(4.21) are, in general, not optimal. Therefore, these precoders are discarded, and the optimal digital precoders for the fixed analog precoding matrix ˆAand the set of selected users K0 are computed by solving the problem
minimize {dk}k∈K0 X k∈K0 dHkdk (4.26a) subject to dHkAˆHRkAdˆ k P m∈K0\kdHmAˆHRkAdˆ m+ 1 ≥ ωk, ∀k ∈ K0. (4.26b)
Similar to problem (4.20), the above problem can be efficiently solved using any stan- dard convex optimization tool [BV04] or the iterative method in [SB04].
4.4
Numerical Results
In this section, we numerically evaluate the proposed hybrid precoding techniques. In the first part, the performances of the proposed uplink-downlink duality-based pre- coding techniques are compared for various system parameters. The computational complexities of various analog beamformer selection algorithms are also numerically analyzed. In the second part, the performances of the proposed joint user selection and hybrid precoding techniques are compared with that of the conventional methods.
4.4.1
Hybrid Precoding Using Uplink-Downlink Duality
For the simulations, we consider a multiuser downlink MIMO system with N = 16 transmit antennas and K = 4 users. A discrete Fourier transform (DFT) codebook C with C = 16 orthogonal analog precoders is adopted. The Rayleigh fading channels are assumed with zero mean and unit variance. The noise power at each user is normalized to unity. The simulation results are averaged over 1000 Monte Carlo runs.
Transmit Power vs. SINR: Figure 4.1 plots the total transmit power (in watts) required by the proposed low-complexity suboptimal algorithm, given in Alg. 2, with three proposed analog beamformer selection algorithms for a range of SINR targets (in dB). In the figure, the total transmit power needed by the optimal exhaustive search
54 Chapter 4: Interference Suppression-Based Hybrid Precoding −4 −2 0 2 4 6 8 10 0 2 4 6 SINR target (dB) T otal transmit p o w er (w atts) Fully-digital precoding Exhaustive search method Deflation alg.
Greedy correction alg. R-critical selection alg.
Figure 4.1. Total transmit power vs. SINR for a system with N = 16, R = 6, K = 4, and 16 × 16 DFT codebook.
method acts as a benchmark for the suboptimal hybrid precoding techniques. The number of RF chains R in the hybrid precoding is set to 6. The figure also includes the total transmit power required for the fully-digital precoding, with the number of RF chains R = N.
The figure shows that the total transmit powers required by the proposed sub- optimal method with deflation algorithm and greedy correction algorithm are almost identical to the optimal power achieved with the exhaustive search method. We notice that even the performance of the R-critical selection algorithm, whose computational complexity is considerably lower than that of the exhaustive search method, is remark- ably close to the optimal performance. The figure also reveals the performance gap between the fully-digital precoding and hybrid precoding. Although the fully-digital precoding leads to considerable transmit power savings when compared to the hybrid precoding, it needs a dedicated RF chain for each antenna element, making it unattrac- tive for large-scale systems.
Transmit Power vs. Number of RF Chains: Figure 4.2 compares the transmit power associated with the proposed suboptimal algorithm with three proposed analog beamformer selection methods for various number of RF chains R. It plots the extra
4.4 Numerical Results 55 6 7 8 9 10 27 28 29 30 Number of RF chains (R) Extra po w er (dBm) Deflation alg.
Greedy correction alg.
R-critical selection alg.
Figure 4.2. Extra power (in dBm) required for the suboptimal method with three proposed analog beamformer selection algorithms compared to the optimal transmit power to achieve an SINR target of 8 dB, in a system with N = 16, K = 4, and 16×16 DFT codebook.
transmit powers (in dBm) required by these algorithms compared to the optimal trans- mit power when the SINR targets of all the users are set to 8 dB. The figure depicts slightly superior performance of the deflation and greedy correction algorithms over the computationally simpler R-critical selection algorithm. Moreover, the figure also illustrates the influence of adding more RF chains on the total transmit power.
Computational Complexity Analysis: Table 4.1 lists the average number of analog beamformer updates and digital beamformer & power updates required for the conver- gence of the suboptimal method with three analog beamformer selection algorithms. In the table, we notice that the proposed method converges in few iterations—only one analog beamformer update is sufficient for the deflation algorithm, whereas an average of four analog beamformer updates is adequate for greedy correction and R- critical selection algorithms. Moreover, the number of digital beamformer and power updates required for the proposed iterative method to converge is less than ten for all the algorithms.
The average CPU time (in seconds) consumed by the exhaustive search method and the suboptimal method are listed in Table 4.2. The simulations are conducted
56 Chapter 4: Interference Suppression-Based Hybrid Precoding
Table 4.1. Average number of updates required in the suboptimal method for conver- gence when N = 16, K = 4, R = 6, SINR target = 5 dB.
— Deflation Greedy correction R-critical selection
Analog beamformer
update 1.00 3.54 3.55
Digital beamformer
& power update 6.76 8.52 9.60
Table 4.2. Average CPU time (in seconds) required by the optimal exhaustive search method and the suboptimal method for N = 16, K = 4, R = 6, SINR target = 5 dB.
Exhaustive search
method Deflation Greedy correctionSuboptimal method R-critical selection
50.06 0.1539 0.1551 0.0427
on a system having the following features: Intel (R) Core (TM) i7-4790K CPU @ 4.00GHz, Arch Linux 4.16.8, MATLAB 2018b. From the table, which also comple- ments the analytical results of Section 4.2.3, we conclude that the proposed suboptimal method substantially reduces the computational complexity of the hybrid precoding when compared to the optimal exhaustive search method, without significant perfor- mance compromise. From Figure 4.1, Figure 4.2, and Table 4.2 we infer that the three analog beamformer selection methods offer performance-complexity trade-off, where the deflation method provides slightly better performance with the cost of higher com- putational time and the R-critical selection method offers the lowest computational time with relatively poor performance.
4.4.2
Joint User Selection and Hybrid Precoding
In this section, we numerically evaluate the proposed joint user selection and hybrid precoding. We compare the performance of the suboptimal method proposed in Sec- tion 4.3.1 with the optimal solution of problem (4.19), which is obtained through the brute-force method. We also evaluate the benefits of the proposed column-norm-based
4.4 Numerical Results 57 3 4 5 6 0.2 0.4 0.6 0.8 1 1.2
No. of selected users (K0)
Total transmit po w er (w atts) Fully-digital precoding Exhaustive search Proposed approach Row-norm-based approach Frobenius-norm-based approach
Figure 4.3. Total transmit power vs. number of selected users (K0) for N = 10, C = 10,
R = 6, K = 6, and SINR = 5 dB.
sparse regression technique in problem (4.21) against the standard Frobenius-norm- based sparse regression in implementing the suboptimal method. In addition, we also consider row-norm-based sparse regression, where we use f(F?− CX) = ||F?− CX||
2,1
in the objective function of problem (4.21).
Transmit Power vs. Number of Selected Users: Figure 4.3 plots the total transmit power (in watts) required by the proposed suboptimal method with column- norm-based sparse regression for various number of selected users K0. The figure
also includes the transmit power resulted when the column-norm is replaced by the Frobenius-norm and the row-norm in the suboptimal method. The performance of the optimal exhaustive search method serves as a benchmark for the suboptimal methods. Moreover, the transmit power needed for the fully-digital precoding (R = N) is also included in the figure for reference. For the simulation, we consider a multiuser MIMO system with N = 10 transmit antennas, R = 6 RF chains, K = 6 users, and SINR target of 5 dB at each user. The analog precoders are selected from a 10 × 10 DFT codebook. The Rayleigh fading channel is assumed between the BS and the users.
The figure illustrates that the suboptimal method yields a slightly superior result when the sparse regression is performed by employing the proposed column-norm func-
58 Chapter 4: Interference Suppression-Based Hybrid Precoding 4 5 6 7 8 9 10 0.2 0.3 0.4 0.5 0.6
No. of selected users (K0)
T otal transmit p o w er (w atts) Proposed approach Row-norm-based approach Frobenius-norm-based approach
Figure 4.4. Total transmit power vs. number of selected users (K0) for N = 32, C = 32,
R = 10, K = 10, and SINR = 5 dB.
tion instead of the standard Frobenius-norm function or row-norm function. We note that the performance of the exhaustive search method is considerably better than that of the suboptimal method. Nonetheless, the exponentially growing complexity of the exhaustive search method makes it impractical for practical systems. Figure 4.4 plots the total transmit power vs. K0 for a relatively larger system with N = 32 transmit
antennas, R = 10 RF chains, and K = 10 users. This figure reaffirms the consistent superiority of the proposed column-norm approach compared to the competing ap- proaches. We found out that the problem (4.19) cannot be solved using the exhaustive search method in an acceptable time duration.
Transmit Power vs. SINR: Figure 4.5 plots the total transmit power (in watts) required by the suboptimal method with the column-norm, the Frobenius-norm and row-norm-based sparse regression techniques over a range of SINR targets (identical at each user). For the simulation we considered N = 32 transmit antennas, R = 10 RF chains, 32 × 32 DFT codebook, K = 10 total users, and K0 = 6 selected users.
The figure reveals that the benefit of the column-norm-based sparse regression is more pronounced for larger SINR requirements.
4.5 Conclusion 59