Channel Correlation Based Zero-Forcing Beamforming in Two-User MISO Broadcast Channel

Abstract

In this paper, a novel zero-forcing beamforming (ZFBF) is presented for the two-user multiple input single output broadcast channel. Based on the knowledge of channel correlations at the transmitter side, a nonlinear optimization problem is formulated to determine the beamformers so that the weighted ergodic sum-rate is maximized subject to avoiding multi-user interference and meeting a total power constraint. By factorizing each beamformer, the original complicated problem can be transformed into a simpler one. Numerical results show that the performance gap between the proposed ZFBF and the traditional instantaneous channel state information based ZFBF is within 1.5 bits per transmission.

Appendix

1.1 Proof of Lemma 1

By denoting $a_{1n}=\gamma \rho \delta _{1n}$, $Z_{1}\,\triangleq\,\sum ^{n_{1}}_{n=1} a_{1n} \Big |[\tilde{{\mathbf {h}}}^{'}_{i}]_n\Big |^2$ is defined. Because the characteristic function of the PDF of the sum of several independent random variables is the product of their respective characteristic functions, given the PDF of $z_{1n}\,\triangleq\,a_{1n} \Big |[\tilde{{\mathbf {h}}}^{'}_{i}]_n\Big |^2$

$$\begin{aligned} f(z_{1n})=\left\{ \begin{array}{ll} \frac{1}{a_{1n}}e^{-\frac{1}{a_{1n}}z_{1n}} &\quad z_{1n}\ge 0\\ {0} &\quad z_{1n}<0,\\ \end{array}\right.\end{aligned}$$

we have the following one-to-one correspondences:

$$\begin{aligned} f(z_{1n}) &\longrightarrow \varphi (t)=\frac{\frac{1}{a_{1n}}}{\frac{1}{a_{1n}}-jt},\\ f(z_{1}) &\longrightarrow \varphi (t)=\prod _{n=1}^{n_{1}} \frac{\frac{1}{a_{1n}}}{\frac{1}{a_{1n}}-jt} \overset{(a)}{=}\sum _{n=1}^{n_{1}}\frac{A_{1n}}{\frac{1}{a_{1n}}-jt}. \end{aligned}$$

where $f(z_{1})$ and $\varphi (t)$ respectively denote the PDF of $Z_{1}$ and their characteristic functions. Here

$$\begin{aligned} A_{1n}=\frac{\prod ^{n_{1}}_{\tilde{n}=1}\frac{1}{a_{1\tilde{n}}}}{\prod ^{n_{1}}_{\tilde{n}\ne n}\big (\frac{1}{a_{1\tilde{n}}}-\frac{1}{a_{1n}}\big )} \end{aligned}$$

in (a) is calculated according to the partial fraction expansion method. Then with the definition of characteristic function,

$$\begin{aligned} f(z_{1})=\left\{ \begin{array}{ll} \sum ^{n_{1}}_{n=1}A_{1n} e^{-\frac{1}{a_{1n}}z_{1}} &\quad z_{1}\ge 0\\ {0} &\quad z_{1}<0\\ \end{array}\right. \end{aligned}$$

(14)

is obtained. Thereafter,

$$\begin{aligned}&{\text {E}}\big \{\log \left( 1+\gamma \rho \alpha _1\right) \big \}=\int ^{+\infty }_{-\infty }\log _2(1+z_{1})f_{z_{1}}(z_{1}) dz_{1}\nonumber \\&\quad =\sum \limits _{n=1}^{n_{1}}\frac{A_{1n}}{\ln 2}\int ^{\infty }_{0}\ln (1+z_{1}) e^{-\frac{1}{a_{1n}}z_{1}}dz_{1}\nonumber \\&\quad =\sum \limits _{n=1}^{n_{1}}\frac{-A_{1n} a_{1n}}{\ln 2}e^{\frac{1}{a_{1n}}}Ei\left( -\frac{1}{a_{1n}}\right) \end{aligned}$$

(15)

where $\int ^{\infty }_{0}e^{-\mu x} \ln (1+x)dx=-\frac{1}{\mu }e^{\mu }Ei(-\mu )$ in [7] is utilized. Similarly, with the definitions $a_{2n}=\left( 1-\gamma \right) \rho \delta _{2n}$ and $A_{2n}=\frac{\prod ^{n_{2}}_{\tilde{n}=1}\frac{1}{a_{2\tilde{n}}}}{\prod ^{n_{2}}_{\tilde{n}\ne n}\big (\frac{1}{a_{2\tilde{n}}}-\frac{1}{a_{2n}}\big )}$, we derive

$$\begin{aligned} {\text {E}}\big \{\log \big (1+(1-\gamma )\rho \alpha _2\big )\big \} =\sum ^{n_{2}}_{n=1}\frac{-A_{2n} a_{2n}}{\ln 2}e^{\frac{1}{a_{2n}}}Ei\left( -\frac{1}{a_{2n}}\right) . \end{aligned}$$

(16)

Thus, combining (15) and (16) leads to (7).

1.2 Proof of Theorem 1

For $i=1,2$, the $P_c \times N_t$ matrix ${\mathbf {W}}_{i}$ ($n_{i}\,\triangleq\,{\text {rank}}({\mathbf {W}}_{i})$) can be always factorized as (10) where ${\varvec{\Psi }}_{i}$ and ${\mathbf {B}}_{i}$ are respectively $P_c \times n_{i}$, $n_{i}\times N_t$ full rank matrices.

Given the factorization of (10), we can prove that

$$\begin{aligned} {\mathbf {W}}^H_{1}{\mathbf {W}}_{2}={\mathbf {0}}_{N_t\times N_t}\Leftrightarrow {\varvec{\Psi }}^H_{1}{\varvec{\Psi }}_{2}={\mathbf {0}}_{n_{1}\times n_{2}}. \end{aligned}$$

Since we have ${\text {rank}}({\varvec{\Psi }}_{i})={\text {rank}}({\mathbf {B}}_{i})={\text {rank}}({\mathbf {W}}_{i})=n_{i}$, $P_c\ge n_{1}+n_{2}$ is derived. To lower the implementation complexity, the optimal $P_c$ should be $n_{1}+n_{2}$.

Next, if we further restrict ${\varvec{\Psi }}_{i}$s satisfying ${\varvec{\Psi }}^H_{i}{\varvec{\Psi }}_{i}={\mathbf {I}}_{n_{i}}$, then ${\mathbf {W}}^H_{i}{\mathbf {W}}_{i}={\mathbf {B}}^H_{i}{\mathbf {B}}_{i}$. Thus, ${\text {tr}}({\mathbf {W}}^H_{i}{\mathbf {W}}_{i})={\text {tr}}({\mathbf {B}}^H_{i}{\mathbf {B}}_{i})=1$. Although ${\varvec{\Psi }}^H_{i}{\varvec{\Psi }}_{i}={\mathbf {I}}_{n_{i}}$ imposes another constraint on ${\varvec{\Psi }}_{i}$, the generality does not lose since this kind of factorization always exists. In this way, (8) is finally cast into (9). The proof of Theorem 1 is completed.