Robust MMSE Design With Asynchronous Interference Mitigation in Cooperative Base Station Systems

Abstract

This paper investigates minimum mean square error (MMSE) with asynchronous interference mitigation in cooperative base station systems. We consider the asynchronous transmission because of the different propagation times between the base station (BS) and mobile stations (BSs). Meanwhile, the channel quantization errors duo to channel quantization is taken into account in our analysis. The proposed scheme is robust to asynchronous interference and channel quantization errors in BSs cooperation systems. Simulations results show that proposed MMSE scheme achieve an improved performance compared with the conventional MMSE in BSs cooperative systems.

Appendix

We can easily get that ${\mathbb {E}}\left\{ {\left\| {s_k } \right\| _F^2 } \right\} =N_t$ and $\beta ^{-2}\mathrm{E}\left\{ {\left\| {n_k } \right\| _F^2 } \right\} =\beta ^{-2}N_t$. For the third term in (6), we substitute the channel quantization model (3) into (6):

$$\begin{aligned}&\beta ^{-2}{\mathbb {E}}\left\{ {\left\| {{\sum \limits _{b=1}^B} {\mathbf{H}_k^{( b)} } {\hat{\mathbf{T}}}_k^{( b )} s_k } \right\| _F^2 } \right\} =\beta ^{-2}{\mathbb {E}}\left\{ tr\left( {{\sum \limits _{b1=1}^B}}{{\sum \limits _{b2=1}^B} {\mathbf{H}_k^{( {b1} )} } } {\hat{\mathbf{T}}}_k^{( {b1} )} s_k s_{_k }^H {\hat{\mathbf{T}}}_k^{( {b2})H} \mathbf{H}_k^{( {b2} )H} \right) \right\} \nonumber \\&=\beta ^{-2}{\mathbb {E}}\left\{ {tr\left( \begin{array}{l} {\sum \limits _{b1=1}^B}{{\sum \limits _{b2=1}^B} {\left( {{\hat{\mathbf{H}}}_k^{( {b1} )} \mathbf{A}_k^{( {b1} )} +\mathrm{B}_k^{( {b1})} \mathbf{S}_k^{( {b1} )} } \right) } } {\hat{\mathbf{T}}}_k^{( {b1} )} s_k \\ *s_{k}^H {\hat{\mathbf{T}}}_k^{( {b2} )} \left( {{\hat{\mathbf{H}}}_k^{( {b2} )} \mathbf{A}_k^{( {b2} )} +\mathrm{B}_k^{( {b2} )} \mathbf{S}_k^{( {b2} )} } \right) ^{H} \\ \end{array} \right) } \right\} \nonumber \\&{\mathop =^{(a)}} \beta ^{-2}{\mathbb {E}}\left\{ \,tr\left( {{\sum \limits _{b1=1}^B}} {{\sum \limits _{b2=1}^B}} \left( {{\hat{\mathbf{H}}}_k^{( {b1} )} \mathbf{A}_k^{( {b1} )} {\hat{\mathbf{T}}}_k^{( {b1})} s_k s_{_k }^H {\hat{\mathbf{T}}}_k^{( {b2})H} \mathbf{A}_k^{( {b2} )H} {\hat{\mathbf{H}}}_k^{( {b2} )H} } \right. \right. \right. \nonumber \\&\left. \left. \left. \quad \quad +\mathrm{B}_k^{( {b1} )} \mathbf{S}_k^{( {b1} )} {\hat{\mathbf{T}}}_k^{( {b1})} s_k s_{_k }^H {\hat{\mathbf{T}}}_k^{( {b2} )H} \mathbf{S}_k^{( {b2} )H} \mathrm{B}_k^{( {b2} )H} \right) \right) \right\} \end{aligned}$$

(25)

(a) follows that $\mathbf{S}$ is independent of $\mathbf{A}$ and $\mathbf{B}$, and ${\mathbb {E}}[ \mathbf{S} ]=0$ There are two cases in (25), so it can be further calculated as:

$$\begin{aligned}&\beta ^{-2}{\mathbb {E}}\left\{ tr \left( \begin{array}{l} {\sum \limits _{b1=1}^B} {{\sum \limits _{b2=1}^B}}{\left( {{\hat{\mathbf{H}}}_k^{( {b1})} \mathbf{A}_k^{( {b1} )} +\mathrm{B}_k^{( {b1})} \mathbf{S}_k^{( {b1} )} } \right) } {\hat{\mathbf{T}}}_k^{( {b1} )} s_k \\ *s_{k }^H {\hat{\mathbf{T}}}_k^{( {b2} )} \left( {{\hat{\mathbf{H}}}_k^{( {b2} )} \mathbf{A}_k^{( {b2})} +\mathrm{B}_k^{( {b2})} \mathbf{S}_k^{( {b2})} } \right) ^{H} \end{array} \right) \right\} \nonumber \\&\quad =\beta ^{-2}{\mathbb {E}}\left\{ {\mathop {tr}\limits _{b1=b2}} \left( {\sum \limits _{b1=b2=1}^B} {\begin{array}{l} \left( {{ \hat{\mathbf{H}}}_k^{( b )} \mathbf{A}_k^{( b)} +\mathrm{B}_k^{( b )} \mathbf{S}_k^{( b )} } \right) {\hat{\mathbf{T}}}_k^{( b)} s_k \\ *s_{_k }^H {\hat{\mathbf{T}}}_k^{( b)} \left( {{ \hat{\mathbf{H}}}_k^{( b )} \mathbf{A}_k^{( b )} +\mathrm{B}_k^{( b )} \mathbf{S}_k^{( b )} } \right) ^{H}\\ \end{array}} \right) \right\} \nonumber \\&\qquad +\beta ^{-2}{\mathbb {E}}\left\{ {\mathop {tr}\limits _{b1\ne b2}} \left( {\begin{array}{l} {\sum \limits _{b1=1}^B} {{\sum \limits _{b2=1}^B}{\left( {{ \hat{\mathbf{H}}}_k^{( {b1} )} \mathbf{A}_k^{( {b1} )} +\mathrm{B}_k^{( {b1} )} \mathbf{S}_k^{( {b1} )} } \right) } } {\hat{\mathbf{T}}}_k^{( {b1} )} s_k \\ *s_{_k }^H {\hat{\mathbf{T}}}_k^{( {b2} )} \left( {{\hat{\mathbf{H}}}_k^{( {b2} )} \mathbf{A}_k^{( {b2} )} +\mathrm{B}_k^{( {b2} )} \mathbf{S}_k^{( {b2} )} } \right) ^{H} \\ \end{array}} \right) \right\} \nonumber \\&\quad =\underbrace{\beta ^{-2}{\mathbb {E}}\left\{ {\mathop {tr}\limits _{b1=b2}} \left( {{\sum \limits _{b=1}^B} {\left( {\begin{array}{l} {\hat{\mathbf{H}}}_k^{( b )} \mathbf{A}_k^{( b )} {\hat{\mathbf{T}}}_k^{( b)} s_k s_{_k }^H {\hat{\mathbf{T}}}_k^{( b )H} \mathbf{A}_k^{( b )H} {\hat{\mathbf{H}}}_k^{( b )H} \\ +\mathrm{B}_k^{( b )} \mathbf{S}_k^{( b)} {\hat{\mathbf{T}}}_k^{( b )} s_{k} s_{k}^H {\hat{\mathbf{T}}}_k^{( b)H} \mathbf{S}_k^{( b)H} \mathrm{B}_k^{( b )H} \\ \end{array}} \right) } } \right) \right\} }_{<7>} \nonumber \\&\qquad +\beta ^{-2}{\mathbb {E}}\left\{ {\mathop {tr}\limits _{b1\ne b2}} \left( {{\sum \limits _{b1=1}^B} {\sum \limits _{b2=1}^B} {\left( {{\hat{\mathbf{H}}}_k^{( {b1})} \mathbf{A}_k^{( {b1} )} {\hat{\mathbf{T}}}_k^{( {b1} )} {\hat{\mathbf{T}}}_k^{( {b2} )H} \mathbf{A}_k^{( {b2} )H} {\hat{\mathbf{H}}}_k^{( {b2} )H} } \right) } } \right) \right\} \end{aligned}$$

(26)

The first term in (26), using $tr(A+B)=tr(A)+tr(B),<7>$ can be rewritten as:

$$\begin{aligned}&<7> \nonumber \\&=\beta ^{-2}{\mathbb {E}}\left\{ \underbrace{tr}_{b1=b2}\left( {{\sum \limits _{b1=b2=1}^B} {\hat{\mathbf{H}}}_k^{( b )} \mathbf{A}_k^{( b)} {\hat{\mathbf{T}}}_k^{( b)} s_k s_{_k }^H {\hat{\mathbf{T}}}_k^{( b)H} \mathbf{A}_k^{( b )H} {\hat{\mathbf{H}}}_k^{( b )H} } \right) \right\} \nonumber \\&+\beta ^{-2}{\mathbb {E}}\left\{ \underbrace{tr}_{b1=b2}\left( {{\sum \limits _{b1=b2=1}^B} {\mathrm{B}_k^{( b )} \mathbf{S}_k^{( b )} {\hat{\mathbf{T}}}_k^{( b)} s_k s_{_k }^H {\hat{\mathbf{T}}}_k^{( b )H} \mathbf{S}_k^{( b )H} \mathrm{B}_k^{( b )H} } } \right) \right\} \nonumber \\&{\mathop =^{(b)}} \beta ^{-2}{\mathbb {E}}\left\{ tr\left( {{\sum \limits _{b1=b2=1}^B} {\mathbf{R}_k (\mathbf{I}_{N_r } -\mathbf{Z}_k^H \mathbf{Z}_k )^{1/2}\mathbf{R}_k^H {\hat{\mathbf{H}}}_k^{( b )} {\hat{\mathbf{T}}}_k^{( b )} {\hat{\mathbf{T}}}_k^{( b )H} {\hat{\mathbf{H}}}_k^{( b )H} } } \right) \right\} \nonumber \\&+\beta ^{-2}{\mathbb {E}}\left\{ \underbrace{tr}_{b1=b2}\left( {{\sum \limits _{b1=b2=1}^B} {\mathrm{B}_k^{( b )} \mathbf{S}_k^{( b )} {\hat{\mathbf{T}}}_k^{( b )} s_k s_{_k }^H {\hat{\mathbf{T}}}_k^{( b )H} \mathbf{S}_k^{( b )H} \mathrm{B}_k^{( b )H} } } \right) \right\} \end{aligned}$$

(27)

(b) follows that $\mathbf{A}_k =(\mathbf{I}_{N_r } -\mathbf{Z}_k^H \mathbf{Z}_k )^{1/2}\mathbf{R}_k$, it further can be expressed as:

$$\begin{aligned}&<7> \nonumber \\&=\beta ^{-2}(Nt-\frac{N_t D}{N_r })tr\left( {{\sum \limits _{b1=b2}^B}} {\hat{\mathbf{H}}}_k^{( b)} {\hat{\mathbf{T}}}_k^{( b )} {\hat{\mathbf{T}}}_k^{( b)H} {\hat{\mathbf{H}}}_k^{( b )H} \right) \nonumber \\&+\beta ^{-2}{\mathbb {E}}\left\{ \underbrace{tr}_{b1=b2}\left( {{\sum \limits _{b1=b2=1}^B} {\mathrm{B}_k^{( b )} \mathbf{S}_k^{( b )} {\hat{\mathbf{T}}}_k^{( b)} s_k s_{_k }^H {\hat{\mathbf{T}}}_k^{( b)H} \mathbf{S}_k^{( b )H} \mathrm{B}_k^{( b)H} } } \right) \right\} \nonumber \\&=\beta ^{-2}(Nt-\frac{N_t D}{N_r })tr\left( {\hat{\mathbf{H}}}_k {\hat{\mathbf{T}}}_k {{\hat{\mathbf{T}}}_k^H {\hat{\mathbf{H}}}_k^H } \right) +\beta ^{-2}\frac{N_t D}{N_t -N_r }tr\left( {\hat{\mathbf{T}}}_k^H {\hat{\mathbf{T}}}_k \right) \nonumber \\&-\beta ^{-2}\frac{N_t N_r D}{N_r (N_t -N_r )}tr\left( {\sum \limits _{b1=b2=b=1}^B} {\hat{\mathbf{T}}}_k^{( b )H} { \hat{\mathbf{H}}}_k^{(b)} {\hat{\mathbf{H}}}_k^{(b)H} {\hat{\mathbf{T}}}_k^{\left( b \right) } \right) \end{aligned}$$

(28)

$<7>$ follows ${\mathbb {E}}\left\{ {\mathbf{RR}^{H}} \right\} =N_t \mathbf{I}_{N_r }$ and ${\mathbb {E}}\left\{ {\mathbf{Z}_k \mathbf{Z}_{_k }^H } \right\} =\frac{D}{N_r }\mathbf{I}_{N_r }$.

The fourth term in (6) can be calculated by:

$$\begin{aligned}&\beta ^{-2}{\mathbb {E}}\left\{ {\left\| {\mathbf{J}_k } \right\| _F^2 } \right\} \nonumber \\&=\beta ^{-2}{\mathbb {E}}tr\left( \mathop {\sum \limits _{j=1}}\limits _{j \ne k}^{k} {\sum \limits _{b1=1}^B} {\sum \limits _{b2=1}^B} {\beta _{jk}^{(b1,b2)} \mathbf{H}_k^{( {b1} )} {\hat{\mathbf{T}}}_j^{( {b1} )} {\hat{\mathbf{T}}}_j^{( {b2} )H} \mathbf{H}_k^{( {b2} )H} } \right) \nonumber \\&=\beta ^{-2}{\mathbb {E}}tr\left( \mathop {\sum \limits _{j=1}}\limits _{j \ne k}^{k} {\sum \limits _{b1=1}^B}{\sum \limits _{b2=1}^B} {\begin{array}{l} \beta _{jk}^{(b1,b2)} \left( {{\hat{\mathbf{H}}}_k^{( {b1} )} \mathbf{A}_k^{( {b1} )} +\mathrm{B}_k^{( {b1} )} \mathbf{S}_k^{( {b1} )} } \right) {\hat{\mathbf{T}}}_j^{( {b1})}\\ *{\hat{\mathbf{T}}}_j^{( {b2} )H} \left( {{\hat{\mathbf{H}}}_k^{( {b2} )} \mathbf{A}_k^{( {b2} )} +\mathrm{B}_k^{( {b2} )} \mathbf{S}_k^{( {b2} )} } \right) ^{H} \\ \end{array}} \right) \nonumber \\&=\beta ^{-2}{\mathbb {E}}tr\left( \mathop {\sum \limits _{j=1}}\limits _{j \ne k}^{k} {\sum \limits _{b1=1}^B}{\sum \limits _{b2=1}^B} {\beta _{jk}^{(b1,b2)} \left( {\hat{\mathbf{H}}}_k^{( {b1} )} \mathbf{A}_k^{( {b1} )} {\hat{\mathbf{T}}}_j^{( {b1} )} {\hat{\mathbf{T}}}_j^{( {b2} )H} \mathbf{A}_k^{( {b2} )H} {\hat{\mathbf{H}}}_k^{( {b2} )H} \right) } \right) \end{aligned}$$

(29)

The fifth and sixth in (6) can be further calculated by:

$$\begin{aligned}&-\beta ^{-1}{\mathbb {E}}\left\{ tr({\sum \limits _{b=1}^B} {\mathbf{H}_k^{( b)} } {\hat{\mathbf{T}}}_k^{( b)} ) \right\} -\beta ^{-1}{\mathbb {E}}\left\{ tr\left( {\sum \limits _{b=1}^B} {\hat{\mathbf{T}}}_k^{(b)H} \mathbf{H}_k^{(b)H} \right) \right\} \nonumber \\&{\mathop =^{(c)}} -\beta ^{-1}\left( {\begin{array}{l} { \mathbb {E}}\left\{ tr\left( {\sum \limits _{b=1}^B} {\left( {{\hat{\mathbf{H}}}_k^{( b )} \mathbf{A}_k^{( b )} +\mathrm{B}_k^{( b )} \mathbf{S}_k^{( b )} } \right) } {\hat{\mathbf{T}}}_k^{( b )} \right) \right\} \\ +{\mathbb {E}}tr\left( {\sum \limits _{b=1}^B} {\hat{\mathbf{T}}}_k^{(b)H} \left( {{\hat{\mathbf{H}}}_k^{( b)} \mathbf{A}_k^{( b )} +\mathrm{B}_k^{( b )} \mathbf{S}_k^{( b )} } \right) ^{H} \right) \end{array}} \right) \nonumber \\&{\mathop =^{(d)}} -\beta ^{-1}\left( {\mathbb {E}} \left\{ tr\left( {\sum \limits _{b=1}^B} {\left( {{\hat{\mathbf{H}}}_k^{( b)} \mathbf{A}_k^{( b )} {\hat{\mathbf{T}}}_k^{( b )} } \right) } \right) \right\} +{\mathbb {E}}tr\left( {\sum \limits _{b=1}^B} {\hat{\mathbf{T}}}_k^{(b)H} \mathbf{A}_k^{( b )H} { \hat{\mathbf{H}}}_k^{( b )H} \right) \right) \nonumber \\&{\mathop =^{(e)}} -2\psi \beta ^{-1}\hbox {Re}\left[ tr\left( {\sum \limits _{b=1}^B} {\left( {{\hat{\mathbf{H}}}_k^{( b )} {\hat{\mathbf{T}}}_k^{( b )} } \right) } \right) \right] \end{aligned}$$

(30)

(c) follows that $\mathbf{H}_{k}={\hat{\mathbf{H}}}_k\mathbf{A}_k +\mathbf{S}_k \mathbf{B}_k$ and (d) comes from $\mathbf{S}$ is independent of $\mathbf{A}$ and $\mathbf{B}$, and ${\mathbb {E}}\left[ \mathbf{S} \right] =0$. (e) holds because ${\mathbb {E}}\left( \mathbf{A} \right) ={\mathbb {E}}\left( {\left( {\mathbf{I}_{N_r } -\mathbf{Z}_k^H \mathbf{Z}_k } \right) ^{1/2}\mathbf{R}_k } \right) =\psi \mathbf{I}_{N_r }$.