# Low Complexity Hybrid Precoding Designs for Multiuser mmWave/THz Ultra Massive MIMO Systems

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## Abstract

**:**

## 1. Introduction

_{s}) is equal to the number of users (N

_{u}), it is shown that the algorithms achieve interesting results when compared to the fully-digital solution. The concept of precoding based on adaptive RF-chain-to-antenna was only introduced in [16] for SU scenarios, but showed promising results. In [17], a nonlinear hybrid transceiver design relying on Tomlinson–Harashima precoding was proposed. Their approach only considers Fully-Connected (FC) architectures but can achieve a performance close to the fully-digital transceiver. A Kalman-based Hybrid Precoding method was proposed for MU scenarios in [18]. While designed for systems with only one stream per user and based on fully-connected structures, the performance of the algorithm is competitive with other existing solutions. A hybrid MMSE-based precoder and combiner design with low complexity was proposed in [19]. The algorithm is designed for MU-MIMO systems in narrowband channels, and it presents lower complexity and better results when compared to Kalman’s precoding. Most of the hybrid solutions for mmWave systems aim to achieve near-optimal performance using FC structures, resorting to phase shifters or switches. However, the difficulty of handling the hardware constraint imposed by the analog phase shifters or by switches in the THz band is an issue that limits the expected performance in terms of SE.

- We propose a hybrid design algorithm with near fully-digital performance, where the digital precoder, analog precoder and multiuser interference mitigation are computed separately through simple closed-form solutions. Even though the hybrid design algorithm is developed independently of a specific channel or antenna configuration, it is particularly suitable for mmWave and THz systems where, on the one hand, very large antenna arrays are required to overcome distance limitations but, on the other hand, current hardware constraints in terms of cost and power consumption make the adoption fully-digital precoders/combiners with one dedicated RF chain per antenna element unviable. Whereas our previous work [10] also proposed a hybrid design algorithm for mmWave, it did not address multiuser systems, and in particular the MIMO broadcast channel. Therefore, it does not include any step for inter-user interference mitigation within its design. As we show here, for this multiuser channel, the hybrid design method must also deal with the residual inter-user interference as it can degrade system performance, particularly at high Signal Noise Ratios (SNRs);
- Due to the separability of the different steps (analog precoder, digital precoder and interference suppression), the proposed algorithm can incorporate different architectures, making it suitable for supporting UM-MIMO in severely hardware-constrained systems typical in the THz band. Unlike [10], where we only considered the adoption of phase shifters, in this paper we present explicit solutions for some of the most common architectures, namely FC, AoSA and DAoSA structures based in either Unquantized Phase Shifters (UPS), Quantized Phase Shifters (QPS), Switches (Swi), Switches and Inverters (SI), Antenna Selection (AS) or Double Phase Shifters (DPS);
- To cope with the large bandwidths available in mmWave/THz bands, where practical MIMO systems likely have to operate in frequency selective channels, the proposed hybrid design considers the application in a multicarrier context, where the same analog precoder is applied at different frequencies;
- We explicitly show how the proposed design can be applied to a DAoSAs approach where a reduced number of switches are inserted at each AoSA panel, which allows the connections to the RF chains to be dynamically adjusted. Through extensive simulations, it is shown that our proposed solution is capable of achieving good trade-offs between spectral efficiency, hardware complexity and power consumption, proving to be a suitable solution for the deployment of UM-MIMO, especially in hardware-constrained THz systems.

## 2. System Model

_{c}is the central frequency and γ is a normalizing factor such that $\mathsf{{\rm E}}\left[{\Vert {H}_{k,u}\Vert}_{F}^{2}\right]={N}_{tx}{N}_{rx}$. Vectors ${a}_{t}({\varphi}_{i,l,u}^{t},{\theta}_{i,l,u}^{t})$ and ${a}_{r}({\varphi}_{i,l,u}^{r},{\theta}_{i,l,u}^{r})$ represent the transmit and receive antenna array responses at the azimuth and elevation angles of $({\varphi}_{i,l,u}^{t},{\theta}_{i,l,u}^{t})$ and $({\varphi}_{i,l,u}^{r},{\theta}_{i,l,u}^{r}),$ respectively. Vectors ${a}_{t}({\varphi}_{u}^{t,LOS},{\theta}_{u}^{t,LOS})$ and ${a}_{r}({\varphi}_{u}^{r,LOS},{\theta}_{u}^{r,LOS})$ have similar meanings but refer to the LOS path angles $({\varphi}_{u}^{t,LOS},{\theta}_{u}^{t,LOS})$ and $({\varphi}_{u}^{r,LOS},{\theta}_{u}^{r,LOS})$. By carefully selecting the parameters of the channel model we can make it depict a mmWave or a THz channel. Considering Gaussian signaling, the spectral efficiency achieved by the system for the transmission to MS-u in subcarrier k is [28]

## 3. Proposed Hybrid Design Algorithm

#### 3.1. Main Algorithm

**R**, combined with the use of the indicator function. The indicator function for a generic set $\mathcal{A}$ is defined as ${I}_{\mathcal{A}}(x)$, returning 0 if $x\in \mathcal{A}$ and +∞ otherwise. A similar approach can be adopted for integrating the other constraints, Equations (11) and (12), also into the objective function. The optimization problem can then be rewritten as

**R**and ${B}_{k}$. The minimization of Equation (18) with respect to

**R**and ${B}_{k}$ can be written as

**A**to denote $A={F}_{\mathrm{RF}}^{(t+1)}{F}_{{\mathrm{BB}}_{k,u}}^{(t+1)}+{Z}_{k,(u-1){N}_{s}+1:u{N}_{s}}^{(t)}$. The procedure to compute the projection of matrix

**A**onto the null-space of ${\overline{H}}_{k,u}$ can be formulated as another optimization problem, which can be expressed as

**X**, one can perform a single value decomposition of ${\overline{H}}_{k,u}$ and then use this to remove the projection of

**A**onto the row space of ${\overline{H}}_{k,u}$. Finally, the expressions for the update of dual variables

**U**,

**W**and

**Z**are given by

#### 3.2. Analog RF Precoder/Combiner Structure

**R**in step 5 of the precoding algorithm has to be implemented according to the specific analog beamformer [6,20,33,34,35,36,37]. This makes the proposed scheme very generic, allowing it to be easily adapted to different RF architectures. In the following, we will consider a broad range of architectures that can be adopted for the RF precoder for achieving reduced complexity and power consumption implementations. We will consider FC, AoSA and DAoSA structures as illustrated in Figure 2, where we assume single phase shifters (SPS). Besides SPS, we will also consider other alternative implementations for these structures, as illustrated in Figure 3 for AoSA. The different solutions either rely on selectors, switches, inverters or phase shifters, or combinations of these. The overall analog structure is defined as a combination of one of the architectures in Figure 2 with either SPS or one of the alternatives illustrated in Figure 3.

- (1)
- Unquantized Phase Shifters (UPS)

- (2)
- Quantized Phase Shifters (QPS)

- (3)
- Double Phase Shifters (DPS)

- (4)
- Switches (Swi)

- (5)
- Switches and Inverters (SI)

- (6)
- Antenna Selection (AS)

**X**as 0 except for ${X}_{{t}_{j},j}=1$, where ${t}_{j}$ is the row position with the highest real component in column j:

- (7)
- Array-of-Subarrays (AoSAs)

_{max}consecutive subarrays. In this case, the RF constraint set comprises matrices where each column has a maximum of L

_{max}blocks of ${N}_{tx}^{SA}$ constant modulus elements, with all the remaining elements being zero. Defining $X={F}_{\mathrm{RF}}^{(t+1)}+{W}^{(t)}$, the projection can be implemented by setting all the elements in

**X**as 0 except for the subblocks in each column j which fulfill

_{max}and $j=1,\dots ,{N}_{RF}$. In this case, the corresponding elements of

**R**are set as ${R}_{i,j}^{(t+1)}=({X}_{i,j}^{})/\left|{X}_{i,j}^{}\right|$, assuming UPS in these connections. Clearly, the phase shifters can be replaced by any of the other alternatives presented previously.

- (8)
- Dynamic Array-of-Subarrays (DAoSAs)

_{max}RF chains (which can be non-adjacent). In this case, the constraint set comprises matrices where each ${N}_{tx}^{SA}\times {N}_{RF}$ component submatrix contains a maximum of L

_{max}columns with constant modulus elements. The rest of the matrix contains only zeros. In this case, starting with X = 0, the projection can be obtained by selecting the L

_{max}columns of

_{1}-norm and setting the corresponding elements of R as ${R}_{i,j}^{(t+1)}=({X}_{i,j}^{})/\left|{X}_{i,j}^{}\right|$, assuming the use of UPS. Care must be taken to guarantee that at least one sub-block will be active in every column of R. Similarly to the AoSA, the phase shifters can be replaced by any of the other presented alternatives.

#### 3.3. Complexity

_{tx}will tend to be very large, it means the algorithms with higher complexity will typically be EBE and the one proposed in this paper due to the terms $\mathcal{O}(Q{N}_{tx}{}^{2})$ and $\mathcal{O}(Q{N}_{tx}{}^{2}{N}_{u}{N}_{s})$. It is important to note, however, that while the computational complexity of these two design methods may be higher, both algorithms can be applied to simple AoSA/DAoSA architectures. In particular, the proposed approach directly supports structures with lower practical implementation complexity (and are more energy-efficient) such as those based on switches. Furthermore, in a single-user scenario, the interference cancellation step of the proposed algorithm is unnecessary, and the complexity reduces to $\mathcal{O}(Q({N}_{u}{N}_{s}{N}_{RF}{N}_{tx}+{N}_{RF}{}^{2}{N}_{tx}))$. Regarding the other algorithms, they have similar complexities. However, the AM-based algorithm is designed for single stream scenarios whereas the others consider multiuser multi-stream scenarios.

## 4. Numerical Results

#### 4.1. Fully-Connected Structures

#### 4.2. Reduced Complexity Architectures

_{max}) in the performance of these schemes. Each subarray has a size of 32 antennas (n

_{t}). Curves assuming the use of SPS as well as of DPS are included. It can be observed that the increase in the number of connections to subarrays, L

_{max}, has a dramatic effect on the performance, resulting in a huge improvement by simply going from L

_{max}= 1 to L

_{max}= 2. Increasing further to L

_{max}= 4, the results become close to the fully-connected case, showing that the DAoSA can be a very appealing approach for balancing spectral efficiency with hardware complexity and power consumption. Combining the increase in L

_{max}with the adoption of DPS can also improve the results but the gains become less pronounced for L

_{max}> 1. It is important to note that the penalty parameters can be fine-tuned for different system configurations.

_{BB}is the power of the baseband block (with N

_{BB}= 1), P

_{DAC}is the power of a DAC, P

_{OS}is the power of an oscillator, P

_{M}is the power of a mixer, P

_{PA}is the power of a power amplifier, P

_{PC}is the power of a power combiner, P

_{PS}is the power of a phase shifter, P

_{SWI}is the power of a switch and P

_{tx}denotes the transmit power. The N

_{x}variable represents the number of elements of each device used in the precoder configuration.

_{BB}= 200 mW, P

_{DAC}= 110 mW, P

_{OS}= 4 mW, P

_{M}= 22 mW, P

_{PA}= 60 mW, P

_{PC}= 6.6 mW, P

_{SWI}= 24 mW and P

_{T}= 100 mW.

_{PS}= 10, 20, 40 and 100 mW for 1, 2, 3 and 4 quantization bits. Considering the same configuration scenario as Figure 7, Figure 8 and Figure 9 with ${N}_{u}=4$, ${N}_{s}=2$, ${N}_{RF}^{tx}={N}_{u}{N}_{s}$, $F=1$ and ${N}_{tx}=256$, we provide the values of power consumption for different precoder configurations in Table 3.

_{PS}= 100 mW, which corresponds to quantized phase shifters with N

_{b}= 4 bits [38]. For the remaining phase-shifter-based precoder structures, we assumed that P

_{PS}= 40 mW, which corresponds to quantized phase shifters with N

_{b}= 3 bits, since with only 3 bits resolution the results are already very close to the UPS curve (see Figure 7). As can be seen from this table, the use of architectures based on DAoSAs allows us to reduce considerably the amount of power that is consumed by the precoder. In fact, we can reduce the amount of consumed power up to 55% if we consider a precoder scheme based on DAoSA with DPS and L

_{max}= 4 versus an FC structure precoder based on UPS, with only a small performance penalty (Figure 9). This reduction increases to 73% if the DPS structure is replaced by an SPS one.

_{max}, is changed. Figure 10 refers to an SU scenario (${N}_{u}=1$) whereas Figure 11 corresponds to an MU scenario with ${N}_{u}=4$. In the SU case, the proposed precoder achieves results very close to the fully-digital precoder, even with only L

_{max}= 2. Compared to the proposed algorithm, EBE shows a wider gap even though it has a smaller complexity (as presented in Table 2 of Section 3.3).

## 5. Conclusions

_{max}= 4 versus an FC structure based on UPS, with only a small performance penalty. This reduction increases to 73% if the DPS structure is replaced by an SPS one.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Alternative implementations to single phase shifters based on array-of-subarrays for a mmWave/THz MIMO system.

**Figure 4.**Spectral efficiency versus SNR achieved by different methods with ${N}_{u}=4$, ${N}_{s}=1$, ${N}_{RF}^{tx}=4$, $F=1$, ${N}_{tx}=100$ and ${N}_{rx}=4$ (only NLOS).

**Figure 5.**Spectral efficiency versus SNR achieved by different methods with ${N}_{u}=4$, ${N}_{s}=1$, ${N}_{RF}^{tx}=8$, $F=64$, ${N}_{tx}=100$ and ${N}_{rx}=4$ (only NLOS).

**Figure 6.**Spectral efficiency versus SNR achieved by different methods with ${N}_{u}=2$, ${N}_{s}=2$, ${N}_{RF}^{tx}=4$, $F=64$, ${N}_{tx}=256$ and ${N}_{rx}=4$ (with LOS component).

**Figure 7.**Spectral efficiency versus SNR achieved by the proposed precoder using different fully-connected architectures for ${N}_{u}=4$, ${N}_{s}=2$, ${N}_{RF}^{tx}=8$, $F=1$, ${N}_{tx}=256$ and ${N}_{rx}=4$ (only NLOS).

**Figure 8.**Spectral efficiency versus SNR achieved by the proposed precoder using different AoSA architectures with L

_{max}= 1, ${N}_{u}=4$, ${N}_{s}=2$, ${N}_{RF}^{tx}=8$, $F=1$, ${N}_{tx}=256$ and ${N}_{rx}=4$ (with LOS component).

**Figure 9.**Spectral efficiency versus SNR achieved by the proposed precoder considering an architecture based on DAoSAs and the variation of the maximum number of subarrays that can be connected to an RF chain (${L}_{max}$) for ${N}_{u}=4$, ${N}_{s}=2$, ${N}_{RF}^{tx}=8$, $F=1$, ${N}_{tx}=256$ and ${N}_{rx}=4$ (with LOS component).

**Figure 10.**Spectral efficiency versus SNR achieved by the proposed precoder and by the EBE algorithm considering an architecture based on DAoSAs and the variation of the maximum number of subarrays that can be connected to an RF chain (L

_{max}) for ${N}_{u}=1$, ${N}_{s}=2$, ${N}_{RF}^{tx}=8$, $F=1$, ${N}_{tx}=256$ and ${N}_{rx}=4$ (with LOS component).

**Figure 11.**Spectral efficiency versus SNR achieved by the proposed precoder and by the EBE algorithm considering an architecture based on DAoSAs and the variation of the maximum number of subarrays that can be connected to an RF chain (L

_{max}) for a mmWave/THz system with ${N}_{u}=4$, ${N}_{s}=2$, ${N}_{RF}^{tx}=8$, $F=1$, ${N}_{tx}=256$ and ${N}_{rx}=4$ (with LOS component).

**Figure 12.**Spectral efficiency versus SNR achieved by different methods for a mmWave/THz MIMO-OFDM system with ${N}_{u}=4$, ${N}_{s}=1$, ${N}_{RF}^{tx}=4$, $F=1$, ${N}_{tx}=100$ and ${N}_{rx}=4$ considering an uncorrelated channel.

1: Input: ${F}_{\mathrm{opt}}^{}{}_{{}_{k}}$, ${F}_{\mathrm{RF}}^{(0)}$, ${F}_{{\mathrm{BB}}_{k}}^{(0)}$, ${R}^{(0)}$, ${B}_{k}^{(0)}$, ${F}_{{\mathrm{aprox}}_{k,u}}^{(0)}$, ρ, Q |

2: for t = 0, 1, …, Q − 1 do |

3: Compute ${F}_{\mathrm{RF}}^{(t+1)}$ using (21). |

4: Compute ${F}_{{\mathrm{BB}}_{k}}^{(t+1)}$ using (24), for all k = 1, …, F. |

5: Compute ${R}^{(t+1)}$ using (25). |

6: Compute ${B}_{k}^{(t+1)}$ using (26), for all k = 1, …, F. |

7: Compute ${F}_{{\mathrm{aprox}}_{k,u}}^{(t+1)}$ using (28), for all k = 1, …, F and u = 1, …, N_{u}. |

8: Update ${U}^{(t+1)}$ using (33). |

9: Update ${W}_{k}^{(t+1)}$ using (34), for all k = 1, …, F. |

10: Update ${Z}_{k}^{(t+1)}$ using (35), for all k = 1, …, F. |

11: end for. |

12: ${\widehat{F}}_{\mathrm{RF}}^{}\leftarrow {R}^{(Q)}$. |

13: ${\widehat{F}}_{{\mathrm{BB}}_{k}}^{}\leftarrow {({\widehat{F}}_{\mathrm{RF}}^{}{}^{H}{\widehat{F}}_{\mathrm{RF}}^{})}^{-1}{\widehat{F}}_{\mathrm{RF}}^{}{}^{H}{F}_{{\mathrm{aprox}}_{k}}^{(Q)}$, for all k = 1, …, F. |

14: ${\widehat{F}}_{{\mathrm{BB}}_{k}}^{}\leftarrow \sqrt{{N}_{u}{N}_{s}}{\Vert {\widehat{F}}_{{\mathrm{BB}}_{k}}^{}{}^{H}{\widehat{F}}_{{\mathrm{BB}}_{k}}^{}\Vert}_{F}^{-1}{\widehat{F}}_{{\mathrm{BB}}_{k}}^{}$. |

15: Output: ${\widehat{F}}_{\mathrm{RF}}^{}$, ${\widehat{F}}_{\mathrm{BB}}^{}$. |

AM—Based | |

Operation | Complexity Order |

Overall [15] | $\begin{array}{c}\mathcal{O}(Q({N}_{u}{N}_{s}{N}_{RF}{N}_{tx}+{N}_{RF}{}^{2}{N}_{u}{N}_{s}+{F}^{-1}{N}_{RF}{}^{3})\\ {F}^{-1}{N}_{RF}{}^{2}{N}_{tx}+{N}_{u}{}^{3}{N}_{s}{}^{3})\end{array}$ |

LASSO—Based Alt-Min (SPS) | |

Operation | Complexity Order |

Overall [14] | $\begin{array}{c}\mathcal{O}(Q({N}_{u}{N}_{s}{N}_{RF}{N}_{tx}+{N}_{RF}{}^{2}{N}_{u}{N}_{s}+{F}^{-1}{N}_{RF}{}^{3})\\ +{N}_{u}{}^{2}{N}_{s}{N}_{RF}{N}_{tx}+{N}_{u}{}^{4}{N}_{s}{}^{3})\end{array}$ |

ADMM | |

Operation | Complexity Order |

Overall [10] | $\mathcal{O}(Q({N}_{s}{N}_{RF}{N}_{tx}+{N}_{RF}{}^{2}{N}_{tx}))$ |

EBE | |

Operation | Complexity Order |

Overall [20] | $\mathcal{O}(Q{N}_{tx}{}^{2})$ |

Proposed | |

Operation | Complexity Order |

${F}_{\mathrm{RF}}^{}$ | $\mathcal{O}(Q{N}_{u}{N}_{s}{N}_{RF}{N}_{tx}+{F}^{-1}Q{N}_{RF}{}^{2}{N}_{tx})$ |

${F}_{\mathrm{BB}}^{}$ | $\mathcal{O}(Q{N}_{u}{N}_{s}{N}_{RF}{N}_{tx}+Q{N}_{RF}{}^{2}{N}_{tx})$ |

R | $\mathcal{O}(Q{N}_{RF}{N}_{tx})$ |

B | $\mathcal{O}(Q{N}_{u}{N}_{s}{N}_{RF}{N}_{tx})$ |

${F}_{\mathrm{aprox}}^{}$ | $\mathcal{O}(Q{N}_{tx}{}^{2}{N}_{u}{N}_{s}+{N}_{u}{}^{3}{N}_{tx}{N}_{rx}{}^{2}+{N}_{u}{}^{4}{N}_{rx}{}^{3})$ |

U, W, Z | $\mathcal{O}(Q{N}_{u}{N}_{s}{N}_{RF}{N}_{tx}+Q{N}_{RF}{}^{2}{N}_{tx})$ |

Overall | $\mathcal{O}(Q({N}_{tx}{}^{2}{N}_{u}{N}_{s}+{N}_{RF}{}^{2}{N}_{tx})+{N}_{u}{}^{3}{N}_{tx}{N}_{rx}{}^{2}+{N}_{u}{}^{4}{N}_{rx}{}^{3})$ |

**Table 3.**Power Consumption for Different Implementations of the Proposed Precoder for ${N}_{u}=4$, ${N}_{s}=2$, ${N}_{RF}^{tx}=8$, $F=1$, ${N}_{tx}=256$.

Precoder | Estimated Power Consumption [W] | |
---|---|---|

Fully-Connected | DPS | 428.04 |

UPS | 223.24 | |

QPS (N_{b} = 2) | 59.4 | |

QPS (N_{b} = 3) | 100.36 | |

SWI | 67.59 | |

SI | 38.92 | |

DAoSA SPS | L_{max} = 1 | 28.87 |

L_{max} = 2 | 39.30 | |

L_{max} = 3 | 49.73 | |

L_{max} = 4 | 60.17 | |

DAoSA DPS | L_{max} = 1 | 39.11 |

L_{max} = 2 | 59.78 | |

L_{max} = 3 | 80.45 | |

L_{max} = 4 | 101.13 |

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**MDPI and ACS Style**

Pavia, J.P.; Velez, V.; Ferreira, R.; Souto, N.; Ribeiro, M.; Silva, J.; Dinis, R.
Low Complexity Hybrid Precoding Designs for Multiuser mmWave/THz Ultra Massive MIMO Systems. *Sensors* **2021**, *21*, 6054.
https://doi.org/10.3390/s21186054

**AMA Style**

Pavia JP, Velez V, Ferreira R, Souto N, Ribeiro M, Silva J, Dinis R.
Low Complexity Hybrid Precoding Designs for Multiuser mmWave/THz Ultra Massive MIMO Systems. *Sensors*. 2021; 21(18):6054.
https://doi.org/10.3390/s21186054

**Chicago/Turabian Style**

Pavia, João Pedro, Vasco Velez, Renato Ferreira, Nuno Souto, Marco Ribeiro, João Silva, and Rui Dinis.
2021. "Low Complexity Hybrid Precoding Designs for Multiuser mmWave/THz Ultra Massive MIMO Systems" *Sensors* 21, no. 18: 6054.
https://doi.org/10.3390/s21186054