1 Introduction

Massive multiple-input multiple-output (MIMO) is a wireless communication technology designed to serve multiple users simultaneously by utilizing a large number of antennas at the base station (BS). Each user is equipped with a single antenna, while the BS is equipped with several hundred or even thousands of antennas [1, 2]. Beamforming, a signal processing technique that employs multi-antenna arrays to enhance the transmission or reception of signals, is an integral part of massive MIMO technology. As the number of users increases and the system’s data rate requirements grow, the use of massive MIMO with beamforming becomes increasingly attractive in wireless communication systems [3,4,5]. Beamforming techniques’ classifications in both uplink (UL) and downlink (DL) transmissions are based on their hardware architecture and implementation approach. The hardware architecture of beamforming can be categorized into three types: analog, digital, and hybrid. The implementation approach of beamforming can be classified as either switched (fixed) beamforming or adaptive beamforming [3, 6].

The millimeter wave band is a range of frequencies that extends from 10 to 300 GHz and is particularly well suited for high-data-rate communication systems. However, the propagation characteristics of this band are poor due to significant free-space path loss and co-channel interference. Nonetheless, the utilization of small antenna sizes and the creation of massive antenna arrays are enabled by the high frequencies and small wavelengths in mm-wave. By employing beamforming procedures that utilize a large number of antennas, it is possible to minimize path loss and co-channel interference, enhance signal-to-interference and noise ratios, and improve system performance in the mm-wave frequencies [3, 7, 8].

Numerous beamforming algorithms are available for massive MIMO communication systems, and several references discuss them. One such reference is a study by authors in [9], which proposes a hybrid beamforming topology for a multiuser massive MIMO system. The authors present the hybrid beamforming design for both the transmitter and the receiver of the system. The study evaluates the system’s performance using the bit error rate and spectral efficiency for each user and compares it to a complete digital beamforming. The study highlights the advantages and weaknesses of the proposed system and shows how it is nearly digital. Another reference, [10], discusses the architecture of a DL multiuser massive MIMO hybrid beamforming communication system with multiple independent data streams per user. The method uses hybrid precoding at the transmitter and hybrid combining at the receiver. To create the precoding and combining weights, the study employs hybrid beamforming in conjunction with a peak search algorithm. The results show a tradeoff between the number of data streams per user and the number of antennas required at the BS. The proposed method recommends a high number of data streams per user to achieve higher order throughputs. In [2], the authors propose a hybrid analog/digital beamforming method for a massive MIMO system. The study suggests alternate antenna array configurations to reduce the cost and power consumption of hybrid beamforming while also reducing the correlations between antennas at the BS. The study also employs both the uniform linear array and the non-uniform linear array. The performance is measured using the bit error rate vs the signal-to-noise ratio and the number of antennas employed at the BS.

In their research work, the authors of reference [11] propose an energy-efficient hybrid beamforming method. The suggested method replaces the conventional phase shifter with a Butler phase shifting matrix in the analog beamformer section, which is dependent on fixed beam directions. The chosen optimum beam directions from the Butler matrices by using either an exhaustive search approach or a suggested low-complexity algorithm are based on known CSI and specific circumstances. Both algorithms start by selecting the analog beamformer and then the digital beamformer. The proposed method’s performance is compared with complete digital beamforming and hybrid beamforming based on phase shifters in the analog section concerning spectrum and energy efficiency. The authors in [12] introduce an adaptive hybrid beamforming approach, where the analog beamformer adapts to multiple users. It generates an estimate for the channel’s angular sparsity as a joint angle-delay sparsity map and power profile. The digital beamformer is responsible for instantaneous channel estimation in small dimensions, and this approach is resistant to the errors in angular estimates. In [6], the authors propose a new hybrid precoding scheme based on geometric mean decomposition. The concept of this scheme is to divide the OFDM channel into sub-channels with equal benefits. The proposed method is compared with the classic singular value decomposition method that separates the channel into sub-channels with significant gain differences. Compared to existing approaches, the proposed method based on geometric mean decomposition in reference [6] reduces computing costs and increases spectral efficiency.

The article titled [13] presents a novel approach for combining a hybrid precoding scheme based on geometric mean decomposition with generalized spatial modulation MIMO systems. The proposed method achieves a high beamforming gain by combining a complete digital geometric mean decomposition with analog beamforming and applying it to the norm-based user selection algorithm. Simulation results of the suggested technique are compared to traditional generalized spatial modulation MIMO systems, and the proposed system is found to outperform existing systems in terms of bit error rate versus signal-to-noise ratio. In [8], the authors introduce a hybrid beamforming transceiver designed for the mm-wave range. Based on the concept of predetermined numbers of dominant orthogonal propagation paths, this hybrid beamforming is expected to be a near-optimum hybrid precoder. It creates analog and digital beamformers using its knowledge of CSI and selects the closest one to the optimal in terms of CSI, with the goal of maximizing the system’s spectral efficiency. The authors demonstrate the outcomes of this technique in terms of boosting the system’s spectral efficiency. In [14], the authors propose a new concept of asymmetric millimeter-wave massive MIMO that goes beyond 5G and 6G communication technologies. The proposed approach achieves all beamforming in the digital domain without using any phase shifters. This technique offers non-reciprocal, fully digital beamforming in massive MIMO systems; it also helps in reducing complexity and expense while retaining the benefits of digital beamforming for a large number of beams. The article also presents the integrated circuit design for a complete digital beamforming based on asymmetric massive MIMO systems.

In their publication, the authors of [15] have proposed a switch-based hybrid beamforming scheme as a cost-effective solution for the analog component of the hybrid beamformer. This online approach leverages a sequence of prior channel realizations and learns from previous iterations, thereby enhancing its spectral efficiency. By increasing the number of trials, this online method can significantly improve spectral efficiency. It is primarily designed for single-user transmit precoding and aims to obtain a suboptimal solution for the analog beamformer. The performance of this approach is evaluated by comparing its spectral efficiency and minimal mean square error to those of a fully digital beamformer as a function of iteration numbers. In contrast, the authors of [16] have proposed an optimal beamforming approach for cell-free massive MIMO that utilizes a large number of randomly distributed access points to serve multiple users simultaneously. The optimal beamforming design is formulated as a max–min problem, with the objective of maximizing the instantaneous signal-to-interference and noise ratio across all users. Second-order cone programming is employed to create the optimal beamformer, which is centrally controlled in a central unit. This approach provides an upper bound on the achievable beamforming performance in cell-free massive MIMO and can serve as a benchmark for developing low-complexity beamformers. The performance of this approach is evaluated in terms of throughput per user and compared with that of conjugate beamforming and zero-forcing beamforming techniques.

As evidenced by prior research, beamforming algorithms are typically presented in the context of hardware design and rely heavily on the estimated channel state information (CSI). However, a novel beamforming technology known as “orthogonal beamforming” has been proposed in this study, which is a variant of switched beamforming. This approach employs a predetermined set of orthogonal beams to optimize user performance. The efficacy of this technique is contingent upon the number of antennas employed at the BS, and with a massive MIMO system, it is capable of serving a greater number of users than current beamforming systems.

The present study centers around the idea of transmit beamforming in a massive MIMO system and proposes an orthogonal beamforming technology. This technology can be categorized as a switched beamforming and a complete digital beamforming. The investigation compares the suggested beamforming approach to the fully digital ZF beamforming technique.

The suggested orthogonal beamforming approach is particularly applicable in the mm-wave spectrum due to its ability to operate solely on the intended user’s angle and not the exact CSI or channel model. Accurately modeling the mm-wave channel is difficult, which is why this approach is more appropriate.

The paper is structured in accordance with the following framework: Sect. 2 provides an overview of the classification of beamforming techniques and the system model that has been employed. Section 3 introduces the proposed orthogonal beamforming approach, along with the associated deduction analysis and algorithm. Section 4 compares the proposed approach to the adaptive beamforming technique. Section 5 illustrates the simulations and results, while Sect. 6 concludes the paper.

2 Beamforming in massive MIMO

2.1 Beamforming techniques

Beamforming is a signal processing technology that employs multi-antenna arrays to facilitate effective communication between the base station (BS) and the users. Transmit and receive beamforming are the two primary types of beamforming, depending on whether the signal processing occurs at the transmitter or receiver end. While receive beamforming is used to differentiate signals from multiple users at the BS, transmit beamforming facilitates the transmission of signals by guiding them towards designated users [1, 3,4,5]. Beamforming techniques’ classification is based on their hardware architecture as analog, digital, or hybrid. Additionally, they can be categorized either as switching or adaptive based on their implementation approach. Figure 1 provides a general classification of beamforming techniques for both uplink (UL) and downlink (DL) transmissions.

Fig. 1
figure 1

Beamforming technique classifications

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Analog beamforming employs low-cost phase shifters to produce beams, making it an economical and straightforward installation process. However, controlling the null orientations and handling multi-stream transmission can present challenges [17, 18]. Conversely, digital beamforming steers both beams and nulls digitally by utilizing appropriate baseband precoding methods, such as MR, ZF, or RZF approaches. This methodology offers greater transmission and reception flexibility and resilience than analog beamforming, and it can handle multi-stream transmission as well. Nonetheless, it can be expensive, as each antenna element requires its own RF chain, particularly when employing a significant number of antennas [17,18,19]. Hybrid beamforming, which merges analog and digital components, strikes a balance between the advantages and drawbacks of both analog and digital approaches [7, 17, 19].

Adaptive beamforming creates a customized beam and directs it towards a specific user based on estimated channel state information (CSI). Its primary objective is to reduce user interference. However, its practical implementation is challenging [3, 5]. Switched beamforming, on the other hand, employs a set of predetermined beams and requires a switching network to select the optimal beam for users. This approach is easy to implement, but distinguishing between users served by the same beam is difficult [3, 5]. The Butler matrix is the most commonly used approach in switched beamforming. It is a passive network where N × N BM comprising \(\frac{N}{2}\)×\({{\text{log}}}_{2}N\) couplers and \(\frac{N}{2}\)×\({{\text{log}}}_{2}(N-1)\) phase shifters that generate N different phase shifts to reflect N separate beams. For instance, a 4 × 4 BM can produce four phases of \({30}^{o}\), \({150}^{o}\), \({120}^{o}\), and \({60}^{o}\) or ± \({45}^{o}\) and ± \({135}^{o}\) [3, 20, 21].

2.2 System model of a transmit beamformer

The system model implements a multiuser MIMO DL transmission, featuring M-antennas at the BS and a single antenna for K users. Figure 2 illustrates the block diagram of the massive MIMO DL transmission. The transmission process commences by generating transmit beamforming vectors utilizing one of the beamforming methods. The signals are then multiplied with these beamforming vectors to direct them towards the intended users. The BS employs a uniform linear array (ULA) antenna design, with the ULA serving a single sector of \({90}^{o}\). This system model will be utilized consistently throughout this paper to scrutinize and authenticate the suggested approach in the massive MIMO system. The forthcoming section will provide a comprehensive description and investigation of the suggested beamforming technology.

Fig. 2
figure 2

Transmit beamforming in downlink transmission

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3 Proposed orthogonal beamforming technique

The proposed orthogonal beamforming is designed to provide users with a set of mutually orthogonal beams. This approach aims to mitigate interference between transmitted beams while enhancing system performance. In the proposed method, a set of predetermined transmit beamforming vectors is employed to generate the orthogonal beams required for the technique. Each orthogonal beam serves one user only, based on their proximity to it.

This section begins by deducing the orthogonal angles utilized to create the transmit beamforming vectors in the proposed orthogonal beamforming approach. Subsequently, the suggested technique’s algorithm is presented, which demonstrates how to serve users by leveraging available orthogonal beams and assigning orthogonal beams to the desired users.

According to the proposed beamforming approach, the number of orthogonal beams available is contingent on the number of antennas utilized at the BS. As such, M/2 orthogonal beams will be available, thereby allowing the suggested orthogonal beamforming approach to serve M/2 users. Accordingly, it is observed that the proposed approach, in conjunction with the massive MIMO system, can serve more users with a higher degree of efficiency relative to previous beamforming systems, as is evident from the simulation section.

3.1 Deducing orthogonal beams’ angles

This section presents an analysis aimed at determining the orthogonal angles necessary to generate the orthogonal beams utilized in the proposed technique. The analysis follows the beamforming procedure outlined in the general DL transmission block diagram depicted in Fig. 2. It entails the generation of the transmit beamforming vectors, which are then multiplied by the requisite signals for transmission through M-antennas at the BS. Finally, the strength of the received signal is utilized at the user end to calculate the orthogonal angles that may be employed in constructing the orthogonal beams in the proposed approach.

The analysis commences with the generation of the transmit beamforming vectors (w), primarily based on the estimated channels between the BS and the users. The study exclusively focuses on the angles of the channels that are crucial for estimating the orthogonal angles. The transmit beamforming vectors are deduced using equations from (1) to (4).

$${w}_{{\text{i}}}= \frac{{\widehat{{\text{h}}}}_{{\text{i}}} }{\Vert {\widehat{{\text{h}}}}_{{\text{i}}}\Vert }$$
(1)

The given equation entails the following variables: \({w}_{{\text{i}}}\) denotes the transmit precoding vector of user i \(({\mathrm{ UE}}_{{\text{i}}})\), \({\widehat{h}}_{{\text{i}}}\) represents the estimated channel between the M-antennas BS and \({{\text{UE}}}_{{\text{i}}}\), and \(\Vert {\widehat{h}}_{{\text{i}}}\Vert\) is the norm of \({\widehat{h}}_{{\text{i}}}\). It should be noted that in an ideal channel estimation method, the estimated channel response is identical to the actual channel response \(\left({\widehat{h}}_{{\text{i}}}= {h}_{{\text{i}}}\right)\), and therefore, \({w}_{{\text{i}}}= \frac{{h}_{{\text{i}}} }{\Vert {h}_{{\text{i}}}\Vert }\). Regarding the ULA channel can be shown as follows [19, 22].

$${h}_{{\text{i}}}= \sqrt{{\beta }_{{\text{i}}}} \left[{e}^{{\text{j}}1\mathrm{\pi sin}{\Phi }_{{\text{i}}}}{ e}^{{\text{j}}2\mathrm{\pi sin}{\Phi }_{{\text{i}}}}\dots { e}^{\mathrm{jM\pi sin}{\Phi }_{{\text{i}}}}\right]$$
(2)

And

$$\Vert {h}_{{\text{i}}}\Vert = \sqrt{{\beta }_{{\text{i}}}+{\beta }_{{\text{i}}}+\dots +{\beta }_{{\text{i}}}} = \sqrt{{\text{M}}{\upbeta }_{{\text{i}}}}$$
(3)

So,

$${w}_{{\text{i}}}= \frac{1}{\sqrt{M}} \left[{e}^{{\text{j}}1\mathrm{\pi sin}{\Phi }_{{\text{i}}}}{ e}^{{\text{j}}2\mathrm{\pi sin}{\Phi }_{{\text{i}}}}{ \dots e}^{\mathrm{jM\pi sin}{\Phi }_{{\text{i}}}}\right]$$
(4)

The value of \({\beta }_{{\text{i}}}\) represents the gain in large-scale fading for \({{\text{UE}}}_{{\text{i}}}\), while M denotes the total number of antennas present at the BS. \({\Phi }_{{\text{i}}}\) stands for the azimuth angle of \({{\text{UE}}}_{{\text{i}}}\).

Then, by multiplying the signal \(\left({S}_{{\text{i}}}\right)\) for \({{\text{UE}}}_{{\text{i}}}\) by its corresponding transmit precoding vector \(\left({w}_{{\text{i}}}\right),\) the transmitted signal looks like this:

$${S}_{\mathrm{i }}\times {w}_{{\text{i}}}= \frac{1}{\sqrt{M}} [{S}_{{\text{i}}} {e}^{{\text{j}}1\mathrm{\pi sin}{\Phi }_{{\text{i}}}} {S}_{{\text{i}}} {e}^{{\text{j}}2\mathrm{\pi sin}{\Phi }_{{\text{i}}}} \dots {S}_{{\text{i}}} {e}^{\mathrm{jM\pi sin}{\Phi }_{{\text{i}}}}]$$
(5)

Finally, the transmitted signals are summed and directed towards users via M-antennas in the BS. Upon reception, the dedicated user (\({{\text{UE}}}_{{\text{i}}}\)) will receive a signal from all M-antennas, which will include the requisite signal as well as interference signals from all other users in the system. This received signal can be described as a superposition of the required signal and the interference signals, as follows:

$$\begin{array}{l}{{S}_{{\text{UE}}}}_{{\text{i}}}= \frac{1}{\sqrt{M}} \sum\nolimits_{k=1}^{{\text{K}}}\sum\nolimits_{m=1}^{{\text{M}}}{S}_{{\text{k}}}{e}^{\mathrm{jm\pi }\left({\text{sin}}{\Phi }_{{\text{k}}}-\mathrm{ sin}{\Phi }_{{\text{i}}}\right)}\\\;\;\;\;\;\;\;\;= \sqrt{M} {S}_{{\text{i}}} + \frac{1}{\sqrt{M}} \sum\nolimits_{\begin{array}{c}k=1\\ , k\ne i\end{array}}^{{\text{K}}}\sum\nolimits_{m=1}^{M}{{\text{S}}}_{{\text{k}}}{{\text{e}}}^{\mathrm{jm\pi }({\text{sin}}{\Phi }_{{\text{k}}}-\mathrm{ sin}{\Phi }_{{\text{i}}})}\end{array}$$
(6)

According to Eq. (6), the received signal consists of a desired signal term \(\left(\sqrt{{\text{M}}} {{\text{S}}}_{{\text{i}}}\right)\) and an interference term \(\left(\frac{1}{\sqrt{{\text{M}}}}\sum_{\begin{array}{c}k=1\\ , k\ne i\end{array}}^{{\text{K}}}{\sum }_{{\text{m}}=1}^{{\text{M}}}{{\text{S}}}_{{\text{k}}}{{\text{e}}}^{\mathrm{jm\pi }({\text{sin}}{\Phi }_{{\text{k}}}-\mathrm{ sin}{\Phi }_{{\text{i}}})}\right)\) that other users have generated. K denotes the number of users served by the system; \({S}_{{\text{i}}}\) represents the transmitted signal of \({{\text{UE}}}_{{\text{i}}}\), and \({{\text{S}}\_{\text{UE}}}_{{\text{i}}}\) indicates the received signal of \({{\text{UE}}}_{{\text{i}}}\).

To simplify the received signal expression depicted in Eq. (6), it is possible to replace the summation of the exponential expression with a Dirichlet kernel function, which is a periodic version of the sinc function [23]. The summation of the exponential expression can be simplified to a periodic sinc by following these steps: First, replace the expression \(\left[\pi ({\text{sin}}{\Phi }_{{\text{k}}}-{\text{sin}}{\Phi }_{{\text{i}}})\right]\) in Eq. (6) with the symbol (a) for simplicity, and then apply the finite geometric series relation [24]. Finally, conduct certain mathematical simplification procedures aimed at simplifying part of Eq. (6), as demonstrated in Eqs. (7)–(9).

$${\sum }_{{\text{m}}=1}^{{\text{M}}}{{\text{e}}}^{{\text{jma}}}= \frac{{e}^{{\text{ja}}(1)} -{e}^{{\text{ja}}({\text{M}}+1)}}{1 -{e}^{{\text{ja}}}}$$
(7)
$$\frac{{e}^{{\text{ja}}(1)} -{e}^{{\text{ja}}({\text{M}}+1)}}{1 -{e}^{{\text{ja}}}}= \frac{{e}^{-{\text{ja}}(\frac{{\text{M}}}{2})}- {e}^{{\text{ja}}(\frac{{\text{M}}}{2})}}{{e}^{-{\text{ja}}(\frac{1}{2})}- {e}^{{\text{ja}}(\frac{1}{2})}} \times \frac{{e}^{{\text{ja}}(\frac{{\text{M}}+2}{2})}}{{e}^{{\text{ja}}(\frac{1}{2})}}= \frac{{e}^{{\text{ja}}(\frac{{\text{M}}}{2})}- {e}^{-{\text{ja}}(\frac{{\text{M}}}{2})}}{{e}^{{\text{ja}}(\frac{1}{2})}- {e}^{-{\text{ja}}(\frac{1}{2})}} \times \frac{2{\text{j}}}{2{\text{j}}} \times \frac{{e}^{{\text{ja}}(\frac{M+2}{2})}}{{e}^{{\text{ja}}(\frac{1}{2})}}= \frac{{\text{sin}}(a\frac{{\text{M}}}{2})}{{\text{sin}}(a\frac{1}{2})} \times {e}^{{\text{ja}}(\frac{M+1}{2})}$$
(8)

So,

$${\sum }_{{\text{m}}=1}^{{\text{M}}}{e}^{{\text{jma}}}= \frac{{\text{sin}}(a\frac{M}{2})}{{\text{sin}}(a\frac{1}{2})} \times [\mathrm{ cos}\left(a\left(\frac{M+1}{2}\right)\right)+j\ \mathrm{ sin}\left(a\left(\frac{M+1}{2}\right)\right)]$$
(9)

The Eq. (9) is simplified and then substituted into Eq. (6) in order to obtain the simplified form of the received signal (\({{\text{S}}\_{\text{UE}}}_{{\text{i}}}\)), as demonstrated in Eq. (10).

$$S\_U{E}_{i}=\frac{1}{\sqrt{M}}{\sum }_{k=1}^{K}{S}_{k}\frac{{\text{sin}}\left(\frac{M}{2}\pi \left(sin{\Phi }_{k}-sin{\Phi }_{i}\right)\right)}{{\text{sin}}\left(\frac{1}{2}\pi \left(sin{\Phi }_{k}-sin{\Phi }_{i}\right)\right)}x\left[{\text{cos}}\left(\left(\frac{M+1}{2}\right)\pi \left({\text{sin}}{\Phi }_{k}-{sin\Phi }_{i}\right)\right)+j\ {\text{sin}}\left(\left(\frac{M+1}{2}\right)\pi \left(sin{\Phi }_{k}-sin{\Phi }_{i}\right)\right)\right]$$
(10)

Consequently, the instantaneous received signal power at \({{\text{UE}}}_{{\text{i}},} \left({{\text{P}}}_{{{\text{S}}\_{\text{UE}}}_{{\text{i}}}}\right)\) denotes the product of the received signal and its complex conjugate value. This relationship is expressed in Eq. (11).

$${{\text{P}}}_{{{\text{S}}\_{\text{UE}}}_{{\text{i}}}}=\frac{1}{M}\sum\nolimits_{{\text{k}}=1}^{{\text{K}}} {({{\text{S}}}_{{\text{k}}}\frac{{\text{sin}}(\frac{M}{2}\pi ({\text{sin}}{\Phi }_{{\text{k}}}-{\text{sin}}{\Phi }_{{\text{i}}}\text{) })}{{\text{sin}}(\frac{1}{2}\pi ({\text{sin}}{\Phi }_{{\text{k}}}-{\text{sin}}{\Phi }_{{\text{i}}}\text{) })})}^{2}$$
(11)

The instantaneous received signal power, as depicted by Eq. (11), bears a resemblance to a squared periodic sinc function, also known as a squared Dirichlet function. The technique for deriving orthogonal angles involves the identification of nulls of this periodic sinc received signal strength. These nulls are critical in estimating the orthogonal angles, which, in turn, determine user positions that will not suffer from interference and appear orthogonal to each other. Therefore, these orthogonal angles are essential in generating the orthogonal beams and are necessary for the proposed beamforming technique as well.

To deduce the orthogonal angles of orthogonal beams, substitute the symbols \({\Phi }_{{\text{k}}}\) and \({\Phi }_{{\text{i}}}\) in Eq. (11) with \({\Phi }_{{\text{orthogonal}}}\) and \({\Phi }_{{\text{ref}}}\), respectively. The nulls of the received signal strength, represented by (\({P}_{{{\text{S}}\_{\text{UE}}}_{{\text{i}}}}\)), can then be calculated using the following method:

$$\frac{M}{2}({\text{sin}}{\Phi }_{{\text{orthogonal}}} -{\text{sin}}{\Phi }_{{\text{ref}}}\text{) = }\mp (1, 2, 3, \dots \dots , \frac{M}{2})$$
(12)
$${\Phi }_{{\text{orthogonal}}}= {{\text{sin}}}^{-1}({\text{sin}}{\Phi }_{{\text{ref}}}\mp \text{ (1, 2, 3, }\dots , \frac{{\text{M}}}{2}\text{) }\frac{2}{M})$$
(13)

where \({\Phi }_{{\text{orthogonal}}}\) is the orthogonal beams’ azimuth angle and \({\Phi }_{{\text{ref}}}\) is the reference angle utilized to initiate the creation of the orthogonal beams.

The number of orthogonal beams in a massive MIMO system is fundamentally dependent on the number of antennas deployed at the BS, as demonstrated in Eq. (13). The proposed orthogonal beamforming approach allows for up to \(\frac{M}{2}\) orthogonal beams to be generated, thereby enabling the system to service up to \(\frac{M}{2}\) users. This approach’s algorithm utilizes these orthogonal beams to allocate and serve the intended users in the system.

3.2 Algorithm of proposed orthogonal beamforming technique

The proposed technique for orthogonal beamforming assigns orthogonal beams to the users by identifying the minimal angular deviation between the intended user’s angle and the angular orientations of all available orthogonal beams. The user is then allocated to the closest orthogonal beam. The assigned orthogonal beam will continue to serve the user as long as the difference in angular orientation between the user and the assigned orthogonal beam is less than or equal to a predetermined reference value. In the event that the angular difference surpasses this reference value, the user will be reassigned to the nearest available orthogonal beam. Equation (14) computes the angular difference \(\left(\Delta\Phi \right)\) between the user’s true angle \(\left({\Phi }_{user}\right)\) and the angles of the orthogonal beams \(\left({\Phi }_{orthogonal}\right)\). Equation (15) specifies the reference level of the angular difference.

$$\Delta \Phi = \left|{\Phi }_{user}-{\Phi }_{orthogonal}\right|$$
(14)
$${\Delta\Phi }_{ref}\ge \pm \frac{step\ between\ 2\ consecutive\ orthogonal\ beam{s}^{\mathrm{^{\prime}}}angles}{2}$$
(15)

Table 1 presents the algorithm for implementing orthogonal beamforming. This algorithm elucidates the process of selecting and allocating orthogonal beams to users while also emphasizing the importance of regularly verifying the angular difference to ensure that the appropriate orthogonal beam is assigned to each user.

Table 1 Proposed orthogonal beamforming algorithm
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Upon the allocation of a user to an orthogonal beam, the corresponding signal is loaded onto the beam and transmitted to the user via the communication channel.

In the proposed beamforming approach, the beams are directed towards the orthogonal position rather than the real location of the user, thereby resulting in a certain level of power for each user based on the angular difference between the assigned orthogonal beam and the actual user location. The high number of potential orthogonal beams in the massive MIMO system ensures a minimal angular difference between the user and the assigned beam, thus providing an adequate power level for the user. Furthermore, the continuous switching operation of the users to the nearest orthogonal beam ensures that the received power level at the user remains acceptable.

3.3 Proposed orthogonal beamforming technique for multiuser with common beam problem

In the proposed beamforming approach, if multiple users are in close proximity to the same orthogonal beam during the beamforming assignment process, it is possible that the algorithm described in Table 1 will assign more than one user to that beam. This situation could lead to a reduced performance due to the heavy interference from multiple users using the shared beam.

To resolve this issue, several potential solutions are available. One possible solution is to increase the number of antennas used at the BS. This would increase the number of orthogonal beams in the suggested approach while reducing the ability to replicate other additional users on the same beam.

Alternatively, another solution is disregarding the users who have issues with the service. This approach would result in a reduction in the number of users served by the system. Another option is assigning problematic users to other available empty orthogonal beams. However, this strategy may not be effective for the users since the newly assigned beams may be far away.

A more practical and effective solution is to separate users allocated to the same orthogonal beam into multiple groups. These groups are then multiplexed, with each set of users serviced at distinct resource blocks (at different times or frequencies). The user grouping mechanism used here is based on the idea that each beam is only assigned to one user. This solution of grouping users is deemed more practical and is summarized in Table 2.

Table 2 Solution by grouping for the problem of common beams in the proposed orthogonal beamforming
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The common beam problem and all suggested solutions for the issue of common beams will be implemented and evaluated at the end of the simulation section.

3.4 System model with proposed orthogonal beamforming

The system model provided in Fig. 2 has been updated to incorporate the proposed orthogonal beamforming approach in the first stage, as depicted in Fig. 3. This new approach enables the creation of transmit beamforming vectors in the overall block diagram using the orthogonal beamforming method.

Fig. 3
figure 3

DL transmission with the proposed orthogonal beamforming technique

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The proposed algorithm, described in Table 1, utilized to select orthogonal beams, results in orthogonal beamforming vectors. These vectors are multiplied with the relevant signals and transmitted through M-antennas. The signals are loaded onto orthogonal beams and directed towards the targeted users. Loading the signals onto orthogonal beams enhances the system’s performance, as evidenced by the simulation.

4 Comparison between the proposed orthogonal beamforming technique and the adaptive beamforming technique

The proposed orthogonal beamforming belongs to the category of digital, switched beamforming. This technique will be compared to the digital adaptive beamforming technique that is based on the ZF precoding method. The proposed orthogonal beamforming approach has both advantages and disadvantages compared to the adaptive beamforming technique. Table 3 provides a comparative analysis between these two technologies.

Table 3 Comparison between the adaptive beamforming and the proposed orthogonal beamforming
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$${\text{SE}}= {{\text{log}}}_{2}(1+{\text{SINR}})$$
(16)
$$\mathrm{sum \;SE}= \sum\nolimits_{\forall \;\mathrm{ users}}{\text{SE}}$$
(17)

5 Simulations and results

This section delves into the simulations and findings of the proposed orthogonal beamforming technique. It presents a comparative analysis of the proposed orthogonal beamforming technique and the digital ZF beamforming approach, highlighting their respective advantages and disadvantages. The simulation commences by demonstrating the orthogonal beams available for use in the proposed approach, represented in both Polar and Cartesian formats. The Polar format shows the orientation of the orthogonal beams, while the Cartesian format illustrates the orthogonality of the beams and the strength of the received power at the user end. The proposed algorithm allocates the intended user to one of the available orthogonal beams based on their availability, as demonstrated in Table 1. To ensure optimal user performance, the user is continually switched between the orthogonal beams based on their position.

Initially, the present study introduces and evaluates the suggested orthogonal beamforming approach for a single user. This is accomplished by displaying the beam assigned to the user and subsequently calculating the resulting spectral efficiency. Subsequently, the study considers the implementation of orthogonal beamforming for multiple users. The efficacy of this approach is assessed through the display of the assigned beams to individual users, the calculation of the received signal intensity at each user, and the evaluation of the system’s sum spectral efficiency.

The simulations will compare the proposed orthogonal beamforming approach with the digital ZF beamforming technique. Through this comparison, it will be demonstrated that the proposed approach has the potential to serve a greater number of individuals than the ZF beamforming technique. Furthermore, the orthogonal beamforming approach will be shown to outperform the ZF beamforming technique in terms of SE. This section will also include simulations of the common beam problem for multiusers in the suggested approach, as well as evaluations of potential solutions to this problem. The simulations will be performed using the MATLAB R2016a software package.

5.1 Orthogonal beams in proposed orthogonal beamforming technique

The proposed method provides a set of orthogonal beams, which were previously derived using Eq. (13) in the analytical section. Assuming that the base station (BS) has a total of 100 antennas (M = 100), the suggested approach provides (M/2 = 50) orthogonal beams. The serviced region’s azimuth angles are confined between \({0}^{o}\) and \({90}^{o}\). Figure 4 displays the orthogonal beams in both Polar and Cartesian forms. Specifically, Fig. 4a represents all 50 orthogonal beams in Polar form, while Fig. 4b portrays the beams in Cartesian form to determine the received and interference signals’ intensity at the targeted users. Notably, Fig. 4b indicates that the received signals exhibit minimal interference levels. This is due to the implementation of orthogonal beams in the transmit beamforming operation. To clarify, Fig. 4b showcases only the orthogonal beams with azimuth angles between \({25}^{o}\) and \({50}^{o}\). Therefore, the suggested approach helps reduce interference levels in the received signals for targeted users.

Fig. 4
figure 4

Orthogonal beams in proposed orthogonal beamforming. a In Polar form. b In Cartesian form

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5.2 Proposed orthogonal beamforming technique for single user

The following section outlines the recommended orthogonal beamforming technique for a single-user instance. This approach assumes that a single user is situated at one of the orthogonal angles pertaining to the orthogonal beams, specifically at an azimuth angle of \({30}^{o}\). It is expected that optimal performance for this user will be achieved by allocating the user to the corresponding orthogonal beam. However, it is important to note that any deviation from the orthogonal angle will negatively affect their performance. Based on the algorithm described in Table 1, after a certain allowable reference shift as specified in Eq. (15), the user will be assigned to another orthogonal beam based on its new location. This section first details the user’s assignment to the orthogonal beam and subsequently evaluates the user’s performance based on spectral efficiency.

Figure 5 shows the process of allocating an orthogonal beam to a user and subsequently moving the user across orthogonal beams due to the mobility. Initially, the user is assigned beam number 26 with an angle of \({30}^{o}\). This beam continues to serve the user until the angular difference between the user’s angle and the angle of the allocated beam exceeds the allowable angular shift, which in this case is \(\Delta\Phi = \pm 0.6661\). When this occurs, or when the user is almost at \({30.6}^{o}\), the user is moved to the next closest orthogonal beam number 27 with an angle of \({30.3323}^{o}\). The aforementioned process is reiterated, and the user is switched between orthogonal beams depending on its altered location. The primary objective of this process is to ensure that the user receives the most feasible power from the allocated beam. It is important to note that Fig. 5 is a subset of Fig. 4b, and it only displays the orthogonal beams discussed in that portion.

Fig. 5
figure 5

Assigning and shifting user among orthogonal beams

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When evaluating the performance of the orthogonal beamforming approach, it is crucial to consider the spectral efficiency during the allocation and movement of users. As depicted in Fig. 5, the three potentially assigned orthogonal beams (beams 26, 27, and 28) are shown in Fig. 6, along with the spectral efficiency of the intended user based on its angular deviation from the allocated orthogonal beam. Figure 6 presents an in-depth analysis of the variation in spectral efficiency as a function of the user’s angular offset from the allocated beam and vice versa.

Fig. 6
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Shifting user between the orthogonal beams and the resultant SE

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In accordance with Eq. (16), the SE is optimized when the user is situated at \({30}^{o}\), which corresponds to the orthogonal beam’s assigned angle, with an angular difference of \(\Delta\Phi = {0}^{o}\). However, the SE diminishes as the angular deviation between the user and the allocated beam increases, reaching an unfavorable value of 2.458 bps/Hz at an angular displacement of \(\Delta\Phi = {0.7}^{o}\). The user is then transferred to beam 27, resulting in an increase in the SE. The SE returns to the optimal state, but with a new value of 8.749 bps/Hz at an angular displacement of \(\Delta\Phi = {0.0323}^{o}\) from the assigned beam. This change in the optimal SE value is due to the user’s misalignment with the allocated beam’s angle, as depicted in Fig. 6.

5.3 Proposed orthogonal beamforming technique for multiuser

This section elucidates the proposed orthogonal beamforming technology for multiuser service. The primary objective of orthogonal beamforming for multiusers is to allocate a unique beam to each user. Suppose that 25 users are randomly positioned, and each user necessitates one orthogonal beam. The beam assignment technique follows the orthogonal beamforming algorithm stated in Table 1. For the anticipated 25 users, Table 4 demonstrates the utilized azimuth angles in the simulation, arranged in ascending order, and their corresponding assigned orthogonal beams, indicating both their azimuth angle \(\left({\Phi }_{orthog}\right)\) and beam number, ranging from 1 to 50.

Table 4 Assigned orthogonal beams for multiuser
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Figure 7 illustrates the orthogonal beams allocated to multiple users in both Polar and Cartesian forms. The signal strength received by each user is dependent on the angle of its assigned beam. As depicted in Fig. 7, user 1 at \({{\Phi }_{1}=1}^{o}\) is proximate to its designated beam (orthogonal beam 1 at \({{\Phi }_{orthog}=1.14}^{o}\)) and, therefore, receives approximately 100% of the transmitted signal power. Conversely, user 10 at \({{\Phi }_{10}=32}^{o}\), which is served by orthogonal beam 26 at \({{\Phi }_{orthog}=31.33}^{o}\), does not receive more than 68% of the signal strength due to its distance from its assigned beam. Consequently, the signal power received by each user serves as a function of its SE, where \({\Phi }_{i}\) represents the azimuth angle of user number “i” as arranged in the table above.

Fig. 7
figure 7

Proposed orthogonal beamforming for multiuser. a In Polar form. b In Cartesian form

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5.4 Comparison between proposed orthogonal beamforming and adaptive ZF beamforming techniques

This section undertakes a comparative analysis of the suggested orthogonal beamforming technique and the adaptive ZF beamforming technique. The performance evaluation is based on two important metrics: the sum spectral efficiency of the system and the maximum number of serviced users.

5.4.1 Comparison from the sum SE point of view

The comparative performance of the suggested orthogonal beamforming and adaptive ZF beamforming techniques in terms of sum spectral efficiency has been studied. The sum spectral efficiency for the suggested beamforming and ZF approaches has been presented in Fig. 8, utilizing the same 25 users as mentioned in Table 4. The study aimed to determine the extent to which the suggested orthogonal beamforming approach outperforms the ZF beamforming technique in terms of sum SE. The sum SE of orthogonal beamforming is 121.6 bps/Hz, while the sum SE of ZF beamforming is 86.44 bps/Hz. These results imply that the suggested orthogonal beamforming technique achieves an improvement of approximately 40.6% in SE compared to ZF beamforming.

Fig. 8
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Sum spectral efficiency for proposed orthogonal and ZF beamforming

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5.4.2 Comparison from the maximum number of served users point of view

The purpose of this section is to compare two techniques based on their respective capacities to serve a large number of users. It involved random selection of users ranging from 10 to 50, and the results were illustrated in Fig. 9. The sum SE for the proposed orthogonal beamforming technique raises with an increase in the number of serviced users. The proposed technique can serve up to 50 users, with a sum SE of approximately 260 bps/Hz. Conversely, the sum SE for the ZF beamforming technique rises until it reaches a specific number of users, which is 25. Beyond this threshold, the sum SE of the ZF technique approximates the same value, indicating performance degradation. The ability of the ZF technique to serve more than 25 users is limited due to the increased interference levels when serving more users with the non-orthogonal beams used in the technique. Consequently, the sum SE remains around 100 bps/Hz in the ZF beamforming.

Fig. 9
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Sum spectral efficiency w.r.t number of served users

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5.5 Orthogonal beamforming with the common beam problem for multiuser

The proposed beamforming approach assigns each orthogonal beam to a single user in all the previously stated scenarios. However, a concern arises when multiple users are positioned in close proximity to one another and allocated to the same beam. This is problematic, as it is preferable to assign each orthogonal beam to a specific user. The current section aims to address this concern by examining the issue and presenting three possible remedies to the problem.

According to the proposed methodology, in the event of 100 antennas being present at the BS, a total of 50 orthogonal beams can be implemented. It is assumed that there are 50 users located randomly between \({0}^{o}\) and \({90}^{o}\). In line with the proposed orthogonal beamforming technique as outlined in Table 1, orthogonal beams are assigned to these users. Table 5 presents the users alongside their respective orthogonal beams, which are identified by numbers ranging from 1 to 50. The table has been arranged in ascending order to ensure a clear presentation of the data.

Table 5 Users served with the proposed technique and their assigned beams with the problem of the common beam
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The data presented in Table 5 highlight the utilization of a single orthogonal beam to serve multiple users. The shaded cells represent the common beam problem, which the adjacent shaded cells refer to the users that are assigned to the same beam. Specifically, orthogonal beam number 3 is allocated to both the user situated at \({\Phi }_{3}=\) \({3}^{o}\) and the user located at \({{\Phi }_{4}=4}^{o}\). Likewise, orthogonal beam number 26 is designated to service users at \({{\Phi }_{24}=31}^{o}\) and \({{\Phi }_{25}=32}^{o}\). To address this challenge, Fig. 10 depicts and elucidates the common beam problems that users encounter. In the given instance, Fig. 10 portrays the positions of users and their respective shared orthogonal beams.

Fig. 10
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The problem of the multiuser with repeated beams in the proposed technique

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The present example exhibits the challenge of a common beam for many users. Nevertheless, despite this difficulty, the system’s sum SE surpasses that of the adaptive ZF approach, measuring at 201.6472 bps/Hz. In contrast, the ZF approach delivers a sum SE of only 120.5 bps/Hz under such circumstances. In the following section, the potential solutions to mitigate the shared beam for multiusers and enhance the system’s sum SE value will be explicated.

5.5.1 Solutions to common beam problem for multiuser

This section aims to address the challenges associated with the proposed orthogonal beamforming technique, which mandates the allocation of each orthogonal beam to a single user, thereby reducing the likelihood of assigning common beams to nearby users.

Solution by increasing the number of antennas used in BS

One potential solution for solving the common beam problem is the augmentation of the number of antennas in the BS. As the number of antennas increases, the number of orthogonal beams in the system increases as well, thus reducing the likelihood of multiple users sharing the same beam. By increasing the number of usable antennas from 100 to 200, for example, the number of possible orthogonal beams doubles to 100 (M/2 = 100). This allows the 50 users in Table 5 to select from a larger pool of orthogonal beams, reducing the probability of multiple users sharing the same beam. Table 6 depicts the allocation of the newly generated orthogonal beams to the 50 users, which results in only two cases of common beams as opposed to the prior scenario, where there were 11. Moreover, the sum SE in this scenario improves to 220.8072 bps/Hz, an improvement over the prior scenario.

Table 6 Solving the problem of common beams by increasing the number of antennas used in BS
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This proposed solution has demonstrated its proficiency in serving 50 users with 200 antennas, equivalent to the adaptive beamforming technique. The suggested approach, however, outperforms the adaptive technique in terms of SE improvement. Moreover, this method is competent enough to handle up to 100 users while increasing the possibility of common beams for multiple users. Consequently, this system possesses the potential to serve more users than the adaptive technique with superior SE improvement.

One of the drawbacks of this particular solution is that it serves a smaller number of users than the maximum permissible limit of users, as recommended by the proposed technique, which is equivalent to M/2. Moreover, while this particular method does indeed reduce the chances of the common beam problem, it does not completely eliminate it.

Solution by ignoring users with problems

The second solution for the common beam problem involves disregarding users who may encounter issues so as to continue allocating a single beam for only one user. This method disregards users having greater angular divergence from the designated common beam. The suggested approach evaluates the angular differences between closely placed users and their assigned shared orthogonal beam. The user experiencing a greater angle difference is ignored as it receives lower signal strength, while the user with the smallest angle difference receives the highest power level. In this manner, beam allocation is optimized by prioritizing users with the least angular deviation from the designated common beam.

In accordance with Table 5, two adjacent users located at \({{\Phi }_{3}=3}^{o}\) and \({{\Phi }_{4}=4}^{o}\) are served by the same orthogonal beam number 3. Angular differences of \(\Delta {\Phi }_{3}= {0.4398}^{o}\) and \(\Delta {\Phi }_{4}={0.5602}^{o}\) have been identified for the respective users. As per the data, the user at \({{\Phi }_{3}=3}^{o}\) will be allocated beam number 3 since it will receive more power, while the user at \({{\Phi }_{4}=4}^{o}\) will be excluded from the allocation process since it will receive less power. This allocation is illustrated in Fig. 10. Similarly, for the two users located at \({{\Phi }_{24}=31}^{o}\) and \({{\Phi }_{25}=32}^{o}\) and assigned to the common beam number 26, the respective angle differences have been identified as \(\Delta {\Phi }_{24}={0.3323}^{o}\) and \(\Delta {\Phi }_{25}={0.6677}^{o}\). Accordingly, the user at \({{\Phi }_{24}=31}^{o}\) will be allocated beam number 26, while the user at \({{\Phi }_{25}=32}^{o}\) will be excluded from the allocation process. This solution is expected to increase the SINR value and, consequently, the sum SE. Nonetheless, it should be noted that the number of served users will decrease as a result.

This solution involves assigning each orthogonal beam to a single user, following the recommended technique. The results indicate, however, that the number of serviced users has diminished from the maximum allowable limit of 50 to 39, as presented in Table 7. Despite this limitation, this solution has heightened the sum spectral efficiency to 229.4975 bps/Hz.

Table 7 The served user by ignoring users with problems
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Solution by assigning users to another unused orthogonal beam

An alternative solution for addressing the issue of a troubled user is to relocate them to an unoccupied orthogonal beam. This method ensures that multiple users are not utilizing the common beam. To accomplish this, one of the two previously mentioned users will be assigned to another vacant beam. The user whose angular deviation from the assigned orthogonal beam is greater will be allocated to another available beam. However, it is important to note that this method has a potential drawback. Users may be assigned to an unoccupied orthogonal beam that is distant, resulting in low received power. Consequently, the user may only receive power from the newly allocated beam’s side lobes, which can reduce its SINR. Nonetheless, the overall system performance improves with this solution as the sum SE rises. Moreover, with the availability of all 50 orthogonal beams, this method can serve all 50 users.

This solution accommodates proximate users located at \({{\Phi }_{3}=3}^{o}\) and \({{\Phi }_{4}=4}^{o}\), characterized by angular disparities of \(\Delta {\Phi }_{3}= {0.4398}^{o}\) and \(\Delta {\Phi }_{4}={0.5602}^{o}\), respectively. The user located at \({{\Phi }_{3}=3}^{o}\), being the closest to beam number 3, is supplied by the aforementioned beam, while the other user located at \({{\Phi }_{4}=4}^{o}\) is served by the nearest vacated orthogonal beam number 8. An analogous scenario can be observed between the users located at \({{\Phi }_{24}=31}^{o}\) and \({{\Phi }_{25}=32}^{o}\), with angular disparities of \(\Delta {\Phi }_{24}={0.3323}^{o}\) and \(\Delta {\Phi }_{25}={0.6677}^{o}\), respectively. In this case, beam number 26 caters to the closest user, whereas beam number 27 serves the other user at the nearest vacated orthogonal beam.

The procedure for assigning a new beam to users is illustrated in Fig. 11. The allocation of other empty beams is dependent on the availability of unused beams in the system. Figure 11 depicts two new assignment scenarios, one of which is considered to be highly effective, whereas the other is relatively ineffective. The former scenario involves assigning a distant beam to the first user at \({{\Phi }_{4}=4}^{o}\), which can supply it with relatively low power from the beam’s side lobe. On the other hand, the later scenario involves the assignment of a neighboring beam to the second user at \({{\Phi }_{25}=32}^{o}\),, which can provide it with an acceptable power level that is neither too high nor too low.

Fig. 11
figure 11

Solving repeated beams by assigning to another beam

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This solution is unlikely deemed satisfactory as certain users are situated at a distance from the recently assigned orthogonal beams. Consequently, these newly assigned beams are unable to offer service to these users. Nonetheless, in comparison to the prior solution, this solution has the potential to improve the SINR and, thus, the sum SE of the system. Table 8 showcases the distribution process of the 50 users to the designated 50 orthogonal beams. Utilizing this solution, the sum SE of the system is measured at 237.4528 bps/Hz.

Table 8 The new assignments of the users by assigning to unoccupied beams
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Solution by separating users in groups

This solution is effective to solve the issue of multiple users sharing beams in close proximity. It involves categorizing the affected users into groups, which can be multiplexed to utilize the same available orthogonal beams. Each group is allocated distinct resource blocks, such as different frequency bands or time slots, to enable them use the same available orthogonal beams. The grouping process aims avoiding common beams for users in the same group.

The data presented in Table 5 and Fig. 10 indicate that users who have azimuth angles of \({{\Phi }_{3}=3}^{o}\) and \({{\Phi }_{4}=4}^{o}\) have been assigned to a common beam, i.e., beam number 3. However, to resolve any common beam issues, the users at \({{\Phi }_{3}=3}^{o}\) and \({{\Phi }_{4}=4}^{o}\) will be allocated to separate groups. This process will be executed for all nearby users who are experiencing similar issues. To address this problem, users can be categorized into distinct groups. A multiplexing technology such as time-division multiplexing (TDM) can be employed to serve these groups differently by multiplexing the distinct groups or serving them in separate time slots.

The present scenario involves two distinct groups of individuals, one comprising 39 users and the other consisting of 11 users. The two groups are serviced by employing a specific type of multiplexing technique ensuring that the two groups do not interfere with each other while sharing beams. This specialized technique allows the efficient servicing of both groups while minimizing the chances of any interference to take place.

The results presented in Table 9 demonstrate that with minimal interference, the 50 users can be served by the nearest 39 beams. The first group of 39 users will have a sum SE of 220.2947 bps/Hz, while the second group of 11 users will have a sum SE of 36.9491 bps/Hz. The sum SE for all 50 consumers will be 257.2438 bps/Hz, which is the highest. However, implementing this solution will require a multiplexing mechanism between the two groups, which will add to the complexity of the system.

Table 9 Solving the problem of common beams with grouping
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Summary of solutions for the common beam problem

In the context of addressing the common beam issue, various solutions were discussed. However, it was observed that the first three methods only provide partial alleviation rather than complete elimination of the problem. On the other hand, the final solution of grouping users is found to be optimal, as it can successfully eradicate the shared beam problem and ensure optimal performance amongst users.

To further elaborate, Table 10 provides a comparative analysis of four potential solutions to the common beam problem for multiusers. The table presents the advantages, drawbacks, sum SE, and SE enhancement of these four solutions in comparison to the common beam issue situation, which has a sum SE of 201.6472 bps/Hz.

Table 10 Summary of solutions for the common beam problem
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5.6 Orthogonal beamforming in the case of an increasing number of users over a number of beams

The current scenario presents an issue where the number of users requiring service surpasses the number of orthogonal beams that can be accessed through the suggested approach. Inevitably, such a situation is likely to result in a common beam problem. For instance, let us consider a scenario where there are 60 users located randomly and 100 utilized antennas at the BS, resulting in the availability of only 50 orthogonal beams. Upon reviewing Table 11, it is apparent that the allocation of these 60 users to the 50 orthogonal beams has led to a common beam problem for some users.

Table 11 The case of increasing served users over the orthogonal beams
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According to Table 11, the prevalence of the common beam issue is more pronounced when the number of serviced users exceeds the available orthogonal beams. To mitigate this issue effectively, it is recommended to segregate the affected users into distinct groups and allocate separate resource blocks to each group, either at different frequencies or at different times. This strategy ensures that each group receives optimal service without interference from other users.

Table 12 presents a viable solution to the common beam problem in that case. The proposed solution involves the division of users into three separate groups, allowing each group to be served seamlessly without interference. However, this approach necessitates a relatively complex multiplexing process. The table categorizes the users into three distinct groups, with the first group comprising 41 users, and the findings indicate a sum SE of 201.05 bps/Hz. The second group is comprised of 16 users, and the sum SE is 74.43 bps/Hz, while the third group includes three users, and the sum SE is 8.87 bps/Hz. The sum SE for all serviced users is therefore 284.36 bps/Hz. This solution provides a feasible option for addressing the common beam problem while ensuring that customers receive optimal service.

Table 12 Solution of the common beam problem in the case of more users than the orthogonal beams
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6 Conclusions

This paper presents orthogonal beamforming, a beamforming approach based on predetermined fixed orthogonal beams. The approach is classified as a switching and an entirely digital beamforming technology that is applied in the context of a massive MIMO system for DL transmission. The fundamental concept behind the proposed approach is determining the available orthogonal beams in the system, which is a function of the number of utilized antennas (M) at the BS. These orthogonal beams are then utilized in the DL transmission procedure leading to identify the approach as “orthogonal beamforming.” The suggested technique involves allocating each orthogonal beam to a specific user, enabling up to M/2 users to be serviced. Each user receives its nearest orthogonal beam based on a proposed algorithm. In comparison to the adaptive ZF beamforming technique, the proposed technique enhances SINR and the sum SE. The adaptive approach can serve up to M/4 users, while the proposed approach can serve up to M/2 users.

The results of the simulations indicate that when utilizing a BS equipped with 100 antennas and serving 25 users, the ZF approach yields a sum SE of approximately 86.44 bps/Hz. In contrast, the proposed orthogonal beamforming technique achieves a sum SE of about 121.6 bps/Hz, resulting in a 40.6% SE boost compared to the ZF approach. Furthermore, the ZF approach can serve 25 users with a sum SE of around 86.44 bps/Hz, whereas the suggested orthogonal beamforming technique may serve up to 50 users with a sum SE of about 260 bps/Hz. The results, therefore, indicate that the suggested technique outperforms the adaptive ZF approach in terms of sum SE and the number of serviced users.

This paper delves into the issue of the common beam problem, which arises when multiple users are assigned to a single beam. The findings indicate that, in such scenarios, the sum SE is only 201.6472 bps/Hz, which is still superior to the adaptive approach. To enhance its performance, the paper proposes four potential remedies to this problem, namely increasing the number of antennas used at the BS, ignoring problematic users, assigning problematic users to unused beams, and employing the user grouping approach. These solutions resulted in SE improvements of 9.5%, 13%, 17%, and 27.5%, respectively. It is worth noting that the suggested approach necessitates knowledge of the user’s angle to assign them to the nearest obtainable orthogonal beam.