**6. User selection algorithms**

**5.5. Condition number**

38 Contemporary Issues in Wireless Communications

of **w***<sup>i</sup>* is not matched to **h***<sup>i</sup>*

defined as [33]:

expression [34]:

where tr(**H**¯(*S*))= **<sup>H</sup>**(*S*) *<sup>F</sup>*

the following combinatorial problem:

The ZF-based beamforming methods are in general power inefficient since the spatial direction

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

> max min | ( ( ))| ( ( )) | ( ( ))| l

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

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

:| | arg min ( ( )) *Nt*

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

> min tr( ( )) ( ( )) ( ( )) *<sup>D</sup>*

channel. By using (30) the set of users that sub-optimally solves (4) is given by the solution of

S

<sup>2</sup> , i.e., the Frobenius norm is related with the overall energy of the

l<sup>=</sup> <sup>S</sup> <sup>S</sup>

**<sup>H</sup> <sup>H</sup>**

ÌW = <sup>=</sup> S S

k

S S **H** (29)

S **H**

l<sup>=</sup> <sup>S</sup> <sup>S</sup>

k

**H**

matrix **A** [20] and (28) generalizes that metric for any matrix **H**(*S*) ∈ ℂ

k

k

number and such set is formally described as:

. Inverting a ill-conditioned channel matrix **H**(*S*) yields a significant

**<sup>H</sup>** (28)


**<sup>H</sup>** (30)

where |*S*|≤ *Nt*.

For MU-MIMO BC systems with ZF-based beamforming the optimum solution to the sum rate maximization problem implies the optimization over the power allocation and user selection as well as the correct dimension and orientation of the signal subspace used by each selected user [9]. Since this optimization is performed over several dimensions (time, space, signal space, power, users) the optimum solution for the MU-MIMO scenario has not been found. Even when the signal space spans over one dimension, i.e. single-antenna users, the optimum solution to the sum rate maximization problem (4) in MU-MISO BC systems is given by an exhaustive search whose computational complexity increases approximately with (*<sup>K</sup> Nt*). In order to find a fair trade-off between complexity and performance, a large number of works have addressed the sum rate maximization problem for ZFBF (e.g., [9], [21], [35] and ZFDP (e.g., [18], [19]) by designing low-complexity algorithms that find a sub-optimal yet acceptable solution to (4). The goal of such algorithms is to construct the solution set *S* of quasi-orthogonal users implementing different iterative approaches. Some works in the literature proposed greedy opportunistic algorithms that exploit the instantaneous channel information. In a greedy selection each new selected user finds a locally maximum for a given global objective function so that the final set of users attempt to converge to a close-to-optimal solution. Another approach to solve (4) is to use channel metrics and reformulate the original problem as an integer program. In this section we present the principles and characteristics of different user selection algorithms proposed in the literature. Our aim is to introduce generic structures of the user selection process and to illustrate under which conditions they can be used to maximize the sum rate and to reduce computational complexity for ZF-based beamforming.

#### **6.1. Metric-based selection**

The objective of the metric-based user selection is to find a set *S* of users with spatially quasiorthogonal channels such that the profit from beamforming is maximized. User selection algorithms can be design to relax the original optimization problem (4) by optimizing a particular channel metric. This kind of optimization may face two main problems: 1) the large search space *L* for the optimum solution, and 2) how to discriminate between metrics of sets with different cardinality without computing the objective function. Both problems are partially solved by imposing a constraint in the optimizations problem, i.e., |*S*|= *Nt* ∀*S* ⊆Ω in problem (4) or by operating in the high SNR regime. This yields a search space of size *L Nt* and metrics that can be easily compared since they are applied over user sets of the same size. In MU-MISO BC systems with ZF-based precoding and optimal power allocation, authors in [5] showed that there exist a critical SNR that depends on **H**(*S*) for which ∀ *P* ≤*P*0 the maximum sum rate is achieved by a subset *S* with cardinality |*S*|<*rank*(**H**(*S*)). For an operation point ∀ *P* >*P*0 the system achieves full multiplexing diversity, i.e., |*S*|=*rank*(**H**(*S*)). The metricbased user selection algorithms solve (4) in two phases. In the first phase *S* is selected in order to optimize a given metric and in the second phase the sum rate is evaluated for the previously defined set. This means that the user selection and the resource allocation problems are decoupled in this approach. The cardinality of *S* is fixed in the first phase and it may be modified during the second phase when the sum rate is evaluated. If power allocation is performed based on water-filling, this might yield a zero power allocation (no data transmis‐ sion) for some users due to the instantaneous value of *P*0.

In the literature of user selection several algorithms solve problem (4) optimizing the NSP by iteratively computing the sum of the effective channel gains each iteration. The optimum set that maximizes the NSP is given by the solution of (9) which is a combinatorial problem that requires the evaluation of *L Nt* matrix product and inversion operations. Therefore, the set *S* is found in a greedy fashion by selecting the user that locally maximizes the total sum of effective channel gains which roughly requires *L <sup>K</sup>* =∑*<sup>k</sup>* =1 *Nt*-1 (*K* - *k*) evaluations of the metric *f* (∙ ). Given *S*(*n* - 1) in the *n*th iteration, the optimum new user *i* ∈Ω achieves the largest effective channel gain when its channel is projected onto the orthogonal subspace spanned by the previously selected users =*Sp*(**H**(*S*(*n* - 1))). In [10], [21] the authors evaluate iteratively at the *n*th iteration the intersection of the null spaces of the previously selected users *S*(*n* - 1) with each new candidate user by means of GSO. Using this technique, it is not necessary to extract the basis of the subspace spanned by the channel matrix (SVD) or to compute the orthogonal projector matrix **Q***<sup>i</sup>* at each iteration [19]. Other user selection algorithms based on the NSP further reduce the required computational complexity by approximating any of the its alternative forms described in Section 5.1.1. The algorithm proposed in [26] approximates (16)

by using only the correlation coefficient between the users in *S*(*n*). The authors in [36] select users based on metric (17) where each user computes its own projector matrix and the algorithm approximates the NSP. The optimization of an approximation of the NSP yields a suboptimal set of users and performance degradation regarding to (8) yet some gains in terms of complexity are attained. The general structure of this kind of algorithms is presented in Alg. 1 and *f* (∙ ) is given by any metric defined in Section 5. Notice that the first selected user is the one with the strongest channel which not necessarily belongs to the optimum set. However, this criterion simplifies the initialization of *S* and does not yield a significant performance degradation, specially in the large *K* regime [21].

Another relevant feature of this generic structure is given by the adjustment of Ω in each iteration. This is relevant in the large *K* regime where the number of operations required to find the next user is roughly *K* which can be computationally demanding. In order to reduce the total number of metric evaluations, an optimization over Ω can be performed. For instance, by performing an optimization similar to (25) per iteration for a given *ε*-orthogonality target whose optimal value depends on *K* and *Nt* [37]. The principle behind this optimization is that the candidate users must satisfy the hyperslab condition Ω(n)={**h***<sup>i</sup>* <sup>∀</sup> *<sup>i</sup>* <sup>∈</sup> <sup>Ω</sup>(n-1): cos *<sup>θ</sup>***h***<sup>i</sup>* **<sup>g</sup>** ≤*ε*} for a given vector **g** [21]. The metric-based user selection does not guarantee full multiplexing gain or sum rate maximization and authors in [10], [25] have proposed a second optimization once that *S* have been found. This is referred in Alg. 1 as *Removal Optimization* and its objective is twofold: to discard inactive users that do not receive data and to maximize the achievable sum rate by optimizing the beamforming weights and powers of the active users.

**Alg 1.** Generic structure of the Metric-Based User Selection
