*5.1.2. Orthogonal projector for ZFDP*

2 *<sup>i</sup> i i*

D = <sup>V</sup> V**h**

. Notice that from (8) and (15) the projection power loss factor of **h***<sup>i</sup>*

22 2

h

(1) ( )| (1) ( 1) 1 (1 ) (1 ) *i i <sup>i</sup> <sup>i</sup>* p

 h

*i* and *η***h***<sup>i</sup>*

effects due to *π*(1)*π*(2)⋯*π*(*k* - 1). Using multiple regression analysis it is possible to evaluate

coefficients choosing one user order *π* out of (|*S*| - 1)! permutations of the users in *S* [17]. A different approach can be applied if for a given set of channels **h** *<sup>j</sup>* ∀ *j* ∈*S* the orthogonal

2 measures how much the vector **h***<sup>i</sup>*

projector matrix of each channel is known so that **Q** *<sup>j</sup>* =**I** - **h** *<sup>j</sup>*

**Figure 2.** The spatial relationship between the components of vector **h***<sup>i</sup>*

, , *<sup>i</sup>*

¹ Î

æ ö <sup>=</sup> ç ÷ ® ¥ ç ÷ è ø

S

*j ij*

<sup>V</sup> Õ

*n*

*n*

*j*

From [24] we have the following result:

onto the null space of *i* is equivalent to 1 - <sup>∆</sup>*i***h***<sup>i</sup>*

where *π*(*k*) is the *k*th ordered element of **H**˜

between the channel vector **h***<sup>i</sup>*

34 Contemporary Issues in Wireless Communications

<sup>2</sup> = **h***<sup>i</sup>* 2 (1 - <sup>∆</sup>*i***h***<sup>i</sup>*

where ∆*i***h***<sup>i</sup>*

**H**˜ *i*

> **h***i* **Q***<sup>i</sup>*

*H i i H i i*

pp

 p*k k*

and the selected vector associated with *π*(*k*) eliminating the

*<sup>H</sup>* (**<sup>h</sup>** *<sup>j</sup>* **h** *j <sup>H</sup>* )-1**<sup>h</sup>** *<sup>j</sup>* .

 and *<sup>i</sup>* .

**Q Q** (17)

¼ - -D = - ¼ - <sup>V</sup>**hh h** (16)

<sup>2</sup> ) by extracting the partial correlation coefficients from the correlation

**h h** (15)

can be predicted from the channel vectors of

*<sup>π</sup>*(*<sup>k</sup>* )|*π*(1)⋯*π*(*<sup>k</sup>* -1) is the partial correlation

2 which can be evaluated as follows [17]:

due to the projection

**hP h**

In Section 4.2 the ZFDP beamforming was introduced and it was mentioned that the received signal of user *i* contains an interference component from all *j* <*i* users, i.e., the previously encoded users. Given an specific encoding order *π*(*i*), *i* ∈{1, ⋯, |*S*|} (a permutation of the users in *S*), the beamforming vectors **w***π*(*i*) are computed either by a QR-type decomposition or by GSO as follows [27]:

$$\mathbf{w}\_{\pi(i)} = \frac{\mathbf{T}\_i \mathbf{h}\_{\pi(i)}^H}{\|\mathbf{T}\_i \mathbf{h}\_{\pi(i)}^H\|}, \qquad \qquad \qquad \mathbf{T}\_i = \begin{cases} \mathbf{I} & \text{for } i = 1 \\ \mathbf{T}\_{i-1} - \mathbf{w}\_{\pi(i-1)} \mathbf{w}\_{\pi(i-1)}^H & \text{for } i = 2, \dots, q \end{cases}$$

and *i* =1, ⋯, *q* with *q* =*rank*(**H**(*S*)). **T***<sup>i</sup>* is the orthogonal projector matrix onto *Sp*(**h***π*( *<sup>j</sup>*<*<sup>i</sup>*)) the subspace spanned by all previously encoded users for which **h***π*( *<sup>j</sup>*<*i*)**w***π*(*i*)=0. Some authors (e.g., [25], [27]) use the following expression as an objective function over the channel matrix **H**(*S*) for user selection and ZFDP beamforming:

$$f(\mathbf{H(S)}) = \sum\_{i=1}^{|S|} \left\| \mathbf{h}\_i \mathbf{T}\_i \right\|^2 \tag{18}$$

Observe that user selection and sum rate maximization based on metric (18) implicitly depend on one particular selected encoding order *π* out of |*S*|! different valid permutations. Since different encoding order yield different values of (18), in [27] it was proposed a method to perform the successive encoding optimizing of the order *π*. Such an optimum order is attained by an iterative algorithm that evaluates (8) each iteration for every successive encoded user. An alternative suboptimal approach can be employed as in [19] where *π* is defined by the descending order of the effective channel gains of the users in *S*.

#### **5.2. Approximation of the NSP**

The objective function of problem (9) can be further relaxed by using a lower bound of tr(**H**˙ (*S*)) in order to avoid the computation of the inverse matrix **H**˙ (*S*). Considering the definition of trace we have that

$$\text{tr}(\dot{\mathbf{H}}(\mathbf{S})) = \sum\_{i} \mathcal{J}\_{i}^{-1}(\tilde{\mathbf{H}}(\mathbf{S})) \tag{19}$$

and using the arithmetic-geometric mean inequality over *λ<sup>i</sup>* (**H**˙ (*S*)) it holds that [28]:

$$\mathbb{I}\mid\mathbb{S}\mid\prod\_{i}\lambda\_{i}^{\wedge\wedge\mathbb{S}}\left(\dot{\mathbf{H}}(\mathbb{S})\right)\leq\mathrm{tr}(\dot{\mathbf{H}}(\mathbb{S}))\tag{20}$$

Since |*S*<sup>|</sup> is constant and independent of the selected channels for all *<sup>L</sup> Nt* possible user sets, the lower bound on the objective function of (9) can be simplified as:

$$\prod\_{i} \lambda\_i(\dot{\mathbf{H}}(\mathbf{S})) = \det(\overline{\mathbf{H}}(\mathbf{S}))^{-1} \tag{21}$$

A suboptimal but less computational demanding way to find a set of users that solves problem (4) is given by the set that solves the following combinatorial problem:

$$\mathbf{S}\_{\tilde{\xi}} = \arg\max\_{\mathbf{S} \subset \Omega : \|\mathbf{S}\| = N\_t} \det(\tilde{\mathbf{H}}(\mathbf{S})) \tag{22}$$

where the optimized objective function only requires to compute the determinant of a matrix product. Observe that the lower bound in (21) is closely related with (14) and the performance degradation for the former metric arises because the terms det(**H**˜ *i* **H**˜ *i <sup>H</sup>* ) ∀*i* ∈*S* are neglected.

In [26] it was presented a greedy algorithm where the metric for user selection is based on an approximation of (16) and the correlation coefficients are used instead of the partial correlation coefficients. Such relaxation neglects the channel gain degradation due to the terms *π*(1)*π*(2)⋯*π*(*k* - 1). Given channel matrix **H**(*S*), the metric that approximates (19) is defined as follows [26]:

$$\Lambda(\mathbf{H(S)}) = \sum\_{i} \left( \left\| \mathbf{h}\_{i} \right\|^{2} \prod\_{j \neq i} \sin^{2} \theta\_{\mathbf{h}\_{i} \mathbf{h}\_{j}} \right) \forall i, j \in \mathbf{S} \tag{23}$$

Using this metric a suboptimal solution to problem (4) is given by the set of users that solve the following combinatorial problem:

$$\mathbf{S}\_{\Lambda} = \arg\max\_{\mathbf{S} \in \Omega : \|\mathbf{S}\| = N\_{\iota}} \Lambda(\mathbf{H}(\mathbf{S})) \tag{24}$$

#### **5.3. ε-orthogonality**

Several user selection algorithms (e.g., [25], [26], [29]) attempt to create groups of quasiorthogonal users based on the information provided by the coefficient of correlation (13). A set of channels **h***<sup>i</sup>* ∀*i* ∈*S* is called *ε*-orthogonal if cos *θ***h***<sup>i</sup>* **h** *j* < , ∀*i*, *j* ∈*S* [29]. Some works addressed problem (4) by scheduling the set of user with minimum *ε*-orthogonality measured either over the normal channels [30] or over the eigenvectors computed by SVD [29]. If the orthogonality among channels in **H**(*S*) is the only metric considered to define *S* (regardless of the channel gains), problem (4) can be sub-optimally solved by the set *S* that minimizes the orthogonality among all *L Nt* possible sets formally described as:

$$\mathbf{S}\_{\mathbf{b}} = \arg\min\_{\mathbf{S} \in \Omega : \|\mathbf{S}\| = N\_t} \left( \max\_{i, j \in \mathbf{S}} \cos \theta\_{\mathbf{h}\_i \mathbf{h}\_j} \right) \tag{25}$$

Some works in the literature define *S* as the set with minimum sum of correlation coefficients ∑ *i* ∑ *j*≠*i* cos *θ***h***<sup>i</sup>* **h** *j* , ∀*i*, *j* ∈*S* or as the set with minimum average correlation coefficient [25], [29]. Observe that these objective functions are based on pairwise metrics and they can be negatively biased by few terms with large values neglecting the remaining coefficients with relatively small values. An alternative utility function that can identify such large terms is the geometric mean over all correlation coefficients since it would assign the same priority to each of term. In MU-MIMO systems the user grouping problem based on (25) for scheduling time slots, subcarriers, or both, can be modeled as a graph coloring problem [13], [31] or graph clique problem [32] and complexity reduction is the main objective of the proposed grouping algorithms.
