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

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 across the matrix which reduces the degree of freedom.

*Ab*

algorithm 1 as follows

1. Input: **H***com* = **H**1; ⋯;**H***<sup>B</sup>* , and σ*<sup>k</sup>*

upper triangular.

3. For *k* =1: *B*

6. End

a Set **Ψ**<sup>=</sup> **<sup>H</sup>**; *Mk*σ*<sup>k</sup>* **<sup>I</sup>***NT* <sup>∈</sup>**<sup>C</sup>**

*<sup>T</sup>* <sup>=</sup>*<sup>Y</sup> <sup>Σ</sup>b*, **<sup>0</sup>** *<sup>Q</sup>* <sup>−</sup><sup>1</sup> and *<sup>A</sup><sup>w</sup>*

where: **D***<sup>b</sup>* =diag(*α*1*α*2⋯*αr*) and **D***<sup>w</sup>* =diag(*β*1*β*2⋯*βr*), and *α<sup>i</sup>* + *β<sup>i</sup>* =1, *i* =1, ⋯, *r*.

**Algorithm 1:** The GSVD based Per-user SLNR Precoding algorithm [58, 60]

users is given by **H**com and the input noise power is given by *σ<sup>k</sup>*

2

2. Output: The algorithm computes the precoding matrices for *B*users.

b Compute the reduced QRD of **Ψ** i.e **ΩHΨ**=**R** where **Ω**∈**C**

4. Compute **V***k* from the SVD of **Ω**((*k* −1)*Mk* + 1:*kMk* , 1: *NT* )

optimum for spatial multiplexing gain extraction.

5. Solve the triangular system **RF***<sup>k</sup>* =**V***<sup>k</sup>* (:, 1:*Mk* )

**leakage-plus-noise ratio**

PA-SLNR given by *γ<sup>k</sup>*

*j*

(*BMk*+*NT* )×*NT*

The authors in [58, 60] made use of this theorem and developed an efficient precoding algorithm to calculate the precoding matrices for multiple users which is summarized in

Assume that the combined channel matrix for MU-MIMO broadcast channel of *B* number of

The Per-user SLNR precoding based on GSVD computation produces better results than all the conventional methods. However, the objective function based on per-user SLNR neglects to take the intra-user antenna interference into account. Hence, this formulation is sub-

**4. The proposed precoding by maximization of per-antenna signal-to-**

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

, is defined as the ratio between the desired signal power of *jth* receive

*<sup>T</sup>* <sup>=</sup>*<sup>Z</sup> <sup>Σ</sup>w*, **<sup>0</sup>** *<sup>Q</sup>* <sup>−</sup><sup>1</sup> (28)

On MU-MIMO Precoding Techniques for WiMAX

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

17

(*BMk*+*NT* )×*NT* orthonormal columns and **R**∈**C***NT* <sup>×</sup>*NT* is

2 .

**Figure 4.** The Definition of the Desired Signal and the Leaked Signal

Two algorithms to solve the objective function of equation (25) are given in [58, 60, 61]. Because of the singularity problem, both algorithms avoid matrix inversion to overcome the compu‐ tational instability. The developed algorithms makes use of the GSVD analysis. Although there are several methods of GSVD formulations in the literature [62-64], the work in [60] and [58] makes use of the least restrictive form of GSVD algorithm due to Paige and Saunders [62] which is now summarized as follows:

**Theorem 1:** Paige and Saunders GSVD:

Consider any two matrices of the form: *Ab*∈**CD**×**C** and *A<sup>w</sup>* <sup>∈</sup>**CD**×*<sup>N</sup>* . The GSVD is given by:

$$\mathbf{Y}^{\top}\mathbf{A}\_{b}^{\top}\mathbf{Q} = \begin{bmatrix} \boldsymbol{\Sigma}\_{b^{\prime}} \ \mathbf{0} \end{bmatrix} \text{and} \mathbf{Z}^{\top}\mathbf{A}\_{w}^{\top}\mathbf{Q} = \begin{bmatrix} \boldsymbol{\Sigma}\_{w^{\prime}} \ \mathbf{0} \end{bmatrix} \tag{26}$$

$$\begin{aligned} \text{where} \quad \begin{bmatrix} \mathbf{I}\_b\\ & \mathbf{D}\_b\\ & & \mathbf{0}\_b \end{bmatrix} \quad \text{and} \begin{bmatrix} \mathbf{I}\_w\\ & \mathbf{D}\_w \end{bmatrix} \end{aligned} \quad \text{and} \begin{bmatrix} \mathbf{I}\_w\\ & \mathbf{D}\_w\\ & & \mathbf{0}\_w \end{bmatrix} \tag{27}$$

*Y* and *Z* are orthogonal matrices and *Q* is the non-singular eigenvectors matrix. Thus, we can write:

$$\mathbf{A}\_b^T = \mathbf{Y} \begin{bmatrix} \boldsymbol{\Sigma}\_{b^\prime} & \mathbf{0} \end{bmatrix} \mathbf{Q}^{-1} \text{ and } \mathbf{A}\_w^T = \mathbf{Z} \begin{bmatrix} \boldsymbol{\Sigma}\_{w^\prime} & \mathbf{0} \end{bmatrix} \mathbf{Q}^{-1} \tag{28}$$

where: **D***<sup>b</sup>* =diag(*α*1*α*2⋯*αr*) and **D***<sup>w</sup>* =diag(*β*1*β*2⋯*βr*), and *α<sup>i</sup>* + *β<sup>i</sup>* =1, *i* =1, ⋯, *r*.

The authors in [58, 60] made use of this theorem and developed an efficient precoding algorithm to calculate the precoding matrices for multiple users which is summarized in algorithm 1 as follows

**Algorithm 1:** The GSVD based Per-user SLNR Precoding algorithm [58, 60]

Assume that the combined channel matrix for MU-MIMO broadcast channel of *B* number of users is given by **H**com and the input noise power is given by *σ<sup>k</sup>* 2 .

1. Input: **H***com* = **H**1; ⋯;**H***<sup>B</sup>* , and σ*<sup>k</sup>* 2

2. Output: The algorithm computes the precoding matrices for *B*users.

a Set **Ψ**<sup>=</sup> **<sup>H</sup>**; *Mk*σ*<sup>k</sup>* **<sup>I</sup>***NT* <sup>∈</sup>**<sup>C</sup>** (*BMk*+*NT* )×*NT*

b Compute the reduced QRD of **Ψ** i.e **ΩHΨ**=**R** where **Ω**∈**C** (*BMk*+*NT* )×*NT* orthonormal columns and **R**∈**C***NT* <sup>×</sup>*NT* is upper triangular.

3. For *k* =1: *B*

4. Compute **V***k* from the SVD of **Ω**((*k* −1)*Mk* + 1:*kMk* , 1: *NT* )

5. Solve the triangular system **RF***<sup>k</sup>* =**V***<sup>k</sup>* (:, 1:*Mk* )

6. End

Two algorithms to solve the objective function of equation (25) are given in [58, 60, 61]. Because of the singularity problem, both algorithms avoid matrix inversion to overcome the compu‐ tational instability. The developed algorithms makes use of the GSVD analysis. Although there are several methods of GSVD formulations in the literature [62-64], the work in [60] and [58] makes use of the least restrictive form of GSVD algorithm due to Paige and Saunders [62]

MRB

MRB-1

NT MR1

Consider any two matrices of the form: *Ab*∈**CD**×**C** and *A<sup>w</sup>* <sup>∈</sup>**CD**×*<sup>N</sup>* . The GSVD is given by:

*b w bb w w*

**I I ΣD Σ D**

éù é ù êú ê ú = =

ëû ë û

*Y* and *Z* are orthogonal matrices and *Q* is the non-singular eigenvectors matrix. Thus, we can

*b w*

**0 0**

*<sup>T</sup> <sup>Q</sup>* <sup>=</sup> *<sup>Σ</sup>w*, **<sup>0</sup>** (26)

U1

U K-1

U B

(27)

*<sup>T</sup> <sup>Q</sup>* <sup>=</sup> *<sup>Σ</sup>b*, **<sup>0</sup>** and*<sup>Z</sup> <sup>T</sup> <sup>A</sup><sup>w</sup>*

which is now summarized as follows:

BS

16 Selected Topics in WiMAX

MU-MIMO single-cell configuration

Desired signal

**Figure 4.** The Definition of the Desired Signal and the Leaked Signal

Leaked signal

**Theorem 1:** Paige and Saunders GSVD:

write:

*<sup>Y</sup> <sup>T</sup> <sup>A</sup><sup>b</sup>*

where and

The Per-user SLNR precoding based on GSVD computation produces better results than all the conventional methods. However, the objective function based on per-user SLNR neglects to take the intra-user antenna interference into account. Hence, this formulation is suboptimum for spatial multiplexing gain extraction.
