Cooperative Spectrum Sensing Algorithm Based on Eigenvalue Fusion

A novel algorithm is introduced to improve collaborative spectrum sensing under low cognitive capabilities and insufficient signal-to-noise ratio. The algorithm is based on the difference of random matrix eigenvalues and uses the theory of random eigenvalues and the extreme distribution of the minimum eigenvalue. It makes use of the average, both arithmetic and geometric, as well as the minimum and maximum values of eigenvalues as the detection metric. It calculates the fusion power parameter through local energy spectrum sensing. Simulation results demonstrate that the algorithm outperforms the DMM algorithm and the NMME algorithm under users with low cognitive capabilities and Insufficient signal-to-noise ratio, making it more suitable for low signal-to-noise ratio environments.


Introduction
The swift progress of the wireless communication industry has resulted in a substantial surge in the proliferation of wireless devices.However, the adopted spectrum allocation strategy is based on an exclusive approach.This so-called exclusive strategy means that even if the primary user does not utilize the spectrum assigned to it, the spectrum remains idle, waiting for the primary user's usage, and other users are unable to access it.While this strategy has facilitated the development of wireless devices in the past, it becomes evidently limited in an era of soaring wireless device numbers.Due to the timebound nature of this exclusive policy and the rapid growth of wireless communication, the availability of wireless spectrum resources has become increasingly scarce and demands a solution.
In a report on spectrum resource utilization, the United States pointed out that a significant portion of allocated spectrum remains underutilized, frequently in idle states without any utilization, thus exacerbating the already scarce spectrum resources.Researchers recognized the urgency of addressing this issue and, in response, cognitive radio technology emerged as a solution.Over recent years, cognitive radio technology has gained substantial attention and research interest.It achieves intelligent spectrum sharing through the detection of "spectrum holes" and enables dynamic access and departure from the spectrum.Importantly, this technology ensures that primary users' spectrum usage is not affected while effectively mitigating the severe wastage of spectrum resources.
Currently, the most commonly used and classic spectrum sensing algorithms are energy spectrum sensing [1], cyclostationary feature spectrum sensing, and eigenvalue-based spectrum sensing [2][3][4].The above algorithm requires multiple cognitive users to perform sensing simultaneously, collecting their respective sampling signals.Then, these sampling signals are used to construct a sampling matrix.Next, the sampling matrix undergoes autocorrelation operation and is subjected to eigenvalue decomposition.
By applying mathematical theory to process these eigenvalues, the results can be obtained as statistical detection parameters.Compared with other classical algorithms, such as energy spectrum sensing, these algorithms can eliminate the impact of noise uncertainty and utilize more sampled signal information to improve detection accuracy.Because of this advantage, this type of algorithm has received widespread attention and research.
Algorithms for spectrum sensing that rely on eigenvalues as a fundamental component include NMME and the DMM.However, the DMM algorithm has low utilization of eigenvalue information, leading to lower detection probability.The NMME algorithm avoids estimating noise by using the eigenvalue ratio, but it does not improve the utilization of eigenvalue information compared to the DMM algorithm.These algorithms do not perform well in low-sampling scenarios.
To address the issues of incomplete utilization of eigenvalues, the need for noise estimation, and low detection probability in low cognitive user scenarios in the aforementioned classic algorithms, this paper combines the ideas in [5][6] to construct a detection statistic using the arithmetic-geometric mean of eigenvalues, minimum eigenvalue, maximum eigenvalue, and fusion power parameters.A new collaborative spectrum sensing algorithm is proposed, which avoids noise estimation by using ratios and improves detection probability while maintaining a low false alarm rate through power function fusion.Whether through mathematical theory or simulations based on real-world scenarios, both demonstrate that the algorithms for spectrum sensing rely on eigenvalues as a fundamental component in scenarios with low sampling and low cognitive users.

Spectrum sensing model
This paper adopts the classic binary hypothesis mathematical model in cognitive radio, that is, assuming that there is only one primary user in a spectrum, M cognitive users collaborate to detect the spectrum and determine whether there is a primary signal.The sampling size of each cognitive user is N.The equation for the sampled signal received by the m-th cognitive user is as follows: In the case of H , that is, when there is no primary signal, cognitive users will only receive noise signal   .In the case of H , that is, when there is a primary signal, cognitive users will receive noise signal   and the signal transmitted through the channel after being modulated by the primary signal. ~ 0,  ,Typically, in conducting such studies, it is assumed that the interference is unrelated to both the primary signal and the noise, under the assumption of additive Gaussian white noise with a normal probability distribution.
X n , n 1,2,3. . .N can be regarded as the sampling vector of the m-th cognitive user.After constructing the sampling vector, the sampled signal vectors received by the M cognitive users can be combined into an M * N matrix as shown in the above equation.
But in reality, Calculating the covariance matrix R of the received signal matrix directly is not a viable option.Generally, it is estimated through limited sampling.
Performing eigenvalue decomposition on R N yields M eigenvalues, which exhibit differences between the cases of H and H . Based on these differences, it is possible to determine whether the main signal exists.

Random matrix theory
According to the random matrix theory, the following two theorems can be proposed regarding the distribution of matrix eigenvalues [7,8]: Theorem1: When lim → c, 0 c 1 , the eigenvalues of a Wishart random matrix can be calculated using the following expression [9]: Theorem 2: When the noise is identical to the actual signal, R (N) is a Wishart matrix, and the R N matrix satisfies the following formula [10]: Then the formula for containing the Minimum eigenvalue closely approximates a first-order 8 The y u is the solution of  Ⅱ nonlinear equation  ,,    2  .At present, researchers have not obtained the continuous function of the T-W distribution, but researchers have obtained many discrete points.We can still use the T-W distribution through these discrete points, and these points compose the discrete distribution function and the inverse distribution function.

Theoretical foundation of the algorithm
The detection statistic for the DMM algorithm is λ λ , and the threshold calculated based on this detection statistic is affected by noise.The detection statistic for the NMME algorithm is , which solves the problem of noise estimation but does not fully utilize the eigenvector information.In this paper, we use as the detection statistic for our algorithm, where the summation embodies the distinctive properties of the arithmetic and geometric means, capturing the essence of the eigenvalues in their collective entirety.
The value of  and in the detection statistic is dynamically determined by the energy spectrum sensing results of each cognitive user in each spectrum sensing iteration.The power parameter is dynamically adjusted.In order to ensure the detection probability while minimizing the probability of false alarms, it is important to strike a balance between accurately detecting the desired signal and avoiding false alarm occurrences.The equation for  is as follows, where  is the number of cognitive users with local energy spectrum sensing results of  .

𝛽 9
The decision criterion is as follows: where γ is the decision threshold, an estimation of this can be obtained by using the maximum eigenvalue and the mean eigenvalue derived from random matrix theory.The decision threshold can then be determined by analyzing the distribution of the minimum eigenvalue.
The equation for the probability of false alarms can be expressed as follows: By using the approximation of the maximum eigenvalue instead of its actual value, and under the condition that  equals  ̅   , it can be obtained that: Then the false alarm probability can be determined based on the distribution of the minimum eigenvalue: Therefore, the decision threshold  can be expressed as: This paper's algorithm adds  ̅ and  to the detection statistics, which makes the utilization of eigenvalues more comprehensive compared to the traditional eigenvalue-based spectrum sensing algorithm, leading to an improvement in detection efficiency.Although the computational complexity has increased, the improvement is not significant compared to the classical eigenvalue algorithm and can be almost ignored.The noise power in the decision threshold is eliminated by using the eigenvalue ratio method.This eliminates the need for noise estimation, achieving a blind algorithm.The power parameter in the detection statistic is dynamically adjusted by the results of the energy spectrum sensing of child nodes.This approach allows for minimizing the false alarm probability of the algorithm by appropriately adjusting the decision threshold while maintaining a high detection probability, thus achieving the goal of minimizing false alarms as much as possible, that is, when most of the child nodes are judged as H0, the judgment threshold can be appropriately reduced, so as to reduce the probability of the false alarm.When the majority of sub-nodes are determined as H1, the algorithm automatically adjusts the decision threshold appropriately to reduce the probability of false negatives.By incorporating a power parameter, the algorithm gains a certain level of adaptive capability.In real-life scenarios, the channel conditions of the Internet of Things (IoT) are constantly changing, leading to variations in signal characteristics and interference conditions.Only algorithms with adaptive capabilities can better accommodate the evolving requirements of IoT in such dynamic environments.This adaptive algorithm can adjust the decision threshold based on the actual circumstances to adapt to the changing channel conditions and the impact of various interferences.As a result, it provides enhanced support and application for IoT deployments in highly variable environments.

Algorithm steps 1)
Cognitive users sample the signal to obtain the sampling signal and perform energy spectrum sensing.
2) The fusion center combines the signal into a matrix and calculates R N based on Equation (3).
3) The eigenvalue decomposition is performed, and detection statistics are constructed.The threshold is calculated based on Equation (15) and the set false alarm probability.
4) The threshold is compared with the detection statistic.The count is accumulated, and the detection probability of the proposed algorithm is calculated.

Simulation Results
In order to evaluate the performance of the algorithm, mere mathematical and logical analysis is insufficient.It is ultimately necessary to test the algorithm in real-world scenarios.However, considering the various limitations of real-world testing, this study adopts a simulation-based comparative approach to assess the performance of the proposed algorithm.During the simulation process, we strive to replicate real-world conditions as closely as possible.For the experimental setup, we assume the primary signal to be a QPSK signal and the noise to follow a Gaussian white noise distribution.We select a sample size of X=2000 and a target false alarm probability of Y=0.05 for the experiments.To compute the detection probability, we conduct 1000 Monte Carlo simulation experiments and compared the results with the NMME and DMM algorithms.Through this simulation-based comparative approach, we can better evaluate the performance of the proposed algorithm in real-world environments.In Figure 1, the performance of different algorithms is clearly observed when the number of cognitive users is M=8.Particularly, under low signal-to-noise ratio conditions, the algorithm proposed in this paper exhibits significantly better detection performance compared to NMME and DMM algorithms, demonstrating superior performance.As the signal-to-noise ratio increases, the detection probabilities of these algorithms also increase.This is because a higher signal-to-noise ratio makes it easier to detect the presence or absence of primary signals.However, in real-world scenarios, high signal-to-noise ratios are not always guaranteed, and most cases involve low signal-to-noise ratios.Nevertheless, even under the same conditions, it is evident that the algorithm proposed in this paper outperforms NMME and DMM algorithms.Therefore, it can be concluded that the algorithm presented in this paper is more suitable for low signal-to-noise ratio environments and exhibits better performance.2 illustrates the relationship between the detection probability of various algorithms and the number of sub-users in a collaborative spectrum sensing network, with a SNR of -10.Upon observation, the curves formed by the different algorithms exhibit similar shapes, indicating an upward trend in detection performance as the number of sub-users increases.Particularly, in scenarios with a lower number of sub-users, the proposed algorithm in this paper demonstrates significant advantages, surpassing the performance of the other two algorithms.Upon further analysis, it is notable that in cases with a higher number of sub-users, the performance of the DMM algorithm is comparable to that of our proposed algorithm.However, it is worth emphasizing that in scenarios with a lower number of subusers, which represents the majority of real-world situations, our algorithm is more applicable.Therefore, our algorithm exhibits superior performance in such environments and is better suited for prevalent reallife scenarios.Based on mathematical theoretical analysis of the algorithm, our research proposes a solution to the detection performance issue that arises when there is a small number of sub-users in low signal-to-noise ratio and collaborative spectrum sensing networks.Additionally, our results have been validated through simulation experiments.

Conclusions
By applying the theory of random matrices, we have improved the detection statistics.The utilization of the algorithm on the feature value information is improved.There is no noise power in the judgment threshold by means of the eigenvalue ratio.There is no need to estimate the noise power, achieving full blind spectral perception.By adding the power parameter, the results of the child node energy spectrum can affect the detection statistics.We prioritize maintaining a high detection rate while actively reducing false alarm probability to minimize instances of false positives.Our algorithm aims to achieve a balance point where efficient detection is ensured while maximizing the reduction of false alarms.Both theoretical analysis and simulation demonstrate the larger noise effect and fewer child nodes in the cooperative spectrum-aware network.This algorithm works better than the DMM and NMME algorithms.This algorithm can be used to improve the detection probability of spectral perception when the cognitive users are small.
Through careful reflection and comprehensive summarization, we have thoroughly analyzed our algorithm and discovered its potential for improvement.In further research and practice, we have realized that despite achieving certain accomplishments, there are still limitations and shortcomings in our algorithm.In future research, we plan to take measures to reduce algorithm complexity and minimize power consumption for sub-user nodes during algorithm execution, making it more suitable for smaller devices and enhancing its convenience.Additionally, we are aware of the shortcomings in considering channel fading.We cannot yet ascertain whether our algorithm can maintain good performance in more complex channel environments.Therefore, we will further enhance our algorithm to address these deficiencies and ensure its reliability and stability in various environments.

Figure 1 .
Figure 1.Curve of Detection Probability vs. Signal-to-Noise Ratio

Figure 2 .
Figure 2. Curve of Detection Probability vs Number of Cognitive Users

Figure
Figure2illustrates the relationship between the detection probability of various algorithms and the number of sub-users in a collaborative spectrum sensing network, with a SNR of -10.Upon observation, the curves formed by the different algorithms exhibit similar shapes, indicating an upward trend in detection performance as the number of sub-users increases.Particularly, in scenarios with a lower number of sub-users, the proposed algorithm in this paper demonstrates significant advantages, surpassing the performance of the other two algorithms.Upon further analysis, it is notable that in cases with a higher number of sub-users, the performance of the DMM algorithm is comparable to that of our proposed algorithm.However, it is worth emphasizing that in scenarios with a lower number of subusers, which represents the majority of real-world situations, our algorithm is more applicable.Therefore,