1 Introduction
With the increase of people’s demand for underwater communication, underwater wireless communication has been widely concerned by researchers. At present, underwater communication usually adopts sound wave as the carrier of signal transmission. Sound wave frequency is low, bandwidth is narrow, and underwater communication environment is very complex. The available communication spectrum of an underwater acoustic channel with serious attenuation of a wireless electromagnetic wave is quite limited. On the other hand, a large amount of communication data and high sampling rate aggravate the scarcity of spectrum resources in underwater acoustic communication. In addition, in underwater communication, more and more communication applications require that channels provide wider frequency bands. How to effectively realize fast perception of spectrum resources and realize broadband information transmission? Some researchers have proposed to introduce compressed sensing technology into spectrum perception. Compressive sensing technology has a huge application prospect. In literature [
1], Zhi et al. first proposed the application of compressive sensing theory to cognitive radio, using wavelet edge detection and base tracking algorithm for spectrum edge detection. Literature [
2] makes full use of L1 norm to solve the problem and reconstructs the signal by minimizing the L1 norm problem. For the L1 norm problem, a simple matching pursuit (MP) algorithm was proposed in literature [
3]. In literature [
4], orthogonal matching pursuit (OMP) algorithm is proposed on the basis of MP. In literature [
5], on the basis of OMP, a piecewise orthogonal matching pursuit (StOMP) algorithm is proposed, which compares selected atoms with a threshold, updates the residual values, and iterates through a loop to get the final solution. The compressed sensing process not only reduces the limitation of signal bandwidth, but also increases the computational complexity and spectrum sensing time of the system due to the complexity of signal reconstruction. The researchers are also working on different ways to reduce computational complexity and improve perceptual performance. In literature [
6], sparse Bayesian learning is used to reconstruct the compressible image under the condition of noise measurement, and it is verified that this method can effectively reduce the time of spectrum perception. In reference [
7], sparse Bayesian model and correlation vector machine learning were studied, and sparse Bayesian regression and classification model were established to reduce the complexity. In literature [
8], CS is used for spectrum sensing, and CS data is studied to conduct joint data fusion of some users’ compressed sensing data under the condition that CS data do not share the same sparse characteristics with super-parameters. Literature [
9] proposes a Bayesian direct spectrum sensing algorithm, that is, Bayesian compressed sensing is used to detect the spectrum without reconstructing the signal, and the information of the main user is estimated directly from the over-parameter in the compressed measured value to complete the detection. This model can estimate not only the signal parameters, but also the error band of the signal.
Compressive sensing technology is a technique proposed for sparse signal sampling and compression at the same time. By sampling a very few characteristic observation values of analog signal, signal acquisition is completed through spatial transformation. The receiver completes the original signal reconstruction from the compressed data by solving an optimization problem. However, the signal reconstruction process in compressed sensing is relatively complex, which makes the computational complexity of the system large and the spectrum perception time long, and affects the overall performance of the system. In the process of spectrum signal detection, cognitive users only care about whether the spectrum is occupied or not, but do not care about which specific signal. In this paper, we use this feature to design a novel algorithm which is spectrum signal detection algorithm based on compressed sensing. In the underwater communication system, each SU is compressed and data acquisition, using Bayesian model to obtain shared super-parameters, fusion center through super-parameters to complete the decision of the spectrum occupancy information. Without signal reconstruction, a single SU completes spectrum signal detection and then popularized the algorithm to the whole underwater cognitive wireless communication system, making full use of the compressed sensing data collected by different cognitive users at different layer and fully considering the correlation of compressed sensing data to complete the signal detection process.
3 Algorithm design
For a hierarchical underwater communication cognitive radio network, assuming that there are
L cognitive radio users in the hierarchical center, the compressed sensing model provided extends to all cognitive radio users in the layer.
$$ \left\{\begin{array}{l}{y}_1={\Phi}_1{X}_1+{w}_1\\ {}\kern1.00em \vdots \\ {}{y}_L={\Phi}_L{X}_L+{w}_L\end{array}\right. $$
(8)
We assume that the variables of the Gaussian distribution are
Xi,
yi,
wi, and Φ
i is the
M ×
N dimension variable, where
i = 1, 2, …
L. The likelihood function of the single cognitive radio user observation value in the hierarchical center can be expressed as:
$$ p\left\{{y}_i|{X}_i,{\lambda}_0\right\}={\left(2\pi /{\lambda}_0\right)}^{-\frac{m_i}{2}}\kern0.5em \exp \left(-\frac{\lambda_0}{2}{\left\Vert {y}_i-{\Phi}_i{X}_i\right\Vert}_2^2\right) $$
(9)
mi——the number of points in the compressed sensing data of the ith cognitive user.
In the underwater communication cognitive wireless network, the data of single hierarchical center perception is combined signal detection through the fusion center [
15]. Based on compress sensing, automatic grouping of compressed perceptual data for hierarchical Bayesian model is realized, different stratified centers may have different hyper-parameter:
λ = {
λi1,
λi2,⋯,
λiM}
, and i = 1, 2, ⋯,
C (
C is the number of stratified centers in CRN). It is assumed that {
λi, .
i = 1, 2, ⋯,
C}
G is independent of the same distribution and
G is one of DP implementations. When the fusion center obtains the super-parametric information
λ−i = {
λ1,
λ2,⋯,
λi − 1,λi + 1,…
λc} of other stratified centers, the underlying distribution G0 will be updated. which is as follows:
$$ p\left({\lambda}_i|{\lambda}^{-i},\xi, {G}_0\right)=\frac{\xi }{\xi +C-1}{G}_0+\frac{1}{\xi +C-1}\sum \limits_{k=1}^K{n}_k^{-i}{\delta}_{\lambda_K} $$
(10)
where \( {\delta}_{\lambda_K} \) is a pulse function. If \( {n}_k^{-i} \) is used to denote the number of subsets with different values in the super-parametric set \( {\left\{{\lambda}_k\right\}}_{k=1}^C \) of hierarchical centers. It is shown that the new hyper-parameter λk is more inclined to be selected the larger membership \( {n}_k^{-i} \) when implemented by DP.
In the data fusion center, if the super-parameter is represented by “
λ∗” and the
K super-parameter is represented by
\( {\lambda}_k^{\ast } \), the probability of the distribution
G is expressed
lk at the point
\( {\lambda}_k^{\ast } \). Moreover,
\( \sum \limits_{k=1}^J{l}_k=1 \),
J is the number of possible values of super-parameters [
16],
J < <
C. In the DP-based hierarchical Bayesian model, the maximum likelihood function corresponding to the data fusion center
λ∗ can be further expressed as
\( l\left({\lambda}^{\ast}\right)=\sum \limits_{k=1}^J{\mathrm{\ell}}_k\left({\lambda}_k^{\ast}\right) \) , if the probability distribution is represented as the
jth stratified center and the
kth
\( {\lambda}_k^{\ast } \) is taken as the maximum super-parameter, then:
$$ {\displaystyle \begin{array}{c}{\mathrm{\ell}}_k\left({\lambda}_k^{\ast}\right)=\sum \limits_{j=1}^C{\gamma}_{j,k}\log p\left({y}_j|{\lambda}_k^{\ast}\right)\Big]\\ {}=\sum \limits_{j=1}^C{\gamma}_{j,k}\log \int p\left({y}_j|{X}_j,{\lambda}_0\right)p\left({y}_{\mathrm{j}}|{\lambda}_k^{\ast },{\lambda}_0\right)p\left({\lambda}_0|a,b\right){dX}_jd{\lambda}_0\Big]\\ {}=-\frac{1}{2}\sum \limits_{j=1}^C{\gamma}_{j,k}\left[\left({m}_j+2a\right)\log \left({y}_j^T{\Lambda}_{j,k}^{-1}{y}_j+b\right)+\log \left|{\Lambda}_{j,k}\right|\right]+\mathrm{Const}\end{array}} $$
(11)
Where
$$ {\Lambda}_{j,k}=\mathbf{E}+\sum \limits_{t=1,t\ne n}^M{\lambda}_{k,t}^{\ast -1}{\Phi}_{j,t}{\Phi_{j,}^T}_t+{\alpha}_{k,n}^{\ast -1}{\Phi}_{j,n}{\Phi_{j,}^T}_n={\Lambda}_{j,k,-n}+{\lambda}_{k,n}^{\ast -1}{\Phi}_{j,n}{\Phi_{j,}^T}_n $$
(12)
where Λ
j, k is not included in the Λ
j, k where is the part of the
n column vector (Φ
j, k, n) that does not contain Φ
j, k:
n = 1, 2⋯,
M. The determinant and inverse matrix of Λ
j, k can be further expressed as:
$$ \mid {\Lambda}_{j,k}\mid =\mid {\Lambda}_{j,k,-n}\Big\Vert 1+{\lambda}_{k,n}^{\ast -1}{\Phi}_{j,n}^T{\Lambda}_{j,k,-n}^{-1}{\Phi}_{j,n}\mid $$
(13)
$$ {\Lambda}_{j,k}^{-1}={\Lambda}_{j,k,-n}^{-1}\hbox{-} \frac{\Lambda_{j,k,-n}^{-1}{\Phi}_{j,n}{\Phi_{j,}^T}_n{\Lambda}_{j,k,-n}^{-1}}{\lambda_{k,n}^{\ast }{\Lambda}_{j,k,-n}^{\hbox{-} 1}{\Phi}_{j,n}} $$
(14)
In this case,
\( {\lambda}_{N,k}^{\ast } \) represents the
Kth stratified center, the
nth super-parameter. When the partial derivative with respect to
\( \frac{\partial {l}_k\left({\lambda}_k^{\ast}\right)}{\partial {\lambda}_{j,k}}=0 \), the extreme point can be obtained, namely:
$$ {\lambda}_{j,k}\approx \frac{L}{\sum \limits_{i=1}^L\frac{\left({m}_i+2a\right){h}_{j,k}^2/{t}_{j,k}-{p}_{j,k}}{p_{j,k}\left({p}_{j,k}-{h}_{j,k}^2/{t}_{j,k}\right)}} $$
(15)
Assuming that:
$$ {p}_{j,k}={\Phi_{\mathrm{j},}^T}_k{\Lambda}_{\mathrm{j},\hbox{-} \mathrm{k}}^{\hbox{-} 1}{\Phi}_{\mathrm{j},\mathrm{k}} $$
(16)
$$ {h}_{j,k}={\Phi_{\mathrm{i},}^T}_k{\Lambda}_{\mathrm{i},\hbox{-} \mathrm{k}}^{\hbox{-} 1}{y}_i $$
(17)
$$ {t}_{j,k}={y}_i^T{\Lambda}_{\mathrm{j},\hbox{-} \mathrm{k}}^{\hbox{-} 1}{y}_i+2\mathrm{b} $$
(18)
If the denominator is 0, then
λj, k = ∞. It needs to be removed Φ
j, k. So we can use the formula (
12) and
λj, k = ∞. Update the super-parameters
λj, k continuously, {
k = 1,2…
L,
j = 1,2...
N}.
In order to obtain the optimal super-parameter, the detailed steps of the proposed algorithm are as follows:
-
① Initialization parameter λj, k and base vector Φi , {k = 1,2… L; i = 1, 2, …, N; j = 1,2... N};
-
② For any k = 1, 2… L, you pick a basis vector Φj, k and calculate the mean μi and the covariance Σi and calculate the following:
$$ {P}_{j,k}={\Phi_{\mathrm{j},}^T}_k{\Lambda}_{\mathrm{j}}^{\hbox{-} 1}{\Phi}_{\mathrm{j},\mathrm{k}} $$
(19)
$$ {H}_{j,k}={\Phi_{\mathrm{i},}^T}_k{\Lambda}_{\mathrm{i}}^{\hbox{-} 1}{y}_i $$
(20)
$$ {T}_{j,k}={y}_j^T{\Lambda}_{\mathrm{j}}^{\hbox{-} 1}{y}_j+2b $$
(21)
If
\( {p}_{j,k}={P}_{j,k}+\frac{{P^2}_{j,k}}{\lambda_k-{P}_{j,k}} \),
\( {h}_{j,k}={H}_{j,k}+\frac{P_{j,k}{H}_{j,k}}{\lambda_n-{P}_{j,k}} \), and
\( {t}_{j,k}={T}_{j,k}+\frac{H_{j,k}^2}{\lambda_n-{P}_{j,k}} \). According to formula (
15) to update the super-parameter
λj, k, when selecting the super-parameter, the increment Δ
lk(
λj, k) of likelihood function
lk(
λj, k) should be maximized during each iteration;
-
③ When the number of iterations is greater than the set threshold or when Δlk(λj, k) is less than a certain value, reach the termination condition of iteration algorithm and stop iteration; otherwise, go back to step ②. The final decision can be made by comparing the obtained optimal over-parameter with the preset threshold value.
The final decision is realized by the iterative algorithm, which initializes parameters \( {\lambda}_{k,n}^{\ast } \) and base vectors Φj, n and constantly updates weight coefficients γj, k and \( {\lambda}_{k,n}^{\ast } \). The super-parameters \( {\lambda}_{k,n}^{\ast } \) were selected in each iteration process to maximize the increment until the calculated increment reaches the iteration termination condition or the iteration number reaches the prescribed upper limit; otherwise, the parameters are updated continuously until the iteration algorithm stops. The binary spectrum decision result is obtained by using the obtained super-parameters and the preset threshold value.
5 Results and discussion
This paper presents an efficient underwater communication signal detection method based on compressed sensing technology. In underwater communication cognitive wireless networks, different cognitive radio users and authorized users may have different sparse spectrum due to their different distributed environments in different spaces and the complexity of underwater channels. In the algorithm design, each CR node uses compressed sampling technology to estimate the frequency spectrum to reduce the rate and overhead of sonar signal sampling. The Bayesian model is used to find the optimal super-parameter to detect the spectrum information. The proposed algorithm effectively solves the high computational complexity of sonar signal reconstruction in compression sensing. The joint cooperative spectrum algorithm utilizes the information of non-parametric grouping mechanism. The multi-layer Bayesian model introduces the Dirichlet process to realize the automatic grouping of compressed perceptual data, deduces the shared hyper-parameter, then selects the best super-parameter to decide the spectrum through the fusion center, and finally transfers it to the SU. The algorithm makes full use of compressed perceptual data collected by different cognitive radio users in different layers and performs fusion and collaboration to complete signal detection and effectively improve the performance of underwater communication spectrum signal detection. Effectively reduce the channel overhead in the underwater communication system; solve the series of problems such as high complexity, slow convergence speed, and low reconstruction precision of the compressed sensing reconstruction algorithm; improve the efficiency; and increase the throughput of the system, which has a great significance for achieving high-quality underwater communication.
From the current research status, there are still the following problems and challenges in the wide-band spectrum compressed sensing technology of underwater communication system: at present, spectrum detection is carried out by using compressed sensing, and the processing method in most literatures is to obtain the compressed sampling first, then reconstruct the sonar signal, and finally carry out spectrum detection on the reconstructed signal. Its perceptual performance is easily affected by the variance uncertainty of noise. The compressed sensing reconstruction algorithm is a np-hard problem with high computational complexity and needs more time, especially for the broadband signal reconstruction process which takes too long and is not conducive to real-time application. Therefore, how to effectively reduce the reconstruction time and the reconstructed signal with a good mean square error is a big difficult problem in compressed sensing.
Secondly, as the sonar signal sparsity in the actual underwater communication network is unknown and variable, it is difficult to determine the signal sparsity for some complex sonar signals, or it is not precisely sparse. In order to ensure the reconstruction accuracy of broadband sonar signals, the sampling rate usually meets the condition of the maximum possible sparsity of the signals, resulting in the waste of sampling resources. Moreover, in CR networks, the cooperative compressed spectrum sensing fusion mechanism is crucial, which will directly affect the performance of underwater cognitive wireless networks. The corresponding efficient and low-complexity fusion mechanism still needs to be further explored.
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