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In a wireless sensor network, the signal received by the terminal processor is usually a complex single channel hybrid chaotic signal. The engineering needs to separate the useful signal from the mixed signal to perform the next transmission analysis. Since chaotic signals are nonlinear and unpredictable, traditional blind separation algorithms cannot effectively separate chaotic signals. Aiming to correct these problems—based on the particle filter estimation algorithm—an extended Kalman particle filter algorithm (EPF) and an unscented Kalman particle filter algorithm (UPF) are proposed to solve the single channel blind separation problem of chaotic signals. Mixing chaotic signals of different intensities performs blind source separation. Using different evaluation indexes carries out the experiment and performance can be analyzed. The results show that the proposed algorithm effectively separates the mixed chaotic signals.

In recent years, the wireless sensor network has been a crucial research directive of both experts and scholars. It plays a very important role in communications, medical treatments, military affairs, and other industries [

For the blind source separation problem and the ICA algorithm, literature [

For the blind separation of chaotic signals, literature [

In view of the above research, based on particle filter, this paper uses the extended Kalman particle filter (EPF) and the unscented Kalman particle filter (UPF) in chaotic signal, separating the general strength of the chaotic signal and strong chaotic signal, respectively. A comparison is then made to separate the effect of EPF and UPF in different scenarios and analyze them under different performance indexes.

For wireless sensor networks, the receiver receives the weighted sum of the source signals.

Due to the fact that the receiver often receives nonlinear chaotic signals in wireless sensor networks, it can be described as (

According to the Bayes theorem, the filtering problem of this model can be regarded as estimating the joint probability density function

In the receiving signal process of the wireless sensor network receiver, the adaptive filter extracts the feature vectors, so the observation equation obtained by the receiver is as follows:

In particle filter, the most important work is to find an optimal proposal density distribution function which can help us transfer the sample points in the prior distribution area to the maximum likelihood region:

In the prediction stage of the basic particle filter, the prediction value

If the likelihood function is bounded, that is,

(1) Firstly, a sample

(2) To determine whether the sample is available, if

Next, we use the particle filter algorithm improved from the suggested density function: the extended Kalman particle filter algorithm (EPF) and the unscented Kalman particle filter algorithm (UPF).

The extended Kalman filter (EKF) uses local linearization to transform the nonlinear functions

According to (

The core of the extended Kalman particle filter is as follows: In the sampling stage, we use EKF algorithm to calculate mean and covariance for each particle and then use every mean and covariance to guide the system sampling. In the process of using the EKF algorithm to calculate the mean and the variance, it uses the posterior filter density function:

In the framework of the particle filter algorithm, the EPF algorithm produces the Gauss recommended density distribution for each particle and updates the

The weights are recalculated and normalized for each particle:

The unscented Kalman filter (UKF) abandoned the processing of linearization of nonlinear functions and adopted the Kalman filtering framework, and for one-step prediction equation, the Unscented Transform (UT) algorithm is used to solve the nonlinear transfer problem of mean and covariance. UKF algorithm approximates the probability density distribution of nonlinear function and uses a series of samples to approximate the posterior probability of the answer. It does not need to approximate nonlinear function; Jacobi derivation of the state transition matrix, and the UKF will not ignore the Taylor higher-order expansions.

Calculate

The Sigma point set obtained by no trace transformation has the following properties:

(1) The Sigma point set around the mean is symmetrically distributed and the symmetric points have the same weight.

(2) The sample variance of the Sigma point set is the same as the random vector sample variance.

The extended Kalman particle filter provides a priori knowledge for sampling based on the extended Kalman filter and the basic particle filter and uses prior knowledge to guide system sampling, so it does not need to predict the preestimation to noise. The unscented Kalman particle filter is used to make symmetrical sampling to the particles around the mean. The number of sampling points compared with that of EPF is

In order to validate the effectiveness of the algorithm and compare it with two different algorithms, the Monte-Carlo experiments were conducted in the MATLAB 2014a simulation platform for 500 times. The nonlinear signals are distinguished by the different expression coefficients of the signal state space. The coefficients close to 1 are general nonlinear systems, and the coefficients close to 0 are strong nonlinear systems. In the experiment, two different mixed signals are selected in the general nonlinear system and the strong nonlinear system, and the noise is the standard Gauss additive white noise. In order to make the difference between two algorithms more obvious, we use the sum of the difference

Estimation result of EPF and UPF for general nonlinear systems.

Estimation results of

Estimation results of

Estimation result of EPF and UPF for strong nonlinear systems.

Estimation results of

Estimation results of

Correlation coefficients at different sampling points for general nonlinear systems.

Sampling Point | 30 | 60 | 90 | 120 | 150 | 180 | 210 | 240 | 270 | 300 | |
---|---|---|---|---|---|---|---|---|---|---|---|

S1 | EPF | 0.9932 | 0.9956 | 0.9968 | 0.9976 | 0.9979 | 0.9981 | 0.9984 | 0.9986 | 0.9987 | 0.9988 |

UPF | 0.9955 | 0.9972 | 0.9981 | 0.9986 | 0.9987 | 0.9989 | 0.9991 | 0.9992 | 0.9992 | 0.9993 | |

| |||||||||||

S2 | EPF | 0.9914 | 0.9917 | 0.9922 | 0.9929 | 0.9934 | 0.9939 | 0.9943 | 0.9951 | 0.9955 | 0.9957 |

UPF | 0.9960 | 0.9967 | 0.9971 | 0.9975 | 0.9977 | 0.9979 | 0.9980 | 0.9983 | 0.9984 | 0.9985 |

Correlation coefficients at different sampling points for strong nonlinear systems.

Sampling Point | 30 | 60 | 90 | 120 | 150 | 180 | 210 | 240 | 270 | 300 | |
---|---|---|---|---|---|---|---|---|---|---|---|

S1 | EPF | 0.5942 | 0.8119 | 0.8941 | 0.9084 | 0.9235 | 0.9358 | 0.9411 | 0.9501 | 0.9547 | 0.9588 |

UPF | 0.9915 | 0.9953 | 0.9974 | 0.9978 | 0.9981 | 0.9984 | 0.9986 | 0.9988 | 0.9989 | 0.9990 | |

| |||||||||||

S2 | EPF | 0.4938 | 0.6310 | 0.7413 | 0.8382 | 0.8654 | 0.8870 | 0.8955 | 0.9053 | 0.9194 | 0.9232 |

UPF | 0.9900 | 0.9916 | 0.9938 | 0.9958 | 0.9965 | 0.9970 | 0.9973 | 0.9975 | 0.9979 | 0.9980 |

Real-time comparison of EPF and UPF algorithms.

Correlation coefficient comparison of EPF and UPF algorithms at different sampling points.

Deviation comparison of EPF and UPF algorithms at different sampling points.

The above experiment analysis shows that, for general nonlinear mixed environment, selecting the EPF algorithm can effectively separate signal and save time and needs to choose the UPF, in a strongly nonlinear mixed environment to some extent, but at the expense of his time. EPF can only be in the filtering error and step prediction error is low and good with the filtering estimation effect. Although UPF solved these problems, on the arithmetic complexity it is far greater than the EPF. Thus, the question of how to effectively achieve separation should be used in assessing which algorithm to use.

To conclude, we have described the extended Kalman particle filter blind separation algorithm and the unscented Kalman particle filter blind separation algorithm based on Kalman filter and particle filter; separation matrix is obtained by system initialization, importance sampling, resampling, and state vector updating of mixed signals, and the two algorithms are used to separate the chaotic signals of two different intensities. In order to prove the applicability of the algorithm in different scenarios, the time effect, deviation, and correlation coefficient are used to analyze the algorithm results.

The advantage of the algorithm is that they can both improve the approximation degree of the posterior probability of the state from different angles and provide an effective basis for the selection of a single channel blind separation. In future work, the particle filter and its extension algorithm can be used to solve the more complicated blind separation problem of mixed communication signals.

This paper uses a nonlinear system model excited by random signals.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was supported by the National Natural Science Foundation of China (no. 61561031).