This section will introduce the proposed spectrum sensing method based on SR and filters.
3.1 Stochastic resonance
In order to recover the periodicity of the original signal furthest from intensive noise, the received signal in each antenna r_{i}(i = 1, …, M) will be processed via the SR system. When the received signal is continuous, the SR output signal is defined as:
$$ \boldsymbol{x}(t)=f\left(\boldsymbol{r}(t)\right)={\left[{x}_1(t),{x}_2(t),\cdots, {x}_M(t)\right]}^T,\kern0.5em \infty <t<\infty, $$
(4)
where f(.) is a nonlinear function representing the physical behavior. The transformation process can be defined by the Langevin equation [11]:
$$ \frac{dx_i(t)}{dt}=\frac{U\left({x}_i\right)}{dx_i}+{\overline{S}}_i(t)+{W}_i(t), $$
(5)
where U(x_{i}) denotes the potential function, the expression and function curve are shown as follows:
$$ U\left({x}_i\right)=\frac{a}{2}{x_i}^2+\frac{b}{4}{x_i}^4, $$
(6)
in which a > 0 and b > 0. Equation (5) indicates the classic and nontrivial SR model. The dynamic and integral characteristics of SR are driven by three basic elements: the bistable nonlinear system, the Gaussian white noise w_{i}(t), and the external excitation \( {\overline{s}}_i(t) \). In Fig. 1, the curve of U(x_{i}) exists three extreme points P_{1}(x_{m}, −U_{0}), P_{2}(−x_{m}, −U_{0}), and O(0, 0). In which, \( {x}_m=\sqrt{a/b} \) and U_{0} = a^{2}/(4b). The two minimum points P_{1} and P_{2} are named as potential wells. Accordingly, the maximum point O is named as potential barrier. The difference between the potential barrier and the potential wells represents the potential barrier height (i.e., U_{0}). Equation (5) is a bistable structure because the two potential wells represent two stable states.
Supposing that a Brown particle lies in a certain point of U(x_{i}) at the initial time instant, and \( {\overline{s}}_i(t) \) is treated as a periodic signal with amplitude A_{m} and carrier frequency f_{c}, there is a critical value \( {A}_c=\sqrt{4{a}^3/(27b)} \) in U(x_{i}). When the SR system is only driven by external signal \( {\overline{s}}_i(t) \) and A_{m} > A_{c}, the particle can jump across the potential barrier, and the balance of U(x_{i}) will be broken. Then, the potential wells will elevate alternatively and periodically with the frequency f_{c} [14]. When only noise w_{i}(t) exists, the Brown particle will switch between two potential wells with the transition speed r_{k}, which is named as Kramers rate [15]:
$$ {r}_k=\frac{a}{\sqrt{2}\pi}\mathit{\exp}\left(\frac{2{U}_0}{\sigma_w^2}\right). $$
(7)
The joint effect of input \( {\overline{s}}_i(t) \) and noise w_{i}(t) will resonate the SR system to achieve as high an amplitude as possible. But the classical SR theory, like adiabatic approximation theory and nonlinear response theory, indicates that SR can only work with small parameters [15]. Firstly, the output signal x_{i}(t) is mainly concentrated on low frequency components rather than higher harmonics, so the input carrier frequency should be a considerably small value (i.e., f_{c} ≪ 1 and f_{c} ≪ r_{k}). Secondly, the amplitude A_{m} and the noise power \( {\sigma}_w^2 \) should also be far less than 1.
The prerequisite of low frequency is a straight conflict of the modulation carrier requirement of high frequency in wireless communication. To solve this problem, we exploit the normalized scale transformation (NST) method to convert high frequency to low frequency [16]. The NST technology exploits the normalization and variable substitutions as follows:
$$ \frac{d{z}_i\left(\tau \right)}{dt}={z}_i{z}_i^3+{A}_0\mathit{\cos}\left(2\pi {f}_0\tau \right)+{w}_0\left(\tau \right), $$
(8)
where \( z=x\sqrt{a/b} \); τ = at is the resample time interval; \( {A}_0={A}_m\sqrt{b/{a}^3} \) is the normalized amplitude; w_{0}(τ) is the normalized noise with expectation 0 and variance \( {\sigma}_0^2={\sigma}_w^2b/{a}^2 \); and f_{0} is the normalized frequency and can be expressed by the original carrier frequency (i.e., f_{0} = f_{c}/a).
It can be found that Eq. (8) is the standard normalized form of Eq. (5), and they have the same dynamic characteristics. However, the main significances and contributions lie in that Eq. (8) can satisfy the preconditions of small parameters according to the adiabatic approximation theory. Note that the condition a ≫ 1 can ensure that the high frequency f_{c} of the carrier signal turns into a low frequency f_{0}. Thus, for algorithm implementation, we can preset small values of f_{0} and A_{0} to obtain a and b. Then, f_{0} and A_{0} can be adjusted based on the output SNR and resonance effect to achieve the desired state.
In general, Eq. (8) is an expression of an ordinary first order differential equation, and no exact solutions have been provided in recent studies. However, it can be approximately solved by the fourth order Runge Kutta (RK) algorithm [13], which is a numerical computation method and includes the process of multistage iteration. Meanwhile, RK algorithm is the form of discrete time system referred to the Langevin Eq. (5), and processes r_{i}(n) rather than r_{i}(t).
3.2 Filters
The output of SR x_{i}(n) will pass through two linear time invariant discrete time systems, that is, low pass, high pass, or band pass filters. The two filters are named as preliminary filter (PF) and additional filter (AF) with the impulse responses h_{p}(m) and h_{a}(m), respectively, which is mentioned in a description of GD in [19]. The processes of PF and AF are described as a convolution form:
$$ \left\{\begin{array}{c}{e}_i(n)=\sum \limits_{m=\infty}^{\infty }{h}_p(m){x}_i\left(nm\right)\\ {}{\eta}_i(n)=\sum \limits_{m=\infty}^{\infty }{h}_a(m){x}_i\left(nm\right)\end{array},\right. $$
(9)
where e_{i}(n), n = 0, …, N_{s} − 1 is the ith antenna and nth sample of the secondary data via SR at the PF output; η_{i}(n) is the corresponding one at AF output; and e and η are the M × N_{s} matrix forms of e_{i}(n) and η_{i}(n).
Assuming that the central frequencies of PF and AF are detuned, the PU signal via SR cannot pass through AF and only appears in the PF output. If the detuning of central frequencies between the AF and PF achieved over four times the PU signal bandwidth, the correlation coefficient between the PF and AF output can be ignored [18]. That means the AF and PF outputs are independent. Then, the amplitude frequency characteristics of the PF and AF can be adjusted to ensure that the noise portions are equal. Hence, it is approximately considered that e ≈ x + η, and the noise power can be estimated by η.
3.3 Detection method
Now that the filtered signal e and η are obtained, this section will introduce three detection algorithms, namely, ED, LRT, and MED.
3.3.1 ED
The ED algorithm is a blind spectrum sensing method used when \( {\overline{s}}_i(n) \) is not known to the CR user. ED employs the sum of the energy at the observed interval. The test statistic is defined as follows [11]:
$$ {T}_{ED}=\sum \limits_{i=1}^M\sum \limits_{n=0}^{N_s1}{\left{e}_i(n)\right}^2, $$
(10)
3.3.2 LRT
Based on the Neyman Pearson (NP) criterion, when the probability of false alarm P_{f} and the noise variance \( {\sigma}_{\eta}^2 \) is given, the LRT will maximize the detection probability. The decision statistic of LRT is determined using the following form:
$$ {T}_L=\frac{p\left(\boldsymbol{e}{H}_1\right)}{p\left(\boldsymbol{e}{H}_0\right)}=\prod \limits_{n=0}^{N_s1}\frac{p\left(\boldsymbol{e}(n){H}_1\right)}{p\left(\boldsymbol{e}(n){H}_0\right)}, $$
(11)
where e is the received signal vector that is the aggregation of e(n), and p(e H_{1}) and p(e H_{0}) denote the likelihood function under the hypotheses H_{1} and H_{0}, respectively. The likelihood function at time instant n can be presented in the following form [13]:
$$ p\left(\boldsymbol{e}(n){H}_1\right)=\frac{\mathit{\exp}\left\{{\boldsymbol{e}}^{\ast }(n){\left({\boldsymbol{R}}_{\overline{s}}+{\boldsymbol{R}}_{\eta}\right)}^{1}\boldsymbol{e}(n)\right\}}{\pi^2\mathit{\det}\left({\boldsymbol{R}}_{\overline{s}}+{\boldsymbol{R}}_{\eta}\right)}, $$
(12)
$$ p\left(\boldsymbol{e}(n){H}_0\right)=\frac{\mathit{\exp}\left\{{\boldsymbol{e}}^{\ast }(n){\boldsymbol{R}}_{\eta}^{1}\boldsymbol{e}(n)\right\}}{\pi^2\mathit{\det}\left({\boldsymbol{R}}_{\eta}\right)}. $$
(13)
Based on the character of the matrix inversion lemma, there is
$$ {\boldsymbol{R}}_{\eta}^{1}{\left({\boldsymbol{R}}_{\overline{s}}+{\boldsymbol{R}}_{\eta}\right)}^{1}=\frac{1}{\sigma_{\eta}^2}{\boldsymbol{R}}_{\overline{s}}{\left({\boldsymbol{R}}_{\overline{s}}+{\boldsymbol{R}}_{\eta}\right)}^{1}. $$
(14)
Hence, Eq. (11) can be simplified as follows:
$$ {\sigma}_{\eta}^2\ln \left({T}_L{\left(\frac{\mathit{\det}\left({\boldsymbol{R}}_{\overline{s}}+{\boldsymbol{R}}_{\eta}\right)}{\mathit{\det}\left({\boldsymbol{R}}_{\eta}\right)}\right)}^{N_s}\right)=\sum \limits_{n=0}^{N_s1}{\boldsymbol{e}}^{\ast }(n){\boldsymbol{R}}_{\overline{s}}{\left({\boldsymbol{R}}_{\overline{s}}+{\boldsymbol{R}}_{\eta}\right)}^{1}\boldsymbol{e}(n). $$
(15)
Note that the left side of Eq. (15) does not contain the signal matrix e and does not relate to constructing the test statistic. In contrast, the right side of Eq. (15) does relate to constructing the test statistic and in fact is defined as a new test statistic: T_{LRT} ≶ γ_{LRT}, where γ_{LRT} is the detection threshold of LRT.
3.3.3 MED
LRT is the optimal and ideal detector based on the likelihood function, in which some parameters, such as noise variance \( {\sigma}_n^2 \) or received source signal covariance \( {R}_{\overline{s}} \), are known. In most practical scenarios, they are blind. This means the probability distribution of the observations or the likelihood functions cannot be obtained. This type of problem can only be solved by GLRT, which estimates the unknown parameters by maximum likelihood estimate (MLE). The test statistics of the GLRT detector have some simple form expressed by the eigenvalue of the sample covariance matrix of the received signal. Thus, MED will be used in this section. The algorithm steps are expressed as follows [5]:

1)
Calculate the sample covariance matrix of filter output signal e(n) as:
$$ {\boldsymbol{R}}_e=\frac{1}{N_s}\sum \limits_{n=0}^{N_s1}\boldsymbol{e}(n){\boldsymbol{e}}^{\ast }(n). $$
(16)

2)
Calculate the eigenvalues of the sample covariance matrix R_{e} and order them as λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{M}.

3)
Use the largest eigenvalues for detection:
$$ {T}_{MED}=\frac{\lambda_1}{\sigma_{\eta}^2}, $$
(17)
3.4 Algorithm summary
The proposed algorithm is summarized in Fig. 2. Next, the computational complexity is analyzed. Due to the RK algorithm, the extra computational cost is produced in contrast to the traditional detector. Since the signals from multi antennas are independent, the receivers can calculate the SR output concurrently. Thus, the RK algorithm needs 5(N_{s} − 1) manipulations. The computational complexity of SR is T_{SR} = O(5(N_{s} − 1)) = O(N_{s}), which is linear order. It indicates that the time cost is acceptable compared to detection probability. Noted that the computational complexity only reflects the gradual change of the time complexity accompany with the problem scale N_{s}. T_{SR} cannot reflect the time frequency, i.e., the actual execution time of the algorithm.