**4. The proposed precoding by maximization of per-antenna signal-toleakage-plus-noise ratio**

The precoding technique originally proposed in [65] maximizes the SLNR for each user, thus the precoder so designed just cancels the inter-user interference. The technique proposed herein, however, utilizes a new cost function that seeks to maximize the Per-Antenna Signalto-Leakage-plus-Noise Ratio (PA-SLNR) which would help minimize even the intra-user antenna interference. Thus, the precoder so designed maximizes the overall SLNR per user more efficiently. This is justified because the PA-SLNR as explained in figure 5, takes into account the intra-user antenna interference cancelation. For *jth* receive antenna of *kth* user, the PA-SLNR given by *γ<sup>k</sup> j* , is defined as the ratio between the desired signal power of *jth* receive antenna to the interference introduced by the signal power intended for *jth* antenna but leaked to all other antennas plus the noise power at that receiving antenna front end. So for the *jth* receive antenna of *kth* user, the PA-SLNR, *γ<sup>k</sup> j* is defined by:

$$\gamma \gamma\_k^j = \frac{\|\|\mathbf{h}\_k^j \mathbf{f}\_k^j\|\|\_F^2}{\sum\_{i=1 \atop i \neq k}^B \|\|\mathbf{H}\_i \mathbf{f}\_k^i\|\|\_F^2 + \sum\_{i=1 \atop i \neq j}^{M\_k} \|\|\mathbf{h}\_k^i \mathbf{f}\_k^i\|\|\_F^2 + \sigma\_{n\_k}^{2j}} \tag{29}$$

where **h***<sup>k</sup> <sup>j</sup>* <sup>∈</sup>**C**1×*NT* is the *kth* user, *jth* antenna received row. If we define an auxiliary matrix **H***<sup>k</sup> j* as the matrix that contains all the received antennas rows of *kth* user except the *jth* row as follows:

$$\mathbf{H}\_{k}^{j} = \begin{bmatrix} h\_{k}^{(1,1)} & h\_{k}^{(1,2)} & \cdots & h\_{k}^{(1,N\_{T})} \\ \vdots & \vdots & \vdots & \vdots \\ h\_{k}^{(j-1,1)} & h\_{k}^{(j-1,2)} & \cdots & h\_{k}^{(j-1,N\_{T})} \\ h\_{k}^{(j+1,1)} & h\_{k}^{(j+1,2)} & \cdots & h\_{k}^{(j+1,N\_{T})} \\ \vdots & \vdots & \vdots & \vdots \\ h\_{k}^{(M\_{k},1)} & h\_{k}^{(M\_{k},2)} & \cdots & h\_{k}^{(M\_{k},N\_{T})} \end{bmatrix} \in \mathbf{C}^{((M\_{k}-1)\times N\_{T})} \tag{30}$$

and the combined channel matrices for all other systems receive antennas except the *jth* desired receive antenna row as as:

$$\mathbf{\widetilde{H}}\_k^{\dot{\prime}} = \mathbf{[H}\_k^{\dot{\prime}T} \mathbf{H}\_1^T \cdots \mathbf{H}\_{k-1}^T \mathbf{H}\_{k+1}^T \cdots \mathbf{H}\_B^T \mathbf{J}^T \tag{31}$$

thenfromequation(31)andequation(30)theoptimizationexpressionin(29) canbe rewrittenas:

$$\gamma \gamma\_k^j = \frac{\|\|\mathbf{A}\_k^j \mathbf{f}\_k^j\|\|^2\_F}{\|\|\widetilde{\mathbf{H}}\_k^j \mathbf{f}\_k^j\|\|^2\_F + \sigma\_{\mathbf{v}\_k^j}^2} \tag{32}$$

The optimization problem in the equation (33) deals with the *jth* antenna desired signal power in the numerator and a combination of total leaked power from desired signal to the *jth* antenna to all other antennas plus noise power at the *jth* antenna front end in the denominator. To calculate the precoding matrix for each user we need to calculate the precoding vector for each receive antenna independently. This requires solving the linear fractional optimization problem in the equation (33) *Mk* × *B* times using either GEVD [65] or GSVD [66] which both leading-to high computational load at the base stations. In the next section, Fukunaga-Koontz Transform (FKT) based solution method for solving such series of linear fractional optimiza‐ tion problems is described and simple computational method for MU-MIMO precoding

*NT* 

 *M2*

Leakage Signal from desired signal to antenna one user one

**s***1*

On MU-MIMO Precoding Techniques for WiMAX

http://dx.doi.org/10.5772/56034

19

**s***2*

*M* **s***<sup>B</sup> <sup>B</sup>*

**GB**

**G2**

**G1**

Desired Signal

Fukunaga-Koontz Transform (FKT) is a normalization transform process which was first introduced in [67] to extract the important features for separating two pattern classes in pattern recognition. Since the time it was first introduced, FKT is used in many Linear Discriminant Analysis (LDA) applications notably in [68, 69]. Researchers in [70, 71] formulate the problem of recognition of two classes as follows: Given the data matrices **ψ**1 and **ψ**2,then from these

algorithm is developed.

**s***B* 

**s***2* 

**F1**

*M1* **<sup>s</sup>***<sup>1</sup>*

**F2**

**FB**

**Figure 5.** System Model depicts all Variables.

**4.1. FKT and FKT based precoding algorithm**

**Problem Formulation:** For any *jth* receive antenna of the *kth* user, select the precoding vector **f***k j* , where *k* =1, ⋯, *B*, *j* =1, ⋯, *Mk* such that the PA-SLNR ratio is maximized:

$$\begin{aligned} \mathbf{f}\_{k}^{j} &= \arg\max\_{\mathbf{f}\_{k}^{j} \in \mathbf{C}^{\mathcal{N}\_{r} \times 1}} \frac{\mathbf{f}\_{k}^{j} (\mathbf{h}\_{k}^{j} \mathbf{h}\_{k}^{j}) \mathbf{f}\_{k}^{j}}{\mathbf{f}\_{k}^{j} (\mathbf{\widetilde{H}}\_{k}^{j} \mathbf{\widetilde{H}}\_{k}^{j} + \sigma\_{n\_{k}^{j}}^{2} \mathbf{I}\_{N\_{r}}) \mathbf{f}\_{k}^{j}} \\ \text{subject to:} \\ & \text{tr}(\mathbf{F}\_{k} \mathbf{F}\_{k}^{\dagger}) = \mathbf{1} \\ \mathbf{F}\_{K} &= \mathbf{f} \mathbf{f}^{1} \quad \cdots \; \mathbf{f}^{M\_{k}} \mathbf{I} \end{aligned} \tag{33}$$

**Figure 5.** System Model depicts all Variables.

antenna to the interference introduced by the signal power intended for *jth* antenna but leaked to all other antennas plus the noise power at that receiving antenna front end. So for the *jth*

is defined by:

*<sup>j</sup>* (29)

*<sup>T</sup> <sup>T</sup>* (31)

*<sup>j</sup>* (32)

*j*

(33)

*j*

*j* **f***k j F* 2

*Mk*

**h***k i* **f***k j F* <sup>2</sup> <sup>+</sup> *<sup>σ</sup><sup>n</sup>* 2 *k*

*<sup>j</sup>* <sup>∈</sup>**C**1×*NT* is the *kth* user, *jth* antenna received row. If we define an auxiliary matrix **H***<sup>k</sup>*

*T*

*N*

*T*

*T*

**H C** (30)

(( 1) )


*k T*

as the matrix that contains all the received antennas rows of *kth* user except the *jth* row as

<sup>=</sup> **<sup>h</sup>***<sup>k</sup>*

**H***i* **f***k j F* <sup>2</sup> + ∑ *i*=1 *i*≠ *j*

(1,1) (1,2) (1, )

L

L L

é ù ê ú

*kk k*

*hh h*

*k j j j N kk k*

*hh h*

*γk j*

**H˜** *k <sup>j</sup>* <sup>=</sup> **<sup>H</sup>***<sup>k</sup> <sup>j</sup> <sup>T</sup>* **<sup>H</sup>**<sup>1</sup>

**f***k*

*<sup>j</sup>* =arg max **f***k j* ∈**C** *NT* ×1

subject to:

tr(**F**k**F**<sup>k</sup> \* )=1

**F**<sup>K</sup> = **f** 1 , ⋯, **f** *Mk*

*hh h hh h*

( 1,1) ( 1,2) ( 1, )

*j j j N j kk k M N*

= Î


M MM M

+ + +

( 1,1) ( 1,2) ( 1, )

( ,1) ( ,2) ( , )

*M M M N kk k*

M MM M

ë û

*k k k T*

L

*<sup>T</sup>* <sup>⋯</sup>**H***<sup>k</sup>* <sup>−</sup><sup>1</sup>

<sup>=</sup> **<sup>h</sup>***<sup>k</sup> j* **f***k j F* 2

**H˜** *k j* **f***k j F* <sup>2</sup> <sup>+</sup> *<sup>σ</sup>***<sup>ν</sup>** 2 *k*

, where *k* =1, ⋯, *B*, *j* =1, ⋯, *Mk* such that the PA-SLNR ratio is maximized:

**f***k j* \* (**H˜** *k j*\***H˜** *k <sup>j</sup>* <sup>+</sup> *<sup>σ</sup><sup>n</sup>* 2 *k <sup>j</sup>***I***NT* )**f***k j*

and the combined channel matrices for all other systems receive antennas except the *jth* desired

*<sup>T</sup>* **<sup>H</sup>***<sup>k</sup>* +1

thenfromequation(31)andequation(30)theoptimizationexpressionin(29) canbe rewrittenas:

**Problem Formulation:** For any *jth* receive antenna of the *kth* user, select the precoding vector

**f***k <sup>j</sup>*\*(**h***<sup>k</sup> <sup>j</sup>*\***h***<sup>k</sup> j* )**f***k j*

*<sup>T</sup>* <sup>⋯</sup>**H***<sup>B</sup>*

receive antenna of *kth* user, the PA-SLNR, *γ<sup>k</sup>*

where **h***<sup>k</sup>*

18 Selected Topics in WiMAX

follows:

**f***k j*

receive antenna row as as:

*γk j*

∑ *i*=1 *i*≠*k*

*B*

The optimization problem in the equation (33) deals with the *jth* antenna desired signal power in the numerator and a combination of total leaked power from desired signal to the *jth* antenna to all other antennas plus noise power at the *jth* antenna front end in the denominator. To calculate the precoding matrix for each user we need to calculate the precoding vector for each receive antenna independently. This requires solving the linear fractional optimization problem in the equation (33) *Mk* × *B* times using either GEVD [65] or GSVD [66] which both leading-to high computational load at the base stations. In the next section, Fukunaga-Koontz Transform (FKT) based solution method for solving such series of linear fractional optimiza‐ tion problems is described and simple computational method for MU-MIMO precoding algorithm is developed.
