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

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 two classes, the autocorrelation matrices **Π**<sup>1</sup> =**ψ**1**ψ**<sup>1</sup> *<sup>T</sup>* and **Π**<sup>2</sup> <sup>=</sup>**ψ**2**ψ**<sup>2</sup> *<sup>T</sup>* are positive semi-definite (p.s.d) and symmetric. For any given p.s.d autocorrelation matrices **Π**1 and **Π**2 the sum **Π** is still p.s.d and can be written as:

$$
\mathbf{II} = \mathbf{II}\_1 + \mathbf{II}\_2 = \begin{bmatrix} \mathbf{U} & \mathbf{U}\_\perp \end{bmatrix} \begin{bmatrix} \mathbf{D} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{U}^T \\ \mathbf{U}\_\perp^T \end{bmatrix} \tag{34}
$$

Without loss of generality the sum **Π** can be singular and *r* =*rank*(**Π**)≤**D***im*(**Π**), where **<sup>D</sup>**=diag (*λ*1, <sup>⋯</sup>*λr*) and also *λ*<sup>1</sup> <sup>≥</sup> <sup>⋯</sup> <sup>≥</sup>*λ<sup>r</sup>* >0.**U**∈**C**Dim(**Π**)×*<sup>r</sup>* is the set of eigenvectors that corre‐ spond to the set of nonzero-eigenvalues and **U**<sup>⊥</sup> <sup>∈</sup>**C**Dim(**Π**)×(Dim(**Π**)−*r*) is the orthogonal comple‐ ment of **U**. From the equation (34), the FKT transformation [71] matrix operator is defined as:

$$\mathbf{P} = \mathbf{U} \mathbf{D}^{-1/2} \tag{35}$$

1 1

2 2

**Π**<sup>1</sup> =**h***<sup>k</sup> <sup>j</sup>*\***h***<sup>k</sup>*

and consequently the sum **Π** of the two covariance matrices becomes:

transformation factor to generate the shared eigenspace matrices **Π˜**

\* 2 2 *<sup>T</sup> jj j k k kN* **Π HH I** = + % % s

> \* 2 com com *<sup>T</sup>*

*k N* **ΠH H I** = +

*j*

s

where **H***com* is the combined channel matrix for all user which is given as

According to FKT analysis, we can calculate the FKT factor and consequently we use this

from equations ( 42-44) for each receive antenna in the system. The shared eigen subspaces are complements of each other such that the best principal eigenvectors of the first transformed

multiplying the FKT factor with the eigenvectors corresponding to the best eigenvalue of the transformed antenna covariance matrix or eigenvector corresponding to the least eigenvalue of the transformed leakage plus noise covariance matrix. The most notable observation is that, for the set of *MK* × *B* receiving antennas in the system, we need to compute the FKT transform factor only once, which cuts down the computation load sharply. Algorithm 2, summarizes the computation steps of the precoding matrix for multiple *B* users in the system using FKT.

<sup>1</sup> are the least principal eigenvector for the second transformed covariance

<sup>2</sup> and vice-versa. Thus, we can find the receive antenna precoding vector by simply

ing eigenvalues matrices. Thus, from these analyses we conclude that FKT gives the best optimum solution for any fractional linear problem without going through any serious matrix inversion step. By relating FKT transform analysis of the two covariance matrices, and the precoding design problem for MU-MIMO (multiple linear fractional optimization problem), we can make direct mapping of the optimization variable from equation (33) to the FKT

Where **V**∈**C***r*×*<sup>r</sup>*

transform as follows:

**H***com* = **H**<sup>1</sup>

matrix **Π˜**

*<sup>T</sup>* **<sup>H</sup>**<sup>2</sup> *<sup>T</sup>* <sup>⋯</sup>**H***<sup>B</sup> <sup>T</sup> <sup>T</sup>* .

covariance matrix **Π˜**

*<sup>T</sup>* = **Π VΛ V** % (39)

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*<sup>T</sup>* = **Π VΛ V** % (40)

**I**=**Λ**<sup>1</sup> + **Λ**<sup>2</sup> (41)

*<sup>j</sup>* (42)

(43)

(44)

<sup>1</sup> and **Π˜**

<sup>2</sup> using the facts

is the matrix that contains all the eigenvectors, and **Λ**1, **Λ**<sup>2</sup> are the correspond‐

By using this FKT transformation factor, the sum p.s.d matrix **Π** can be whitened such that the sum of the two Sub-matrices **Π˜** <sup>1</sup> and **Π˜** <sup>2</sup> gives the identity matrix as follows:

$$\mathbf{P}^T \mathbf{I} \mathbf{I} \mathbf{P} = \mathbf{P}^T (\Pi\_1 + \Pi\_2) \mathbf{P} = \widetilde{\Pi}\_1 \, | \, + \, \widetilde{\Pi}\_2 = \mathbf{I}^{r \times r} \tag{36}$$

Where **Π˜** =1 **<sup>P</sup>***<sup>T</sup>* **<sup>Π</sup>**<sup>1</sup> **<sup>P</sup>**, **Π˜** <sup>2</sup> <sup>=</sup>**P***<sup>T</sup>* **<sup>Π</sup>**<sup>2</sup> **<sup>P</sup>** are the transformed covariance matrices for **Π**1 and **Π**<sup>2</sup> respectively, and **I** *r*×*r* is an identity matrix. Suppose that **ν** is an eigenvector of **Π˜** <sup>1</sup> with corresponding eigenvalue *λ*1, then **Π˜** <sup>1</sup> **<sup>ν</sup>**=*λ*1**ν** and from the equation (36) we have **Π˜** <sup>1</sup> <sup>=</sup>**I**−**Π˜** 2 . Thus, the following results can be pointed:

$$(\mathbf{I} - \widetilde{\mathbf{II}}\_2)\mathbf{v} = \boldsymbol{\lambda}\_1 \mathbf{v} \tag{37}$$

$$
\tilde{\mathbf{II}}\_2 \mathbf{v} = (1 - \mathbb{A}\_1)\mathbf{v} \tag{38}
$$

This means that **Π˜** <sup>2</sup> has the same eigenvectors as **Π˜** <sup>1</sup> with corresponding eigenvalues related as *λ*<sup>2</sup> =(1−*λ*1). Thus, we can conclude that the dominant eigenvectors of **Π˜** <sup>1</sup> is the weakest eigenvectors of **Π˜** <sup>2</sup> and vice versa. Based on the FKT transform analysis we conclude that the transformed matrices **Π˜** <sup>1</sup> and **Π˜** <sup>2</sup> share the same eigenvectors and the sum of the two corresponding eigenvalues are equal to one. Thus, the following decomposition is valid for any positive definite and positive semi-definiate matrices:

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$$
\tilde{\mathbf{II}}\_1 = \mathbf{V} \mathbf{A}\_1 \mathbf{V}^T \tag{39}
$$

$$
\tilde{\mathbf{II}}\_2 = \mathbf{V} \mathbf{A}\_2 \mathbf{V}^T \tag{40}
$$

$$\mathbf{I} = \mathbf{A}\_1 + \mathbf{A}\_2 \tag{41}$$

Where **V**∈**C***r*×*<sup>r</sup>* is the matrix that contains all the eigenvectors, and **Λ**1, **Λ**<sup>2</sup> are the correspond‐ ing eigenvalues matrices. Thus, from these analyses we conclude that FKT gives the best optimum solution for any fractional linear problem without going through any serious matrix inversion step. By relating FKT transform analysis of the two covariance matrices, and the precoding design problem for MU-MIMO (multiple linear fractional optimization problem), we can make direct mapping of the optimization variable from equation (33) to the FKT transform as follows:

$$\mathbf{I}\mathbf{1}\mathbf{1}\_1 = \mathbf{1}\_k^{\prime}\mathbf{1}\mathbf{1}\_k^{\prime}\tag{42}$$

$$\mathbf{II}\_2 = \mathbf{\tilde{H}}\_k^T \mathbf{\tilde{H}}\_k^\prime + \sigma^2 \mathbf{\tilde{I}}\_k \mathbf{I}\_{N\_\Gamma} \tag{43}$$

and consequently the sum **Π** of the two covariance matrices becomes:

two classes, the autocorrelation matrices **Π**<sup>1</sup> =**ψ**1**ψ**<sup>1</sup>

1 2

spond to the set of nonzero-eigenvalues and **U**<sup>⊥</sup> <sup>∈</sup>**C**Dim(**Π**)×(Dim(**Π**)−*r*)

**<sup>D</sup>**=diag (*λ*1, <sup>⋯</sup>*λr*) and also *λ*<sup>1</sup> <sup>≥</sup> <sup>⋯</sup> <sup>≥</sup>*λ<sup>r</sup>* >0.**U**∈**C**Dim(**Π**)×*<sup>r</sup>*

still p.s.d and can be written as:

20 Selected Topics in WiMAX

the sum of the two Sub-matrices **Π˜**

*r*×*r*

Thus, the following results can be pointed:

corresponding eigenvalue *λ*1, then **Π˜**

Where **Π˜** =1 **<sup>P</sup>***<sup>T</sup>* **<sup>Π</sup>**<sup>1</sup> **<sup>P</sup>**, **Π˜**

respectively, and **I**

This means that **Π˜**

eigenvectors of **Π˜**

transformed matrices **Π˜**

*<sup>T</sup>* and **Π**<sup>2</sup> <sup>=</sup>**ψ**2**ψ**<sup>2</sup>

*T*

^

**0 0 <sup>U</sup>** (34)


<sup>2</sup> gives the identity matrix as follows:

is the set of eigenvectors that corre‐

*<sup>r</sup>*×*<sup>r</sup>* (36)

<sup>1</sup> with

2 .

<sup>1</sup> <sup>=</sup>**I**−**Π˜**

<sup>1</sup> is the weakest

is the orthogonal comple‐

^ *T*

é ù é ù

ë û ê ú ë û

(p.s.d) and symmetric. For any given p.s.d autocorrelation matrices **Π**1 and **Π**2 the sum **Π** is

=+= é ù ê ú ë û ê ú

Without loss of generality the sum **Π** can be singular and *r* =*rank*(**Π**)≤**D***im*(**Π**), where

ment of **U**. From the equation (34), the FKT transformation [71] matrix operator is defined as:

By using this FKT transformation factor, the sum p.s.d matrix **Π** can be whitened such that

<sup>1</sup> <sup>+</sup> **<sup>Π</sup>˜**

is an identity matrix. Suppose that **ν** is an eigenvector of **Π˜**

<sup>2</sup> and vice versa. Based on the FKT transform analysis we conclude that the

<sup>2</sup> =**I**

<sup>2</sup> <sup>=</sup>**P***<sup>T</sup>* **<sup>Π</sup>**<sup>2</sup> **<sup>P</sup>** are the transformed covariance matrices for **Π**1 and **Π**<sup>2</sup>

<sup>1</sup> **<sup>ν</sup>**=*λ*1**ν** and from the equation (36) we have **Π˜**

<sup>2</sup> )**ν**=*λ*1**ν** (37)

<sup>1</sup> with corresponding eigenvalues related

% (38)

<sup>2</sup> share the same eigenvectors and the sum of the two

<sup>1</sup> and **Π˜**

**<sup>P</sup>***<sup>T</sup>* **<sup>Π</sup>P**=**P***<sup>T</sup>* (**Π**<sup>1</sup> <sup>+</sup> **<sup>Π</sup>**<sup>2</sup> )**P**=**Π˜**

(**I**−**Π˜**

<sup>2</sup> has the same eigenvectors as **Π˜**

<sup>1</sup> and **Π˜**

any positive definite and positive semi-definiate matrices:

as *λ*<sup>2</sup> =(1−*λ*1). Thus, we can conclude that the dominant eigenvectors of **Π˜**

2 1 = - (1 ) l**Πv v**

corresponding eigenvalues are equal to one. Thus, the following decomposition is valid for

**D 0 <sup>U</sup> Π Π Π UU**

*<sup>T</sup>* are positive semi-definite

$$\mathbf{II} = \mathbf{H}\_{\text{com}}^{\*} \mathbf{H}\_{\text{com}} + \sigma^{2}{}\_{k}^{\prime} \mathbf{I}\_{N\_{\text{I}}} \tag{44}$$

where **H***com* is the combined channel matrix for all user which is given as **H***com* = **H**<sup>1</sup> *<sup>T</sup>* **<sup>H</sup>**<sup>2</sup> *<sup>T</sup>* <sup>⋯</sup>**H***<sup>B</sup> <sup>T</sup> <sup>T</sup>* .

According to FKT analysis, we can calculate the FKT factor and consequently we use this transformation factor to generate the shared eigenspace matrices **Π˜** <sup>1</sup> and **Π˜** <sup>2</sup> using the facts from equations ( 42-44) for each receive antenna in the system. The shared eigen subspaces are complements of each other such that the best principal eigenvectors of the first transformed covariance matrix **Π˜** <sup>1</sup> are the least principal eigenvector for the second transformed covariance matrix **Π˜** <sup>2</sup> and vice-versa. Thus, we can find the receive antenna precoding vector by simply multiplying the FKT factor with the eigenvectors corresponding to the best eigenvalue of the transformed antenna covariance matrix or eigenvector corresponding to the least eigenvalue of the transformed leakage plus noise covariance matrix. The most notable observation is that, for the set of *MK* × *B* receiving antennas in the system, we need to compute the FKT transform factor only once, which cuts down the computation load sharply. Algorithm 2, summarizes the computation steps of the precoding matrix for multiple *B* users in the system using FKT.

**Algorithm 2:** PA-SLNR MU-MIMO precoding based on FKT for multiple *B* independent MU-MIMO users.

• Input: Combined channel matrix for all *B*users and the input noise variance

#### **H***com* = **H**<sup>1</sup> *<sup>T</sup>* **<sup>H</sup>**<sup>2</sup> *<sup>T</sup>* <sup>⋯</sup>**H***<sup>B</sup> <sup>T</sup> <sup>T</sup>* , σ*<sup>k</sup>* 2


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

4. For *j*=1 to *Mk*

○ Transform the *j th* receive antenna covariance matrix **Π**1 using the FKT factor **P**to **Π˜** <sup>1</sup> and select the first eigenvector **ν***<sup>k</sup> j* of **Π˜** <sup>1</sup>

symbol vectors are modulated and spatially multiplexed at the base station. At the receivers, matched filter is used to decode each user's data. Detailed summary of the MU-MIMO-BC

• 2000 System transmission running for SINR outage calculation

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Each user channel Matrix elements generated as zero-mean and unit-variance i.i.d complex Gaussian random variables

• Reference SLNR-GEVD, SLNR-GSVD

In this scenario we consider single cell transmission where implicitly we assume that the multicell interference is zero. Figure 6. shows the average received BER performance of the proposed PA-SLNR-FKT and the reference methods of SLNR-GSVD and SLNR-GEVD precoding schemes for the MU-MIMO-BC system configurations of *NT* =14, *B* =3*Mk* =4. In this configu‐ ration, the numbers of the base station antennas are more than the sum of all receiving antennas which also signifies more degree of freedom in MU-MIMO transmission. In this simulation also, the base station utilizes 4-QAM modulation to modulate, spatially multiplexes and precodes a vector of length 4 symbols to each user. The average BER is calculated over 5000 MU-MIMO channel realization for each algorithm. The proposed method outperforms SLNR-GSVD and SLNR-GEVD. At BER equal to 10−<sup>4</sup> there is approximately 4dB performance gain

Figure 7 compares the received output SINR outage performance of the proposed PA-SLNR-FKT precoding and the reference SLNR-GSVD and SLNR-GEVD precoding methods. MU-MIMO system with full rate configuration of *B* =3, *Mk* =4, *NT* =12 and 2dB input SNR is considered in the simulation. The proposed method outperforms the SLNR-GSVD method by

Figure 8 compares the received output SINR outage performance of the proposed PA-SLNR-FKT precoding and the reference SLNR-GSVD and SLNR-GEVD precoding methods. MU-MIMO system with full rate configuration of *B* =3, *Mk* =4, *NT* =12 and 10dB input SNR are considered in the simulation. The proposed method outperforms the SLNR-GSVD method by

system configuration parameters are given in table (1).

**Table 1.** Narrowband MU-MIMO System Configuration Summary

Precoding methods • Proposed method (PA-SLNR-FKT)

Performance metrics Received BER and Received SINR outage

Number of Iterations • 5000 System transmission for BER calculation

*4.2.1. Scenario 1: Single cell MU-MIMO*

**Parameter Configuration** System configuration MU-MIMO-BC

Modulation 4-QAM

MIMO Decoder Matched filter.

1 dB gain at 10% received output SINR outage.

approximately 1.5 dB gain at 10% received output SINR outage.

over SLNR-GSVD.

○ The precoding vector corresponding to the *j th*receive antenna at the *kth* user is: **f***<sup>k</sup> <sup>j</sup>* <sup>=</sup>**Pν***<sup>k</sup> j*

End

```
○ Synthesize the kth user precoding matrix is Fk = fk
                                                      1 ⋯fk
                                                          Mk
```
End

The algorithm takes the combined MU-MIMO channel matrix as well as the value of the noise variance as an input and outputs *B* users precoding matrices. It computes the FKT factor in step one and two and iterates *B* times (step three to six) to calculate the precoding matrices for *B* number of users. For each user, there are *Mk* sub-iteration operations (step four to five) to calculate each individual user precoding matrix in vector by vector basis.
