Robust collaborative relay beamforming design for two-way relay systems with reciprocal CSI

Abstract

In this paper, we propose a robust collaborative relay beamforming scheme in the presence of imperfect channel state information (CSI) for a multiple single-antenna two-way relays network with reciprocal channels. Many existing beamforming schemes have been investigated using perfect CSI for relay-aided systems. However, in practical communication systems the channel estimation is always inaccurate at relay nodes, leading to the deviation between the true CSI and the estimated CSI. Aiming to tackle the performance degradation caused by the CSI mismatch, a constrained optimization problem corresponding to the robust relay beamforming vector is formulated, in which the channel estimation error is characterized by the stochastic error model. Herein the commonly used performance metric, expected weighted sum mean squared error, is adopted as design objective to evaluate the overall link reliability. It is found that due to the intractable expression of the cost function, we can not resort to the traditional methods for solving the involved optimization problem. Whereas, by exploring the lower bound of the cost function using Jensen’s inequality and the Taylor series expansion, the original optimization problem is recast to a convex optimization in the form of Rayleigh–Ritz ratio, which results in a closed-form robust relay beamforming solution. Numerical results verify the robustness and superiority of the proposed scheme against CSI errors in terms of system sum rate and average BER.

Appendix

Here we will give the derivation of $\mathop E\limits _{\Delta _{\mathbf{h}},\Delta _{\mathbf{g}}} {\left\{ {\mathrm{{SIN}}{\mathrm{{R}}_{{s_i}}}} \right\} _{appr}}$ in (15). Let us take $\mathop E\limits _{\Delta _{\mathbf{h}},\Delta _{\mathbf{g}}} {\left\{ {\mathrm{{SIN}}{\mathrm{{R}}_{{s_1}}}} \right\} _{appr}}$ for example and the rest can be done in the same manner. Now, we expand $\mathop E\limits _{\Delta _{\mathbf{h}},\Delta _{\mathbf{g}}} {\left\{ {\mathrm{{SIN}}{\mathrm{{R}}_{{s_1}}}} \right\} _{appr}}$ as

$$\begin{aligned} \mathop E\limits _{{\Delta _{\mathbf{h}}},{\Delta _{\mathbf{g}}}} {\left\{ {\mathrm{{SIN}}{\mathrm{{R}}_{{s_1}}}} \right\} _{appr}}&= \frac{{\mathop E\limits _{{\Delta _{\mathbf{h}}},{\Delta _{\mathbf{g}}}} \left( {{P_2}{{\left| {{{\mathbf{h}}^T}({\mathbf{w}} \odot {\mathbf{g}})} \right| }^2}} \right) }}{{\mathop E\limits _{{\Delta _{\mathbf{h}}},{\Delta _{\mathbf{g}}}} \left( {\sigma _1^2 + {{\left\| {{\mathbf{w}} \odot {\mathbf{h}} \odot \mathbf {u}} \right\| }^2}} \right) }}\end{aligned}$$

(25)

$$\begin{aligned}&= \frac{{\mathop {\mathrm{{ }}E}\limits _{{\Delta _{\mathbf{h}}},{\Delta _{\mathbf{g}}}} \left\{ {{P_2}{{\left| {\sum \limits _{i = 1}^N {{{\widehat{h}}_i}{w_i}{{\widehat{g}}_i}} + \sum \limits _{i = 1}^N {{\delta _{{h_i}}}{w_i}{{\widehat{g}}_i}} + \sum \limits _{i = 1}^N {{{\widehat{h}}_i}{w_i}{\delta _{{g_i}}}} + \sum \limits _{i = 1}^N {{\delta _{{h_i}}}{w_i}{\delta _{{g_i}}}} } \right| }^2}} \right\} }}{{\mathop {\mathrm{{ }}E}\limits _{{\Delta _{\mathbf{h}}},{\Delta _{\mathbf{g}}}} \left\{ {\sum \limits _{i = 1}^N {\left( {{{\left| {{{\widehat{h}}_i}} \right| }^2} + {{\left| {{\delta _{{h_i}}}} \right| }^2} + \widehat{h}_i^*{\delta _{{h_i}}} + {{\widehat{h}}_i}\delta _{{h_i}}^*} \right) {{\left| {{w_i}} \right| }^2}\sigma _{{r_i}}^2} } \right\} + \sigma _1^2}}\end{aligned}$$

(26)

$$\begin{aligned}&\mathop = \limits ^{(a)} {P_2}\frac{{\mathop {\mathrm{{ }}E}\limits _{{\Delta _{\mathbf{h}}},{\Delta _{\mathbf{g}}}} \left\{ {{{\left| {\sum \limits _{i = 1}^N {{{\widehat{h}}_i}{w_i}{{\widehat{g}}_i}} } \right| }^2}} \right\} \mathrm{{ + }}\mathop {\mathrm{{ }}E}\limits _{{\Delta _{\mathbf{h}}},{\Delta _{\mathbf{g}}}} \left\{ {{{\left| {\sum \limits _{i = 1}^N {{\delta _{{h_i}}}{w_i}{{\widehat{g}}_i}} } \right| }^2}} \right\} \mathrm{{ + }}\mathop {\mathrm{{ }}E}\limits _{{\Delta _{\mathbf{h}}},{\Delta _{\mathbf{g}}}} \left\{ {{{\left| {\sum \limits _{i = 1}^N {{{\widehat{h}}_i}{w_i}{\delta _{{g_i}}}} } \right| }^2}} \right\} \mathrm{{ + }}\mathop {\mathrm{{ }}E}\limits _{{\Delta _{\mathbf{h}}},{\Delta _{\mathbf{g}}}} \left\{ {{{\left| {\sum \limits _{i = 1}^N {{\delta _{{h_i}}}{w_i}{\delta _{{g_i}}}} } \right| }^2}} \right\} }}{{\sigma _1^2 + \sum \limits _{i = 1}^N {\left( {{{\left| {{{\widehat{h}}_i}} \right| }^2} + \mathop {\mathrm{{ }}E}\limits _{{\Delta _{\mathbf{h}}},{\Delta _{\mathbf{g}}}} \left\{ {{{\left| {{\delta _{{h_i}}}} \right| }^2}} \right\} } \right) {{\left| {{w_i}} \right| }^2}\sigma _{{r_i}}^2} }}\end{aligned}$$

(27)

$$\begin{aligned}&\mathop = \limits ^{(b)} {P_2}\frac{{{{\left| {\sum \limits _{i = 1}^N {{{\widehat{h}}_i}{w_i}{{\widehat{g}}_i}} } \right| }^2}\mathrm{{ + }}\sum \limits _{i = 1}^N {{{\left| {{w_i}} \right| }^2}{{\left| {{{\widehat{g}}_i}} \right| }^2}\sigma _{{h_i}}^2} \mathrm{{ + }}\sum \limits _{i = 1}^N {{{\left| {{w_i}} \right| }^2}{{\left| {{{\widehat{h}}_i}} \right| }^2}\sigma _{{g_i}}^2} \mathrm{{ + }}\sum \limits _{i = 1}^N {{{\left| {{w_i}} \right| }^2}\sigma _{{h_i}}^2\sigma _{{g_i}}^2} }}{{\sigma _1^2 + \sum \limits _{i = 1}^N {\left( {{{\left| {{{\widehat{h}}_i}} \right| }^2} + \sigma _{{h_i}}^2} \right) {{\left| {{w_i}} \right| }^2}\sigma _{{r_i}}^2} }} \end{aligned}$$

(28)

where in steps $(a)$ and $(b)$ the predefined properties that the elements of $\Delta _{\mathbf{h}}$ and $\Delta _{\mathbf{g}}$ are i.i.d. zero-mean complex Gaussian and they are independent to each other should be used.

Similarly, we can get $\mathop E\limits _{\Delta _{\mathbf{h}},\Delta _{\mathbf{g}}} {\left\{ {\mathrm{{SIN}}{\mathrm{{R}}_{{s_2}}}} \right\} _{appr}}$ as follows.

$$\begin{aligned} \mathop E\limits _{{\Delta _{\mathbf{h}}},{\Delta _{\mathbf{g}}}} {\left\{ {\mathrm{{SIN}}{\mathrm{{R}}_{{s_2}}}} \right\} _{appr}} = {P_1}\frac{{{{\left| {\sum \limits _{i = 1}^N {{{\widehat{h}}_i}{w_i}{{\widehat{g}}_i}} } \right| }^2}\mathrm{{ + }}\sum \limits _{i = 1}^N {{{\left| {{w_i}} \right| }^2}{{\left| {{{\widehat{h}}_i}} \right| }^2}\sigma _{{g_i}}^2} \mathrm{{ + }}\sum \limits _{i = 1}^N {{{\left| {{w_i}} \right| }^2}{{\left| {{{\widehat{g}}_i}} \right| }^2}\sigma _{{h_i}}^2} + \sum \limits _{i = 1}^N {{{\left| {{w_i}} \right| }^2}\sigma _{{h_i}}^2\sigma _{{g_i}}^2} }}{{\sigma _2^2 + \sum \limits _{i = 1}^N {\left( {{{\left| {{{\widehat{g}}_i}} \right| }^2} + \sigma _{{g_i}}^2} \right) {{\left| {{w_i}} \right| }^2}\sigma _{{r_i}}^2} }} \end{aligned}$$

(29)

End of derivation.