho _{d-1,d} \mathbf{I}_{R\times R} } \\ {\rho _{d,1} \mathbf{I}_{R\times R} }&{} \cdots &{} {\rho _{d,d-1} \mathbf{I}_{R\times R} }&{} {\rho _{d,d} \mathbf{I}_{R\times R} } \\ \end{array} }} \right] \left[ {{\begin{array}{l} {\mathbf{W}_\mathbf{1} } \\ {\mathbf{W}_\mathbf{2} } \\ \vdots \\ {\mathbf{W}_\mathbf{d} } \\ \end{array} }} \right] \end{aligned}$$

The kernel of the above form can be expressed as a Kronecker products as follow

$$\begin{aligned} \mathbf{Q}=\mathbf{W}^{H}\left( {\Upsilon \otimes \mathbf{I}_{R\times R} } \right) _{Rd\times Rd} \mathbf{W} \end{aligned}$$

(43)

where $\Upsilon \triangleq \left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\rho _{1,1} }&{} {\rho _{1,2} }&{} {...}&{} {\rho _{1,d} } \\ {\rho _{2,1} }&{} \ddots &{} &{} \vdots \\ \vdots &{} &{} &{} \\ {\rho _{d,1} }&{} \cdots &{} &{} {\rho _{d,d} } \\ \end{array} }} \right]$.

Applying Lemma 1 to (42), the MIMO-Relay total transmit power (15) will be obtained.

Appendix B: Derivation of (18)

We can rewrite (11) as

$$\begin{aligned} {\varvec{\Gamma }} ={\varvec{\Gamma }}_{-\mathbf{k}} +{\varvec{\Gamma }}_\mathbf{k} +{\varvec{\Gamma }}_{{\varvec{\upvarepsilon }}_\mathbf{n} } ={\varvec{\Gamma }}_{-{\varvec{\upvarepsilon }}_\mathbf{n} } +{\varvec{\Gamma }}_{{\varvec{\upvarepsilon }}_\mathbf{n} } \end{aligned}$$

(44)

The first term of the right hand side of the above formula can be expressed as

$$\begin{aligned} {\varvec{\Gamma }}_{-{\varvec{\varepsilon }}_\mathbf{n} }&= \left[ {\left( {\sum _{l=1}^d {\mathbf{f}_l s_l \rho _{l,1} } } \right) ,...,\left( {\sum _{l=1}^d {\mathbf{f}_l s_l \rho _{l,d} } } \right) } \right] ^{T}=\sum _{l=1}^d {\left[ {\left( {\mathbf{f}_l s_l \rho _{l,1} } \right) ,...,\left( {\mathbf{f}_l s_l \rho _{l,d} } \right) } \right] } ^{T}\nonumber \\&= \sum _{l=1}^d \left( {\mathbf{r}_l^{T}\otimes \mathbf{f}_l } \right) s_l \end{aligned}$$

(45)

Then the correlation matrix of ${\varvec{\Gamma }}_{-\varepsilon _n}$ can be calculated as

$$\begin{aligned}&E\left( {{\varvec{\Gamma }}_{-\varepsilon _n } {\varvec{\Gamma }}_{-\varepsilon _n } ^{H}} \right) \nonumber \\&\qquad = E\left( {\sum _{n=1}^d {\sum _{l=1}^d {\left( {\mathbf{r}_l^{T}\otimes \mathbf{f}_l } \right) \left( {\mathbf{r}_n^{T}\otimes \mathbf{f}_n } \right) ^{H}s_l s_n^*} } } \right) {\mathop {=}\limits ^{\left( a \right) }} E\left( {\sum _{l=1}^d {\left( {\mathbf{r}_l^{T}\otimes \mathbf{f}_l } \right) \left( {\mathbf{r}_l^{T}\otimes \mathbf{f}_l } \right) ^{H}P_l } } \right) \nonumber \\&\qquad {\mathop {=}\limits ^{\left( b \right) }} E\left( {\sum _{l=1}^d {\left( {\mathbf{r}_l^{T}\otimes \mathbf{f}_l } \right) \left( {\mathbf{r}_l^{*}\otimes \mathbf{f}_l ^{H}} \right) P_l } } \right) {\mathop {=}\limits ^{(c)}} \sum _{l=1}^d {\left( {\mathbf{r}_l^{T}\mathbf{r}_l^{*}\otimes E\left( {\mathbf{f}_l \mathbf{f}_l ^{H}} \right) } \right) P_l } \end{aligned}$$

(46)

where $(a)$ holds because $s_n$ and $s_l$ are uncorrelated for $l\ne n, (b)$ holds because $\left( {\mathbf{x}\otimes \mathbf{y}} \right) ^{H}=\mathbf{x}^{H}\otimes \mathbf{y}^{H}$ and $(c)$ holds because $\left( {\mathbf{A}\otimes \mathbf{B}} \right) \left( {\mathbf{C}\otimes \mathbf{D}} \right) =\mathbf{AC}\otimes \mathbf{BD}$ for any conforming matrices $\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}$ reference [57].

By using the following definition

$$\begin{aligned} \mathbf{u}(t)\triangleq [u_1 (t),u_2 (t),...,u_d (t)]^{T} \end{aligned}$$

and using ${\varvec{\upvarepsilon }}_n = =\int \limits _{-\infty }^{\infty }u_n^{*}(t)\mathbf{v}(t)dt{\overset{\Delta }{=}}\left\langle {u_{n} \left( t \right) ,{\mathbf {v}}\left( t \right) } \right\rangle$ from (6) in the definition of (11), ${\varvec{\Gamma }}_{\varepsilon _n }$ can be written as

$$\begin{aligned} {\varvec{\Gamma }}_{\varepsilon _n} = {\mathop {\int }\limits _{-\infty }^{\infty }}\left[ u_{1}^{*}(t)\mathbf{v} (t)^{T}, u_{2}^{*}(t)\mathbf{v}(t)^{T},..., u_{d}^{*}(t)\mathbf{v}(t)^{T}\right] ^{T} dt= {\mathop {\int }\limits _{-\infty }^{\infty }} \mathbf{u}^{*}(t)\otimes \mathbf{v}(t)dt \end{aligned}$$

(47)

then the correlation of ${\varvec{\Gamma }}_{\varepsilon _n }$ is thus:

$$\begin{aligned}&E\left( {\,{{\varvec{\Gamma }} _{{\varepsilon _n}}}{\varvec{\Gamma }} _{{\varepsilon _n}}^H} \right) = E\left( {\mathop {\int }\limits _{-\infty }^{\infty }} (\mathbf{u}^{*}(t)\otimes \mathbf{v}(t))dt\left[ {\mathop {\int }\limits _{-\infty }^{\infty }} (\mathbf{u}^{*}(t)\otimes \mathbf{v}(t)) dt\right] ^{H} \right) \nonumber \\&\qquad {\mathop {=}\limits ^{(a)}} E\left( {\mathop {\int }\limits _{-\infty }^{\infty }} {\left( {{u^T}(t) \otimes {v^H}(t)} \right) dt} \right) {\mathop {=}\limits ^{( b)}} \left( {\mathop {\int }\limits _{-\infty }^{\infty }}\left( \mathbf{u}^{*}(t)\mathbf{u}^{T}(t)\right) \otimes E\left( \mathbf{v}(t)\mathbf{v}^{H}(t)\right) \right) dt\nonumber \\&\qquad {\mathop {=}\limits ^{(c)}} \Upsilon \, \otimes \sigma _v^2{I_{R \times R}} \in {^{Rd \times Rd}} \end{aligned}$$

(48)

where $(a)$ holds because $\left( {\mathbf{x}\otimes \mathbf{y}} \right) ^{H}=\mathbf{x}^{H}\otimes \mathbf{y}^{H},(b)$ holds because $\left( {\mathbf{A}\otimes \mathbf{B}} \right) \left( {\mathbf{C}\otimes \mathbf{D}} \right) =\mathbf{AC}\otimes \mathbf{BD}$ for any conforming matrices $\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}$ [58] and $(c)$ holds because

$$\begin{aligned} {\mathop {\int }\limits _{-\infty }^{\infty }} \left( \mathbf{u}^{*}(t)\mathbf{u}^{T}(t)\right) dt= \Upsilon , {\mathop {\int }\limits _{-\infty }^{\infty }} E \left( \mathbf{v}(t)\mathbf{v}^{H}(t)\right) dt=\sigma _{v}^{2}I_{R \times R} \end{aligned}$$

As a result, the correlation matrix of ${\varvec{\Gamma }}$ will be obtained as

$$\begin{aligned} E\left( {{\varvec{\Gamma }}{\varvec{\Gamma }}^{H}} \right) =E\left( {{\varvec{\Gamma }}_{\varepsilon _n } {\varvec{\Gamma }}_{\varepsilon _n } ^{H}} \right) +E\left( { {\varvec{\Gamma }}_{-\varepsilon _n } {\varvec{\Gamma }}_{-\varepsilon _n } ^{H}} \right) =\sum _{l=1}^d {\left( {\mathbf{r}_l^{T}\mathbf{r}_l^{*}\otimes E\left( {\mathbf{f}_l \mathbf{f}_l ^{H}} \right) } \right) P_l } +\Upsilon \otimes \sigma _v^2 \mathbf{I}_{R\times R} \end{aligned}$$

(49)

Appendix C: Derivation of (26)

From (11), ${\varvec{\Gamma }}_{-k}$ can be rewritten as

$$\begin{aligned} {\varvec{\Gamma }}_{-k}&= [{\varvec{\upgamma }}_{1,-k}, {\varvec{\upgamma }}_{2,-k} ,...,{\varvec{\upgamma }}_{d,-k} ]^{T} =\left[ {\left( {\mathop {\sum _{l=1}}\limits _{l\ne k}^{d}} {\mathbf{f}_l \rho _{l,1} } s_l \right) , \left( {\mathop {\sum _{l=1}}\limits _{l\ne k}^{d}} {\mathbf{f}_l s_l \rho _{l,2} }\right) ,.... , \left( {\mathop {\sum _{l=1}}\limits _{l\ne k}^{d}} {\mathbf{f}_l s_l \rho _{l,d} } \right) } \right] ^{T} \nonumber \\&= {\mathop {\sum _{l=1}}\limits _{l\ne k}^{d}} {\left[ {\left( {\mathbf{f}_l s_l \rho _{l,1} } \right) , \left( {\mathbf{f}_l s_l \rho _{l,2} } \right) ,.... , \left( {\mathbf{f}_l s_l \rho _{l,d} } \right) } \right] ^{T}} ={\mathop {\sum _{l=1}}\limits _{l\ne k}^{d}} {\left( {\mathbf{r}_l^{T}\otimes \mathbf{f}_l } \right) s_l } \end{aligned}$$

(50)

then the correlation of ${\varvec{\Gamma }}_{-k}$ i.e., ${\varvec{\upbeta }}_k \triangleq E\left( {{\varvec{\Gamma }}_{-k} {\varvec{\Gamma }}_{-k}^H } \right)$ can be easily constructed as below

$$\begin{aligned} {\varvec{\upbeta }}_k&= E\left( {{\varvec{\Gamma }}_{-k} {\varvec{\Gamma }}_{-k}^H } \right) =E\left( {\mathop {\sum _{n=1}}\limits _{n\ne k}^{d}} {\mathop {\sum _{l=1}}\limits _{l\ne k}^{d}} \left( {\mathbf{r}_l ^{T}\otimes \mathbf{f}_l } \right) \left( {\mathbf{r}_n^{T}\otimes \mathbf{f}_n } \right) ^{H}s_l s_n^*\right) \nonumber \\&= E\left( {\mathop {\sum _{l=1}}\limits _{l\ne k}^{d}} \left( {\mathbf{r}_l^{T}\otimes \mathbf{f}_l } \right) \left( {\mathbf{r}_l^{T}\otimes \mathbf{f}_l } \right) ^{H}P_l \right) =E\left( {\mathop {\sum _{l=1}}\limits _{l\ne k}^{d}} \left( {\mathbf{r}_l ^{T}\otimes \mathbf{f}_l } \right) \left( {\mathbf{r}_l^{*}\otimes \mathbf{f}_l ^{H}} \right) P_l \right) \nonumber \\&= {\mathop {\sum _{l=1}}\limits _{l\ne k}^{d}} \left( \mathbf{r}_l ^{T}\mathbf{r}_l^{*}\otimes E\left( {\mathbf{f}_l \mathbf{f}_l ^{H}} \right) \right) P_l \end{aligned}$$

(51)