Blind Demodulation of Chaotic Direct Sequence Spread Spectrum Signals Based on Particle Filters

Applying the particle filter (PF) technique, this paper proposes a PF-based algorithm to blindly demodulate the chaotic direct sequence spread spectrum (CDS-SS) signals under the colored or non-Gaussian noises condition. To implement this algorithm, the PFs are modified by (i) the colored or non-Gaussian noises are formulated by autoregressive moving average (ARMA) models, and then the parameters that model the noises are included in the state vector; (ii) the range-differentiating factor is imported into the intruder’s chaotic system equation. Since the range-differentiating factor is able to make the inevitable chaos fitting error advantageous based on the chaos fitting method, thus the CDS-SS signals can be demodulated according to the range of the estimated message. Simulations show that the proposed PF-based algorithm can obtain a good bit-error rate performance when extracting the original binary message from the CDS-SS signals without any knowledge of the transmitter’s chaotic map, or initial value, even when colored or non-Gaussian noises exist.

known, and even that the parameters are given; or (ii) chaotic maps and their time series are only affected by white noise, in particular Gaussian noise.
In this study, the particle filtering technique is employed to solve the problems mentioned above. The particle filter (PF) is the state-of-the-art solution to nonlinear and non-Gaussian problem, following the Bayesian filtering framework, which uses a sequential Monter Carlo method to approximate the optimal filtering by representing the probability density function with a swarm of particles [22][23][24]. Due to this sample-based representation, particle filters are able to represent a wide range of probability densities, allowing online, real-time estimation of nonlinear, colored or non-Gaussian dynamic systems. Li and Feng [25] prove that the PF has better performance in convergence rate and estimation accuracy than the extended Kalman filter (EKF) and unscented Kalman filter (UKF).
In this paper, a novel PF-based algorithm is proposed to blindly demodulate the CDS-SS communication system without any knowledge of the transmitter's chaotic map, parameters, or initial value, and even under colored or non-Gaussian noise conditions. To begin with, the formulation of this problem is introduced. Then, autoregressive moving average (ARMA) models are utilized to describe the colored noises, so that the demodulation could be formulated as a mixed parameter and state estimation problem. Accordingly, the message signal can be estimated by using two alternate PFs, and the state variable of one PF is the chaotic state, and the other one is the message signal. Finally, the proposed algorithm is implemented to blindly demodulate the CDS-SS signals.

Problem Formulation
Consider a CDS-SS communication system, in which the transmitter's chaotic system is described by: The chaotic spreading sequence   k x is supposed to be a simple one that is generated by one chaotic map that is doubly symmetrical, i.e., the range of its values and its probability distribution are both symmetrical [5,19]. Each original binary message where N is the spreading factor. Namely, n b is encoded as the string: . Since the transmitter's chaotic map is unknown, the intruder uses another chaotic map ) ( g instead of ) ( f , and its chaotic system equation is given by: Because the transmitted CDS-SS signal is 1 , the CDS-SS signal estimated by the intruder could be , where b and x are the estimation of b and x, respectively.
Considering the model error or noise, the following system and measurement equations [15] can be obtained: (1) 1 where 1 k z  is the observation, (1) 1 k v  is the measurement noise with zero-mean and arbitrary distribution, (1) k w is the colored or non-Gaussian process noise that drives the dynamic systems through the nonlinear state transition function. On the other hand, since the message k b in CDS-SS varies more slowly than the chaotic sequence   k x , 1k approximately. Thus, the system and measurement equations can also be written as: 1 k v  is the measurement noise with zero-mean and arbitrary distribution, (2) k w is the colored or non-Gaussian process noise that drive the dynamic systems through the nonlinear state transition function.
The colored noises (1) k w and (2) k w can be formulated by ARMA models [26] as: Let: Furthermore, let: where k x  satisfies: and k b  satisfies: Then, the state vectors are augmented as . Therefore, the augmented state-space models corresponding to Equations (4) and (5) can be rewritten as:

Particle Filter
To begin with, a brief review of the PF is described first [22][23][24]. Let us consider a dynamic system represented by: where k x is the state vector, k z is the measurement vector, ) ( f and ) ( h are system and measurement equations, respectively, and k w and k v are process and measurement noises, respectively. Unlike the conventional analytical approximation methods, PFs are commonly used for the approximation of intractable integrals and rely on the ability to draw random samples (or particles) from a probability distribution. The main idea is to represent the posterior probability density function of the target state by a set of random particles where k is the time step, j is the particle index, and M is the particle number, i.e.: is the set of accumulated measurements up to the kth time step. Then, one can compute an optimal estimate based on these particles and weights. In the processing of PFs, generally there are six important steps: x which is the initial probability for the state; (ii) Prediction/sampling is the proposed distribution. The most popular choice of the proposed distribution is the prior transition where ) ( v p denotes the distribution of the measurement noise v .
Followed by normalization 1 (iv) State estimation (v) Resampling The basic idea of resampling is to eliminate particles which have small weights, and to replicate particles with large weights. Draw new particles from the above set of particles

Blind Demodulation of CDS-SS Signals under the White Gaussian Noises Condition
Firstly, consider the situation that the signal is influenced by the white Gaussian noises, i.e., (1) k w in Equation (4) and (2)  will inevitably obviously result in a chaos fitting error. Therefore, the best method is to utilize the chaos fitting error. The chaos fitting method [15] employs a range-differentiating factor  in Equation (5) to differentiate the vibration range of the amplitude of ˆk b corresponding to −1 and 1. Accordingly, the following state-space equations can be obtained corresponding to Equations (4) and (5), respectively: Equations (22) and (23) , and the chaos fitting is . Then, the chaos fitting error is: Let ' , then: b   are expressed as follows:  Initiate ˆk x and ˆk b , then operate circularly starting from 0 k  . One cycle of the PF chaos fitting includes two steps: Step 1: Do PF chaos fitting and estimate 1k b  according to Equation (23) and ˆk x . The state equation is , and the measurement equation is Step 2: Do PF chaos fitting and estimate 1k x  according to Equation (22) and 1k b  . The state equation is , and the measurement equation is

Blind Demodulation of CDS-SS Signals under Colored/Non-Gaussian Noises Condition
Under the condition that the noises is colored or non-Gaussian as described in Equations (15) and (16), the range-differentiating factor  is imported into Equation (16) as shown in Equation (27): Then, the proposed algorithm of alternately estimating the message signal { } k b and chaotic state { } k x according to Equations (15) and (27) by two PFs is illustrated in Figure 2 and described as follows.  (1) ( , respectively. It should be mentioned that the distribution of the noises is obtained by the priori knowledge in practice. Assume that the measurement noises (1) k v and (2) k v with zero-mean and arbitrary distribution are both white Gaussian noises for simplicity, the spreading factor N is 127, and the range-differentiating factor  is 0.9. Figure 3 shows the estimated message ˆk b without the range-differentiating factor at a signal-to-noise ratio (SNR) of 7dB, i.e.,    In [5], under a condition of cooperative communication (known transmitter structure), several data blocks containing up to 200 bits were transmitted at a SNR of about 7-8 dB, and no error was noticed at the authorized receiver. In this paper, under the conditions of non-cooperative communication (unknown transmitter structure) and colored or non-Gaussian noises, the intercepted CDS-SS signals containing 200 bits at SNR = 7 dB are blindly demodulated by the proposed PF-based algorithm, and no error is found. Figure 5(a) shows the message ˆk b estimated by PF at SNR = 7dB; Figure 5(b) shows the result of lowpass filtering; Figure 5  In [25], the PF is used to estimate the parameter of the chaotic system with the requirement that the structure of the chaotic map is known. For comparison in the simulation, the algorithm in [25] is also modified by the fact that the colored or non-Gaussian noises are formulated by ARMA models, and then the parameters that model the noises are included in the state vector. A nonlinear RPROP neural network is developed to blindly demodulate (unknown transmitter structure) the CDS-SS signals in [16]. Figure 6 gives the bit-error rate (BER) performances of the algorithm in [25], the nonlinear RPROP neural network algorithm in [16], the UKF-based algorithm in [15] and the proposed PF-based algorithm with the noises in Equations (40) and (41). The BER of the proposed algorithm is about 5 10  at SNR = 7dB. It can be concluded that the BER performance of the proposed PF-based algorithm is better than the nonlinear RPROP neural network and the UKF-based algorithm on the non-cooperative communication condition (unknown transmitter's structure), and approaches the level the algorithm in [25] that is belong to the cooperative communication (known transmitter's structure) could achieve. Figure 6. Comparison of BER performance among the algorithm in [25], the nonlinear RPROP neural network algorithm in [16], the UKF-based algorithm in [15] and the proposed PF-based algorithm. UKF-based algorithm in [15] (unknown transmitter's structure) nonlinear RPROP neural network in [16] (unknown transmitter's structure) algorithm in [25] (known transmitter's structure)

Conclusions
In order to blindly demodulate the CDS-SS signals under the colored or non-Gaussian noises condition, a PF-based algorithm is proposed in this paper. To implement this algorithm, the intruder uses a different chaotic system equation to fit the transmitter's chaotic system equation. Moreover, the colored or non-Gaussian noises are formulated by ARMA models. In addition, the range-differentiating factor is imported into the intruder's chaotic system equation. Therefore, two modified PFs are obtained, which are implemented in reciprocal interaction to estimate the message and the chaotic state. Since the range-differentiating factor is able to make the inevitable chaos fitting error advantageous, the CDS-SS signals can be blindly demodulated according to the range of the estimated message signals. Simulations show that the proposed algorithm is robust to both colored or non-Gaussian noises, and the original binary message can be retrieved from the CDS-SS signals with satisfactory BER performance without any knowledge of the transmitter's chaotic map, parameters, or initial value.