CN103795481A - Cooperative spectrum sensing method based on free probability theory - Google Patents

Abstract

The invention discloses a cooperative spectrum sensing method based on free probability theory. The method is suitable for MIMO communication environment. Firstly, the received signals of multiple antennas of each secondary base station are sampled, and the sampled signals are processed centrally; then, according to all received The noise variance of the sampling signal and the channel, with the help of the asymptotic free characteristics of the random matrix and the Wishart distribution characteristics, uses the algorithm based on free deconvolution to solve the average received signal power of all receiving antennas That is, the detection statistic; then, according to the target false alarm probability p _f , use Monte Carlo simulation to calculate the detection threshold τ under the condition that only noise exists; finally, the Compared with τ, it is judged whether the main base station is sending a signal. The invention can obtain accurate received power from the received signal of the secondary base station, and can effectively improve the spectrum sensing performance, especially in the case of low signal-to-noise ratio and small sample.

Description

A Cooperative Spectrum Sensing Method Based on Free Probability Theory

technical field

The invention relates to the technical field of computer communication of cognitive radio spectrum sensing, in particular to a cooperative spectrum sensing method based on free probability theory.

Background technique

Under the current spectrum fixed allocation method commonly adopted by various countries, the utilization rate of most licensed frequency bands is very low, which has caused a great waste of wireless spectrum resources. On the other hand, with the rapid development of the wireless communication industry, available wireless spectrum resources are increasingly scarce. How to meet the explosive growth of wireless spectrum demand has become a common problem faced by global mobile communications. Therefore, cognitive radio, as one of the effective ways to alleviate the shortage of wireless spectrum resources, has attracted extensive attention from academia and industry in recent years. The basic idea of cognitive radio is spectrum reuse or spectrum sharing, which allows cognitive users to use the frequency band for communication when the primary user frequency band is idle. In order to do this, cognitive users need to frequently perform spectrum sensing, that is, to detect whether the primary user is using the frequency band. Once the primary user re-uses the frequency band, the cognitive user must detect the primary user with a high detection probability and exit the frequency band quickly within a specified time. As the core and basis of cognitive radio technology, spectrum sensing technology has become a hot research topic. At present, research in this field has made great progress. Various research institutions or individuals have conducted in-depth research on spectrum sensing from multiple aspects, and initially established a theoretical framework for spectrum sensing, which has been applied to the corresponding international standard. For example, the IEEE802.22 standard is the first international standard that explicitly adopts spectrum sensing. It stipulates that fixed wireless area networks and TVs work in the same frequency band, and can automatically detect and utilize idle TV bands to improve spectrum efficiency.

The existing spectrum sensing methods mainly include energy detection (Energy Detection, ED), matched filter detection (Matched Filter Detection, MFD), cyclostationary feature detection (Cyclostationary Feature Detection, CFD), interference temperature detection, etc. Various multi-node cooperative detection. The main purpose of using multi-node cooperative detection is to overcome the influence of wireless communication fading and shadows, and to prevent hidden terminal problems. Recently, some scholars have proposed a scheme based on Random Matrix Theory (RMT), such as the MME algorithm. Unlike previous work in this area, such schemes do not require knowledge of the noise statistics and variance, only the largest and smallest eigenvalues of the random matrix.

The above spectrum sensing methods have their own advantages and disadvantages and applicable conditions, but there is a problem in common, that is, their performance cannot meet the actual needs in the case of low signal-to-noise ratio and small samples. Therefore, finding a high-performance spectrum sensing method has become an urgent problem to be solved.

In the 1980s, under the pioneering work of Voiculescu, free probability theory developed into an entire research field. As an important branch of random matrix theory, free probability theory is a powerful tool to describe the asymptotic properties of random matrices. It establishes a strong connection between two random matrices and their sum or product matrices, which can be used to construct Modular digital communication system. In recent years, it has been applied to the field of spectrum sensing to further improve the performance in the case of low signal-to-noise ratio, which essentially separates the real signal power matrix from the sample covariance matrix containing signal and noise power. And the present invention can well solve the above problems.

Contents of the invention

The purpose of the present invention is to provide a cooperative spectrum sensing method based on free probability theory, which is suitable for MIMO communication environment, can obtain accurate received power from the received signal of the secondary base station, and solve the problem of low signal-to-noise ratio and small sample The problem of low spectrum sensing performance.

The technical solution adopted by the present invention to solve the technical problem is: in Fig. 1 of the cooperative spectrum sensing system model targeted by the present invention, K secondary base stations BS ₁ , BS ₂ ,..., BS _K distributed in different geographic locations Cooperate to sense the channel, and judge whether the main base station is sending a signal. The primary base station and each secondary base station are equipped with N _t and N _r antennas respectively, and there is a MIMO Rayleigh fading channel between them. The specific steps of the collaborative spectrum sensing method based on free probability theory are as follows:

其中(·)^Η表示共轭转置；1. The received signals of multiple antennas of each sub-base station are sampled, and the sampled signals are processed centrally. The sampling rate is 1/T _s , then the sampling signal output by the kth receiver at time n is k=1,2,...,K. Then the sampled signal collected by all receiving antennas at time n is

Wherein ( ) ^Η represents conjugated transposition;

也就是检测统计量。具体步骤为：2. According to the noise variance of all received sampling signals and channels, with the help of the asymptotic free characteristics of the random matrix and the Wishart distribution characteristics, an algorithm based on free deconvolution is used to solve the average received signal power of KN _r antennas

That is the detection statistic. The specific steps are:

(1) Input: receive sampling signals {y(n), n=1, 2, ..., M _S } and channel noise variance σ ² . Where M _S is the total number of samples of the received signal;

和它的特征值{λ_i,i＝1,2,…,KNr}；(2) Calculate the sample covariance matrix of the received sampling signal

And its eigenvalue {λ _i , i=1,2,...,KNr};

的k阶矩

和

的k阶矩

(3) calculation

k-order moment

and

k-order moment

步骤如下：(4) calculation

Proceed as follows:

①

② $\{m_k^{\mathrm{\ν}}={\mathrm{\γ}}_k/\left(\frac{{\mathrm{KN}}_r}{m_S}\right)\};$

步骤如下：(5) calculation

Proceed as follows:

① ${\mathrm{\α}}_k^{\mathrm{\ν}}=\mathrm{mom\; cum}\left(\right\{m_k^{\mathrm{\ν}}\left\}\right)$

② ${\mathrm{\α}}_k^{{\mathrm{\μ}}_{{\mathrm{\σ}}^{2}I}}=\mathrm{mom\; cum}\left(\right\{{\left({\mathrm{\σ}}^2\right)}^k\left\}\right)$

③ ${\{\mathrm{\α}}_k^{\mathrm{\η}}={\mathrm{\α}}_k^{\mathrm{\ν}}-{\mathrm{\α}}_k^{{\mathrm{\μ}}_{{\mathrm{\σ}}^{2}I}}\}$

④ $\left\{m_k^{\mathrm{\η}}\right\}=\mathrm{cummom}\left(\right\{{\mathrm{\α}}_k^{\mathrm{\η}}\left\}\right);$

(6) calculation Proceed as follows:

① $\left\{{\mathrm{\ρ}}_k\right\}=\mathrm{mom\; cum}\left(\right\{\frac{{\mathrm{KN}}_r}{N_t}\·m_k^{\mathrm{\η}}\left\}\right)$

② $\{m_k^P={\mathrm{\ρ}}_k/\left(\frac{{\mathrm{KN}}_r}{N_t}\right)\};$

也就是μ_P的一阶矩。(7)

That is, the first moment of μ _P.

和

分别表示自由概率理论中的自由加法解卷积算子和自由乘法解卷积算子；

和μ_P分别表示矩阵σ²I（I为单位矩阵）和P的极限概率分布；和

对应于

律μ_c，c分别取

和

momcum和cummom是根据分布μ的矩-累积量公式得到的，momcum的输入为矩序列，输出为累积量序列，cummom的输入为累积量序列，输出为矩序列。Explanation of symbols in the above steps:

and

represent the free additive deconvolution operator and the free multiplicative deconvolution operator in the free probability theory, respectively;

and μ _P respectively denote the matrix σ ² I (I is the identity matrix) and the limiting probability distribution of P; and

corresponds to

Law μ _c , c respectively take

and

Momcum and cummom are obtained according to the moment-cumulant formula of the distribution μ. The input of momcum is a moment sequence and the output is a cumulant sequence, and the input of cummom is a cumulant sequence and the output is a moment sequence.

经过很多次这样的仿真后计算

的均值θ_P和方差

近似满足高斯分布。然后依据目标虚警概率p_f，计算阈值τ=υ_PQ^-1(p_f)+θ_P，其中Q^-1(·)为逆Q-函数， 3. According to the target false alarm probability p _f , Monte Carlo simulation is used to calculate the detection threshold τ under the condition that only noise exists. First perform step 2 with only noise samples to obtain

After many such simulations, the calculation

The mean θ _P and variance of

approximately satisfy a Gaussian distribution. Then according to the target false alarm probability p _f , calculate the threshold τ=υ _P Q ^-1 (p _f )+θ _P , where Q ^-1 (·) is the inverse Q-function,

和τ进行比较，判断主基站是否在发送信号。当时，主基站在发送信号；当

时，主基站没有发送信号。4. will

Compared with τ, it is judged whether the main base station is sending a signal. when When , the master base station is sending signals; when

, the main base station does not send a signal.

Beneficial effect:

1. In the MIMO communication environment, the present invention uses an algorithm based on free deconvolution to calculate detection statistics, uses Monte Carlo simulation to obtain detection thresholds, realizes spectrum sensing, and can obtain accurate received power from received signals of secondary base stations.

2. The present invention can effectively improve the performance of spectrum perception, especially in the case of low signal-to-noise ratio and small samples.

Description of drawings

FIG. 1 is a model diagram of the cooperative spectrum sensing system of the present invention.

Fig. 2 is a flow chart of the method of the present invention.

的归一化直方图。Figure 3 is obtained by using Monte Carlo simulation in the presence of only noise

The normalized histogram of .

FIG. 4 is a schematic diagram of the comparison of perceptual performance between the method of the present invention and the detection method based on eigenvalues.

Detailed ways

The present invention will be described in further detail below in conjunction with the accompanying drawings and embodiments.

Example 1

As shown in FIG. 1 , K secondary base stations BS ₁ , BS ₂ , ..., BS _K distributed in different geographical locations cooperate to sense the channel, and judge whether the primary base station is sending signals. The primary base station and each secondary base station are equipped with N _t and N _r antennas respectively, and there is a MIMO Rayleigh fading channel between them. When the main base station sends a signal, the sampling rate is 1/T _s , then the sampling signal output by the kth receiver at time n can be expressed as

${\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\sqrt{\frac{{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}}{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}}}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

是在时刻n的发送符号矢量，其元素满足零均值独立同分布，方差为1；是在发射机和第k个接收机之间的MIMO信道矩阵，它的元素是均值为0，方差为1的复高斯变量，即服从N_c(0,1)；P_k是第k个接收机的每个天线的接收信号功率；

是在第k个接收机上的复高斯噪声矢量，

in,

is the transmitted symbol vector at time n, whose elements satisfy the zero-mean independent and identical distribution, and the variance is 1; Is the MIMO channel matrix between the transmitter and the kth receiver, and its elements are complex Gaussian variables with mean value 0 and variance 1, that is, subject to N _c (0,1); P _k is the kth receiver The received signal power of each antenna of the machine;

is the complex Gaussian noise vector at the kth receiver,

All secondary base stations sense the same frequency band and their received signals are processed collectively. Define the following symbols:

The received signal model (1) can also be written as

$\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}^{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}}\mathrm{<span\; class="notranslate">\; Hs</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

The spectrum sensing problem is therefore the following binary hypothesis testing problem

$\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\\ \mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>& {\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

in

$\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>\stackrel{\mathrm{<span\; class="notranslate">\; \&Delta;</span>}}{<span\; class="notranslate">\; =</span>}{\mathrm{<span\; class="notranslate">\; P</span>}}^{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}}\mathrm{<span\; class="notranslate">\; Hs</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

通过将

与阈值τ进行比较，在Η₀和Η₁之间做出判决，也就是如果

为Η₁，否则为Η₀。H ₀ means that the master BS is not sending a signal, and H ₁ means that the master BS is sending a signal. In this invention, we propose a new spectrum sensing method based on free probability theory, the basic idea of which is to estimate the distribution of the elements of the diagonal matrix P, so that the average received signal power can be estimated

by putting

Compared with the threshold τ, a decision is made between H ₀ and H ₁ , that is, if

is H ₁ , otherwise it is H ₀ .

是一种简单易行的办法。自由概率理论的数学背景为：Here, using free probability theory to solve

It is a simple and easy way. The mathematical background of free probability theory is:

Let A _N be an N×N-dimensional Hermitian matrix with only real eigenvalues, and the empirical probability distribution on its eigenvalue set λ _i (A _N ), i=1,2,...,N is

${\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

where 1(·) is an indicator function. We are interested in the limiting spectral distribution μ _A at N→∞, which is given by the moment

Unique description, where Ε[·] represents the expectation, and tr(·) represents the trace of the matrix.

律，它的密度函数为In particular, if the elements of the N×M matrix H satisfy the zero-mean independent and identical distribution, and the variance is 1/M, then when N,M→∞ and N/M→c, A _N =HH ^H 's limiting spectral distribution μ _c yes

law, its density function is

${\mathrm{<span\; class="notranslate">\; f</span>}}^{{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; +</span>}\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;cx</span>}}\sqrt{{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; +</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; +</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

特别地，μ_c描述了Wishart矩阵的渐近特征值分布，这里H的元素满足独立同分布，服从于

where (z) ⁺ = max{0,z},

In particular, μ _c describes the asymptotic eigenvalue distribution of the Wishart matrix, where the elements of H satisfy independent and identical distribution, obeying

A_NB_N的极限概率分布可由μ_A与μ_B的自由乘法卷积得到，表示为换句话说，当A_N和B_N满足渐近自由时，A_N+B_N和A_NB_N的矩可由A_N和B_N的矩求得。If two random matrices A _N and B _N are given, and their limit probability distributions are μ _A and μ _B respectively, we hope to obtain the limit probability distributions _of A _N + B _N and A _N BN according to μ _A and μ _B. To this end, we introduce a concept similar to "independence" in classical probability theory, called "asymptotic freedom", to calculate these distributions. When A _N and B _N satisfy asymptotic freedom, the limit probability distribution of A _N + B _N can be obtained by the free additive convolution of μ _A and μ _B , expressed as

The limit probability distribution of A _N B _N can be obtained by the free multiplication convolution of μ _A and μ _B , expressed as In other words, when A _N and B _N satisfy asymptotic freedom, the moments of A _N + B _N and A _N B _N can be obtained from the moments of A _N and B _N.

和

并且定义为自由加法解卷积，即如果

那么

类似地，定义为自由乘法解卷积，即如果

那么

Both free additive and multiplicative convolutions are commutative, that is,

and

and define Deconvolution for free addition, i.e. if

So

Similarly, define deconvolution for free multiplication, that is, if

So

The conditions for A _N and B _N to be asymptotically free are very abstract. However, we know that two IID Gaussian matrices, two IID Hermitian matrices, an IID Gaussian or Hermitian matrix, and a definite diagonal matrix are asymptotically free.

的具体理论依据与方法为：Then, using free probability theory to solve

The specific theoretical basis and method are as follows:

A. Limit distribution μ _P of power matrix P

Assume M _S samples y(1),...,y(M _S ) of the received signal are used for sensing the spectrum. The sample covariance matrix of the received signal is

When the master base station is sending a signal, there is y(n)=x(n)+v(n). The sample covariance matrix of the signal component x(n) is

$\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; x</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

和满足以下等式For the signal-Gaussian noise model, using free probability theory, the above two sample covariance matrices

and satisfy the following equation

in

$\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; KN</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}}{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; S</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 11</span>}<span\; class="notranslate">\; )</span>$

On the other hand, the covariance matrix of the signal component x(n) obtained from formula (4) is

$\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; x</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}^{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; HH</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; P</span>}}^{\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$

Z是KN_r×M_S维矩阵，它的元素满足零均值独立同分布，方差为1/M_S。使用式(9)得到definition

Z is a KN _r ×M _S dimensional matrix, its elements satisfy zero-mean independent and identical distribution, and the variance is 1/M _S . Using formula (9) to get

${\mathrm{ZZ}}^h={\mathrm{\&Sigma;}}^{-\frac{1}{2}}\widehat{\mathrm{\&Sigma;}}{\mathrm{\&Sigma;}}^{-\frac{1}{2}}$ or $\widehat{\mathrm{\&Sigma;}}={\mathrm{\&Sigma;}}^{\frac{1}{2}}{\mathrm{ZZ}}^h{\mathrm{\&Sigma;}}^{\frac{1}{2}}---\left(13\right)$

律下它们对应的极限分布为

和由式(12)和(13)，我们得到That is, the Wishart distribution characteristic. It is worth noting that the elements of ZZ ^Η satisfy the zero-mean independent and identical distribution, and the variance is 1; HH ^Η is a Wishart matrix, and its elements also satisfy the zero-mean independent and identical distribution, and the variance is 1. Thus, in

Their corresponding limiting distributions under the law are

and From equations (12) and (13), we get

Substituting Equation (14) into Equation (10), the limiting distribution of the signal power matrix P is obtained as

B. Numerical calculation process of μ _P

的自由加法解卷积以及两个与

律μ_c的自由乘法解卷积，它们都可以由下文介绍的矩-累积量方法有效地实现。In Equation (15), the expression of μ _P includes a and

Additive-free deconvolution of and two AND

The free multiplicative deconvolution of the law μ _c can be efficiently realized by the moment-cumulant method described below.

的自由加法解卷积：概率分布μ的R-变换定义为1) Calculate with

Additive-free deconvolution of : The R-transform of the probability distribution μ is defined as

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}{\mathrm{<span\; class="notranslate">\; z</span>}}^{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 16</span>}<span\; class="notranslate">\; )</span>$

in is the nth order cumulant of μ. The importance of the R-transform is reflected in the additive nature of the free additive convolution, that is

It is equivalent to the cumulant under the free additive convolution is additive, that is,

The relationship between the moment and the cumulant of the distribution μ is as follows:

${\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; k</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}{\mathrm{<span\; class="notranslate">\; coef</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; (</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}{\mathrm{<span\; class="notranslate">\; z</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 19</span>}<span\; class="notranslate">\; )</span>$

得到矩序列

反之亦然。定义函数momcum以矩序列为输入量，累积量序列为输出量；函数cummom以累积量序列为输入量，矩序列为输出量。where coef _n (·) represents the coefficient of z ⁿ . From formula (19), we can get from the cumulant sequence

get sequence of moments

vice versa. The definition function momcum takes moment series as input and cumulant series as output; function cummom takes cumulant series as input and moment series as output.

的矩，我们首先计算式(8)中的样本协方差矩阵

的特征值

接着计算矩in order to achieve

Moments of , we first calculate the sample covariance matrix in equation (8)

The eigenvalues of

Then calculate the moments

的矩为On the other hand, the eigenvalues of σ ² I are all σ ² . therefore

The moment of

${\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}^{{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; I</span>}}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; one</span>}<span\; class="notranslate">\; )</span>$

和μ_A的矩有如下关系：2) Compute the free multiplicative deconvolution with μ _c :

and the moment of μ _A have the following relationship:

in

$\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}}\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}}{\mathrm{<span\; class="notranslate">\; z</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; no</span>}}$

替换，矩由替换。因此，使用函数momcum，其输入量为

相应的输出量为

来计算

的矩。Comparing formulas (22) and (19), it is found that formula (22) is the moment-cumulant formula, but the cumulant is given by

replace the moment by replace. Therefore, using the function momcum, its input is

The corresponding output is

to calculate

moment.

如前所述，频谱感知的判决规则是基于平均接收功率

的估计，也就是μ_P的一阶矩，即

因此，我们可以总结出基于自由解卷积计算

的步骤为：After performing free multiplication and additive deconvolution in (15), we obtain the moment of μ _P

As mentioned earlier, the decision rule for spectrum sensing is based on the average received power

The estimate of , which is the first moment of μ _P , is

Therefore, we can conclude that based on the free deconvolution computation

The steps are:

(1) Input: receive sampling signal {y(n), n=1, 2,..., M _S } and channel noise variance σ ² ;

和它的特征值{λ_i,i＝1,2,…,KN_r}；(2) Calculate the sample covariance matrix in formula (8)

and its eigenvalue {λ _i , i=1,2,...,KN _r };

(3) Calculate the moment in formula (20) and the moments in (21)

步骤如下：(4) calculation

Proceed as follows:

①

② $\{m_k^{\mathrm{\&nu;}}={\mathrm{\&gamma;}}_k/\left(\frac{{\mathrm{KN}}_r}{m_S}\right)\};$

步骤如下：(5) calculation

Proceed as follows:

① ${\mathrm{\&alpha;}}_k^{\mathrm{\&nu;}}=\mathrm{mom\; cum}\left(\right\{m_k^{\mathrm{\&nu;}}\left\}\right)$

② ${\mathrm{\&alpha;}}_k^{{\mathrm{\&mu;}}_{{\mathrm{\&sigma;}}^{2}I}}=\mathrm{mom\; cum}\left(\right\{{\left({\mathrm{\&sigma;}}^2\right)}^k\left\}\right)$

③ ${\{\mathrm{\&alpha;}}_k^{\mathrm{\&eta;}}={\mathrm{\&alpha;}}_k^{\mathrm{\&nu;}}-{\mathrm{\&alpha;}}_k^{{\mathrm{\&mu;}}_{{\mathrm{\&sigma;}}^{2}I}}\}$

④ $\left\{m_k^{\mathrm{\&eta;}}\right\}=\mathrm{cummom}\left(\right\{{\mathrm{\&alpha;}}_k^{\mathrm{\&eta;}}\left\}\right);$

步骤如下：(6) calculation

Proceed as follows:

① $\left\{{\mathrm{\&rho;}}_k\right\}=\mathrm{mom\; cum}\left(\right\{\frac{{\mathrm{KN}}_r}{N_t}\&CenterDot;m_k^{\mathrm{\&eta;}}\left\}\right)$

② $\{m_k^P={\mathrm{\&rho;}}_k/\left(\frac{{\mathrm{KN}}_r}{N_t}\right)\};$

(7)

但是在Η₀情况下，由上面介绍的基于自由解卷积的算法得到的

没有概率分布的解析表达式。于是我们采用Monte Carlo仿真来获得在Η₀情况下

的直方图，如图3所示，经验证直方图近似于高斯分布。因此，在只有噪声样本的情况下，我们可以使用估计的噪声方差并且执行基于自由解卷积的算法来获得我们运行很多次这样的仿真，然后以此计算的均值θ_P和方差

得到

则目标虚警概率p_f的检测阈值为Next, for the selection of the detection threshold τ, it is usually necessary to satisfy the target false alarm probability p _f , that is

But in the case of Η ₀ , obtained by the algorithm based on free deconvolution introduced above

There is no analytical expression for the probability distribution. So we use Monte Carlo simulation to obtain in the case of Η ₀

The histogram of , as shown in Figure 3, has been verified that the histogram approximates a Gaussian distribution. Therefore, in the case of only noisy samples, we can use the estimated noise variance and perform a free deconvolution based algorithm to obtain We run this simulation many times, and then calculate The mean θ _P and variance of

get

Then the detection threshold of target false alarm probability p _f is

τ=υ _P Q ^-1 (p _f )+θ _P (23) where Q ^-1 ( ) is the inverse Q-function,

和τ进行比较，判断主基站是否在发送信号。当

时，主基站在发送信号；当

时，主基站没有发送信号。Finally, the

Compared with τ, it is judged whether the main base station is sending a signal. when

When , the master base station is sending signals; when

, the main base station does not send a signal.

In order to better describe the effect of the present invention, it will be further proved by the following simulation examples:

1. Simulation parameter setting

其中

There is a MIMO Rayleigh fading channel between the primary base station and each secondary base station. The secondary base stations are distributed in different geographical locations, so _Pk is different. Also average

in

2. Simulation method

Existing detection methods based on eigenvalues and the method of the present invention.

和它的特征值然后比较

和阈值τ_EV，如果

则为Η₁，否则为Η₀。阈值计算公式为 ${\mathrm{\&tau;}}_{\mathrm{EV}}={\left(\frac{\sqrt{{\mathrm{KN}}_r}+\sqrt{M_S}}{\sqrt{{\mathrm{KN}}_r}-\sqrt{M_S}}\right)}^2(1+\frac{{(\sqrt{{\mathrm{KN}}_r}+\sqrt{M_S})}^{-2/3}}{{\left({\mathrm{KN}}_r{M}_S\right)}^{1/6}}{F}_{\mathrm{TW}2}^{-1}(1-p_f)),$ 其中

为逆Tracy-Widom第二累计分布函数。在我们的仿真中

Introduction to eigenvalue-based detection methods: The eigenvalue-based detection method does not require noise variance, and it uses the asymptotic characteristics of the ratio of the largest and smallest eigenvalues of the sample covariance matrix. To be precise, it first computes the sample covariance matrix in equation (8)

and its eigenvalues then compare

and threshold τ _EV , if

Then it is Η ₁ , otherwise it is Η ₀ . The threshold calculation formula is ${\mathrm{\&tau;}}_{\mathrm{EV}}={\left(\frac{\sqrt{{\mathrm{KN}}_r}+\sqrt{m_S}}{\sqrt{{\mathrm{KN}}_r}-\sqrt{m_S}}\right)}^2(1+\frac{{(\sqrt{{\mathrm{KN}}_r}+\sqrt{m_S})}^{-2/3}}{{\left({\mathrm{KN}}_r{m}_S\right)}^{1/6}}{f}_{\mathrm{TW}2}^{-1}(1-p_f)),$ in

is the inverse Tracy-Widom second cumulative distribution function. In our simulation

3. Simulation results

As shown in FIG. 4 , the spectrum sensing performance is compared between the method of the present invention and the detection method based on eigenvalues. It can be seen that the performance of the method of the present invention is better than the method based on eigenvalues under all SNR values and total number of samples, especially in the case of low signal-to-noise ratio and small samples.

Example 2

As shown in Figure 2, the present invention provides a cooperative spectrum sensing method based on free probability theory, which is suitable for MIMO communication environments, and specifically includes the following steps:

Step 1: Sampling the received signals of multiple antennas of each secondary base station, and the sampled signals will be processed centrally;

也就是检测统计量；Step 2: According to the noise variance of all received sampling signals and channels, with the help of the asymptotic free characteristics of the random matrix and the Wishart distribution characteristics, the algorithm based on free deconvolution is used to solve the average received signal power of all receiving antennas

That is, the detection statistic;

Step 3: According to the target false alarm probability p _f , use Monte Carlo simulation to calculate the detection threshold τ under the condition that only noise exists;

和τ进行比较，判断主基站是否在发送信号。当时，主基站在发送信号；当

时，主基站没有发送信号。Step 4: Put

Compared with τ, it is judged whether the main base station is sending a signal. when When , the master base station is sending signals; when

, the main base station does not send a signal.

Step 1 in the method of the present invention comprises:

k＝1,2,…,K。那么所有接收天线在时刻n集中起来的采样信号为

其中(·)^Η表示共轭转置。K secondary base stations BS ₁ , BS ₂ , ..., BS _K distributed in different geographical locations cooperate to sense the channel, and determine whether the primary base station is sending signals. The primary base station and each secondary base station are equipped with N _t and N _r antennas respectively, and there is a MIMO Rayleigh fading channel between them. The sampling rate is 1/T _s , then the sampling signal output by the kth receiver at time n is

k=1,2,...,K. Then the sampled signal collected by all receiving antennas at time n is

Wherein (·) ^Η represents conjugate transposition.

Step 2 in the method of the present invention comprises:

(1) Input: receive sampling signals {y(n), n=1, 2, ..., M _S } and channel noise variance σ ² . Where M _S is the total number of samples of the received signal;

和它的特征值{λ_i,i＝1,2,…,KN_r}；(2) Calculate the sample covariance matrix of the received sampling signal

and its eigenvalue {λ _i , i=1,2,...,KN _r };

的k阶矩

和

的k阶矩

(3) calculation

k-order moment

and

k-order moment

步骤如下：(4) calculation

Proceed as follows:

①

② $\{m_k^{\mathrm{\&nu;}}={\mathrm{\&gamma;}}_k/\left(\frac{{\mathrm{KN}}_r}{m_S}\right)\};$

步骤如下：(5) calculation

Proceed as follows:

① ${\mathrm{\&alpha;}}_k^{\mathrm{\&nu;}}=\mathrm{mom\; cum}\left(\right\{m_k^{\mathrm{\&nu;}}\left\}\right)$

② ${\mathrm{\&alpha;}}_k^{{\mathrm{\&mu;}}_{{\mathrm{\&sigma;}}^{2}I}}=\mathrm{mom\; cum}\left(\right\{{\left({\mathrm{\&sigma;}}^2\right)}^k\left\}\right)$

③ ${\{\mathrm{\&alpha;}}_k^{\mathrm{\&eta;}}={\mathrm{\&alpha;}}_k^{\mathrm{\&nu;}}-{\mathrm{\&alpha;}}_k^{{\mathrm{\&mu;}}_{{\mathrm{\&sigma;}}^{2}I}}\}$

④ $\left\{m_k^{\mathrm{\&eta;}}\right\}=\mathrm{cummom}\left(\right\{{\mathrm{\&alpha;}}_k^{\mathrm{\&eta;}}\left\}\right);$

步骤如下：(6) calculation

Proceed as follows:

① $\left\{{\mathrm{\&rho;}}_k\right\}=\mathrm{mom\; cum}\left(\right\{\frac{{\mathrm{KN}}_r}{N_t}\&CenterDot;m_k^{\mathrm{\&eta;}}\left\}\right)$

② $\{m_k^P={\mathrm{\&rho;}}_k/\left(\frac{{\mathrm{KN}}_r}{N_t}\right)\};$

也就是μ_P的一阶矩。(7)

That is, the first moment of μ _P.

和

分别表示自由概率理论中的自由加法解卷积算子和自由乘法解卷积算子；

和μ_P分别表示矩阵

σ²I（I为单位矩阵）和P的极限概率分布；

和

对应于律μ_c，c分别取

和

momcum和cummom是根据分布μ的矩-累积量公式得到的，momcum的输入为矩序列，输出为累积量序列，cummom的输入为累积量序列，输出为矩序列。Explanation of symbols in the above steps:

and

represent the free additive deconvolution operator and the free multiplicative deconvolution operator in the free probability theory, respectively;

and μ _P respectively denote the matrix

σ ² I (I is the identity matrix) and the limiting probability distribution of P;

and

corresponds to Law μ _c , c respectively take

and

Momcum and cummom are obtained according to the moment-cumulant formula of the distribution μ. The input of momcum is a moment sequence and the output is a cumulant sequence, and the input of cummom is a cumulant sequence and the output is a moment sequence.

Step 3 in the method of the present invention comprises:

经过很多次这样的仿真后计算

的均值θ_P和方差

近似满足高斯分布。然后依据目标虚警概率p_f，计算阈值τ=υ_PQ^-1(p_f)+θ_P，其中Q^-1(·)为逆Q-函数，

First perform step 2 with only noise samples to obtain

After many such simulations, the calculation

The mean θ _P and variance of

approximately satisfy a Gaussian distribution. Then according to the target false alarm probability p _f , calculate the threshold τ=υ _P Q ^-1 (p _f )+θ _P , where Q ^-1 (·) is the inverse Q-function,

Claims (5)

1. A collaborative spectrum sensing method based on free probability theory, characterized in that, the method comprises the steps of: Step 1: Sampling the received signals of multiple antennas of each secondary base station, and the sampled signals will be processed centrally; Step 2: According to the noise variance of all received sampling signals and channels, with the help of the asymptotic free characteristics of the random matrix and the Wishart distribution characteristics, the algorithm based on free deconvolution is used to solve the average received signal power of all receiving antennas

That is, the detection statistic; Step 3: According to the target false alarm probability p _f , use Monte Carlo simulation to calculate the detection threshold τ under the condition that only noise exists; Step 4: Put Compare with τ to judge whether the main base station is sending a signal; when

When , the main base station is sending signals; when

, the main base station does not send a signal. 2. A kind of collaborative spectrum sensing method based on free probability theory according to claim 1, is characterized in that, step 1 in the described method comprises: K secondary base stations BS ₁ , BS ₂ ,...,BS _K distributed in different geographic locations cooperate to sense the channel, and judge whether the primary base station is sending signals; the primary base station and each secondary base station are equipped with N _t and N _r antennas respectively, Between them is a MIMO Rayleigh fading channel; the sampling rate is 1/T _s , then the sampling signal output by the kth receiver at time n is

k=1,2,...,K; then the sampled signals collected by all receiving antennas at time n are Wherein (·) ^Η represents conjugate transposition. 3. A kind of collaborative spectrum sensing method based on free probability theory according to claim 1, is characterized in that, step 2 in the described method comprises: (1) Input: receive sampling signal {y(n),n=1,2,...,M _S } and channel noise variance σ ² , where M _S is the total number of samples of the received signal; (2) Calculate the sample covariance matrix of the received sampling signal

and its eigenvalue {λ _i , i=1,2,...,KN _r }; (3) calculation

k-order moment and

k-order moment

(4) calculation

Proceed as follows: ① ② $\{m_k^{\mathrm{\&nu;}}={\mathrm{\&gamma;}}_k/\left(\frac{{\mathrm{KN}}_r}{m_S}\right)\};$ (5) calculation

Proceed as follows: ① ${\mathrm{\&alpha;}}_k^{\mathrm{\&nu;}}=\mathrm{mom\; cum}\left(\right\{m_k^{\mathrm{\&nu;}}\left\}\right)$ ② ${\mathrm{\&alpha;}}_k^{{\mathrm{\&mu;}}_{{\mathrm{\&sigma;}}^{2}I}}=\mathrm{mom\; cum}\left(\right\{{\left({\mathrm{\&sigma;}}^2\right)}^k\left\}\right)$ ③ ${\{\mathrm{\&alpha;}}_k^{\mathrm{\&eta;}}={\mathrm{\&alpha;}}_k^{\mathrm{\&nu;}}-{\mathrm{\&alpha;}}_k^{{\mathrm{\&mu;}}_{{\mathrm{\&sigma;}}^{2}I}}\}$ ④ $\left\{m_k^{\mathrm{\&eta;}}\right\}=\mathrm{cummom}\left(\right\{{\mathrm{\&alpha;}}_k^{\mathrm{\&eta;}}\left\}\right);$ (6) calculation Proceed as follows: ① $\left\{{\mathrm{\&rho;}}_k\right\}=\mathrm{mom\; cum}\left(\right\{\frac{{\mathrm{KN}}_r}{N_t}\&CenterDot;m_k^{\mathrm{\&eta;}}\left\}\right)$ ② $\{m_k^P={\mathrm{\&rho;}}_k/\left(\frac{{\mathrm{KN}}_r}{N_t}\right)\};$ (7)

That is, the first moment of μ _P ; Explanation of symbols in the above steps:

and

represent the free additive deconvolution operator and the free multiplicative deconvolution operator in the free probability theory, respectively;

and μ _P respectively denote the matrix

σ ² I (I is the identity matrix) and the limiting probability distribution of P;

and

corresponds to

Law μ _c , c respectively take

and

Momcum and cummom are obtained according to the moment-cumulant formula of the distribution μ. The input of momcum is a moment sequence and the output is a cumulant sequence, and the input of cummom is a cumulant sequence and the output is a moment sequence. 4. A kind of collaborative spectrum sensing method based on free probability theory according to claim 1, is characterized in that, step 3 in the described method comprises: First perform step 2 with only noise samples to obtain

After many such simulations, the calculation

The mean θ _P and variance of Approximately satisfy the Gaussian distribution, and then calculate the threshold τ=υ _P Q ^-1 (p _f )+θ _P according to the target false alarm probability p _f , where Q ^-1 (·) is the inverse Q-function,

5. A cooperative spectrum sensing method based on free probability theory according to claim 1, characterized in that: said method is applicable to a MIMO communication environment.

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