**7. Numerical examples**

there exist more degrees of freedom (DoF) at the BS to mitigate inter-users interference. This approach may face two drawbacks: 1) the achievable fairness at low SNR and low *K* regimes since less users are scheduled; 2) the complexity in evaluating at most *L <sup>K</sup>* times the achievable sum rate. The fairness issue can be tackled by changing the global utility function, e.g., maximizing the weighted sum rate in (3) using the proportional fairness approach [21], [25], [39]. The complexity problem can be alleviated by exploiting the properties of water-filling in order to reduce the computational load of some matrix operations. Further complexity gains can be obtained if the set of candidate users Ω is optimized each iteration, reducing the total number of sum rate evaluations. This refinement proposed in [10], [18] is called *Search Space Pruning* and is based on the fact that if water-filling allocates *pi* =0 to the *i*th user in Ω(*n*) at the *n*th iteration, such user cannot achieved a nonzero power allocation in future iterations.

The sum rate maximization formulation in (4) is a combinatorial problem subject to mixed constraints. Some works in the literature reformulate the original problem similarly to the metric-based approaches, performing user selection (*S*) and resource allocation (**w***<sup>i</sup>* and *pi* ∀ *i* ∈*S* ) as independent processes [25], [26]. The NSP in Section 5.1 can be used to refor‐ mulate the utility-based selection presented in Alg. 1 as an integer program with linear objective function and linear constraint [26]. Let us consider that for each user *i* ∈Ω there exists

that computes the NSP assuming pairwise orthogonality with user

**Alg. 2.** Generic structure of the Utility-Based User Selection

42 Contemporary Issues in Wireless Communications

**6.3. User selection via integer optimization**

a function *f <sup>i</sup>*

In this section we compare the average sum rate achieved when optimal and suboptimal user selection is performed over different metrics of **H**(*S*). The simulations consider perfect CSIT, fading channels are generated following a complex Gaussian distribution with unit variance and the average sum rate is given in (bps/Hz). Since we evaluate system performance via Shannon capacity expressions in (6) and (7), the results are independent of the specific implementation on the coding and modulation schemes, which provides us a general design insight**.**

The curves displayed in Fig. 3 are obtained by optimally solving the combinatorial problems introduced in Section 5 whose optimum user sets are employed to evaluate (4) with ZFBF. Notice that these results are upper bounds of the average sum rate for each metric, which implies that any greedy user selection algorithm can achieve at most the same performance for its optimized metric. The sum rate achieved by *S<sup>ω</sup>* in (9) has a negligible performance gap regarding to the optimum set *S* <sup>⋆</sup> for low values of *K*, and such gap rapidly vanishes as *K* →*∞* with a cost of matrix product and inversion for each possible set *S* ⊆ Ω. Due to the properties of water-filling the sum rate is maximized when the terms *bi* have larger and uniform values. For low values of *K* there exists a performance gap between *<sup>S</sup>* <sup>⋆</sup> and *Sω* because the probability that the terms *bi* have large uniform values is small. As *K* grows this probability increases and the performance gap vanishes. The average sum rate achieved by *S<sup>ξ</sup>* in (22) computes a lower bound of the NSP metric (8) with a performance gap not larger than 1% regarding to *S* <sup>⋆</sup> for *K* =6 and such gap vanishes as *K* grows. This result suggest that (21) is a good candidate metric for user selection since it achieves a good trade-off between complexity and performance. The fact that *δ*(**H**(*S*)) cannot identify with accuracy the set that maximizes the sum rate is because this metric only reflects the degradation of the product of the eigenvalues of **H**¯(*S*) with respect to the channel gains. Consider that for a given channel instance the optimum solution of (27) yields *δ*(**H**(*Sδ*)) =1 which may indicate that **H**(*Sδ*) forms an orthogonal basis yet this results does not provide information regarding the channel magnitudes. Observe that the optimum channel matrix **H**(*S* <sup>⋆</sup>) does not achieve perfect *ε*-orthogonality or pairwise orthogonality among its components. The sum rate attained with the solution of problems (27), (29), and (31) degrades as *K* grows. This is due to the fact that for large multiuser diversity the probability of finding a set of users that minimizes the dispersion of the eigenvalues of **H**¯(*S*) increases but this does not imply that the maximum sum of effective channel gains or transmitted power is achieved. Metric (23) improves its performance when *K* →*∞* since the probability that **H**(*S*Λ) achieves both large sum of effective channel gains and pairwise uncorrelated channels is a function of the multiuser diversity. The performance gap between *S*Λ regarding *<sup>S</sup>* <sup>⋆</sup> ranges from 12% for *K* =6 to less than 1% for *K* =100. The advantage of metric (23) is that only the computation of correlation coefficients in (13) is required avoiding matrix operations. The performance gap between the solutions of (24) and (9) depends on the spatial resources at the transmitter since the probability that two independent channels **h***<sup>i</sup>* and **h** *<sup>j</sup>* are correlated is a decreasing function of *Nt*. For the *ε*-orthogonality optimization in (25), the set *S* may be not unique since two different sets may containing the same maximum correlation coefficient but their achievable sum rates may be quite different. Due to the fact that channel magnitudes are neglected, problem (4) is solved without fully exploiting multiuser diversity.

User Selection and Precoding Techniques for Rate Maximization in Broadcast MISO Systems http://dx.doi.org/10.5772/58937 45

**Figure 3.** Average sum rate versus the number of users *K* with SNR=18(*dB*) and *Nt* =4.

and the average sum rate is given in (bps/Hz). Since we evaluate system performance via Shannon capacity expressions in (6) and (7), the results are independent of the specific implementation on the coding and modulation schemes, which provides us a general design

The curves displayed in Fig. 3 are obtained by optimally solving the combinatorial problems introduced in Section 5 whose optimum user sets are employed to evaluate (4) with ZFBF. Notice that these results are upper bounds of the average sum rate for each metric, which implies that any greedy user selection algorithm can achieve at most the same performance for its optimized metric. The sum rate achieved by *S<sup>ω</sup>* in (9) has a negligible performance gap regarding to the optimum set *S* <sup>⋆</sup> for low values of *K*, and such gap rapidly vanishes as *K* →*∞* with a cost of matrix product and inversion for each possible set *S* ⊆ Ω. Due to the properties

For low values of *K* there exists a performance gap between *<sup>S</sup>* <sup>⋆</sup> and *Sω* because the probability

the performance gap vanishes. The average sum rate achieved by *S<sup>ξ</sup>* in (22) computes a lower bound of the NSP metric (8) with a performance gap not larger than 1% regarding to *S* <sup>⋆</sup> for *K* =6 and such gap vanishes as *K* grows. This result suggest that (21) is a good candidate metric for user selection since it achieves a good trade-off between complexity and performance. The fact that *δ*(**H**(*S*)) cannot identify with accuracy the set that maximizes the sum rate is because this metric only reflects the degradation of the product of the eigenvalues of **H**¯(*S*) with respect to the channel gains. Consider that for a given channel instance the optimum solution of (27) yields *δ*(**H**(*Sδ*)) =1 which may indicate that **H**(*Sδ*) forms an orthogonal basis yet this results does not provide information regarding the channel magnitudes. Observe that the optimum channel matrix **H**(*S* <sup>⋆</sup>) does not achieve perfect *ε*-orthogonality or pairwise orthogonality among its components. The sum rate attained with the solution of problems (27), (29), and (31) degrades as *K* grows. This is due to the fact that for large multiuser diversity the probability of finding a set of users that minimizes the dispersion of the eigenvalues of **H**¯(*S*) increases but this does not imply that the maximum sum of effective channel gains or transmitted power is achieved. Metric (23) improves its performance when *K* →*∞* since the probability that **H**(*S*Λ) achieves both large sum of effective channel gains and pairwise uncorrelated channels is a function of the multiuser diversity. The performance gap between *S*Λ regarding *<sup>S</sup>* <sup>⋆</sup> ranges from 12% for *K* =6 to less than 1% for *K* =100. The advantage of metric (23) is that only the computation of correlation coefficients in (13) is required avoiding matrix operations. The performance gap between the solutions of (24) and (9) depends on the spatial resources at the

have large uniform values is small. As *K* grows this probability increases and

have larger and uniform values.

and **h** *<sup>j</sup>* are correlated is a

of water-filling the sum rate is maximized when the terms *bi*

transmitter since the probability that two independent channels **h***<sup>i</sup>*

neglected, problem (4) is solved without fully exploiting multiuser diversity.

decreasing function of *Nt*. For the *ε*-orthogonality optimization in (25), the set *S* may be not unique since two different sets may containing the same maximum correlation coefficient but their achievable sum rates may be quite different. Due to the fact that channel magnitudes are

insight**.**

44 Contemporary Issues in Wireless Communications

that the terms *bi*

In Fig. 4 and Fig. 5 we compare several user selection algorithms that find sub-optimal solution to (4), namely the metric-based selection using the NSP metric proposed in [21] using GSO, [19], [35] using SVD, and the NSP approximation based on (23) proposed in [26]. The optimal solution of (4) is found by exhaustive search subject to |*S* | = *Nt*, i.e., maximum multiplexing gain. In order to highlight the contribution of multiuser diversity we compare performance with respect to two simplistic user selection approaches, one based on the maximum channel gain (MCG) criterion (selecting the *Nt* users with higher channels norms), and a second approach performing round robin user scheduling (RRS) policy. We compare the performance of the metric-based selection in Section 6.1 with two greedy utility-based (sum rate) algorithms in Section 6.2 [18] and [40], and with the integer linear program (ILP) selection described in Section 6.3.

In Fig. 4 we compare the average sum rate as a function of the number of users for different user selection strategies for both precoding techniques. For the case of ZFBF Fig. 4 (a) and *K* <10, the solution set of the utility-based selection in [18] is achieved with a cardinality less than *Nt* compared to the optimum selection. This suggest that when *K* ≈ *Nt* is more efficient to select the users in a greedy fashion as described in Section 6.2, than to per‐ form the selection based on channel metrics. When the number of competing users is large *K* ≫*Nt*, the performance gap between the three user selection approaches described in Section 6 decreases and all techniques achieve maximum multiplexing gain. For the large K regime, the metric-based selection achieves a performance close to the utility-based selection with less computational effort. Moreover, the performance of the NSP approxima‐ tion in [26] converges to one of [19], [21], [35] with more gains in terms of computational complexity. In the case of ZFDP Fig. 4 (b) all approaches in Section 6 achieve full multiplex‐ ing diversity regardless the regime of *K*. This is due to the fact that ZFDP can efficiently use all spatial resources at the transmitter for the high SNR regime. The utility and metric based selection techniques achieve the same performance over all range of *K*. The NSP approximation metric in [26] achieves more than 95% of the optimum capacity and the performance gap reduces as the number of users increases. Notice that the approach in Section 6.3 optimally solves problem (24) and its performance is an upper bound for the user selection in [26]. The performance of all approaches presented in Section 6 is suboptimal, yet they represent different and acceptable trade-offs between sum rate perform‐ ance, computational complexity, and multiplexing gain for different values of *K* and the SNR regime.

In Fig. 5 the average sum rate is a function of the SNR for ZFBF and *K* =10. The results show that for *P* <*P*<sup>0</sup> ≈10(dB) the utility-based selection is more efficient than the metricbased selection in the low SNR regime. In the high SNR regime the performance gap between the optimum selection and the approaches in Section 6 is less than 10%. Observe that even the NSP approximation metric in [26] can efficiently exploit multiuser diversity and the approaches based exclusively on channel magnitudes (MCG) and random selec‐ tion incur in a large performance degradation. Comparing the performance of the utilitybased versus the metric-based user selection is in general not fair since the former will optimize directly its utility function while the latter attempts to reduce computational complexity by optimizing a simpler utility function. The performance gap between the two approaches is large for low SRN and *K* regimes. The reason is that the former directly optimizes the set of selected users while the latter indirectly estimates if a given set can maximize the sum rate. In the high SNR regime, the gap between both approaches diminishes and results in [18] show that the achievable sum rate in both approaches is proportional to the number spatial resources *Nt* at the transmitter.

**Figure 4.** Average sum rate versus the number of users (*K*) with SNR =18 (*dB*) and *Nt* =4 for (a) ZFBF and (b) ZFDP.

User Selection and Precoding Techniques for Rate Maximization in Broadcast MISO Systems http://dx.doi.org/10.5772/58937 47

**Figure 5.** Average sum rate versus the SNR for the with *K* =10, *Nt* =4, and ZFBF.
