Chaotic signal reconstruction with application to noise radar system

In this paper, we present a new method that is accurate and has low computational complexity to estimate the initial condition of chaotic signal. Then the estimation method is applied to a noise radar system using noncoherent reception scheme. In the new scheme, the received signal correlates with the estimated transmitted signal by the estimation method proposed in this paper. The new scheme avoids a difficult problem that how to accomplish the transmitted signal delayed by delay lines when the signal band is large. Finally, computer simulations are performed to illustrate the effectiveness of the proposed scheme.


INTRODUCTION
Recently, there has been a growing amount of research interest in chaotic signal estimation, which has been applied in the field of communication [1], radar imaging [2] and encoding [3]. Initial condition estimation is an important way to estimate the chaotic signal, because the whole signal will be known once the bifurcating parameters have been given. Initial condition estimation has been investigated by many researchers and a variety of algorithms have been proposed for estimating chaotic signal [4][5][6][7][8][9]. These methods perform well in many applications, but none of them are suitable for the transmitted signal reconstruction for noise radar due to the inaccuracy or the high computational complexity. In this paper, we propose a new method for initial condition estimation, which is accurate enough and has low computational complexity. Then method is used to reconstruct the estimated signal.
In conventional noise radar scheme, the transmitted signal is first delayed by delay lines and then mixed with the received signal. When the transmitted signal is wideband signal, the delay process is hard to be realized [10]. To solve the delay lines problem, the author in [11] propose to use a low-loss linear phase shifter to delay the transmitted signal, but it is still a physical process, which requires expensive instruments on delay process and is not easy to realize. In this paper a new noise radar scheme, called noncoherent reception noise radar scheme, is proposed based on the initial condition estimation. In the new scheme, we use the initial condition estimation method to reconstruct the transmitted signal and use it to 978-1-4244-7371-7/10/$26.00 ©20 10 IEEE mix with the received signal. The proposed scheme avoids the physical delay process of the transmitted signal, which provides a new way to solve the wideband signal delay lines problem.
This paper is organized as follows. In section II we present the new chaotic signal initial condition estimation method and compare its performance with the other methods. In Section III we present the noncoherent reception noise radar system based on the piecewise method. Simulation results of range solution are also given to validate the theory analysis and to compare performance of the proposed system with that of the conventional noise radar system. Brief conclusion of this paper is drawn in Section IV.

CONDITION ESTIMATION
In this section we propose a new initial condition method which is a combination of the maximum-likelihood estimation method (MLEM) [5] and the symbolic-dynamics method (SDM) [6]. Here we use the one-dimensional (I-D) chaotic maps to validate its accuracy and computational complexity, since the I-D map is potentially useful and well understood. Our presentation here is mainly focused on I-D chaotic systems. However, the proposed method is not restricted to I-D systems. Next we will show the new method.
Let fO be a chaotic map defmed on some certain closed interval and let the initial condition X o lie . · ,N be a chaotic signal generated by f(·) and {yen)} be its noisy observation. That is where {w (n)}, n = 0,1"", N is a zero-mean additive white Gaussian noise (AWGN) process with variance a;. The estimation problem considered here is to estimate the initial condition 2). Using the MLEM to get the optimal value Xo over the small interval T.
where a is a small constant. It is shown in [6] that when using Eq. (2), the estimating error of SDM is about 2-N • If we choose an interval T' = [x� -2-N , x� + 2-N ] , then the estimation error is greater than 2-N and is even worth at low SNR. In order to improve the estimation accuracy, the interval should be longer than T'.
3).Repeat step 1 and step 2 L times to Here we can see that the if the computational complexity of MLEM is q, and the computational complexity of the proposed method is O2 , then O2 = aq . Chebyshev map is not topologically conjugate to the Tent map and HM is only viable to those maps which are topologically conjugate to the Tent map, the curve for the HM is not provided here. The comparison of CRLB which is computed by Eq.(17) in [12], the proposed method and SDM is shown in Figure.  To compare our method with the conventional noise radar scheme, we first give the basic schematic diagram of the conventional noise radar scheme which is illustrated in Fig.S. We notice that in the conventional noise radar system, the transmitted signal is delayed by the delay lines to mix with the received signal. The basic schematic diagram of the proposed noncoherent reception noise radar scheme is illustrated in Fig.9. We can see the main difference between the conventional noise radar system and the proposed radar system is that the later one uses the reconstructed transmitted signal by initial condition estimation instead of the transmitted signal replication delayed by delay lines to mix with the received signal. Thus it can avoid the difficult physical delay process.
In order to illustrate the effect of the proposed radar system, we do simulations on a stationary target which is nearly 10km (l0303m) far from the radar. We let the signal be chaotic signal generated by the Chebyshev map. The SNR is 20dB. The signal bandwidth is 200MHz, the sampling frequency is 1 GMHz and the pulse length is Ips .
So every pulse contains 1000 points. Here we divide the whole part into 125 parts. (The reason in doing so is to improve the estimation accuracy especially when the initial condition estimation is not exact. Since chaos has the deterministic characteristic, it can be predicted in short time).
The simulation result is in Fig.lO. It indicates that the target range information can be obtained. Here we also use the ideal conventional system to do the same simulation for comparison which is given in Fig.l2. We note that the proposed system has nearly the same effect in range solution as the ideal condition of the conventional noise radar system. Thus it offers a simple way to solve the delay lines problem when the signal band is wide.  4. CONCLUSION In this paper we developed an improved MLE method for estimating the initial condition of chaotic signal in noise. It is demonstrated to work effectively at low SNR environment. A novel method called piecewise estimation method is then proposed based on it. The piecewise estimation method develops a new way to reconstruct the chaotic signal.
Based on the piecewise estimation method a noncoherent reception noise radar system is developed. It uses the reconstructed signal to mix with the received signal. The simulation result for the new scheme has nearly the same effect as that of the ideal conventional noise radar system. This offers a simple way to solve the delay lines problem in the conventional noise radar system.