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Resource Allocation in Cognitive Radio 5G Massive MIMO NOMA 

Systems 

 

Sahar Mohammed Madhloom Al-Inizi1,2, Mohsen Nickray1 

1Department of computer and information technology, Faculty of engineering Qom University, Qom, Iran. 

2Najaf technical institute, Al-Furat Al-Awsat Technical University (ATU), Najaf, Iraq. 

Corresponding email author: Sahar.mohammed.inj124@atu.edu.iq 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  

Mobile telecommunications systems and services are widely deployed across global 

markets. The proliferation of frequency bands and high-speed services underscores the 

challenge of managing limited frequency resources in next-generation wireless 

communication systems. Cognitive radio technology presents a promising solution to 

efficiently utilize spectrum resources. Additionally, emerging technologies like massive 

multiple-input-multiple-output (MIMO) and non-orthogonal multiple access (NOMA) 

enhance spectrum efficiency in these systems. This paper focuses on cluster head 

selection, user allocation to clusters, and power allocation within clusters in a cognitive 

radio network employing massive MIMO and power domain NOMA. Cluster head 

decisions are influenced by primary user interference considerations. User clustering is 

based on the correlation coefficient between each user's channel correlation matrix and 

the desired cluster head's spatial vector. Power allocation is formulated as a non-convex 

optimization problem, simplified for practical implementation using the CVX toolbox in 

MATLAB. Furthermore, the paper introduces a modified zero-forcing (ZF) beamforming 

technique that nulls interference toward primary users in downlink connections, 

demonstrating improved spectrum efficiency compared to conventional interference 

thresholding methods, as validated through simulation results. 

Keywords: Radio Cognitive Networks, Non-orthogonal Multiple Access, Power 

Allocation, massive MIMO and NOMA Systems, Sum Rate, Convex Optimization, 

Downlink connection, Spectrum Efficiency, Modified ZF, CVX. 

 

1- Introduction 

Cognitive radio (CR) and non-orthogonal multiple access (NOMA) have emerged as significant 

technologies to improve spectrum efficiency in wireless networks, especially when combined with 

massive multiple-input multiple-output (MIMO) systems. Zhou et al. [1] provide a comprehensive 

overview of NOMA's role in cognitive radio networks, detailing the challenges and potential solutions 

to improve spectrum usage. This review is paralleled by research by Wang and Liu [3], who explore 

the advancements in cognitive radio networks and suggest that integrating NOMA with these networks 

could significantly enhance their efficiency. The application of massive MIMO in cognitive radio 

systems, particularly with NOMA, is considered a transformative approach to address spectrum 

scarcity. Senel et al. [5] discuss the implications of NOMA in massive MIMO settings, arguing that it 

could dramatically increase network capacity. Akyildiz et al. [6] provide a broader perspective on next-

mailto:Sahar.mohammed.inj124@atu.edu.iq


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generation dynamic spectrum access and cognitive radio networks, underscoring the importance of 

advanced spectrum sharing techniques to meet the growing demand for wireless communication 

services. Goldsmith et al. [7] further this discussion by offering an information-theoretic perspective 

on breaking spectrum gridlock through cognitive radios, which is fundamental in developing strategies 

for efficient spectrum utilization. Tao et al. [8] survey several technologies of non-orthogonal 

transmission for 5G, which includes NOMA, indicating its critical role in future wireless networks. 

Tragos et al. [9] provide a comprehensive survey on spectrum assignment in cognitive radio networks, 

which is crucial for effective spectrum management and avoiding interference with incumbent users. 

This is supported by the work of Ding et al. [31], who explore the application of MIMO technologies 

to NOMA, indicating potential efficiency gains in terms of throughput and user fairness. Resource 

allocation in these systems is a critical area of study. Higuchi and Benjebbour [13] highlight the 

efficiency of successive interference cancellation (SIC) in NOMA, which can be crucial for optimizing 

resource allocation in cognitive radio systems. Further, the work of El Ayach et al. [57] on spatially 

sparse precoding in millimeter wave MIMO systems points to significant efficiency gains through 

beamforming techniques, which could be effectively combined with NOMA strategies. Power 

allocation is another crucial aspect. Shahrokh and Mohamed-pour [2] address sub-optimal power 

allocation strategies in MIMO-OFDM-based cognitive radio networks, underscoring the trade-offs 

involved in such systems. Similarly, Zhou et al. [24] and Glei and Chibani [25] discuss the rate-optimal 

power domain NOMA, emphasizing the need for efficient power allocation to maximize system 

throughput without causing undue interference to primary users. On the interference management 

front, Li et al. [32] explore the outage performance of cooperative NOMA networks, emphasizing the 

need for robust strategies to manage interference in dense network environments. This is crucial in 

cognitive radio settings where the primary and secondary users must coexist without detrimental 

interference, as detailed by Proakis [35] in his work on spread spectrum signals for digital 

communications. The integration of these technologies requires careful consideration of practical 

limitations and implementation strategies. Srinivasa and Jafar [16] provide a theoretical perspective on 

the throughput potential of cognitive radios, which could inform practical deployment strategies. 

Meanwhile, the survey by Islam et al. [23] on power-domain NOMA in 5G systems reviews the 

potentials and challenges, offering insights into future research directions that could enhance the 

deployment of cognitive radio massive MIMO NOMA systems. Resource allocation in such integrated 

systems often requires sophisticated algorithms to handle the dynamic nature of cognitive radio 

environments. Wang et al. [63] discuss spectrum and energy-efficient beamspace MIMO-NOMA 

configurations that utilize lens antenna arrays to enhance signal directionality and reception quality. 

This approach, which strategically allocates resources in millimeter-wave communications, signifies 

the potential for increased spectrum efficiency and network performance in high-demand scenarios. 

Furthermore, the need for advanced beamforming techniques, as discussed by El Ayach et al. [57] and 

Dai et al. [53], is evident in maintaining effective communication in dense network environments 

where cognitive radios operate. These techniques help manage interference and optimize the spatial 

distribution of network resources, which are critical for supporting the high data rates required in 

modern wireless systems. Theoretical studies, such as those by Srinivasa and Jafar [16] and Proakis 

[35], provide foundational knowledge that aids in understanding the limits and capabilities of these 

complex systems. Practical implementations, however, must also consider the real-world variabilities 



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and imperfections that affect system performance. This gap between theoretical potential and practical 

application is a focal point of ongoing research. The integration of NOMA with cognitive radio-

enabled massive MIMO systems also presents unique opportunities for innovation in power allocation 

strategies. Researchers like Zhou et al. [24] and Glei and Chibani [25] have explored power allocation 

mechanisms that aim to maximize throughput while adhering to the strict power constraints and 

interference management requirements of cognitive radios. These studies underscore the delicate 

balance required between achieving high throughput and maintaining fair access to the spectrum for 

all users. On the regulatory and standardization front, efforts by Chen et al. [20] and Mohammadi et 

al. [21] aim to guide the development of frameworks that accommodate the complexities of NOMA in 

cognitive radio and massive MIMO contexts. These efforts are crucial for ensuring that technological 

advancements align with global standards and regulations, facilitating wider adoption and 

interoperability across different networks and regions. Lastly, the evolution of 5G and beyond 

networks brings additional layers of complexity and opportunity. The work by Sun et al. [52] and Liu 

et al. [19] on advanced MIMO-NOMA systems and cognitive relaying highlight the ongoing 

innovation in this space, pushing the boundaries of what is possible in terms of network capacity and 

efficiency. 

A key distinction in conducting this paper compared to other studies is the proposition of an optimal 

method for resource allocation in a cognitive radio network based on massive MIMO and NOMA 

technologies. The goal is to maximize network throughput or sum rate in the presence of primary users. 

This key objective considers the interference introduced by secondary users on the primary user, the 

total transmitted power constraint in the network, and ensuring the desired SNR for the users. The 

issue of resource management and efficiency in cognitive radio networks was addressed in Chapter 2, 

followed by a review of relevant references in the continuation of this chapter. In Chapter 3, focusing 

on the selected reference, the problem of maximizing the network sum rate along with four constraints 

was introduced. The objective function of the problem was non-convex and the constraints were linear 

and convex, based on which we proposed methods to enhance the spectral efficiency and efficiency of 

the CRMN communication system. Chapter 4 analyzed and examined the results obtained from the 

simulations conducted based on our proposed methods, comparing the strengths and weaknesses of 

each method through several graphs. Finally, in the concluding chapter of this paper, we intend to 

provide a brief summary of the proposed methods and the work carried out in this research. 

2- Research background 

2-1 Proposed System Model of the Problem 

As mentioned in the previous chapter, this chapter of the paper will delve into the investigation of 

power allocation in a cognitive radio network based on NOMA technology with multiple massive 

antennas. Reference [66] has been selected as the primary source due to its relevance to the theme of 

this paper in certain aspects such as NOMA discussion, user clustering, multi-antenna base station, 

and optimization parameters. Unlike other articles, in articles [53] and [66], the number of users in a 

NOMA-based system within each cluster is more than 2 users. 

As depicted in the following system model, in this cognitive radio network, a cognitive radio base 

station with multiple antennas, having 𝑁 transmit antennas serving a primary user and 𝑈 secondary 



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users with a single antenna each, based on NOMA and SDMA technologies, and is considered 

according to the downlink communication state in a cell. Each user occupies a specific spectrum in the 

available frequency spectrum. The spectral sharing between secondary and primary users is considered 

in an underlay access pattern. The proposed system model's primary network includes a primary user 

with a single transmit and receive antenna. The 𝑈 users are grouped in 𝐺 clusters (𝐺<𝑈). The number 

of users assigned to cluster 𝑔 is being considered as 𝑁𝑔. 

 

Fig. 3: Proposed CRMN (Cognitive-Radio-MIMO-NOMA) System Model 

 

In the system model shown in Fig. 1-3, the minimum number of users in each cluster is one user, and 

the number of users in each cluster is different from the number of users in other clusters. On the other 

hand, due to the fact that the system model is considered to be a massive multi-antenna, our assumption 

is that the number of CBS transmitting antennas is much more than the number of receiving antennas 

of all users, so the relationship 𝑁≫𝑈 is established. 

2-2 Signal Model and Mathematical Relationships 

In order to beamforming, the transmitter of a base station employs a Digital Pre-coder when 

transmitting signals to users. The pre-coding matrix for digital pre-coding is denoted by 𝐷. The 

columns of this matrix are normalized, and its dimensions are 𝑁 × 𝐺, defined as follows: 

𝑫 = [𝒅1. 𝒅𝟐. ⋯ . 𝒅𝑮 ] (1-3) 

In equations (1-3), the normalized digital precoding vector corresponding to cluster (g) is represented 

as 𝒅𝒈 ∈ 𝐶
𝑁×1 and is utilized in the primary user channel and secondary users. The transmitted signal 

vector from the antennas of the base station is 𝒙 ∈ 𝐶𝑁×1 and is formulated as follows: 

𝒙 = 𝑫𝒔 (2-3) 

The transmitted symbol vector for each cluster, as shown in equation (2-3), 𝒔 ∈ 𝐶𝑁×1 is defined as 𝒔 =

[𝑠1. 𝑠2. ⋯ . 𝑠𝐺  ]
𝑇. The transmitted symbols of the primary and secondary users from the base station for 



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g cluster are denoted as 𝑠𝑔, and the application of the superposition coding scheme in the power domain 

is as follows: 

𝑠𝐺−𝑆𝑈 = ∑ √𝑝𝑔∙𝑚

𝑁𝑔

𝑚=1

𝑠𝑔∙𝑚−𝑆𝑈 

𝑠𝐺−𝑃𝑈 = ∑ √𝑝𝑔∙𝑚

𝑁𝑔

𝑚=1

𝑠𝑔∙𝑚−𝑃𝑈 

(3-3) 

In Eq. (3-3), 𝑝𝑔∙𝑚 represents the power allocated to user 𝑚 in cluster g, and 𝑠𝑔∙𝑚denotes the message 

symbol. The transmitted message symbol 𝑠𝑔∙𝑚 is normalized, hence the relation 𝐸 {|𝑠𝑔∙𝑚|
2
} = 1 is 

definitely satisfied.. 

The information about the channel status between the base station, primary user, and cognitive users 

for the base station is specified. Therefore, the allocation of power to users by the base station, 

clustering of users, and beamforming related to clusters can be implemented. If the channel vector 

between the base station and the m-th secondary user in cluster g is defined as 𝒉𝑠𝑢, and the channel 

vector between the base station and a single primary user in cluster g is defined as 𝒉𝑝𝑢, the 

𝒉𝑠𝑢 . 𝒉𝑝𝑢𝐶
1×𝑁, which represent millimeter-wave MIMO communication channels, are modeled as 

stated in [57]: 

𝒉𝒔𝒖 = √
𝑵

𝑳𝒈.𝒎
∑𝜶𝒈.𝒎

(𝒍)

𝑳𝒈.𝒎

𝒍=𝟏

∙ �⃗⃗� 𝒂𝒛 (𝝋𝒈.𝒎
(𝒍)
) 

𝒉𝒑𝒖 = √
𝑵

𝑳𝒈.𝒎
∑𝜶𝒈.𝒎

(𝒍)

𝑳𝒈.𝒎

𝒍=𝟏

∙ �⃗⃗� 𝒂𝒛 (𝝋𝒈.𝒎
(𝒍)
) 

(4-3) 

In Eq. (3-4), the number of paths between the base station and user m within cluster g is denoted as 

𝑳𝒈.𝒎. 𝜶𝒈.𝒎
(𝒍)

 is also a random variable and represents the complex channel coefficient of path l where 

𝝋𝒈.𝒎
(𝒍)

is a random variable and  azimuth angle of l path. On the other hand, �⃗⃗� 𝒂𝒛 (𝝋𝒈.𝒎
(𝒍) ) signifies the 

conductor vector of azimuth angle of antennas arrayed in the direction of user m within cluster g along 

path l. These parameters are also defined for the primary channel, but with the distinction that there is 

only one primary user. Furthermore, a uniform linear array is used such that it is assumed that all 

antennas in the array are equally spaced, and the utilization of all N base station antennas in the 

horizontal direction is in accordance with the following relation [58]. 

𝒂𝑎𝑧(𝑛)(𝜑) =
1

𝑁
𝑒𝑥𝑝(𝑗2𝜋𝑛(

𝑑

𝜆
)𝑠𝑖𝑛(𝜑)) 

𝑛 = 0.1.2.⋯ .𝑁 − 1 

(5-3) 



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In Eq. (5-3), d is the horizontal distance between each antenna, 𝜆 is the wavelength of the transmitted 

signal, and 𝒂𝑎𝑧(𝑛)(𝜑)is the n-th element of conductor vector�⃗⃗� 𝒂𝒛 (𝝋𝒈.𝒎
(𝒍)
). If the sum of additive white 

Gaussian noise at the primary user receiver and the m-th secondary user in cluster g are 𝑣𝑠𝑢 and 𝑣𝑝𝑢 

respectively, assuming it as a complex circularly symmetric Gaussian noise with variance 𝒞𝒩(0. σ𝑣
2), 

then the received signal at the m-th user receiver in cluster g and the received signal at the primary user 

in cluster g are as follows:  

𝑦𝑠𝑢 = 𝒉𝑠𝑢
𝐻 𝒙 + 𝑣𝑠𝑢 = 𝒉𝑠𝑢

𝐻 𝑫𝒔 + 𝑣𝑠𝑢 = 𝒉𝑠𝑢
𝐻 ∑∑𝒅𝑖

𝑁𝑔

𝑗=1

𝐺

𝑖=1

√𝑝𝑖.𝑗𝑠𝑖.𝑗 + 𝑣𝑠𝑢 

𝑦𝑝𝑢 = 𝒉𝑝𝑢
𝐻 𝒙 + 𝑣𝑝𝑢 = 𝒉𝑝𝑢

𝐻 𝑫𝒔 + 𝑣𝑝𝑢 = 𝒉𝑝𝑢
𝐻 ∑∑𝒅𝑖

𝑁𝑔

𝑗=1

𝐺

𝑖=1

√𝑝𝑖.𝑗𝑠𝑖.𝑗 + 𝑣𝑝𝑢 

(6-3) 

Considering the proposed system, the received signal by each primary and secondary user's receiver 

can be represented as the desired signal corresponding to the user, intra-cluster interference signal, 

inter-cluster interference signal, and channel noise. As indicated in Eq. (7-3), the desired signal 

corresponding to the secondary user m in cluster g, denoted as 𝑠𝑔∙𝑚−𝑃𝑈. Therefore, the remaining 

signals received by the mentioned user's receiver are considered as interference signals. The key point 

is to note that the intra-cluster interference signal is the same as the interference signal related to the 

NOMA scheme. Hence, to combat this type of interference and neutralize its effects, as mentioned in 

Chapter 1, sequential interference cancellation is utilized by users with stronger channel conditions. 

This relationship holds for the received signal of the primary user based on its channel conditions as 

well. 

𝑦𝑔.𝑚−𝑆𝑈 = (𝒉𝑠𝑢
𝐻 𝒅𝒈√𝑝𝑔∙𝑚 𝑠𝑔∙𝑚−𝑆𝑈) 

                +( 𝒉𝑠𝑢
𝐻 𝒅𝒈 ∑√𝑝𝑔.𝑗𝑠𝑔.𝑗

𝑁𝑔

𝑗=1
𝑗≠𝑚⏟            

𝒊𝒏𝒕𝒓𝒂−𝒄𝒍𝒖𝒔𝒕𝒆𝒓 𝒊𝒏𝒕𝒆𝒓𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒔𝒊𝒈𝒏𝒂

) (7-3) 

                  +

(

 
 
 

𝒉𝑠𝑢
𝐻 𝒅𝒈∑∑𝒅𝒊√𝑝𝑖.𝑗𝑠𝑖.𝑗

𝑁𝑖

𝑗=1
 

𝐺

𝑖=1
𝑖≠𝑔⏟                

𝒊𝒏𝒕𝒆𝒓−𝒄𝒍𝒖𝒔𝒕𝒆𝒓 𝒊𝒏𝒕𝒆𝒓𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒔𝒊𝒈𝒏𝒂)

 
 
 

+ 𝑣𝑠𝑢⏟
𝑐ℎ𝑎𝑛𝑛𝑒𝑙 𝑛𝑜𝑖𝑠𝑒

 

The important and fundamental assumption based on the following relationship is applied to users 

present in each of the clusters: 

|𝒉𝑔.1
𝐻 𝒅𝒈| ≥ |𝒉𝑔.2

𝐻 𝒅𝒈| ≥ ⋯ ≥ |𝒉𝑔.𝑁𝑔
𝐻 𝒅𝒈|     . 𝑔 = 1.2.⋯ . 𝐺 (8-3) 



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The practical assumption mentioned in relation (8-3) illustrates the process of ordering the primary 

and secondary user numbers present in a specific cluster based on the channel conditions of users 

within that cluster. As the user number increases within each cluster, the channel condition of that user 

in the cluster becomes weaker. In other words, User 1 in each cluster has the strongest channel, and 

User 𝑁𝑔in each cluster has the weakest effective channel. Considering the concepts based on the 

NOMA scheme mentioned in Chapter 1, users with better channel conditions will have less power 

allocation, while users with worse channel conditions will have more power allocation. Therefore, for 

each user belonging to a specific cluster, a user with a weaker channel in that cluster will receive a 

stronger signal compared to other users in that cluster.  In other words, the weakest user in a specific 

cluster receives their signal stronger than other users in that cluster and therefore only decodes the 

signal relevant to themselves. It is worth noting that this scenario is only feasible in practice when the 

difference in power levels between strong user signals and weak user signals in a specific cluster is 

considerable. Hence, for all j falling within the inequality interval 𝑚 < 𝑗 ≤ 𝑁𝑔, the m-th user in the g-

th cluster has the capability to eliminate the interfering signal related to the mentioned users in the g-

th cluster by employing the SIC method [59].  

According to equation (3-9), if 𝛾𝑔.𝑚 represents the Signal-to-Noise-plus-Interference Ratio (SINR) for 

user m in cluster g, then we have: 

𝛾𝑔.𝑚 =
𝑝𝑔.𝑚‖𝒉𝑔.𝑚

𝐻 𝒅𝒈‖2
2

(‖𝒉𝑔.𝑚𝐻 𝒅𝒈‖2
2
∑ 𝑝𝑔.𝑗 +
𝑚−1
𝑗=1
 

∑ ∑ 𝑝𝑔.𝑗‖𝒉𝑔.𝑚𝐻 𝒅𝒈‖2
2
+

𝑁𝑖
𝑗=1
 

𝐺
𝑖=1
𝑖≠𝑔

𝜎𝑣2)
    (9-3) 

In Eq. (9-3), for convenience, if we consider the denominator of the fraction to be equal to the scalar 

quantities 𝜉𝑔.𝑚, we will have: 

𝜉𝑔.𝑚 = (‖𝒉𝑔.𝑚
𝐻 𝒅𝒈‖2

2
∑ 𝑝𝑔.𝑗 +

𝑚−1

𝑗=1
 

∑∑𝑝𝑔.𝑗‖𝒉𝑔.𝑚
𝐻 𝒅𝒈‖2

2
+

𝑁𝑖

𝑗=1
 

𝐺

𝑖=1
𝑖≠𝑔

𝜎𝑣
2)    (10-3) 

According to Shannon's equation, the maximum achievable information rate accessible to the m-th 

secondary user in cluster g, considering BW as the system's communication bandwidth, will be 

calculated as follows: 

𝑅𝑔.𝑚 = 𝐵𝑊 × 𝑙𝑜𝑔2(1 + 𝛾𝑔.𝑚)    (11-3) 

Therefore, the total achievable sum rate for all users in the system is calculated as the following 

equation: 

𝑅𝑠𝑢𝑚 =∑∑ 𝑅𝑔.𝑚

𝑁𝑔

𝑚=1

𝐺

𝑔=1

    (12-3) 

Moreover, the system spectral efficiency will also be obtained according to the following relationship: 

𝑆𝐸 =
𝑅𝑠𝑢𝑚
𝐵𝑊

    (13-3) 



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According to Eq. (13-3), in order to enhance the spectral efficiency of the system, efficient user 

clustering, effective pre-coding for each cluster, and optimal power allocation of NOMA to each user 

must be utilized. As we know, it is very challenging to simultaneously obtain the optimal method and 

solution for each of the parameters mentioned above. In the following, we will thoroughly investigate 

the optimization problem for NOMA power allocation in the proposed CRMN network.  

The summary of our work process is as follows: Initially, we incorporate cognitive radio interference 

constraints by defining a threshold from the primary user side to the issue so that the base station, 

managed by power control, can manage incoming interferences from secondary users to primary users. 

Subsequently, we propose a method for selecting clusters in cognitive radio networks, and finally, an 

amended beamforming method is introduced as an alternate approach. 

3- Proposed Method 

The primary goal of next-generation telecommunications systems is to achieve high-quality services 

for all users in the network, each requiring its own specific strategies. Resource allocation aiming to 

maximize the total sum rate of all users with NOMA power optimization variable is formulated as a 

𝑷𝟏 problem: 

𝑷𝟏:

{
 
 
 
 

 
 
 
 

max
{𝑝𝑔.𝑚}

∑∑ 𝑅𝑔.𝑚

𝑁𝑔

𝑚=1

𝐺

𝑔=1

𝑺𝒖𝒔𝒃𝒋𝒆𝒄𝒕 𝒕𝒐.        𝐶1
(𝑝1): 𝑝𝑔.𝑚 ≥ 0  .     ∀𝑔.𝑚

𝐶2
(𝑝1):∑ ∑ 𝑝𝑔.𝑚 ≤ 𝑃𝑚𝑎𝑥

𝑁𝑔

𝑚=1

𝐺

𝑔=1

           𝐶3
(𝑝1): 𝑝𝑔.𝑚+1 ≥ 𝛼 × 𝑝𝑔.𝑚  .     ∀𝑔.𝑚    

 

In problem 𝑷𝟏, 𝑅𝑔.𝑚 represents an argument of the objective function, essentially indicating the 

maximum achievable throughput of the cognitive radio network and the achievable rate of information 

by user m in cluster g. The optimization variable {𝑝𝑔.𝑚}is the power allocated by the NOMA scheme 

to users in each cluster, thereby maximizing the objective function. The parameters 𝑃𝑚𝑎𝑥, 𝛼, and 𝐼𝑡ℎare 

known as inputs to the problem, representing respectively the maximum power from the transmitter's 

power amplifier, the Scale Ratio factor greater than 1 , and the maximum tolerable interference level 

for the primary user. The constraint 𝐶1
(𝑝1)is a non-negativity constraint on the power allocated by each 

user for each m, g. The constraint 𝐶2
(𝑝1)limits the power level that can be transmitted by the base station 

or, in other words, at best, it corresponds to the total power allocated to all users, which is equivalent 

to the power transmitted by the base station. The constraint  𝐶3
(𝑝1) defines the power allocation among 

users close and far from the base station in all clusters. By satisfying this constraint, fairness in the 

allocation of power levels to weak and strong users is ensured, the SIC method is correctly 

implemented in the user receivers, and severe SINR degradation for weak users is prevented. By 

imposing this constraint, the problem of power allocation among users within each cluster in the 

reference [53], which was in conflict with the nature of NOMA, was corrected [66]. By generalizing 

the issue to the cognitive radio domain, one of the most significant challenges we will face is the 



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interferences affecting the primary user. Hence, the proposed methods discussed below are considered 

as solutions for minimizing these interferences. Continuing, we define the optimization problem 𝑷𝟐. 

We introduce the proposed methods in each subsection and then formulate the problem for optimizing 

the total system rate of the proposed system. 

3-1 First Proposed Method: Power Management by Interference Threshold Setting in CRMN 

System 

Problem 𝑷𝟐is an optimization problem of resource allocation in the CRMN system, with the objective 

function of maximizing the sum rate of cognitive radio network. The constraints of problem 𝑷𝟐include 

total transmit power of cognitive radio network, power allocation for ensuring fairness among users 

with different channel conditions, and the interference constraint introduced by secondary users on 

primary users. Ultimately, the problem is formulated to optimize NOMA power as follows: 

𝑷𝟐:

{
 
 
 
 
 
 

 
 
 
 
 
 

max
{𝑝𝑔.𝑚}

∑∑ 𝑅𝑔.𝑚

𝑁𝑔

𝑚=1

𝐺

𝑔=1

𝑺𝒖𝒔𝒃𝒋𝒆𝒄𝒕 𝒕𝒐.        𝐶1
(𝑝2): 𝑝𝑔.𝑚 ≥ 0  .     ∀𝑔.𝑚

𝐶2
(𝑝2):∑ ∑ 𝑝𝑔.𝑚 ≤ 𝑃𝑚𝑎𝑥

𝑁𝑔

𝑚=1

𝐺

𝑔=1

           𝐶3
(𝑝2): 𝑝𝑔.𝑚+1 ≥ 𝛼 × 𝑝𝑔.𝑚  .     ∀𝑔.𝑚

           𝐶4
(𝑝2):∑[‖𝒉𝒑𝒖𝒅𝒈‖2

2
∑ 𝑝𝑔.𝑚

𝑁𝑔

𝑚=1

] ≤ 𝐼𝑡ℎ

𝐺

𝑔=1
 

  .     ∀𝑔.𝑚
    

 

The difference between the optimization problem 𝑷𝟏and the problem 𝑷𝟐lies in the addition of a 

constraint 𝐶4
(𝑝2)that represents the interference constraint imposed by cognitive radio users on the 

primary user; since no interference should occur for the primary user when using the channel, the base 

station should control interference by reducing its power to a level below the interference threshold 

that is tolerable for the primary user and prevent interference on the PU channel. The objective function 

of problem 𝑷𝟏, as indicated by the definition of sum rate and Shannon's Law in Eq. (11-3), is a 

logarithmic function and essentially a non-convex optimization problem with respect to the 

optimization variables {𝑝𝑔.𝑚} that satisfy convexity conditions; all constraints of the problem are linear 

and convex. The subscription of four 𝑷𝟏problem constraints forms a convex feasible set, hence for a 

problem to be convex with linear constraints, the objective function must be convex. Therefore, by 

simplifications and utilizing several lemmas, a convex optimization problem can be transformed into 

one, and ultimately, its decomposition and analysis are conducted in MATLAB software using the 

CVX toolbox. This toolbox is highly practical and important for solving complex convex optimization 

problems effectively. It is essential to note that before using the CVX toolbox, the problem must be 

transformed into a standard Disciplined Convex Programming in order to be definable for this toolbox. 

According to interpretations, we need to address the simplification of the optimization problem 𝑷𝟐; to 

this end, we first introduce the following lemma.  



Communications on Applied Nonlinear Analysis 

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Lemma 1: Considering the Sherman-Morrison-Woodbury matrix relation [60]: 

(𝐀 + 𝐁𝐂𝐃)−1 = 𝐀−1  −  𝐀−1𝐁(𝐈 + 𝐂𝐃𝐀−𝟏𝐁) −𝟏𝐂𝐃𝐀−𝟏 (14-3) 

Given the relationships (3-9) and (3-10), we have the following equation: 

𝛾𝑔.𝑚 =
𝑝𝑔.𝑚‖𝒉𝑔.𝑚

𝐻 𝒅𝒈‖2
2

𝜉𝑔.𝑚
= 𝑝𝑔.𝑚‖𝒉𝑔.𝑚

𝐻 𝒅𝒈‖2
2
𝜉𝑔.𝑚
−1  (15-3) 

Based on relation (3-15) and using Lemma 1, the following relation is established:  

(1 + 𝛾𝑔.𝑚)
−1 = 1 − 𝑝𝑔.𝑚‖𝒉𝑔.𝑚

𝐻 𝒅𝒈‖2
2
 (𝑝𝑔.𝑚‖𝒉𝑔.𝑚

𝐻 𝒅𝒈‖2
2
+ 𝜉𝑔.𝑚)

−1 (16-3) 

On the other hand, for the sake of simplicity, relation (3-17) can be rewritten as follows: 

�̃�𝑔.𝑚−𝑆𝑈 = 𝒉𝑠𝑢
𝐻 𝒅𝒈√𝑝𝑔.𝑚𝑠𝑔.𝑚−𝑆𝑈 + 𝒉𝑠𝑢

𝐻 𝒅𝒈  ∑ √𝑝𝑔.𝑗𝑠𝑔.𝑗

𝑚−1

𝑗=1

 (17-3) 

                  +𝒉𝑠𝑢
𝐻 ∑∑𝒅𝒈√𝑝𝑔.𝑗𝑠𝑔.𝑗 +

𝑁𝑖

𝑗=1
 

𝐺

𝑖=1
𝑖≠𝑔

𝑣𝑔.𝑚 

If the Minimum Mean Square Error (MMSE) method is employed by user m's receiver in cluster g for 

detecting the desired signal 𝑠𝑔.𝑚, which corresponds to user m in cluster g, then the Mean Squared 

Error is defined as follows: 

𝑒𝑔.𝑚 = 𝐸{|𝑠𝑔.𝑚 − 𝑐𝑔.𝑚�̃�𝑔.𝑚|
2
} (18-3) 

𝑐𝑔.𝑚is the Channel Equalization Coefficient in equation (3-18), and its optimal value is calculated as 

follows: 

𝑐𝑔.𝑚
𝑜𝑝𝑡 = 𝑎𝑟𝑔min

𝑐𝑔.𝑚
𝑒𝑔.𝑚 (19-3) 

By substituting equation (3-17) into equation (3-18), we will have: 

𝑒𝑔.𝑚 = 1 − 2𝑅𝑒{𝑐𝑔.𝑚√𝑝𝑔.𝑚𝒉𝑔.𝑚
𝐻 𝒅𝒈} + |𝑐𝑔.𝑚|

2
(𝑝𝑔.𝑚‖𝒉𝑔.𝑚

𝐻 𝒅𝒈‖2
2
+ 𝜉𝑔.𝑚) (20-3) 

Based on the principles of optimization, in order to minimize the objective function with respect to the 

optimal unknown variable optimization, we must take the partial derivative of the objective function 

𝑒𝑔.𝑚 with respect to the unknown variable 𝑐𝑔.𝑚, and set the resulting partial derivative equal to zero to 

find the optimal value of 𝑐𝑔.𝑚 that minimizes the objective function. 

After this step, according to equations (3-19) and (3-20), the optimal value of the channel equalization 

coefficient will be obtained as follows: 

𝑐𝑔.𝑚
𝑜𝑝𝑡 = (√𝑝𝑔.𝑚𝒉𝑔.𝑚

𝐻 𝒅𝒈)(𝑝𝑔.𝑚‖𝒉𝑔.𝑚
𝐻 𝒅𝒈‖2

2
+ 𝜉𝑔.𝑚)

−1 (21-3) 

Then by substituting Equation (3-21) into Equation (3-20), the optimal value of the least squares mean 

error is calculated as follows: 



Communications on Applied Nonlinear Analysis 

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𝑒𝑔.𝑚
𝑜𝑝𝑡 = 1 − (𝑝𝑔.𝑚‖𝒉𝑔.𝑚

𝐻 𝒅𝒈‖2
2
)(𝑝𝑔.𝑚‖𝒉𝑔.𝑚

𝐻 𝒅𝒈‖2
2
+ 𝜉𝑔.𝑚)

−1 (22-3) 

Where Equation (3-21) is equivalent to Equation (3-16), therefore we have: 

(1 + 𝛾𝑔.𝑚)
−1 = min

𝑐𝑔.𝑚
𝑒𝑔.𝑚 (23-3) 

After substituting Equation (3-23) into Equation (3-11) assuming 𝐵𝑊 = 1𝐻𝑧, Equation (3-11) is 

rewritten as follows: 

𝑅𝑔.𝑚 = 𝑙𝑜𝑔2((1 + 𝛾𝑔.𝑚) = −𝑙𝑜𝑔2 (min
𝑐𝑔.𝑚

𝑒𝑔.𝑚) = min
𝑐𝑔.𝑚

(−𝑙𝑜𝑔2𝑒𝑔.𝑚) (24-3) 

To simplify Equation (3-24), we introduce a new lemma. 

Lemma 2: We consider the following assumption [61]: 

𝑓(𝑎) −
𝑎𝑏

𝑙𝑛2
+ 𝑙𝑜𝑔2(𝑎) +

1

𝑙𝑛2
 (25-3) 

If the optimal value of 𝑎 is in the form of 𝑎𝑜𝑝𝑡 =
1

𝑏
, then for  𝑎 > 0, the following relationship holds 

true: 

𝑚𝑎𝑥
𝑎
𝑓(𝑎) = − 𝑙𝑜𝑔2(𝑎𝑏) (26-3) 

Based on Lemma 2, Equation (3-24) is rewritten as follows: 

𝑅𝑔.𝑚 = 𝑚𝑎𝑥
𝑐𝑔.𝑚

𝑚𝑎𝑥
𝑐𝑔.𝑚>0

(−
𝑎𝑔.𝑚𝑒𝑔.𝑚

𝑙𝑛2
+ 𝑙𝑜𝑔2(𝑎𝑔.𝑚) +

1

𝑙𝑛2
 (27-3) 

Where 𝑎𝑔.𝑚in Equation (3-27) is calculated as follows: 

𝑎𝑔.𝑚 =
1

𝑒𝑔.𝑚
𝑜𝑝𝑡 =

1

1 − (1 − (𝑝𝑔.𝑚‖𝒉𝑔.𝑚𝐻 𝒅𝒈‖2
2
)(𝑝𝑔.𝑚‖𝒉𝑔.𝑚𝐻 𝒅𝒈‖2

2
+ 𝜉𝑔.𝑚)−1)

 (28-3) 

By substituting equation (3-27) into the optimization problem 𝑷𝟐, since the value 
1

𝑙𝑛2
is a constant 

numerical value, it is therefore irrelevant in the analysis of the optimization problem and we can 

exclude it from the objective function argument. Subsequently, the optimization problem 𝑷𝟐 is 

transformed into a new optimization problem 𝑷𝟑 and formulated as follows: 

𝑷𝟑:

{
 
 
 
 
 
 

 
 
 
 
 
 

max
{𝑝𝑔.𝑚}

∑∑𝑚𝑎𝑥
𝑐𝑔.𝑚

𝑚𝑎𝑥
𝑐𝑔.𝑚>0

(−
𝑎𝑔.𝑚𝑒𝑔.𝑚

𝑙𝑛2
+ 𝑙𝑜𝑔2(𝑎𝑔.𝑚))

𝑁𝑔

𝑚=1

𝐺

𝑔=1

𝑺𝒖𝒔𝒃𝒋𝒆𝒄𝒕 𝒕𝒐.        𝐶1
(𝑝3): 𝑝𝑔.𝑚 ≥ 0  .     ∀𝑔.𝑚

𝐶2
(𝑝3):∑ ∑ 𝑝𝑔.𝑚 ≤ 𝑃𝑚𝑎𝑥

𝑁𝑔

𝑚=1

𝐺

𝑔=1

           𝐶3
(𝑝3): 𝑝𝑔.𝑚+1 ≥ 𝛼 × 𝑝𝑔.𝑚  .     ∀𝑔.𝑚

           𝐶4
(𝑝3):∑[‖𝒉𝒑𝒖𝒅𝒈‖2

2
∑ 𝑝𝑔.𝑚

𝑁𝑔

𝑚=1

] ≤ 𝐼𝑡ℎ

𝐺

𝑔=1
 

  .     ∀𝑔.𝑚
    

 



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Based on the optimization problem formulation 𝑷𝟑, first the problem needs to be related to the 

variables {𝑐𝑔.𝑚}and {𝑎𝑔.𝑚}. Then, the problem should be optimized with respect to the optimal values 

of these variables relative to the variable {𝑝𝑔.𝑚}, while according to equations (21-3) and (3-28), both 

variables {𝑐𝑔.𝑚}and {𝑎𝑔.𝑚} are dependent on the values of the variable {𝑝𝑔.𝑚}. An iterative 

optimization algorithm is employed to analyze the 𝑷𝟑 problem, in which the variable {𝑝𝑔.𝑚} is 

optimized and then subsequently, the variables {𝑐𝑔.𝑚}and {𝑎𝑔.𝑚} are optimized together 

simultaneously. The details of this process are described as pseudo code in the following algorithm.  

 

Algorithm for calculating the combination of 𝒄𝒈.𝒎and 𝒂𝒈.𝒎according to the values of 

𝒑𝒈.𝒎in the previous iteration 

The known inputs of the algorithm 

𝑡∶𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑙𝑔𝑜𝑟𝑖𝑡ℎ𝑚 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛 

𝐶𝑜𝑢𝑛𝑡𝑒𝑟∶𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝑉𝑋 𝑇𝑜𝑜𝑙𝑏𝑜𝑥 𝐸𝑥𝑒𝑐𝑢𝑡𝑖𝑛𝑔 

𝑝𝑔.𝑚
(0)

 ∶ 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑓𝑜𝑟 𝑝𝑔.𝑚 

휀∶ 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 

𝑡 = 1      . 𝑝𝑔.𝑚
(0)

=
𝑃𝑚𝑎𝑥
𝑈

  .         휀 = 10−5 

Algorithm implementation steps 

1- While    𝑡 < 𝐶𝑜𝑢𝑛𝑡𝑒𝑟           |𝑝𝑔.𝑚
(𝑡)

− 𝑝𝑔.𝑚
(𝑡−1)

| >휀 

2- Compute optimal value of 𝑝𝑔.𝑚
(𝑡)
 by CVX Toolbox based on Problem 𝑷𝟑 

3- 𝑡= 𝑡+1 

4- Update value of 𝑐𝑔.𝑚
(𝑡)

 & 𝑎𝑔.𝑚
(𝑡)

  based on Problem 𝑷𝟑 

5- End While 

Termination of the algorithm 

 

In order to clarify the matter, it is assumed that according to the optimal values obtained for the 

variables {𝑝𝑔.𝑚}in iteration t-1 of the algorithm, the optimal values of the variables {𝑐𝑔.𝑚}and {𝑎𝑔.𝑚} 

in the t-th iteration of this algorithm will be calculated as follows: 

𝑐𝑔.𝑚
(𝑡)

= (√𝑝𝑔.𝑚
(𝑡−1)

 𝒉𝑔.𝑚
𝐻 𝒅𝒈) (𝑝𝑔.𝑚

(𝑡−1)
‖𝒉𝑔.𝑚

𝐻 𝒅𝒈‖2
2
+𝜉𝑔.𝑚

(𝑡−1)
)−1 (29-3) 



Communications on Applied Nonlinear Analysis 

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𝑎𝑔.𝑚
(𝑡)

=
1

𝑒𝑔.𝑚
𝑜𝑝𝑡(𝑡)

 (30-3) 

In equations (3-29) and (3-30), the variables 𝜉𝑔.𝑚
(𝑡−1)

 and 𝑒𝑔.𝑚
𝑜𝑝𝑡(𝑡)

are calculated as follows: 

𝜉𝑔.𝑚
(𝑡−1)

= ‖𝒉𝑔.𝑚
𝐻 𝒅𝒈‖2

2
∑ 𝑝𝑔.𝑗

(𝑡−1)
+

𝑚−1

𝑗=1
 

∑∑𝑝𝑔.𝑗
(𝑡−1)

‖𝒉𝑔.𝑚
𝐻 𝒅𝒈‖

2

2
+

𝑁𝑖

𝑗=1
 

𝐺

𝑖=1
𝑖≠𝑔

𝜎𝑣
2    (31-3) 

𝑒𝑔.𝑚
𝑜𝑝𝑡(𝑡)

= 1 − (𝑝𝑔.𝑚
(𝑡−1)

‖𝒉𝑔.𝑚
𝐻 𝒅𝒈‖2

2
)(𝑝𝑔.𝑚

(𝑡−1)
‖𝒉𝑔.𝑚

𝐻 𝒅𝒈‖2
2
+ 𝜉𝑔.𝑚

(𝑡−1)
)−1 (32-3) 

 Moreover, 𝑒𝑔.𝑚 is considered as one of the arguments of the objective function of the optimization 

problem 𝑷𝟑. The optimal values calculated for the variable 𝑝𝑔.𝑚 in the  t-1  iteration of the above 

algorithm are dependent, and in the  t  iteration of the algorithm, calculations are written as: 

𝑒𝑔.𝑚
(𝑡)

= 1 − 2𝑅𝑒 {𝑐𝑔.𝑚
(𝑡) √𝑝𝑔.𝑚

(𝑡)
𝒉𝑔.𝑚
𝐻 𝒅𝒈} + |𝑐𝑔.𝑚

(𝑡)
|
2

(𝑝𝑔.𝑚
(𝑡)
‖𝒉𝑔.𝑚

𝐻 𝒅𝒈‖2
2
+ 𝜉𝑔.𝑚

(𝑡)
) (33-3) 

After the insertion of relation (3-10) into relation (3-33), we will have: 

�̃�𝑔.𝑚
(𝑡) = 1 − 2𝑅𝑒{𝑐𝑔.𝑚

(𝑡) 𝒉𝑔.𝑚
𝐻 𝒅𝒈}√𝑝𝑔.𝑚

(𝑡)
 

           +|𝑐𝑔.𝑚
(𝑡)
|
2

(‖𝒉𝑔.𝑚
𝐻 𝒅𝒈‖2

2
∑𝑝𝑔.𝑗

(𝑡)
+

𝑚

𝑗=1
 

∑∑𝑝𝑔.𝑗
(𝑡)
‖𝒉𝑔.𝑚

𝐻 𝒅𝒈‖
2

2
+

𝑁𝑖

𝑗=1
 

𝐺

𝑖=1
𝑖≠𝑔

𝜎𝑣
2 

(34-3) 

In the end, the objective function of relation (3-35) is a convex function: 

min
{𝑝𝑔.𝑚
(𝑡)

}
= ∑∑ 𝑎𝑔.𝑚

(𝑡)

𝑁𝑔

𝑚=1

𝐺

𝑔=1

�̃�𝑔.𝑚
(𝑡)

 (35-3) 

Using the objective function relationship (3-53), the new problem 𝑷𝟒 is a convex standard optimization 

problem because it has a convex objective function and four convex constraints, all of which jointly 

form a convex feasible region for the problem. The convex optimization formulation of problem 𝑷𝟒 is 

as follows: 

By convexifying the 𝑷𝟒optimization problem to its standard form in the MATLAB simulation 

software, this problem can be defined in the CVX toolbox and solved using numerical analysis 

methods. The objective function of this problem is expected to reach its minimum value due to the 

interference nature of the signal types through optimization. The variable 𝑝𝑔.𝑚, representing the power 

allocation factor for the NOMA scheme in cognitive radio networks, is optimized through simulating 

this complex problem in the CVX toolbox, yielding the output of the 𝑷𝟒optimization problem. By 

implementing this method, the interference caused by secondary users on the primary user channel is 

significantly mitigated. This is achieved by reducing and controlling the power of secondary users to 

meet the threshold 𝐼𝑡ℎ set by the primary user, thereby managing the interferences. Consequently, with 



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the desirable conditions in the primary user channel, the primary user will receive the signal with 

minimal interference at their receiver. All results will be detailed, interpreted, and evaluated in the 

form of figures and graphs in Chapter Four. 

𝑷𝟒:

{
 
 
 
 
 
 

 
 
 
 
 
 

max
{𝑝𝑔.𝑚
(𝑡)

}
∑∑ 𝑎𝑔.𝑚

(𝑡)

𝑁𝑔

𝑚=1

𝐺

𝑔=1

�̃�𝑔.𝑚
(𝑡)

𝑺𝒖𝒔𝒃𝒋𝒆𝒄𝒕 𝒕𝒐.        𝐶1
(𝑝3): 𝑝𝑔.𝑚 ≥ 0  .     ∀𝑔.𝑚

𝐶2
(𝑝3):∑ ∑ 𝑝𝑔.𝑚 ≤ 𝑃𝑚𝑎𝑥

𝑁𝑔

𝑚=1

𝐺

𝑔=1

           𝐶3
(𝑝3): 𝑝𝑔.𝑚+1 ≥ 𝛼 × 𝑝𝑔.𝑚  .     ∀𝑔.𝑚

           𝐶4
(𝑝3):∑[‖𝒉𝒑𝒖𝒅𝒈‖2

2
∑ 𝑝𝑔.𝑚

𝑁𝑔

𝑚=1

] ≤ 𝐼𝑡ℎ

𝐺

𝑔=1
 

  .     ∀𝑔.𝑚
    

 

3-2 Second Proposed Method: Choosing Cluster Heads Considering Initial User Status 

To initiate user clustering, it is necessary to designate one user as the cluster head for each cluster. The 

objective of this task is to align the beam forming direction of the array of transmitter antennas of the 

Base Station towards the cluster head of each cluster. In the previous sections, we discussed the CHS 

algorithm for selecting cluster heads. According to this algorithm, user clustering is achieved based on 

the correlation between the channels of the remaining users and the cluster heads. A cognitive radio 

network consists of a number of secondary users and one primary user; therefore, in the context of 

selecting cluster heads, attention must be paid to the fact that the remaining users who are around the 

Primary User (PU) and meet the criteria for becoming cluster heads should not be chosen as cluster 

heads in order to have the least impact on the PU. As mentioned in the user clustering section, to avoid 

problem complexity, we assign only one beam to each cluster. Since we use ZF beamforming method, 

if a user present around the initial user is selected as the master user, the beam direction focuses towards 

that user, allowing them to receive significant benefits from the antenna array. Consequently, the 

received signal becomes very strong, enabling a substantial decrease in antenna array utilization 

towards the initial user to prevent interference within the master user cluster. Therefore, we should 

consider this approach as another proposed method in the context of investigating the correlation of 

the master channel with other lingering cognitive users. By implementing this technique, when 

calculating the correlation of users with master clusters, we introduce a new factor to reduce the impact 

of other cognitive users on the Primary User (PU). Furthermore, based on the favorable outcomes of 

utilizing special channel matrix values as referenced in [62], when assigning users to each cluster, we 

consider the coefficient of correlation between the eigenvector corresponding to the largest eigenvalue 

of the channel matrix and the correlation of each user's channel with the target master cluster. This 

approach is employed in Algorithm 1 as described below. All inputs of the algorithm must be specified, 

such as the number of users in the system, the channel vector of each user, the number of clusters for 

user clustering, and the initial adaptive threshold value, denoted as T. Additionally, Set represents the 

set of users selected as cluster heads, and 𝑆𝑒𝑡𝑐 denotes the set of users that are candidates for cluster 



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head selection at each stage. The algorithm iterates until cluster heads are determined for all clusters, 

meaning the number of iterations is proportional to the number of clusters considered. Ultimately, in 

the algorithm output, we have a number of users equal to the number of clusters selected as cluster 

headsets for the proposed CRMN network. It is worth mentioning again that at each stage the proposed 

algorithm is executed, the degree of correlation between each user's channel and the channels of cluster 

heads selected in the previous stage is determined by calculating the inner product of their normalized 

channel vectors. 

 

Algorithm 1: Selecting the cluster head for all clusters with the least impact on PU in the 

CRMN system 

The inputs of the algorithm 

Number of Users = 𝐾, Number of Clusters = 𝐺;  

User Channel Vector=𝒉𝑘 , 𝑘 = 1 .2 . … . 𝐾 ; 

 Initial Threshold = 𝑇= 0 , 𝑃𝑟𝑖𝑚𝑎𝑟𝑦 𝑢𝑠𝑒𝑟 𝑐ℎ𝑎𝑛𝑛𝑒𝑙 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡=𝛽=0.5 

Algorithm implementation steps 

𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑠 𝐻𝑒𝑎𝑑 𝑆𝑒𝑡∶ 𝑆𝑒𝑡 

1- 𝑎𝑘 = ‖ℎ𝑘‖2 ; % users channel vector’s 2-norm 

2- 𝐴 = [ 𝑎1 , 𝑎2 , … , 𝑎𝐾 ] ; 

3- ℎ̃𝑘 = ℎ𝑘/𝑎𝑘; % secondary users channel vector’s normalization 

4- 𝐴𝑠𝑜𝑟𝑡 = 𝑠𝑜𝑟𝑡 ( 𝐴 , ′ 𝑑𝑒𝑠𝑐𝑒𝑛𝑑 ′ ) ; % users ordering from strong to weak 

5- 𝑆𝑒𝑡=𝐴𝑠𝑜𝑟𝑡 (1); 

6- 𝑆𝑒𝑡𝑐 = 𝐴𝑠𝑜𝑟𝑡 / 𝛤 ; 

7- Ω =  Φ ; 

8- 𝑔=2; 

 9- 𝐰𝐡𝐢𝐥𝐞 𝑔 ≤ 𝐺 

10- 𝐢𝐟 𝛺 ==  𝛷 

11- 𝐰𝐡𝐢𝐥𝐞 𝛺 ==  𝛷 

12-  𝑇 =  𝑇 +
1−𝑇

10
 ; % increase adaptive threshold level 

13- Ω =  { 𝑖 ∈   𝑆𝑒𝑡𝑐  ||𝒆𝒊𝒈𝒗𝒆𝒄 𝒉𝒊
𝑯. 𝒆𝒊𝒈𝒗𝒆𝒄 𝒉𝒋| + 𝜷|�̃�𝒊

𝑯�̃�𝒑𝒖
𝑻 | < 𝛿 . ∀𝑗 ∈ 𝑆𝑒𝑡}; 

14-𝐞𝐧𝐝 𝐰𝐡𝐢𝐥𝐞 

15-𝐞𝐧𝐝 𝐢𝐟 



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16- Ω =  { 𝑖 ∈   𝑆𝑒𝑡𝑐  ||𝒆𝒊𝒈𝒗𝒆𝒄 𝒉𝒊
𝑯. 𝒆𝒊𝒈𝒗𝒆𝒄 𝒉𝒋| + 𝜷|�̃�𝒊

𝑯�̃�𝒑𝒖
𝑻 | < 𝛿 . ∀𝑗 ∈ 𝑆𝑒𝑡}; 

17- 𝑆𝑒𝑡=𝑆𝑒𝑡 ∪ Ω (1); 

18- 𝑆𝑒𝑡𝑐=O/𝑆𝑒𝑡; 

19- Ω= Φ;  

20- 𝑇= 0; 

21- 𝑔=𝑔+1; 

22- 𝐞𝐧𝐝 𝐰𝐡𝐢𝐥𝐞 

23- 𝐫𝐞𝐭𝐮𝐫𝐧 𝑆𝑒𝑡 . 

 

Termination of the algorithm1 

Assigning Users to Clusters 

After selecting the cluster heads, user clustering is performed. All remaining users, except for the 

cluster heads that have already been selected, must be assigned to one of the clusters related to the 

chosen cluster heads. User clustering is based on the correlation coefficient₁ of each user with the 

selected cluster heads [66]. According to the following relationship, which represents a definition of 

the correlation coefficient, the maximum value of this coefficient is set to 1. Assigning m-th user to 

cluster �̃� in such a way that the following relationship holds: 

�̃� = argmax
𝑔

|𝒆𝒊𝒈𝒗𝒆𝒄 𝒉𝒎
𝑯 . 𝒆𝒊𝒈𝒗𝒆𝒄 𝒉𝒔𝒆𝒕(𝒈)|

‖𝒆𝒊𝒈𝒗𝒆𝒄 𝒉𝒎‖2 ∙ ‖𝒊𝒈𝒗𝒆𝒄 𝒉𝒔𝒆𝒕(𝒈)‖2

 

𝑔 ∈ {1.2. .⋯ . 𝐺} 

(36-3) 

According to equation (36-3), for clustering users, correlation coefficients of the desired user channel 

m, who is not a member of Set, with each of the sub-clusters selected in the previous stage are 

separately calculated. Following this approach, all users present in a specific cluster will have high 

channel correlation coefficients with each other, while on the other hand, users in other clusters will 

have lower channel correlation coefficients with each other. Therefore, this method efficiently helps 

eliminate interference signals between clusters. According to this equation, the user channel cluster 

matrix is computed using the calculated channel eigenvalues. This user clustering method based on 

channel correlation coefficients has been a very common approach that does not consider the system's 

state. It is commonly used for clustering users in various system models. The graphs and simulation 

results of this method will also be thoroughly examined in Chapter Four. 

3-3 Third Proposed Method 

Modified ZF beamforming with zero-forcing in the beamforming direction towards the primary 

user (Null Method). 

After clustering users and before sending signals for each cluster according to the OMA scheme, 

initially, digital precoding operation using the BF matrix on symbols of each cluster is applied. 



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Following this, according to equation (3-3), the transmitter BS sends the signal corresponding to each 

of the clusters. To eliminate interference signals between clusters, focusing on the sub-cluster with the 

strongest user in each cluster, digital precoding at the base station is performed using the beamforming 

with zero-forcing method [63]. In the ZF method, the transmission beams from the transmitting 

antennas of the BS are focused on users within each cluster that have the best channel conditions. As 

a result, the interference effect on other users is almost minimized to zero to prevent interference with 

the primary user. Since the system model of this paper is based on cognitive radio networks and the 

primary user in this network is of particular importance, we modify the beamforming in the ZF method 

and introduce a zero-forcing beamforming approach aligned with the primary user in a downlink 

communication as the third proposed method. 

This method is similar to the ZF method, with the difference that we employ the initial user information 

in ZF planning in cognitive radio networks. In defining the BF vector, we exactly place the PU channel 

vector in the last column of the channel vector. By doing this on the PU efficiency, it will have a very 

positive impact. The reason for this is that when the BF vector is directed towards PU, practically no 

interference from SUs will affect PU. It is true that by implementing this method, the efficiency of 

secondary users will slightly decrease, which is acceptable, but interference directed towards the 

primary user (null method) will be significantly reduced. The impact of this method on PU in low 

interference thresholds injected from the base station to the network is highly noticeable, and by 

implementing this method, the achievable information rate for the primary user will greatly improve. 

In this context, it is assumed that the best channel conditions for the primary user 𝑝𝑢𝑔 located in the 

specific cluster 𝑔 are achievable. In other words, due to having the best channel conditions, this user 

(the primary user) is selected as the primary user for cluster 𝑔. Therefore, based on the channel vectors 

of strong users (cluster heads) in the G cluster, the aggregation matrix associated with millimeter wave 

MIMO communication channels is defined as follows: 

𝑯𝟏 = [𝒉𝑚1 . 𝒉𝑚2 . ⋯ . �̃�𝑝𝑢
𝑇  ] (37-3) 

In general, and according to equation (3-37), the matrix forming method in this approach forms the 

pseudo-inverse matrix of the user channel aggregation matrix at the base station. Therefore, the digital 

pre-encoding matrix �̅�  ∈ 𝐶𝑁×𝐺  will be equivalent to the pseudo-inverse of the system MIMO channel 

aggregation matrix [64]: 

𝑫𝟏̅̅ ̅̅ = 𝑯𝟏(𝑯𝟏𝑯𝑯𝟏)−𝟏 = [�̅�1. �̅�1. ⋯ . �̅�𝐺 ] (38-3) 

In equation (3-38), the vector representing the pre-encoding of digital cluster 𝑔 is denoted by 𝑔, where 

the normalized version of this vector is calculated as follows: 

𝒅𝒈 =
�̅�𝒈

‖�̅�𝒈‖2
2 (39-3) 

The parameter 𝒅𝒈in equation (3-39) is the column g of the digital pre-encoding matrix introduced in 

equation (1-3). 

 



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4- Results and Analysis 

4-1 Introduction 

In the previous chapter, we thoroughly discussed the system model and its mathematical relationships. 

By employing several simplifications, we transformed the non-convex problem into a convex 

optimization problem. Due to the complex nature of convex optimization problems, we will analyze 

and decompose it using the CVX toolbox in the MATLAB simulation software. The CVX toolbox 

provides highly practical solvers suitable for solving complex and standard convex optimization 

problems. In light of the aforementioned discussions, this chapter will focus on analyzing and 

interpreting the simulation results of the proposed methods in the form of comparative graphs. 

4-2 Numerical Parameters Used in Simulation 

Before conducting the simulation, this section introduces the input parameters of the problem, which 

are defined as a set of parameters with specific values in the MATLAB environment. The values of 

these parameters should be reasonably defined to ensure the practical efficiency of modern 

communication systems is not compromised. To this end, we initially address the generation of random 

communication channels and, in a tabular format, determine the numerical values of the input 

parameters for optimization problem analysis and simulation in the MATLAB software. 

The frequency of transmitted signals in fifth-generation telecommunication systems is very high. Since 

the proposed system model in this paper is based on 5G, the frequency ranges from 30 to 40 GHz. As 

mentioned in Chapter 3, we define a telecommunication channel as a random channel with millimeter 

wave MIMO, whose size and phase have statistically Rayleigh and uniform probability distributions, 

respectively. The telecommunication channel exhibits flat fading and only considers the small-scale 

fading effect within it. According to reference [66], 𝛼𝑔.𝑚
(𝑙)

is a random variable representing the channel 

coefficient through which each receiver receives the transmitted signal along that path. Statistically, 

for both direct and non-direct paths, they have different probability distributions. Therefore, the 

assumption is that there exists one direct path and two non-direct paths among the possible paths for 

signal reception. The reason for considering only two non-direct paths is that in millimeter-wave 

communication systems, the transmission frequency is very high, typically in the range of several tens 

of gigahertz, causing significant signal attenuation in Non-Line-of-Sight (NLoS) paths, hence the 

effective number of non-direct paths is not significant. Statistically, 𝛼𝑔.𝑚
(𝑙)

  follows  Complex Gaussian 

distribution 𝒞𝒩(0.1) in the direct path and a Complex Gaussian distribution 𝒞𝒩(0. 10−1)  in both 

non-direct paths. 

The 𝜑𝑔.𝑚
(𝑙)

is a random variable and considered as azimuth angle of departure of the BS transmitter 

antenna array. Statistically, 𝜑𝑔.𝑚
(𝑙)

 has a uniform distribution of  𝒰(−𝜋. 𝜋). According to reference [58], 

the horizontal distance of each transmitter antenna (𝑑) from each other is considered equal to half the 

wavelength of the transmitted signal. The numerical values of the input parameters of the optimization 

problem for analysis and simulation in MATLAB software are given in Table 1-4  

 

 



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Table 4-1 Numerical Values of Simulation Parameters. 

Parameter Description Numerical value 

𝑵 The number of antennas at the transmitter  base station 64 

𝑼 Number of single antenna secondary users 24 

𝑳𝒈.𝒎 
The possible paths for receiving the transmitted signal 

by each primary and secondary receiver 
3 

𝑮 The number of clusters 4 

𝑰𝒕𝒉 
Interference threshold determined from the primary user 

side 
-60 dB w 

𝑺𝑵𝑹 Signal-to-noise ratio 10dB 

𝝈𝒗
𝟐 The received noise power in the antenna of each receiver -50 dB w 

𝑷𝒎𝒂𝒙 
The maximum transmit power from the base station 

transmitter. 𝝈𝒗
𝟐 × (𝟏𝟎)

𝑺𝑵𝑹
𝟏𝟎  

𝜷 
The impact factor of the primary user in selecting the 

cluster head in clustering discussion 
0.5 

𝜶 
The scaling factor of NOMA power for users within 

each cluster 
10 

𝑰𝒕𝒆𝒓𝒂𝒕𝒊𝒐𝒏 

The number of occurrences of random 

telecommunication channel realizations or channel state 

changes. 

100 

𝑪𝒐𝒖𝒏𝒕𝒆𝒓 

The number of iterations required for the convergence 

of the CVX solver algorithm to the optimal point in the 

optimization problem. 

10 

 

4-3 Analysis of Simulation Results 

All simulations were conducted using version R2015a of the MATLAB software on a computer system 

with the following specifications. 

Table 4-2: Complete specifications of the computer system used for conducting the simulations. 

Manufacturer ASUSTeK Computer Inc 

Windows Edition Windows 10 Enterprise 

Processor Intel® – Core™ i7-6700K – CPU@ 

4.00 GHz 



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RAM 32.0 GB 

System Type 64-bit Operating System , x64-based 

Processor 

 

4-3-1 Performance Results of Power Management with Interference Threshold in CRMN System 

By extending the convex optimization problem in the radio cognitive domain and adding the 

interference threshold constraint under the optimized sub-problem as discussed in the previous chapter, 

the optimization problem in 𝑷𝟒 form was obtained. 

 𝐶4
(𝑝3):∑[‖𝒉𝒑𝒖𝒅𝒈‖2

2
∑ 𝑝𝑔.𝑚

𝑁𝑔

𝑚=1

] ≤ 𝐼𝑡ℎ

𝐺

𝑔=1
 

  .     ∀𝑔.𝑚 (1-4) 

In equation (1-4), the interference threshold 𝐼𝑡ℎis defined by the primary user. By defining this 

threshold, a significant amount of incoming interference to the primary user will be prevented; because 

the base station manages the power to satisfy the 𝐼𝑡ℎthreshold and the interference constraint, and the 

user equipment, based on having favorable conditions in its channel, will receive the signal with 

minimal interference at the receiver. The range of changes in the interference threshold is between 

10−10and 10−6. 

PRO1 graph represents the first proposed method in this research. In this method, due to the paper 

topic, the energy harvesting process and SWIPT reference [66] were not considered, and the cognitive 

radio interference constraint (1-4) was introduced into the problem. ORG [66] illustrates the reference 

graph [66]. 

 

Fig 4-1: Comparison of the spectral efficiency curve obtained as a function of the interference 

threshold and the reference [66]. 



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By generalizing the problem and bringing it into the cognitive radio domain, as evident in the graphs, 

the system efficiency decreases significantly due to the presence of the limiting power factor, which is 

the interference constraint. Additionally, as we utilize zero-forcing beam-forming in shaping the beam, 

the decrease in system efficiency, especially at low interference thresholds, will be more pronounced, 

and the impact of the interference constraint will decrease involuntarily with an increase in the 

threshold. We acknowledge the reduced system throughput compared to the reference [66] because 

our network is a cognitive radio network, and to prevent incoming interferences to the primary user, it 

is necessary to define an interference threshold so that the primary user can receive its signal without 

interference in their receivers. However, the decrease in spectral efficiency of the system, especially 

at low thresholds, leads to a loss of significant data rates. Therefore, in the proposed methods discussed 

later, our aim is to enhance the spectral efficiency of the system as much as possible. 

4-3-2 Results of Cluster Head Selection Considering the Initial User Status 

The cognitive radio network in this paper is composed of several secondary users and one primary 

user. Therefore, it is important not to select users who are in the vicinity of the primary user and meet 

the cluster heading conditions as cluster heads, in order to minimize the impact on PU. As mentioned 

in Algorithm 1 and also alluded to in Chapter 3, we consider the presence of PU in the correlation 

relationship among cluster head channel and other cognitive users. By doing so, when calculating the 

correlation of users with cluster heads, the influence of other cognitive users on PU is reduced. In the 

simulations, we utilized zero-forcing precoding during beam-forming. Now, we aim to investigate the 

impact of Algorithm 1 on the performance and behavior of the system from various aspects; hence, we 

consider the interference constraint of cognitive radio for all graphs, and compare this proposed 

method with three graphs that will be introduced subsequently. It is worth mentioning that at each 

stage of the proposed methods, the influence of the previous proposed method is taken into account in 

the simulation process. 

 

Fig. 4-2: compares the spectral efficiency curve obtained as a function of the interference 

threshold in the reference method [66] for selecting the cluster head with our proposed new 

method to prevent neighboring users of the initial user from being selected as the cluster head. 



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The PRO1 chart is an identifier of the first proposed method. In this section, our attempt is to explain 

in detail the steps of adding sub-steps to the first proposed method and reaching the second proposed 

method, which is Algorithm 1. 

Step 1: In this stage, as mentioned above, in the cluster head selection section, attention is paid to the 

problem of neighboring users of the initial user in a way that these users are not selected as cluster 

heads. According to Fig. 4-2 of PRO1+ Betta, a diagram is presented considering the correlation 

between cluster heads and other remaining users, where 𝛽 =  0.5. As clearly shown in the Fig., at low 

interference thresholds, due to channels not being in a favorable state and being weak channels, the 

signal-to-noise ratio, plus interference, is low, and the use of the zero-forcing criterion has reinforced 

channel noise in these thresholds; thus, the presence of noise and interference causes an undesirable 

effect on the initial user's channel. Therefore, in these thresholds where the dB is less than -70 in both 

diagrams, the spectral efficiency curve decreases, and with an increase in the interference threshold 

(>-70 dB), the spectral efficiency curve gradually ascends, meaning that the interference constraint 

considered becomes less effective. Consequently, the achievable information rate for users increases, 

which is logically justifiable based on the explanations provided. 

Step 2: In this stage, the problem of assigning users to each cluster is calculated considering the 

correlation coefficient between the eigenvector corresponding to the largest eigenvalue of the 

correlation matrix and the channel correlation of each user with the desired cluster head. The graph 

(PRO1+eigvec), as shown in Fig. 4-2, illustrates the significant impact of this step on improving the 

spectral efficiency of the proposed method. 

Step 3: The second proposed method in this paper, presented as Algorithm 1, is added to the first 

proposed method. PRO2 also represents the graph of this proposed method. By comparing the first and 

second proposed methods, it is observed that the addition of Algorithm 1 results in a doubling of the 

spectral efficiency of the system at low thresholds and an almost threefold growth at high thresholds. 

Therefore, the outcome of this method compared to the previous one is very promising. However, the 

issue of low capacity has not been fully resolved, especially at low thresholds, as in both the first and 

second methods, zero-forcing beam-forming criterion has been employed. ZF intensifies noise at low 

thresholds (weaker channels), significantly reducing network information rate and causing the system 

capacity to be lost in these thresholds. 

Fig. 4-3 illustrates the spectral efficiency curve of the system in the second proposed method based on 

a fixed number of 24 users and a variable number of clusters from 4 clusters to 10 clusters with a step 

length of 1. It is evident that as the number of clusters increases, users are distributed among all 

clusters, making the clusters more vacant. This phenomenon indicates the same effect as beam-

forming, as with an increasing number of clusters, more beams are formed, and if previously only one 

of the busy clusters benefited well from a narrow beam, here members of other clusters will also benefit 

more from these beams. Fig. 4-3 indicates that the trend of the system's spectral efficiency curve is 

almost linearly increasing; therefore, as clusters become more vacant, intra-cluster interference due to 

NOMA design decreases, leading to an increase in achievable information rate and the signal-to-noise-

plus-interference ratio. 



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Fig. 4-3: The spectral efficiency curve of the system in accordance with the second proposed 

method as a function of varying the number of clusters with a fixed number of users. 

 

Fig. 4-4: The elapsed time graph of the processor in a computer system for performing 

computations and solving mathematical problems in the second proposed method based on the 

increase in complexity in the structure of the telecommunication system. 

In Fig. 4-4, we depict the duration it takes for a computer system processor to address a problem in 

terms of the number of variable clusters. As evident, with an increase in the number of clusters, the 

complexity of computations and mathematical analysis at the base station (BS) transmitter increases. 

This issue is justified in a way that as the number of clusters increases, during beam-forming, the 

number of antenna array elements at the transmitter also increases, thus making the calculation of 

inverse matrices used in beam-forming based on the zero-forcing criterion more complicated. As 

shown in Figs. 4-3 and 4-4, it is clear that we have considered the same horizontal axis range for both 

plots to evaluate the time spent on necessary mathematical analyses for increasing computational 

complexity based on the change in the number of clusters. According to Fig. 4-3, as the computational 

complexity increases, the spectral efficiency curve is almost 2.5 times greater, In Fig. 4-4, as the 



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computational complexity increases, the duration spent has approximately doubled. Beyond a certain 

point, the trend of curve changes remains relatively constant. Through this method, in a time period 

that is almost twice as long, we achieve a spectral efficiency nearly 2.5 times greater in the system. 

 

Fig. 4-5: the trend charts of attainable information rates for the weakest user in each cluster in 

the second proposed method at the interference threshold of -60 dB 

The bar graph Fig. 4-5 presents a comparative analysis of achievable information rates for the weakest 

user in each cluster under the second proposed method at an interference threshold of -60 dB. The blue 

bar graph represents the second proposed method, while the orange bar graph illustrates this method 

without considering the effect of the primary user in cluster selection. Simulation of this aspect has 

been conducted to demonstrate the impact of the third constraint in our proposed method optimization. 

This constraint pertains to the influence of the power scaling factor of NOMA allocation on cluster 

users for fairness considerations. The simulation was run for 4 clusters and 24 users. As evident in Fig. 

4-5, in both bar graphs, at high interference thresholds, the weakest users in each of the 4 clusters 

achieve significantly noticeable information rates. This implies that even if no attention is given to the 

users surrounding the primary user in cluster head selection (orange bar graph), and a cluster head is 

chosen from among them, it does not affect power allocation to weaker users, ensuring fairness. This 

means that not only is fairness not violated by this approach, but more power is allocated to the weaker 

users in each cluster, resulting in improved achievable information rates for these users. 

4-3-3 Modified ZF-based Beamforming Results by Introducing Zero in Beamforming towards 

the Primary User 

In the previous section, we have extensively explained this method; therefore, here we refrain from 

repeating it and directly delve into analyzing the results obtained from the diagrams using this 

approach.  



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Fig. 4-6: The spectral efficiency curves of all three proposed methods as a function of the 

interference threshold. 

Fig. 4-6 illustrates the simulations conducted in all three proposed methods of this paper. The diagram 

(PRO3(NULL)) represents the chart resulting from the simulation of the proposed third method. As 

mentioned earlier, as it is obvious from the diagrams, the weaknesses of the first and second proposed 

methods are significantly reduced in the efficiency spectrum chart. Consequently, there is a decrease 

in the capacity at low interference thresholds. As illustrated in Fig. 4-6, in the second proposed method, 

the weaknesses of the first method were somewhat improved, but these weaknesses were not 

completely rectified at low thresholds. With the proposal of the third method, the spectral efficiency 

graph of the system at thresholds below -70 dB will significantly improve compared to higher 

thresholds. Therefore, in this method, we attempted to prevent capacity reduction at lower interference 

thresholds and enhance our simulation results noticeably. Utilizing this method involves adding the 

primary user channel in the last column of the channel matrix. This action leads to narrowing the beam 

and orienting the antenna array towards the primary user, assuming that the base station knows the 

status information of the primary user channel. As a result, the base station behaves like the cluster 

head towards the primary user, enabling the user to receive a good beam and recognize a better channel 

status compared to other users.  By implementing this method, the interference constraint added to the 

primary user to prevent interference from cognitive users becomes significantly less effective. This is 

why the graph of this method remains almost constant. This method prevents interference on the 

primary user side, compensates for the decrease in achievable information rate and capacity at lower 

thresholds, and weakens the influence of secondary users to the extent that there is no longer a need to 

impose restrictions on the transmit power. Therefore, with this method, we will conclude that not 

considering the neighboring users of the primary user as cluster heads using the ZF criterion does not 

lead us to the desired outcome of zero beam-forming alignment towards the primary user. Hence, the 

use of the third proposed method for enhancing spectral efficiency, total rate, and system capacity, 

especially for the primary user, is highly recommended. 



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The conditions for implementing simulations and all parameters of this method are similar to the 

second proposed method. With this approach, the trend of changes in the efficiency curve spectrum 

increases linearly with an increase in the number of clusters. By comparing Figs. 4-3 and 4-7, the trend 

of changes in the 4-7 curve had more than approximately twice the growth. 

 

Fig. 4-7: The spectral efficiency curve of the system in accordance with the third proposed 

method based on varying the number of clusters with a fixed number of users. 

 

Fig. 4-8: The elapsed time diagram by the computer system processor for performing 

computations and solving mathematical problems based on the modified ZF in terms of 

increasing complexity in the telecommunications system structure. 

In accordance with Fig. 4-7, with increasing complexity of mathematical computations, the spectral 

efficiency curve has nearly tripled. As depicted in Fig. 4-8, with further complexity in mathematical 

computations, the time taken has increased to slightly less than approximately twice and from a certain 



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point onwards, the trend of time reduction has significantly decreased. Hence, by comparing Fig. 4-4 

and 4-8, it is observed that the third method is worthwhile as it can achieve a better spectral efficiency 

(almost triple) in the system within a time frame almost similar to the previous method. 

 

Fig. 4-9: A bar chart comparing the achievable information rates for the weakest user in each 

cluster in the second and third proposed methods. 

Fig. 4-9 provides a comparative analysis of the achievable information rate for the weakest users within 

each cluster between the second and third proposed methods. The orange bar indicator represents the 

second proposed method, while the blue bar indicator depicts the third proposed method. As mentioned 

in the bar chart analysis in Fig. 4-5, the purpose of simulating this Fig. is to demonstrate the impact of 

the third constraint in the optimization problem to ensure fairness in the NOMA scheme in our 

proposed methods.  This simulation, similar to the previous method, was conducted for 4 clusters and 

24 users. As evident in Fig. 4-9, in both methods, the weakest users in each cluster have achieved a 

considerable information rate. Therefore, our proposed methods not only do not violate the nature of 

NOMA technology based on ensuring fairness and proper implementation of SIC but also provide 

weaker users in each cluster with more capacity. Consequently, SIC operates correctly, and as a result, 

the achievable information rate for each user also improves. 

 

Fig. 4-10: The system information rate with respect to the increase in the number of antennas. 



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Fig. 4-10 illustrates the evolution of the achievable data rate for users in a massive MIMO system 

(CRMN). As the number of antennas increases, antenna gain rises, signals are received more strongly, 

the data rate of received signals improves, and the system exhibits scalability properties. However, at 

a certain point, the data rate curve does not increase significantly as the number of antennas grows and 

the beams sharpen because some users within NOMA lose coverage. 

.5- Conclusion 

The optimization problem concerning the CRMN system aiming to maximize the sum user rate 

network has been formulated. This problem includes an objective function and several constraints, 

where the objective function is non-convex and the constraints of the problem are all linear and convex. 

Therefore, the formulated optimization problem is also non-convex. By employing simplifications and 

several lemmas, the problem was transformed into the form of a convex optimization problem. The 

convexities form of the problem was then analyzed and deconstructed using the CVX toolbox in the 

MATLAB computational simulation software. The optimization variables {𝑝𝑔.𝑚𝑔} were optimized 

using the CVX toolbox. In the analysis of simulation results based on our proposed methods, which 

involve power management in the CRMN network by introducing interference constraints from the 

primary user side, cluster selection considering the status of the primary user, and beamforming based 

on modified ZF, we have demonstrated that by generalizing the problem to a cognitive radio problem 

and adding interference constraints, the efficiency graph decreases compared to a problem without 

interference constraints, which is intuitive. While presenting a second proposed method, although the 

efficiency graph showed significant improvement in a spectral domain, at low interference thresholds, 

the information rate and throughput remained low. Consequently, at these low interference thresholds, 

we were unable to achieve a good network capacity. Ultimately, utilizing a third method, the issue of 

reduced information rate and spectral efficiency of the system at low thresholds noticeably improved. 

However, a crucial point found in the analysis of the graphs is that the simultaneous use of NOMA 

and massive MIMO technology in a network up to a certain stage enhances the overall system rate. 

Excessive increase in antennas not only fails to enhance network capacity but also leads to the 

exclusion of some users within the NOMA scheme, exacerbating the system's capacity reduction.  

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