**5.4. Orthogonality defect**

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

Since |*S*<sup>|</sup> is constant and independent of the selected channels for all *<sup>L</sup> Nt*

£ & & <sup>S</sup>

<sup>1</sup> ( ( )) det( ( ))

A suboptimal but less computational demanding way to find a set of users that solves problem

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

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

:| | arg max det( ( )) *Nt*

<sup>2</sup> <sup>2</sup> ( ( )) sin ,

¹

:| | arg max ( ( )) *Nt*

<sup>L</sup> ÌW = = L S S

set of channels **h***<sup>i</sup>* ∀*i* ∈*S* is called *ε*-orthogonal if cos *θ***h***<sup>i</sup>*

*i j i*

*i j i*

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

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

addressed problem (4) by scheduling the set of user with minimum *ε*-orthogonality measured

æ ö L = ç ÷ " Î è ø

q*i j*

S S å Õ **h h hH** (23)

S S **H** (24)

**h** *j*

S S

S SS **H H** (20)


S S **H** (22)

*i* **H**˜ *i* possible user sets,

*<sup>H</sup>* ) ∀*i* ∈*S* are neglected.

< , ∀*i*, *j* ∈*S* [29]. Some works

*i* Õl

36 Contemporary Issues in Wireless Communications

the lower bound on the objective function of (9) can be simplified as:

*i i* l

(4) is given by the set that solves the following combinatorial problem:

ÌW =

x

=

degradation for the former metric arises because the terms det(**H**˜

follows [26]:

**5.3. ε-orthogonality**

the following combinatorial problem:

The orthogonality defect derived from Hadamard's inequality [28] measures how close a basis is to orthogonal. Given the matrix **H**(*S*) this metric is given by:

$$\delta(\mathbf{H(S)}) = \frac{\prod\_{i=1} \left\| \mathbf{h}\_i \right\|^2}{\prod\_{i=1} \lambda\_i(\mathbf{\tilde{H}(S)})} \tag{26}$$

and *δ*(**H**(*S*))=1 if and only if the elements of **H**(*S*) are pairwise orthogonal. The metric (26) reflects the orthogonality of the set {**h***<sup>i</sup>* }*i* ∈*S* and has been used to evaluate the compatibility between wireless channels in order to maximize the spatial multiplexing gain [29]. The original problem (4) can be sub-optimally solved by the set that minimizes the orthogonality defect which is formally described as:

$$\mathbf{S}\_{\delta} = \arg\min\_{\mathbf{S} \in \Omega : \|\mathbf{S}\| = N\_t} \delta(\mathbf{H}(\mathbf{S})) \tag{27}$$

Observe that problem (27) uses a weighted version of the utility function of problem (22) where the weight is defined by the inverse of ∏ *i*=1 **h***i* 2. The orthogonality defect can be seen as a scaled version of the lower bound of metric (19).

#### **5.5. Condition number**

The ZF-based beamforming methods are in general power inefficient since the spatial direction of **w***<sup>i</sup>* is not matched to **h***<sup>i</sup>* . Inverting a ill-conditioned channel matrix **H**(*S*) yields a significant power penalty and a strong SINR degradation at the receivers. In numerical analysis a metric to measure the invertibility of a matrix is given by the condition number. In MIMO system this metric is used to measure how the eigenvalues of a channel matrix spread out and to indicate multipath richness for a given channel realization. The less spread of the eigenvalues, the larger the achievable capacity in the high SNR regime. For the matrix **H**(*S*) the condition number is defined as [33]:

$$\kappa(\mathbf{H(S)}) = \frac{|\mathcal{A}\_{\text{max}}(\bar{\mathbf{H(S)}})|}{|\mathcal{A}\_{\text{min}}(\bar{\mathbf{H(S)}})|} \tag{28}$$

and *κ*(**H**(*S*)) is an indication of the multiplexing gain of a MIMO system. Another definition of the condition number is given by the product **A A**-1 for a given non-singular square matrix **A** [20] and (28) generalizes that metric for any matrix **H**(*S*) ∈ ℂ |*S* |×*Nt* where |*S*|≤ *Nt*. If the condition number is small, the matrix **H**(*S*) is said to be well-conditioned which implies that as *κ*(**H**(*S*))→1 the total achievable sum rate in the MISO systems under ZF-based beamforming can achieve a large portion of the sum rate of the inter-user interference free scenario. Problem (4) can be sub-optimally solved by a set of users with the minimum condition number and such set is formally described as:

$$\mathbf{S}\_{\kappa} = \arg\min\_{\mathbf{S} \in \Omega : |\mathbf{S}| = N\_t} \kappa(\mathbf{H}(\mathbf{S})) \tag{29}$$

Another important metric to estimate matrix condition is given by the Demmel condition number. For such metric several applications in MIMO systems have been proposed in recent years, e.g., link adaptation, coding, and beamforming [34]. The Demmel condition number can be seen as the ratio between the total energy of the channels of **H**(*S*) over the magnitude of the smallest eigenvalue of **H**¯(*S*) in the current channel realization and is given by the following expression [34]:

$$\kappa\_D(\mathbf{H(S)}) = \frac{\text{tr}(\mathbf{H(S)})}{\lambda\_{\text{min}}(\bar{\mathbf{H(S)}})} \tag{30}$$

where tr(**H**¯(*S*))= **<sup>H</sup>**(*S*) *<sup>F</sup>* <sup>2</sup> , i.e., the Frobenius norm is related with the overall energy of the channel. By using (30) the set of users that sub-optimally solves (4) is given by the solution of the following combinatorial problem:

User Selection and Precoding Techniques for Rate Maximization in Broadcast MISO Systems http://dx.doi.org/10.5772/58937 39

$$\mathbf{S}\_{\kappa\_D} = \arg\min\_{\mathbf{S} \subset \Omega : \|\mathbf{S}\| = N\_t} \kappa\_D(\mathbf{H}(\mathbf{S})) \tag{31}$$
