Transmit Beamforming for MIMO Optical Wireless Communication Systems

Abstract

This paper focuses on transmit beamforming for multiple-input multiple-output optical wireless communication (OWC) systems with intensity modulation and direct detection (IM/DD). OWC with IM/DD requires the transmitted signals to be nonnegative, for which existing beamforming schemes developed for radio frequency systems cannot be applied directly. We propose effective schemes for OWC over frequency flat and frequency selective channels. For frequency flat fading, the property of the beamforming vector is derived. For frequency selective fading, bit-error rate performances of the proposed scheme with zero-forcing and minimum mean-square error frequency domain equalization receivers are derived, and a suboptimal beamforming vector for frequency selective fading channels is proposed. Compared with asymmetrically clipped optical orthogonal frequency division multiplexing based frequency domain beamforming, the proposed scheme needs much less feedback information and has a better error performance.

Appendices

Appendix 1

Setting of the beamforming vector can be viewed as the following optimization problem. The object function

$$\begin{aligned} \mathop {\max }\limits _{\varvec{b}_f } \{ ||\varvec{H}\varvec{b}_f ||^2_2 \} = \max \left\{ \sum \limits _{i = 1}^{n_r } {\left( \sum \limits _{j = 1}^{n_t } {\eta _{i,j} b_{f,j} }\right) ^2 } \right\} \end{aligned}$$

(39)

subject to

$$\begin{aligned} {\left\{ \begin{array}{ll} \sum \limits _{j = 1}^{n_t } {b_{f,j} } = 1 \\ b_{f,j} \ge 0\quad j = 1,\cdots ,n_t. \end{array}\right. } \end{aligned}$$

(40)

The object function $g(\varvec{b}_f ) = \sum \limits _{i = 1}^{n_r } {\left( \sum \limits _{j = 1}^{n_t } {\eta _{i,j} b_{f,j}} \right) ^2 }$, when the constraints are removed, is a multivariate quadratic function, and the gradient of this function is

$$\begin{aligned} {{\varvec{r }}}= \left[ \frac{{\partial g(\varvec{b}_f )}}{\partial {b_{f,1} }},\;\frac{{\partial g(\varvec{b}_f )}}{\partial {b_{f,2} }}, \cdots ,\frac{{\partial g(\varvec{b}_f )}}{\partial {b_{f,n_t } }}\right] ^T, \end{aligned}$$

(41)

where the $k$-th entry of ${{\varvec{r}}}$ is $r_k= 2\sum \limits _{i = 1}^{n_r } {\eta _{i,k} \sum \limits _{j = 1}^{n_t }\eta _{i,j}b_{f,j} }$. In the constraint region, all components of ${{\varvec{r}}}$ are nonnegative. The Hessian matrix of the object function is

$$\begin{aligned} {{\varvec{D}}} = \left[ \begin{array}{l} \frac{{\partial ^2 g(\varvec{b}_f )}}{{\partial (b_{f,1} )^2 }}\quad \quad \frac{{\partial ^2 g(\varvec{b}_f )}}{{\partial b_{f,1} \partial b_{f,2} }}\quad \cdots \quad \frac{{\partial ^2 g(\varvec{b}_f )}}{{\partial b_{f,1} \partial b_{f,n_t } }} \\ \quad \vdots \\ \frac{{\partial ^2 g(\varvec{b}_f )}}{{\partial b_{f,n_t } \partial b_{f,1} }}\quad \quad \cdots \quad \quad \quad \; \cdots \quad \;\frac{{\partial ^2 g(\varvec{b}_f )}}{{\partial (b_{f,n_t } )^2 }} \\ \end{array} \right] \end{aligned}$$

(42)

where the $(m,k)$-th entry of $\varvec{D}$ is $(\varvec{D})_{m,k} = 2\sum \limits _{i = 1}^{n_r } {\eta _{i,m} \eta _{i,k} }$. Define a matrix ${{\varvec{A}}}$ whose $(i,j)$-th entry is ${{\varvec{A}}}_{i,j}=\eta _{i,j}$. Then $\varvec{D}= 2\varvec{A}^T\varvec{A}$. For any nonzero vector ${{\varvec{z}}}$, it can be shown

$$\begin{aligned}&{{\varvec{z}}}^T \varvec{D}{{\varvec{z}}} \nonumber \\&= 2(\varvec{A}{{\varvec{z}}})^T \varvec{A}{{\varvec{z}}} > 0. \end{aligned}$$

(43)

Therefore, $\varvec{D}$ is a positive definite matrix, which means that the object function is convex region and there exists a minimum value for the object function. The maximum value of the optimization problem exists in the boundary of the constraints. The constraints are linear, and the boundary points are

$$\begin{aligned} {{\varvec{a}}}_k = [0,0,\cdots ,\underbrace{\;1\;}_{k\mathrm{{ - th}}},\cdots ,0]^T\quad k = 1,\cdots ,n_t. \end{aligned}$$

(44)

Thus, the optimal beamforming vector takes the form as

$$\begin{aligned} \varvec{b}_{f,\mathrm{opt}} \in \{ {{\varvec{a}}}_k = [0,0,\cdots ,\underbrace{\;1\;}_{k\mathrm{{ - th}}},\cdots ,0]^T\quad k = 1,\cdots ,n_t \}. \end{aligned}$$

(45)

Appendix 2

In frequency selective fading channels, the suboptimal algorithm is

$$\begin{aligned} \mathop {\max }\limits _{b_{f,j},j=1,\cdots ,n_t } \left\{ { \sum \limits _{n}{\left( \sum \limits _{i = 1}^{n_r }\sum \limits _{j = 1}^{n_t } {b_{f,j} \tilde{h}_{i,j} (n)}\right) ^2 } }\right\} \end{aligned}$$

(46)

subject to

$$\begin{aligned} {\left\{ \begin{array}{ll} \sum \limits _{j = 1}^{n_t } {b_{f,j} } = 1 \\ b_{f,j} \ge 0\quad j = 1,\cdots ,n_t. \end{array}\right. } \end{aligned}$$

(47)

The object function can be expressed as

$$\begin{aligned} g(\varvec{b}_f )={ \sum \limits _{n}{\left( \sum \limits _{i = 1}^{n_r }\sum \limits _{j = 1}^{n_t } {b_{f,j} \tilde{h}_{i,j} (n)}\right) ^2 } } \end{aligned}$$

(48)

which is a quadratic function of $b_{f,j}$. The $k$-th component of the gradient vector is $r_k={\sum \limits _{n}2{\left( \sum \limits _{i = 1}^{n_r }\sum \limits _{j = 1}^{n_t } {b_{f,j} \tilde{h}_{i,j} (n)}\right) {\left( \sum \limits _{i = 1}^{n_r }\tilde{h}_{i,k} (n)\right) }}}$, which is nonnegative in the constraint region. The $(m,k)$-th entry of the Hessian matrix $\varvec{D}$ is $(\varvec{D})_{m,k} = \sum \limits _{ n } {2\sum \limits _{i = 1}^{n_r } {\tilde{h}_{i,m} (n)}\sum \limits _{j = 1}^{n_r }{\tilde{h}_{j,k} (n)} }$. Define

$$\begin{aligned} \varepsilon [x(n)] = \sum \limits _{ n } {x(n)} \end{aligned}$$

(49)

and a vector ${{\varvec{m}}}(n)$ with the $k$-th entry

$$\begin{aligned} ({{\varvec{m}}}(n))_{k} = \sum \limits _{j = 1}^{n_r }{\tilde{h}_{j,k} (n)}. \end{aligned}$$

(50)

The Hessian matrix is $\varvec{D}= 2\varepsilon [{{\varvec{m}}}(n){{\varvec{m}}}(n)^T]$. For any nonzero vector z, it can be shown that

$$\begin{aligned}&{{\varvec{z}}}^T \varvec{D}{{\varvec{z}}} \nonumber \\&= 2{{\varvec{z}}}^T \varepsilon [{{\varvec{m}}}(n) {{\varvec{m}}}(n)^T]{{\varvec{z}}} \nonumber \\&= 2\varepsilon [({{\varvec{m}}}(n)^T{{\varvec{z}}})^T ({{\varvec{m}}}(n)^T{{\varvec{z}}})]>0. \end{aligned}$$

(51)

Thus the Hessian matrix is a positive definite matrix, and the object function is convex and has a minimum value. The maximum value of the suboptimal problem exists in the boundary of the constraints. The suboptimal beamforming vector is

$$\begin{aligned} \varvec{b}_{f,\mathrm{sopt}} \in \{ {{\varvec{a}}}_k = [0,0,\cdots ,\underbrace{\;1\;}_{k\mathrm{{ - th}}},\cdots ,0]\quad k = 1,\cdots ,n_t \}. \end{aligned}$$

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