**3.1. Precoding by signal-to- leakage-plus-noise ratio maximization based on GEVD computation**

This precoding method is based on maximizing Signal-to-Leakage-plus-Noise Ratio (SLNR) proposed by [34, 59]. In MU-MIMO-BC, recall the system description of section 2.2 and figure 2. The received signal at the *kth* user is given by:

$$\mathbf{y}\_k = \mathbf{H}\_k \mathbf{F}\_k \mathbf{s}\_k + \mathbf{H}\_k \sum\_{\substack{j=1 \\ j \neq k}}^B \mathbf{F}\_j \mathbf{s}\_j + \mathbf{n}\_k \tag{22}$$

users in the system ∑

*M***R***<sup>k</sup> σn* 2

**computation**

*j*=1 *j*≠*k* **F***k* plus the noise power at the *kth* user front end which is given by

**H F** (23)

On MU-MIMO Precoding Techniques for WiMAX

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

15

*<sup>k</sup>* as:

**F HH I F** (25)

*k kk B* - + <sup>=</sup> **H H HH H** % L L (24)

. Thus the SLNR objective function for the *kth* user can be written as:

*k*

*M* s

the precoding matrix **F***k* obtained by per-user SLNR maximization is defined as:

<sup>=</sup> <sup>+</sup> % % **<sup>F</sup>**

number of transmitting antennas and has better BER performance.

across the matrix which reduces the degree of freedom.

=

*k B*

*SLNR*

By defining the *kth* user auxiliary interference domain matrix **H˜**

*k*

**F**

2

<sup>2</sup> <sup>2</sup> 1

*k*

*R n jk <sup>F</sup> <sup>j</sup> j k*

= ¹

<sup>+</sup> å

1 11 [ ]*<sup>T</sup>*

\* \* \* \* 2 ( ) arg max ( ) *<sup>k</sup> k T*

Closed form solution is developed to solve this fractional rational mathematical optimization problem by making use of the Generalized Eigenvalue Decomposition (GEVD) technique. Unlike the conventional precoding formulations, this method relaxes the constraints on the

One important point of observation in GEVD computation is that it is sensitive to the matrix singularity. Thus, the resulting computation accuracy is low. To resolve the singularity problem in the computation of the multi-user precoding matrices from the Per-user SLNR performance criteria, the work in [58, 60, 61] proposes a Generalized Singular Valve Decom‐ position (GSVD) and QR- Decomposition (QRD) based methods that both overcome the singularity problem and produce numerically better results. Basically, both the GSVD algorithm and QRD based methods optimize the same Per-user SLNR objective function of equation (25) but they handle the singularity problem in the covariance matrix differently. Thus, the final computation result is accurate and the calculated precoder is more efficient in inter-user interference mitigation. The reason behind the singularity problem in the leakage power plus the noise covariance matrix is that at the high SNR value, the power dominates

**3.2. Precoding by signal-to- leakage-plus-noise ratio maximization based on GSVD**

*k k kk*

**F HHF**

*k k k R nN k M* s

**H F**

*<sup>k</sup> <sup>F</sup>*

*B* **H** *<sup>j</sup>*

In this received signal expression, the first term represents the desired signal to the *kth* user, while the second term is the Multi-User Interference (MUI) from the other users to the *kth* user and the third term is the additive white Gaussian noise at the *kth* user antenna front end. In the per-user SLNR precoding method, various variables used in the method are depicted in figure 4. The objective function is formulated such that the desired signal component to the *kth* user, **H***k***F***<sup>k</sup> <sup>F</sup>* 2 is maximized with respect to both the signal leaked from the *kth* user to all other users in the system ∑ *j*=1 *j*≠*k B* **H** *<sup>j</sup>* **F***k* plus the noise power at the *kth* user front end which is given by

*M***R***<sup>k</sup> σn* 2 . Thus the SLNR objective function for the *kth* user can be written as:

used to develop these linear precoding methods that are hard to deeply survey in one chapter. Generally, one can divide the MU-MIMO linear precoding methods in the literature into two categories - methods that formulate the design objective function for both the precoder and decoder independently such as the methods in [30, 46, 47] and methods that jointly design both the precoding and decoding matrices at the transmitter site (also called iterative method), such as the work in [20, 47-52]. In spite of the good performance joint precoding design obtains relative to the independent formulation methods, the downlink channel overload and complexity are the main drawbacks of this kind of design. One more possible classification is to distinguish between formulations that lead to a closed-form solution expressions such as the works in [53-55] versus those that lead to iterative solutions such as the works in [47, 50, 56, 57]. For comparison, formulations leading to iterative solutions tends to have higher computational complexity than closed form solution methods that are linear. Among the stateof-the-art methods in recent research works, the precoding method originally proposed by Mirette M. Sadek in [34] and based on Per-User Signal to Leakage plus Noise Ratio- General‐ ized Eigenvalue Decomposition (SLNR-GEVD) and its computationally stable extended version that appeared in [58] which is based on Per-User Signal to Leakage plus Noise Ratio-Generalized Singular Value Decomposition (SLNR-GSVD) are the best in performance. In the next section, we will review these state-of-the-art linear precoding method that seek to maximize Per-user Signal to Leakage plus Noise Ratio (SLNR), which will then be followed by a detailed derivation of our proposed precoding method which is based on maximizing Per-Antenna Signal to Leakage plus Noise Ratio (PA-SLNR) followed by simulation results

under WiMAX Physical layer assuming the TDD mode of operation.

2. The received signal at the *kth* user is given by:

**computation**

14 Selected Topics in WiMAX

**H***k***F***<sup>k</sup> <sup>F</sup>*

**3.1. Precoding by signal-to- leakage-plus-noise ratio maximization based on GEVD**

<sup>1</sup> *k j*

This precoding method is based on maximizing Signal-to-Leakage-plus-Noise Ratio (SLNR) proposed by [34, 59]. In MU-MIMO-BC, recall the system description of section 2.2 and figure

> *B k kk k j k j J k* = ¹

In this received signal expression, the first term represents the desired signal to the *kth* user, while the second term is the Multi-User Interference (MUI) from the other users to the *kth* user and the third term is the additive white Gaussian noise at the *kth* user antenna front end. In the per-user SLNR precoding method, various variables used in the method are depicted in figure 4. The objective function is formulated such that the desired signal component to the *kth* user,

2 is maximized with respect to both the signal leaked from the *kth* user to all other

**y HFs H Fs n** =+ + å (22)

$$SLNR\_k = \frac{\left\| \mathbf{H}\_k \mathbf{F}\_k \right\|\_F^2}{M\_{R\_k} \sigma\_n^2 + \sum\_{\substack{j=1 \\ j \neq k}}^B \left\| \mathbf{H}\_j \mathbf{F}\_k \right\|\_F^2} \tag{23}$$

By defining the *kth* user auxiliary interference domain matrix **H˜** *<sup>k</sup>* as:

$$\tilde{\mathbf{H}}\_k = \left[ \mathbf{H}\_1 \cdots \mathbf{H}\_{k-1} \mathbf{H}\_{k+1} \cdots \mathbf{H}\_B \right]^T \tag{24}$$

the precoding matrix **F***k* obtained by per-user SLNR maximization is defined as:

$$\mathbf{F}\_{k} = \arg\max\_{\mathbf{F}\_{k}} \frac{(\mathbf{F}\_{k}^{\ast}\mathbf{H}\_{k}^{\ast}\mathbf{H}\_{k}\mathbf{F}\_{k})}{\mathbf{F}\_{k}^{\ast}(\tilde{\mathbf{H}}\_{k}^{\ast}\tilde{\mathbf{H}}\_{k} + M\_{R\_{k}}\sigma\_{n}^{2}\mathbf{I}\_{N\_{T}})\mathbf{F}\_{k}} \tag{25}$$

Closed form solution is developed to solve this fractional rational mathematical optimization problem by making use of the Generalized Eigenvalue Decomposition (GEVD) technique. Unlike the conventional precoding formulations, this method relaxes the constraints on the number of transmitting antennas and has better BER performance.
