**5. Complexity analysis**

This section aims to examine the implementation complexities of the proposed hybrid precoding and combining algorithms for various architectures. To simplify the analysis,

we use the following notations: *N* ¼ max *Nt* f g , *Nr* represents the maximum number of antennas, *N*RF ¼ maxf g *NtRF*, *NrRF* represents the maximum number of RF chains, *<sup>N</sup>*<sup>g</sup> <sup>¼</sup> max *Ntg* , *Nrg* � � represents the maximum number of RF groups, and *K* denotes the maximum number of iterations for the proposed IFA hybrid design, ISA hybrid design, and IHA hybrid design algorithms. Moreover, we denote the number of antennas for each subarray in the SA design as *N*SA. Our analysis is based on the total number of floating-point operations (flops) for each hybrid precoding and combining method. **Table 1** shows that the computational complexities of the proposed IFA hybrid design, ISA hybrid design, and IHA hybrid design algorithms are much lower compared to that of the FA sparse hybrid precoding method, which has a complexity of O(*N*<sup>2</sup> *NRFNS*). Furthermore, the computational complexities of the proposed IHA hybrid design and ISA hybrid design algorithms are lower than that of the IFA Hybrid design, particularly for larger numbers of groups, *N*g. When *N*<sup>g</sup> >1, the proposed IHA hybrid design and ISA hybrid design algorithms have lower hardware costs than the sparse hybrid design and the proposed IFA hybrid design. To summarize, the proposed IHA hybrid design has lower computational and hardware complexities than the proposed IFA hybrid design and is comparable to that of the proposed ISA hybrid design when *N*RF ¼ *N*g.


**Table 1.**

*Complexity of the proposed algorithms.*

### **6. Simulation results**

This section presents the numerical results to show the performance advantages of the proposed IFA, ISA, and IHA hybrid precoding/combining algorithms. We consider the case where there are only one BS and one MS at a distance of 100 m. The spacing between antenna elements is equal to *λ=*2. The system is assumed to operate at a 28 GHz carrier frequency in an outdoor scenario, and with a path loss exponent *n* ¼ 3*:*4. The channel model is described in (1), with Pα,i ¼ 1 for all clusters. The azimuth and elevation angles of arrival and departure (AoAs/AoDs) of the rays within a cluster are assumed to be randomly Laplacian distributed. The AoAs/AoDs azimuths and elevations of the cluster means are assumed to be uniformly distributed. We use the AoD/AoA beamforming codebooks (the exact array response of the mmWave channel) at the BSs and MSs, respectively, for the sparse hybrid design [2]. The signalto-noise ratio (*SNR*) in all the plots is defined as *SNR* <sup>¼</sup> <sup>ρ</sup>*=σ*2. We assume perfect channel estimation at the BS and MS. For fairness, the same total power constraint is enforced on all precoding/combining solutions. The maximum number of iterations *K* for the proposed IHA hybrid precoder/combiner, the IFA hybrid precoder/combiner, and the ISA hybrid precoder/combiner is equal to 10 for all data cases.

In this section, we show the spectral efficiencies achieved by the proposed IFA, ISA, and IHA hybrid precoding/combining algorithms, FA sparse hybrid design [2], and the optimal unconstrained digital method at both the BS and the MS.

**Figure 3** shows the spectral efficiencies achieved by the proposed IHA hybrid precoding/combining, the FA sparse hybrid precoding/combining [2], the optimal unconstrained digital design, the proposed IFA hybrid precoding/combining, and the proposed ISA hybrid precoding/combining in a 256 x 64 uniform planar arrays (UPAs) mmWave system for different *SNR* values with *NS*∊f g 2, 8 , *NtRF* ¼ *NrRF*∊f g 4, 16 , and *Ng*∊f g 1, 2, 4, 8, 16 . The spectral efficiency performance of the proposed IFA hybrid precoder/combiner is close to that of the unconstrained digital one and better than those of other methods for all cases. The proposed IHA hybrid precoding/combining method outperforms the ISA hybrid precoder, regardless of the number of data streams *NS* and the number of groups *Ng*. Also, the proposed IHA hybrid precoding/combining design outperforms the FA sparse hybrid design when *Ng*∊f g 1, 2 for *NS* ¼ 2 and 8. The performance of the proposed IHA hybrid precoding/combining is degraded with the increase of *Ng* , which is equivalent to the decrease of phase shifters, leading to an increase of the interference between data streams. However, when *Ng*∊f g 4, 16 , the proposed IHA hybrid

*Architectures for Hybrid Precoding and Combining Techniques in Massive MIMO Systems… DOI: http://dx.doi.org/10.5772/intechopen.112113*

**Figure 3.**

*Average spectral efficiency achieved by the proposed iterative hybrid array (IHA) precoding/combining with K = 10, compared to the full array (FA) sparse hybrid precoding/combining design [2], the optimal unconstrained digital precoding/combining, iterative full array (IFA) hybrid precoding/combining design, and the iterative subarray (ISA) hybrid precoding/combining, for a* 256 x 64 *uniform planar arrays (UPAs) mmWave system for different signal-to-noise ratio (SNR) values with NS*∊f g 2, 8 , and *NtRF* ¼ *NrRF*∊f g 4, 16 *.*

precoding/combining becomes similar to the SA architecture with better performance compared to the proposed ISA hybrid precoding/combining. Also, when *Ng* ¼ 1, the performance of IHA hybrid precoding/combining is close to that of the proposed IFA hybrid precoding/combining.

In **Figure 4**, we use the same methods as they were used in **Figure 3** in a 64 x 16 UPAs mmWave system for different SNR values, with *NS*∊f g 2, 4 , *NtRF* ¼ *NrRF*∊f g 4, 8 , and *Ng*∊f g 1, 2, 4, 8 . We obtain the same results as in **Figure 3**. However, the proposed IHA hybrid precoding/combining method overlaps with the proposed ISA hybrid precoding/combining when *Ng* ¼ 8 and 4 for *NS* ¼ 4 and 2, respectively, where the numbers of BS and MS antennas are reduced compared to **Figure 3**.

**Figure 5** shows the performance when the number of RF chains *NtRF* ¼ *NrRF* is greater than the number of data streams, where *NS*∊f g 2, 4 , *Ng*∊f g 1, 2, 4, 8 , and the SNR is fixed to 0 dB over the whole range of RF chains in a 256 x 64 UPAs mmWave system. The spectral efficiency of the proposed IFA hybrid precoding/combining is close to that of the unconstrained digital one with the increase of the RF chains. The performance of the IHA hybrid precoding/combining becomes worse with the increase of *Ng*, where the interference of data streams increases. The performance of the IHA hybrid precoding/combining is much better than that of the proposed ISA hybrid precoding/combining, regardless of the number of *Ng* . Also, the proposed IHA hybrid precoding/combining outperforms the FA sparse hybrid design when *Ng*∊f g 1, 2 for any data stream *NS*; however, the FA sparse hybrid design outperforms the proposed IHA hybrid precoding/combining when *Ng*∊f g 4, 8 , but the performance

**Figure 4.**

*Average spectral efficiency achieved by the proposed iterative hybrid array (IHA) precoding/combining with K = 10, compared to the full array (FA) sparse hybrid precoding/combining design [2], the optimal unconstrained digital precoding/combining, the iterative full array (IFA) hybrid precoding/combining design, and the iterative subarray (ISA) hybrid precoding/combining, for a* 64 x 16 *UPAs mmWave system for different signal-to-noise ratio (SNR) values with NS*∊f g 2, 4 , and *NtRF* ¼ *NrRF*∊f g 4, 8 *.*

gap between them reduces with the increase of the RF chains. Also, when *Ng* ¼ 1, the performance of IHA hybrid precoding/combining is close to that of the proposed IFA hybrid precoding/combining.

**Figure 6** shows the spectral efficiency achieved by the same methods when the number of RF chains equals the number of data streams, varying from 2 to 16, in a 256 x 64 UPAs mmWave system with *Ng*∊f g 1, 2, 4, 8 . The *SNR* is fixed to 0 dB for any number of RF chains. When *Ng* ¼ 1, the performance of IHA hybrid precoder overlaps with that of the proposed IFA hybrid precoding/combining, and both are close to the unconstrained digital one. The performance of the proposed IHA hybrid precoding/combining becomes worse with the increase of *Ng*, where the interference of data streams becomes higher. As seen in **Figure 6**, the proposed IHA hybrid precoding/combining outperforms the FA sparse hybrid precoding/combining and the proposed ISA hybrid precoding/combining, especially for a large number of *Ng*, and *NS* ¼ *NtRF* ¼ *NrRF*.

In conclusion, although we use the proposed IHA hybrid design in both transmitter and receiver, its performance is acceptable, especially for 2 ≤ *Ng* < *NtRF* and 2≤ *Ng* < *NrRF*, when compared to the higher hardware complexity of FA hybrid designs, such as the FA sparse hybrid design and the proposed IFA hybrid design. All FA hybrid designs require a higher hardware complexity in the BS and MS, with a higher number of phase shifters in the BS and MS, which is equal to *NtNtRF* þ *NrNrRF*, whereas the number of phase shifters for the IHA hybrid precoder/combiner is equal

$$\text{to } \left(\frac{N\_i N\_{iBF}}{N\_\sharp}\right) + \left(\frac{N\_r N\_{rRF}}{N\_\sharp}\right). \text{ The constraint of the analog and baseband preceding}$$

*Architectures for Hybrid Precoding and Combining Techniques in Massive MIMO Systems… DOI: http://dx.doi.org/10.5772/intechopen.112113*

#### **Figure 5.**

*Average spectral efficiency achieved by the proposed iterative hybrid array (IHA) precoding/combining with K* ¼ 10 *compared to the full array (FA) sparse hybrid precoding/combining [2], the optimal unconstrained digital precoding/combining, the iterative full array (IFA) hybrid precoding/combining with K =10, and the iterative subarray (ISA) hybrid precoding/combining with K* ¼ 10 *for* 256 x 64 *uniform planar arrays (UPA) mmWave systems for signal-to-noise ratio (SNR) = 0 dB with NS*∊f g 2, 4 *and different radio frequency (RF) chains.*

#### **Figure 6.**

*Average spectral efficiency achieved by the proposed hybrid array (HA) precoding/combining using Algorithm 2 with K* ¼ 10 *compared to the full array (FA) sparse hybrid precoding/combining [2], the optimal unconstrained digital precoding/combining, iterative full array (IFA) hybrid precoding/combining with K* ¼ 10*, and the iterative subarray (ISA) hybrid precoding/combining with K* ¼ 10 *for* 256 x 64 *uniform planar arrays (UPAs) mmWave systems for signal-to-noise ratio (SNR) = 0 dB with NS* ¼ *NtRF* ¼ *NtRF.*

combining matrices helps to build the structure of block diagonal matrices in the proposed IHA hybrid precoder/combiner, yielding higher gains compared to the other methods. When *Ng* ¼ *NtRF*, the proposed HA structure becomes similar to the SA one; the performance of the proposed IHA hybrid precoding/combining design gives higher gains compared to the proposed ISA hybrid precoding/combining design, especially for a large number of BS antennas.

Also, when *Ng* ¼ 1, the proposed HA structure becomes similar to the FA one, and the performance of the proposed IHA hybrid precoding/combining design is comparable to that of the proposed IFA hybrid precoding/combining design. The number of iterations should be 10 or less because the gain after that will be very small, which is confirmed by our results that we did not include in this chapter.
