Design of Low Complex Linear Precoding Scheme for MU-MIMO Systems

Abstract

The block diagonalization (BD) is a linear precoding technique for multiuser interference elimination in multi-user multiple-input multiple-output (MU-MIMO) systems. Though various methods of block diagonalization have been identified, they are convoluted regarding the computations involved. In this paper we have modeled a new paradigm for the performance improvement and complexity reduction in BD based precoding algorithms. This is consummated using principal component analysis (PCA). The traditional highly complex singular value decomposition is replaced by QR decomposition in PCA (QR-PCA) for complexity reduction. The PCA along with QR decomposition and minimum mean squared error (MMSE) channel inversion technique parallelize the MU-MIMO channel into independent and proportionate single user MIMO channel. The simulation result shows that the proposed QR-PCA based precoding algorithm in comparison with the existing algorithm achieves comparatively better sum-rate, lower BER, and lower computational complexity. The PCA along with MMSE channel inversion avoids the decoder structure, which makes the receiver system simple.

Appendices

Appendix 1

Let us consider a complex matrix ${\mathbf{X}} \in {\mathbb{C}}^{M \times M}$. Normalizing each row of ${\mathbf{X}}$ to zero mean,

$${\bar{\mathbf{X}}}_{\text{k}} = {\mathbf{X}}_{\text{k}} - {\mathbf{X}}_{\text{k}}^{ '}$$

(37)

where ${\mathbf{X}}_{k}^{'}$ represents the mean of kth row of ${\mathbf{X}}$. Now define the unitary matrix ${\mathbf{P}}$ such that

$${\mathbf{Y}} = {\mathbf{PX}}$$

(38)

while the covariance of ${\mathbf{Y}}$ matrix is diagonalized. The matrix ${\mathbf{P}}$ denotes the principal components of ${\mathbf{X}}$ and each row of P denotes the Eigen vectors of covariance matrix ${\mathbf{XX}}^{\text{H}}$. If ${\mathbf{z}}_{\text{k}}$ denotes the set of Eigen vector corresponding to Eigen value $\lambda_{k}$ for the symmetric matrix XX ^H, then

$$\left( {{\mathbf{XX}}^{\text{H}} } \right){\mathbf{z}}_{\text{k}} = {\varvec{\uplambda}}_{\text{k}} {\mathbf{z}}_{\text{k}}$$

(39)

where $\varvec{z}_{k} = \left( {\varvec{z}_{1} ,\varvec{z}_{2} , \ldots ,\varvec{z}_{m} } \right) \in {\mathbb{C}}^{M \times 1}$ and $\lambda_{k} = \left( {\lambda_{1} ,\lambda_{2} , \ldots ,\lambda_{m} } \right) \in {\mathbb{R}}^{M \times 1}$

Let the covariance of $\bar{\varvec{X}}_{k}$ be ${\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}}$ and ${\bar{\mathbf{X}}}_{\text{k}} = {\mathbf{QR}}$ is the QR decomposition of ${\bar{\mathbf{X}}}_{\text{k}}$ matrix. Now the covariance matrix can be rewritten as,

$${\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}} = \left( {{\mathbf{QR}}} \right)\left( {{\mathbf{QR}}} \right)^{\text{H}}$$

(40)

where ${\mathbf{Q}} \in {\mathbb{C}}^{M \times M}$ is a unitary matrix; ${\mathbf{R}} \in {\mathbb{C}}^{M \times M}$ is an upper triangular matrix.

The matrix ${\mathbf{R}}^{\text{H}}$ can be expanded as ${\mathbf{R}}^{\text{H}} = {\mathbf{UDV}}^{\text{H}}$ using SVD decomposition. So (40) can be rewritten as,

$$\begin{aligned} & {\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}} = {\mathbf{Q}}\left( {{\mathbf{UDV}}^{\text{H}} } \right)^{\text{H}} \left( {{\mathbf{UDV}}^{\text{H}} } \right){\mathbf{Q}}^{\text{H}} \\ & {\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}} = {\mathbf{QVD}}\left( {{\mathbf{U}}^{\text{H}} {\mathbf{U}}} \right){\mathbf{DV}}^{\text{H}} {\mathbf{Q}}^{\text{H}} \\ & {\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}} = \left( {{\mathbf{QV}}} \right){\mathbf{D}}^{2} \left( {{\mathbf{QV}}} \right)^{\text{H}} \left( { \because {\mathbf{U}}^{\text{H}} {\mathbf{U}} = 1} \right) \\ & \left( {{\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}} } \right) \left( {{\mathbf{QV}}} \right) = \left( {{\mathbf{QV}}} \right){\mathbf{D}}^{2} \\ \end{aligned}$$

(41)

Comparing (39) and (41), the Eigen vector matrix which diagonalizes $\left( {{\bar{\mathbf{X}}}_{\text{k}} {\bar{\mathbf{X}}}_{\text{k}}^{\text{H}} } \right)$ is ${\mathbf{QV}}$ and their corresponding Eigen value is ${\mathbf{D}}^{2}$. Thus the principal component,

$${\mathbf{P}} = {\mathbf{QV}}$$

(42)

Therefore the PCA extracted matrix is given as, ${\mathbf{Y}} = {\mathbf{PX}}.$

Appendix 2

The received signal without PCA transform at the receiver is given by,

$${\mathbf{y}} = {\boldsymbol{\mathcal{H}}}{\acute{\mathbf{W}}}{\mathbf{b}} + {\mathbf{n}}$$

(43)

Let ${\acute{\mathbf{W}}} = \left({\frac{1}{{\sqrt {\updelta}}}} \right){\acute{\mathbf{W}}}^{1} {\acute{\mathbf{W}}}^{2}$ be the precoding matrix without PCA transform and $\delta = \frac{{\parallel {\acute{\mathbf{W}}} {\mathbf{b}}\parallel_{F}^{2}}}{{P_{T}}}$ is the scaling factor.

Here we now rearrange (43) in terms of QR-PCA transform. In PCA-MMSE-BD, the PCA is applied only in the manipulation $\varvec{W}^{2}$. Hence (23) can be rewritten using (22) as,

$${\mathbf{W}}^{2} = {\mathbf{E}} ({\mathbf{PC}} {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}} )^{\text{H}} \left( {\left( {{\mathbf{PC}} {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}} } \right)({\mathbf{PC}} {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}} )^{\text{H}} } \right)^{ - 1}$$

(44)

By using the matrix properties [33], ${\mathbf{X}}^{\text{H}} = {\mathbf{X}}^{ - 1}$ for any unitary matrix ${\mathbf{X}}$ and $\left( {{\mathbf{AB}}} \right)^{ - 1} = \left( {\mathbf{B}} \right)^{ - 1} \left( {\mathbf{A}} \right)^{ - 1}$ for any two matrices ${\mathbf{A}}$ and ${\mathbf{B}}$; (44) can be rewritten as

$$\begin{aligned} {\mathbf{W}}^{2} &= {\mathbf{E}}\left( {{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}}^{\text{H}} \left( {{\mathbf{PC}}^{\text{H}} {\mathbf{PC}}} \right)\left( {{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}} {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}}^{\text{H}} } \right)^{ - 1} {\mathbf{PC}}^{\text{H}} } \right) \hfill \\ \mathbf{W}^{2} &= \mathbf{E}\left( {{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}}^{\text{H}} \left( {{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}} {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}_{\text{ext}}^{\text{H}} } \right)^{ - 1} {\mathbf{PC}}^{\text{H}} } \right)\quad \left\{ {\because \mathbf{PCPC}^{\text{H}} = {\mathbf{I}}} \right\} \hfill \\ {\mathbf{W}}^{2} &= {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H} }}^{2} {\mathbf{PC}}^{\text{H}} \hfill \\ \end{aligned}$$

(45)

where ${\acute{\mathbf{W}}}^{2} = {\mathbf{E}}\left({{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H}}}_{\text{ext}}^{\text{H}} \left({{\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H}}}_{\text{ext}} {\mathbf{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{H}}}_{\text{ext}}^{\text{H}}} \right)^{- 1}} \right)$ . The scaling factor after PCA transform is,

$$\delta = \frac{{\parallel {\mathbf{PCWb}}\parallel_{F}^{2} }}{{P_{T} }}$$

(46)

Since $\parallel {\mathbf{UB}}\parallel_{\text{F}}\, =\, \parallel {\mathbf{B}}\parallel_{\text{F}}$ for any unitary matrix ${\mathbf{U}}$ [33], (46) can be written as,

$$\delta = \frac{{\parallel {\mathbf{W}} {\mathbf{b}}\parallel_{F}^{2} }}{{P_{T} }}$$

(47)

The received signal at UE after PCA transform could be obtained by substituting (45) and (47) in (43),

$${\mathbf{y}} = {\boldsymbol{\mathcal{H}}}{\mathbf{PCWb}} + {\mathbf{n}}$$

(48)

As ${\mathbf{PC}}$ is a unitary matrix it will not lead any variation in statistical property of ${\mathbf{n}}$ [33]. So (48) can be re-written as,

$${\mathbf{y}} = {\mathbf{PC}}\left( {{\boldsymbol{\mathcal{H}}}{\mathbf{Wb}} + {\mathbf{n}}} \right)$$

(49)