Hybrid Precoder Using Stiefel Manifold Optimization for Mm-Wave Massive MIMO System

Abstract

Due to the increasing demand for fast data rates and large spectra, millimeter-wave technology plays a vital role in the advancement of 5G communication. The idea behind Mm-Wave communications is to take advantage of the huge and unexploited bandwidth to cope with future multigigabit-per-second mobile data rates, imaging, and multimedia applications. In Mm-Wave systems, digital precoding provides optimal performance at the cost of complexity and power consumption. Therefore, hybrid precoding, i.e., analog–digital precoding, has received significant consideration as a favorable alternative to digital precoding. The conventional methods related to hybrid precoding suffer from low spectral efficiency and large processing time due to nested loops and the number of iterations. A manifold optimization-based algorithm using the gradient method is proposed to increase the spectral efficiency to be near optimal and to speed up the processing speed. A comparison of performances is shown using the simulation outcomes of the proposed work and those of the existing techniques.

1. Introduction

Even with the improvements of 4G LTE, the network sufferers from a lack of bandwidth. The solution to this problem is to increase the bandwidth using a 30 GHz -300 GHz band of millimeter wave as shown in Figure 1. This solution was proposed by the scientists behind 5G and is expected to be capable of providing extremely high data rates and almost zero latency in phone calls [1,2,3]. Along with these merits of Mm-Wave due to high carrier frequency (10 times greater than that of the previous solution), it also suffers from more path losses in the system. However, a large number of antenna arrays can be placed in millimeter-wave systems due to the very small wavelength of antennas [4,5]. Figure 1 shows the frequency spectrum of 5G communication.

Digital precoding achieves better performance by adjusting the signal’s amplitude as well as the phase at the expense of power consumption due to the huge number of RF chains. It also increases the hardware cost. Initially, analog beamforming was established for P-T-P communication systems [6] with fewer RF chains using mixers and phase shifters [7,8,9]. The weakness of analog beamforming is that it can only control the phase.

Hybrid precoding was introduced to tackle the issues of analog precoding and digital precoding [10]. Hybrid precoding is categorized into optimal precoding and analog precoding. The interference among different data streams is reduced using an optimal precoder, while the antenna gain is maximized using an analog precoder. The hybrid beamforming structure is of two types: spatially sparse hybrid precoding and successive interference cancelation (SIC). Spatially sparse precoding is a fully connected structural design [11] in which every RF chain from the analog precoder is linked to all base station antennas through a different or the same value of phase shifts. SIC is a partially connected architecture [12] in which the RF chain from the analog precoder is connected to a subgroup of antennas.

The design of the Mm-Wave system is configured as the design of two sub-problems, i.e., precoding on the transmitter side and combining on the receiver side [11,12,13,14]. The performance of the millimeter-wave system in terms of spectral efficiency can be maximized by solving a convex problem in which the Euclidean distance is to be minimized between the hybrid precoder (F_RF ∗ F_BB) and the digital precoder (F_opt). F_RF represents the analog RF precoder matrix; F_BB is the digital precoder matrix at the baseband; and F_opt is the optimal precoder matrix obtained by applying singular value decomposing on the channel matrix. The analog RF precoder makes use of phase shifters that inflict supplementary element-wise unit modulus constraints. To solve this problem, an algorithm, known as the orthogonal matching pursuit (OMP) algorithm [11,12], is employed for the spatial structure of Mm-Wave systems having a large array of antennas, thus transforming it into a sparse reconstruction problem. However, this solution deprecates the performance of Mm-Wave systems due to the limitation on space of analog precoding. Lastly, an algorithm based on the gradient approach was proposed to maximize mutual information at the cost of high complexity [15,16]. Ref. [14] proposed a manifold optimization (MO) technique based on the alternating minimization (AltMin) algorithm for improving the spectral efficiency to be near optimal. This solution has a nested-loop structural design between F_BB and F_RF that reduces the speed of convergence and also increases complexity. Therefore, ref. [17] proposed PE-AltMin, an alternative low-complexity algorithm. However, this algorithm decreases the spectral efficiency by increasing the number of transmitting RF chains. Mm-Wave is a recent technology in the field of wireless communication. To prevent severe loss of propagation of the Mm-Wave channel, a hybrid precoder and combining transceivers are used for Mm-Wave massive MIMO systems [18,19,20,21]. The RF signal in analog precoding is used for energy-harvesting purposes [22,23,24,25,26]. The concept of cognitive radio can be used for the detection of primary users to improve the energy efficiency [27].

Due to slow variation in the spatial-channel matrix (SCM), as compared with the instantaneous value of complete CSI, ref. [28] proposed a unique spatial-channel algorithm for sub-6 GHz using an SCM. The SCM is a hard problem in the massive MIMO framework due to the massive antenna at the base station. To approximate the behavior to digital precoding, limited CSI is required in the spatial channel to estimate the SCM using an acceptable number of RF chains. This technique can be utilized for single and multiple users, and it can be used in both TDD and FDD systems, as well as at single and multiple carrier frequencies and in LOS and NLOS channels. The current scheduling of users in downlink transmission and sub-carrier power allocation algorithms for massive MIMO is considered in frequency selective channels [29]. The performance of a hybrid precoder can reach that of a digital precoder under the condition of $N_p=2N_{RFT}\left(N_T-N_{RFT}+1\right)$, where $N_{RFT}<N_T$. Furthermore, a unique approach is applied to realize hybrid precoding by employing an $N_{RFT}$ RF chain and a phase shifter less than Np. Three ways are used to solve the problem: (1) the $N_T$ number of RF chains in the digital precoder, (2) antenna selection precoding, and (3) the proposed hybrid precoding method, which combines zero-forcing and the proposed algorithm for user scheduling and sub-carrier power distribution. Ref. [30] proposed an optimal hybrid precoder design for large antenna systems, in which the $N_{RFT}$ required is equal to or greater than the total number of data streams used to achieve optimal precoding performance. If this requirement is not encountered, two techniques for the maximization of spectral efficiency are provided. The hybrid precoding algorithm is designed for large MIMO systems with the following requirements: (1)$N_{RFT}$ = $N_s$ and phase shifters with infinite resolution and (2) $N_s$ < $N_{RFT}$ < 2$N_s$ and phase shifters with limited resolution. The algorithm approximates the maximum capacity for $N_{RFT}$ = $N_s$. The JSDM scheme was discussed for FDD systems in [31] to achieve significant throughput and for the simplification of system operation. The covariance matrix is formed by correlating the channel vectors from the BS to the MSs. To limit the quantity of feedback necessary, this characteristic is utilized to group the MSs with comparable covariance matrices. To eliminate inter-group interferences, a precoding matrix based on DFT is used based on second-order channel statistics. A framework was presented in [32] to minimize the dimension of the precoding matrix and pilot sequence, enabling a low-complexity system using hybrid precoding for massive MIMO systems. A Mm-Wave MIMO system with hybrid precoding was investigated in [33] to improve the spectral efficiency in downlink transmission by jointly optimizing the system using a hybrid precoder at the base station and using a digital precoder in user equipment. An RF precoder was developed for MIMO systems to minimize the overhead in backward and forward cases. A codebook-based hybrid precoder is used to minimize the inter-beam interference and overhead of scheduled users. The sum rate can be increased by increasing the number of users. Hybrid precoding was discussed in [34]; it updates one column using the Lagrange multiplier of the analog precoder and the same row using the least square method of the digital precoder. The hybrid precoder can also be designed using feedback-supported codebooks to reduce the mean error between the optimal beam and vector beam. The Kalman-based filter concept and manifold optimization with MMSE design were also used for the design of hybrid precoder [35,36].

A novel hybrid precoder design is proposed in this article using the manifold optimization algorithm, namely, Stiefel manifold optimization, to increase the performance of Mm-Wave massive communication systems. A manifold optimization-based algorithm using the gradient method is proposed to increase the spectral efficiency to be near optimal and to speed up the processing speed. A comparison of performances is shown using the simulation results of the proposed work with those of existing work. The Stiefel manifold optimization algorithm makes use of the Riemannian gradient method to solve the optimization problem. The algorithm complexity is much lower than that of the linear optimization problem, which is measured with the calculation of gradient, total energy, and projection on the Stiefel manifold. It is also easy to parallelize.

In this paper, a fast optimization algorithm based on the Stiefel manifold is proposed to increase the spectral efficiency. Manifold optimization has superior performance in terms of spectral efficiency using a nested-loop architecture. The proposed technique increases the convergence speed by decreasing the nested loop between F_BB and F_RF. The proposed technique also increases spectral efficiency when compared with PE-AltMin.

The remainder of this paper is organized as follows: The channel and the system model for the hybrid precoding Mm-Wave system are presented in Section 2. Section 3 shows the proposed Stiefel manifold-based optimization algorithm for the hybrid precoding problem and the computational complexity in detail. Numerous simulation results are shown in Section 4; they reveal the performance of the proposed hybrid precoding algorithm. At last, Section 5 consists of the conclusion of this paper.

The following symbolizations are used in this paper: (A) i, j is a matrix with i rows and j columns; bold, upper-case ‘A’ is a vector; upper-case ‘A’ is a scalar; 𝒜 is a set; |A| is the determinant of (A) i, j; ||A||F denotes the Frobenius norm of A; (A) i, j T, (A) i, j *, and (A) i, j-1 represent the transpose of (A) i, j, the conjugate transpose of (A) i, j, and the inverse of (A) i, j, respectively.

2. System Model

Let us consider a hybrid precoding millimeter-wave system for a single user as shown in Figure 2. It consists of N_T antennas and $N_{RFT}$ RF chains on the transmitter side and N_R antennas and $N_{RFR}$ RF chains on the receiver side. Each user transmits Ns data streams from the base station on each subcarrier with the help of N_T antennas. The user symbols, RF chains, and antenna terminals are bounded as

$$N_S\le N_{RFT}\le N_T \mathrm{and} N_S\le N_{RFR}\le N_R$$

The hardware structure consists of a digital baseband precoder, ${\mathit{F}}_{BB}$ ($N_{RFT}$ × $N_S$), that is followed by an analog RF precoder, ${\mathit{F}}_{RF}$ ($N_S$ × $N_{RFT}$), at the transmitting end and an analog RF combiner, ${\mathit{W}}_{RF}$ ($N_R$ × $N_{RFR}$), followed by a digital baseband combiner, ${\mathit{W}}_{BB}$ ($N_{RFR}$ × $N_S$). Therefore, the transmitted signal of each user can be expressed as

$$x(N_{\mathrm{T}}\times N\mathrm{s})=({\mathit{F}}_{RF}\ast {\mathit{F}}_{BB})\ast S,$$

where S($N_S$ × 1) is the symbol vector of each user. For the given system, the normalized transmit power constraint is imposed and is given by $\left|\right|({\mathit{F}}_{RF}\ast {\mathit{F}}_{BB})|{|}_F^2=N_S$.

More significantly, the analog precoder consists of several phase shifters, so there is a need to adjust the phase of the transmitted signals. Hence, ${\mathit{F}}_{RF}$ should gratify the element-wise unit modulus constraints given by

$$\left|{({\mathit{F}}_{RF})}_{i,j}\right|=1$$

Channel matrix H of size $N_R$ × $N_T$ having an L propagation path using the clustered channel model is modeled as

$$\mathit{H}=\sqrt{\frac{N_T{N}_R}{L}} {{\displaystyle \sum }}_{l=1}^L{\alpha }_l{\mathit{a}}_R\left({\theta }_{l,}^R{\varnothing }_l^R\right){\mathit{a}}_T^H\left({\theta }_l^T,{\varnothing }_l^T\right)$$

(1)

where ${\alpha }_l$ is the complex gain of the lth propagation path and vectors ${\mathit{a}}_R\left({\theta }_{l,}^R{\varnothing }_l^R\right){\mathit{a}}_T^H\left({\theta }_l^T,{\varnothing }_l^T\right)$ are the normalized array response vectors, with $\left({\theta }_{l,}^R{\varnothing }_l^R\right) \mathrm{and} \left({\theta }_l^T,{\varnothing }_l^T\right)$ being the azimuth (elevation) angles of the receiver and of the transmitter, respectively. These parameters depend on the structure of the antenna array.

In matrix form, H is represented as

$$\mathit{H}=\sqrt{\frac{N_T{N}_R}{L}} [a_R\left({\theta }_1^R\right) a_R\left({\theta }_2^R\right)\dots \dots ..a_R\left({\theta }_l^R\right)\ast \left[\begin{array}{ccc}{\alpha }_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & {\alpha }_l\end{array}\right] \ast \begin{array}{c}\left[\begin{array}{c}{a}_T\left({\theta }_1^T\right)\\ .\\ .\\ a_T\left({\theta }_1^T\right)\end{array}\right]\end{array}$$

(2)

3. Problem Formulation

To maximize the spectral efficiency, R (${\mathit{F}}_{RF}$, ${\mathit{F}}_{BB}$), and the maximum mutual information, ${\mathit{F}}_{RF}$ and ${\mathit{F}}_{BB}$ are designed using Gaussian signaling over the Mm-Wave channel in the following manner:

$$\mathrm{R} \left({\mathit{F}}_{RF}, {\mathit{F}}_{BB}\right)=lo{g}_2\left(\left|\mathrm{I}+\frac{\rho }{N_{S }{\sigma }_n^2}\mathit{H}{\mathit{F}}_{RF}{\mathit{F}}_{BB}{\mathit{F}}_{BB}^H{\mathit{F}}_{RF}^H{\mathit{H}}^H \right|\right)$$

(3)

To maximize the spectral efficiency (3), the precoder and the combiner are jointly optimized. This paper concentrates on the design of the precoder, so there is the need to decouple the precoder section and the combiner section. The performance of the optimal hybrid precoder is nearly close to that of the optimal precoder of matrix ${\mathit{F}}_{opt}(N$_T$\times N$_s$)$, which is obtained from channel matrix H. ${\mathit{F}}_{opt}$ is the first Ns column of the eigenvectors of V generated by $\mathit{H}=U{\displaystyle \sum }{\mathit{V}}^{\mathit{H}}$, i.e., the SVD of channel matrix H. So, the formulated problem for a narrowband system (4) and an OFDM system (5) based on ${\mathit{F}}_{opt}$ can be written as

$$\begin{gathered} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\underset{{\mathit{F}}_{RF}{\mathit{F}}_{BB}}{\min } \Vert {\mathit{F}}_{opt}-{\mathit{F}}_{RF}{\mathit{F}}_{BB}{\Vert }_F \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t. |{\left({\mathit{F}}_{RF}\right)}_{i,j}=1 \\ \mathrm{and} \Vert {\mathit{F}}_{RF}{\mathit{F}}_{BB}{\Vert }_F^2=N_s \\ \hspace{1em}\hspace{1em}\Vert {\mathit{F}}_{RF}{\mathit{F}}_{BB}{\Vert }_F^2=N_s \\ \hspace{1em}\hspace{1em}={\left|\left|{\mathit{F}}_{BB}^H{\mathit{F}}_{RF}^H{\mathit{F}}_{RF} {\mathit{F}}_{BB}\right|\right|}_F \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\surd Tr({\mathit{F}}_{BB}^H{\mathit{F}}_{RF}^H{\mathit{F}}_{RF} {\mathit{F}}_{BB}{\mathit{F}}_{BB}^H{\mathit{F}}_{RF}^H{\mathit{F}}_{RF}{\mathit{F}}_{BB}) \end{gathered}$$

(4)

Narrowband systems have poor performance due to the effect of multipath fading. Therefore, the multi-carrier concept is used in the above problem for the OFDM millimeter-wave system and is expressed as

$$\begin{gathered} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\underset{{\mathit{F}}_{RF}{\mathit{F}}_{BB\left[k\right]}}{\min } {{\displaystyle \sum }}_{k=0}^{K-1}\Vert {\mathit{F}}_{opt}\left[k\right]-{\mathit{F}}_{RF}{\mathit{F}}_{BB}\left[k\right]{\Vert }_F \\ s.t. |{\left({\mathit{F}}_{RF}\right)}_{i,j}=1 \\ \mathrm{and} \Vert {\mathit{F}}_{RF}{\mathit{F}}_{BB}\left[k\right]{\Vert }_F^2=N_s \end{gathered}$$

(5)

where K signifies the number of subcarriers and k$\in \left[0, K-1\right]$ is the index of subcarriers. The spectral efficiency can be increased by transmitting the number of data streams from different users. These data streams should verify that the condition of existence is less than the number of RF chains in the analog section.

3.1. Proposed Methodology

Let us consider that f is a real-valued function on set $ℳ$ of various points from $n\times p$ matrices in real time without any particular structure. Here, p represents the orthonormal vectors in space ${ℝ}^n.$ So, an optimization problem can be formed for $\mathit{X}$ of the $n\times p$ matrix that satisfies the condition of ${\mathit{X}}^T\mathit{X}=I$.

$$\begin{gathered} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\underset{\mathit{X}\mathsf{ℇ}{ℝ}^{n\times p}}{\min } f\left(X\right) \\ \mathrm{s}.\mathrm{t} {\mathit{X}}^T\mathit{X}=I \end{gathered}$$

(6)

The set of p orthonormal vectors in space ${ℝ}^n$ is known as the Stiefel manifold, represented as St(p, n).

$$St\left(p, n\right)=\{\mathit{X}\in {ℝ}^{n\times p}: {\mathit{X}}^T\mathit{X}=I\}$$

(7)

This manifold becomes a unit sphere, $S^{n-1}$, i.e., a simple nonlinear manifold, when p becomes 1.

Now, let us consider $ℳ$ as a unit sphere for retractions as a Riemannian sub-manifold of ${ℝ}^n$. For the above problem (1), vec(${\mathit{F}}_{RF}\in {ℝ}^{N_T\times N_{RFT}}$ forms a complex Stiefel manifold on sphere St(p, ℂ), where St(p, $ℂ$) = {X= [x1, ………xp] $\in {ℂ}^{n\times p}$, with $p=N_T\times N_{RFT}$.

A retraction on the Stiefel manifold is given by:

$$R_X\left(\xi \right)=q f\left(\mathit{X}+\xi \right)$$

where $q f\left(\mathit{X}+\xi \right)$ indicates the Q factor of the decomposition of $\left(\mathit{X}+\xi \right)$ ∈ $R^{n\times p}$.

This retraction on the Stiefel manifold is used in the proposed algorithm as discussed in Section 3.3. The proposed algorithm 1 uses the Riemannian Steepest Descent for its simplicity.

3.2. Gradient on Sub Manifolds

Let us assume that $ℳ$ represents the Riemannian submanifold of $\overline{ℳ}$ and that $T_Xℳ$ is a unique element that is known as a gradient of f at $\mathit{X}$, represented by grad f($\mathit{X}$), with its definition depending on the Riemannian metric such that

$$\begin{gathered} grad f\left(\mathit{X}\right)\in T_{\mathit{X}}ℳ \\ i.e.=P_{\mathit{X}} grad \overline{f_{\mathit{X}}} \end{gathered}$$

where $P_{\mathit{X}}$ is the projection operator that belongs to $T_{\mathit{X}}ℳ$.

3.3. Algorithm for Precoder Design

This Algorithm 1 can be used for narrowband systems and also for wideband systems.

Algorithm 1 RSD (Riemannian Steepest Descent)

Input: 1: Assuming s is Gaussian distributed: Singular Value Decomposition: 2: $\mathbf{H}=\mathbf{U} \sum {\mathbf{V}}^{\mathrm{H}}, {\mathit{F}}_{opt}$= VH Proposed with RSD (Riemannian Steepest Descent): 3: $\mathrm{Require} {\mathit{F}}_{opt}$ 4: For 0 ≤ k ≤ K − 1 5: $\mathrm{Set} {\mathit{F}}_{RF}$$\mathrm{using} \mathrm{inverse} \mathrm{vectorization} \mathrm{of} {\mathit{X}}^k$ 6: $\hspace{1em}\hspace{1em}\mathrm{Compute} {\widehat{\mathit{F}}}_{BB}={\mathit{F}}_{RF}{\mathit{F}}_{opt}$ for each k 7: $\hspace{1em}\hspace{1em}\mathrm{Compute} \mathrm{Euclidean} \mathrm{gradient} \mathrm{of} \nabla f\left(\mathit{X}\right)=vec(-2\left({\mathit{F}}_{opt}-{\mathit{F}}_{RF}{\mathit{F}}_{BB}\right){\mathit{F}}_{BB}^H$ 8: $\hspace{1em}\hspace{1em}\mathrm{Compute} \mathrm{Riemannian} \mathrm{gradient} \xi =P_x\nabla f\left(\mathit{X}\right)$$, \mathrm{where} P_X$ is the projection operator onto tangent space. 9: $\hspace{1em}\mathrm{Find} \mathrm{step} \mathrm{size} \mathsf{\alpha } \mathrm{and} \mathrm{update} {\mathit{X}}^{k+1}=R_{\mathit{X}}\left(-\mathsf{\alpha } \xi \right)$ 10: End for 11: $\mathrm{output} {\mathit{F}}_{RF}=ve{c}^{-1}\left({\mathit{X}}^{k+1}\right)$ $\mathrm{and} {\mathit{F}}_{BB}={\frac{\surd N_s}{{\Vert \mathit{F}}_{RF}{\mathit{F}}_{BB}\Vert }}_F{\widehat{\mathit{F}}}_{BB}$

4. Simulation Results

The performance of the proposed algorithm and its comparison with those of existing algorithms is compared in this section [14]. The system model is designed as an environment having 8 clusters and 10 rays per cluster. The system environment is considered to have a 10-degree angular speed that is uniformly distributed from 0 to 360 degrees [22]. The experiment considers an OFDM system with 128 multicarrier paths. The experimental setup considers an equal number of RF chains at the BS as well as the MS. This section also shows the comparison among the proposed algorithm, the OMP algorithm [11], the PE-AltMin algorithm [14], and the optimal digital precoder for a certain level.

Figure 3 firstly presents that the spectral efficiency of the proposed algorithm is superior to that of the fast precoding algorithm for a range of SNRs [−15:10], with Ns = 4 and $N_{RFT}$ = 6. The size of the antenna array taken for this simulation is 144 × 36. This shows that the proposed algorithm gives a better result than the fast precoding algorithm [2].

The spectral efficiency of the proposed algorithm is also compared with those of the optimal digital precoder for reference, PE-AltMin [14], and OMP [11] for different values of the number of RF chains that are equal at the transmitting and receiving ends. This comparison is shown in Figure 4. From this figure, it can be seen that the proposed algorithm produces steadily higher results than the PE-AltMin and OMP algorithms and is nearly close to the optimal precoder. The value of the number of symbols (Ns) is kept lower than twice the number of RF chains. This comparison is made with the SNR value of 0 dB. The performance of the proposed algorithm increases in proportion to the number of RF chains and is nearly close to that of the optimal precoder [24]. The performance of OMP also improves as compared with PE-AltMin upon increasing the number of RF chains. Figure 5 shows the spectral efficiency for a range of SNRs having different values of the number of RF chains.

Comparison of Proposed Model with Existing Techniques

Table 1. Comparison of spectral efficiency and complexity for different algorithms for different values of the SNR.

Algorithm	Spectral Efficiency (b/s/Hz)			Complexity
	SNR = −10 dB	SNR = 0 dB	SNR = 10 dB
OMP [11]	15.4	24.07	40.8	$\mathcal{O}\left(N_T^2{N}_{RFT}Ns\right)$
MO-AltMin [14]	15.9	25.3	42.4	$\mathcal{O}\left(N_{RFT}{N}_T^3{N}_s^2\right)$
PE-AltMin [14]	15.7	25.1	41.9	$\mathcal{O}\left(N_{RFT}{N}_s^2\right)$
CO- Manifold [12]	16.4	25.3	42.7	$\mathcal{O}\left(N_{RFT}{N}_{T}Ns\right)$
MO-MMSE [36]	16.8	25.5	43.9	$\mathcal{O}\left(N_{RFT}{N}_T{N}_s^2\right)$
Proposed	16.9	25.5	44.1	$\mathcal{O}\left({\mathit{N}}_{\mathit{R}\mathit{F}\mathit{T}}{\mathit{N}}_{\mathit{T}}\mathit{N}\mathit{s}\right)$

shows the comparison of the proposed system with existing system models on the basis of performance parameters such as spectral efficiency (measured using −10 dB, 0 dB, and 10 dB SNRs) and computational complexity. As the optimized digital precoder and the weight matrix have closed-form expressions and have much lower dimensions than the analog precoder, we ignore their computational complexity. Additionally, as both the analog precoder and combiner can be solved in the same procedure, we focus on the complexity analysis of the analog precoder. The proposed algorithm needs to calculate the Euclidean gradient defined in step 7 of the proposed algorithm, which is the dominant calculation cost of the proposed algorithm. Hence, the computational complexity for the gradient calculation is $\mathcal{O}\left(N_{RFT}{N}_{T}Ns\right)$. Meanwhile, the calculation cost of the MO-AltMin algorithm is $\mathcal{O}\left(N_{RFT}{N}_T^3{N}_s^2\right)$ [14]. Furthermore, since PE-AltMin requires min{$\mathcal{O}(N_{RFT}{N}_s^2$, ($N_s\left(N_{RFT}\right)^2)$} for the SVD calculation per iteration, the proposed algorithm competes with the low-complexity PE-AltMin algorithm. Consequently, as summarized in Table 1, the proposed algorithm is comparable to or much faster at processing than state-of-the-art algorithms, and it produces near-optimal spectral efficiency.

5. Conclusions

In this paper, a fast algorithm based on Stiefel manifold optimization on the sphere is proposed in terms of improved performance. This algorithm has high spectral efficiency, low complexity, and fast processing speed compared with existing state-of-the-art techniques such as the complex oblique manifold, the OMP algorithm, MO-AltMin, and PE-AltMin. The simulation results and the numerical observation show that the proposed algorithm is superior in terms of spectral efficiency and algorithm complexity aimed to improve the processing speed. Many research papers show that performance does not always improve with the increase in RF chains. Therefore, future work could extend research in the field of the optimization of the number of RF chains.