## 1 Introduction  
The underwater acoustic communication has always been a crucial research topic [1; 2; 3; 4; 5; 6]. In some special applications, high-performance communication is required, resulting that the system must satisfy the constraints of reliability, security and efficiency simultaneously. The direct-sequence spread spectrum (DSSS) communication is one of the most effective potential solutions to the problem of confidential underwater acoustic communication. It spreads the frequency spectrum of the carrier wave with a spreading code sequence, so the modulated signal is hard to be detected by a third party within the underwater noise, possessing greater concealment [7; 8]. Conventional DSSS technique is used to modulating the carrier wave with pseudo-noise (PN) sequences, such as m-, Gold and Kasami sequences [9; 10; 11]. However, these PN sequences can only provide finite keysings and limited code resources, (e.g., binary phase-shift modulation keying, quadrature amplitude modulation, frequency shift keying), so the transmitted DSSS signals usually possess binary-value and periodic characters. Estimating the modulation parameters from the signals of this type is not very hard even with the blind methods [12; 13; 14; 15]. For example, Figure 1a plotsthe power spectrum density reprocessing \(\mathsf{S}_{rep}\) of the m-sequence spread spectrum signal computed as follows:  
\[\mathsf{S}_{rep}(g)=|\mathcal{F}[\mathsf{S}(f)]|^{2}=|\int_{-\infty}^{+\infty} \mathsf{S}(f)e^{-i2\pi gf}df|^{2} \tag{1}\]  
where \(\mathsf{S}\) is the power spectrum of the received signal, and \(\mathcal{F}\) refers to the Fourier Transform. Due to the periodicity of m-sequence, regular spikes occur at the integer multiple of the sequence period along the time axis, the period of the transmitted signals can be, therefore, accurately estimated by measuring the interval width between two spikes [15]. Over the recent twenty years, chaos theory has been incorporated into the conceptions of PN sequences for the benefits of broad band, irregular complexity and orthogonality [16; 17; 18]. Usually, chaos (deterministic chaos) refers to irregular, unpredictable behavior in deterministic, dissipative, and nonlinear dynamical systems [19]. Within DSSS, it can be considered as a kind of disorder sensitive to initial conditions, allowing to generate a large number of orthogonal chaos sequences by setting different initial values. This method is helpful to overcome the binary-value and periodic problems of old code sequences. As the simulation results shown in Figure 1b, adopting chaos sequence significantly increases the difficulties of the parameter retrieving. Up till now, a series of ingenious chaos maps have been proposed (e.g., logistic [20], tent [21], cubic [22], Chebyshev [23], Bernoulli [24], multi-segment piecewise linear maps [25]), considerably improving communication security. Despite of multiple advantages, adopting chaos spread spectrum into real-life underwater acoustic communication is far from easy. The de-spreading and demodulation of chaos spread spectrum signals at the receiver is challenged by the signal distortions caused by time and space variations of underwater environments, reverberation, multi-path effect, low signal-to-noise ratio, etc. [26]. Matched filter is one of the most widely used demodulators in underwater acoustic communication [27]:  
\[\mathsf{s}_{out}(t)=\int_{-\infty}^{+\infty}\mathbf{H}(f)\mathsf{S}(f)e^{i2\pi ft}df \tag{2}\]  
where \(\mathsf{s}_{out}\) is the instantaneous output of the filter, \(\mathbf{H}\) is its transfer function and \(\mathsf{S}\) is the spectrum density function of the given signal \(\mathsf{s}\). When the channel impairment is only the white gaussian noise, according to the Cauchy-Bunyakovsky-Schwarz inequality, the optimal matched filter is, therefore, \(\mathbf{H}(f)=\alpha\mathsf{S}^{*}(f)e^{-2\pi ft_{o}}\), where \(\alpha\) and \(t_{o}\) are, respectively, a constant value and the time delay, implying that its impulse response is actually the replica of \(\mathsf{s}\).