# Research on Passive Ranging Technology of Moving Ship Based on Vertical Hydrophone Array
## 2 Basic Theory
### DEMON Spectrum Analysis for A Single Hydrophone  
As mentioned above, the DEMON spectral analysis is widely used in the feature extraction of ship-radiated noise. Generally speaking, the high-frequency band of the vessel-radiated noise has a periodic modulation, and the modulation frequency is composed of low-frequency characteristic signals such as the shaft frequency and blade frequency. The shaft frequency and blade frequency contain the information on the propeller speed and blade number. Normally, the ship-radiated noise is a periodic and locally stable process, whose expression in the time domain can be expressed as:  
\[x(t)=[1+f(t)]\cdot g(t) \tag{1}\]  
where \(f(t)\) represents the modulation signal and is a low-frequency, slowly varying periodic function. \(g(t)\) is the carrier frequency and is a stable White Gaussian random process with a limited bandwidth. \(x(t)\) is a periodic stationary stochastic Gaussian process for ship-radiated noise. The modulation signal \(f(t)\) can be obtained by demodulating the ship noise \(x(t)\). This paper adopts the absolute value demodulation technology of the incoherent demodulation method. The specific flow diagram has been presented in Figure 2.  
The process includes four steps: band-pass filtering, the operation of taking the absolute value, the low-pass filter, the spectrum analysis.  
We take a simple situation as an example to explain the operation principle. In this case, the amplitude of the high-frequency carrier is modulated, and the carrier signal is a single frequency signal. The amplitude modulation (AM) signal is expressed as:  
\[x(t)=A_{m}\cos(2\pi f_{m}t)[1+A_{c}\cos(2\pi f_{c}t)] \tag{2}\]  
where \(A_{m}\) and \(A_{c}\) are the amplitude of the carrier signal and modulation signal, respectively. Moreover, we assume that \(A_{m}\) and \(A_{c}\) are both greater than one. \(f_{m}\) and \(f_{c}\) are the frequency of the carrier signal and modulation signal, respectively.  
Taking the absolute value of Equation (2), the AM signal can be written as:  
\[\left|x(t)\right|=A_{m}\big{|}\cos(2\pi f_{m}t)\big{|}1+A_{c}\cos(2\pi f_{c}t) \big{|} \tag{3}\]  
Then, the content of the absolute value is expanded into a Fourier series, that is:  
\[\left|\cos(2\pi f_{m}t)\right|=A_{0}+A_{1}\sin(2\pi f_{m}t)+A_{2}\sin(2\pi\cdot 2 f_{m}t)+\cdots \tag{4}\]  
\[\left|1+A_{c}\cos(2\pi f_{c}t)\right|=B_{0}+B_{1}\sin(2\pi f_{c}t)+B_{2}\sin(2 \pi\cdot 2f_{c}t)+\cdots \tag{5}\]  
Figure 1: Flow diagram of the whole process.  
Figure 2: Flow diagram of the DEMON spectrum analysis for a single hydrophone.  
where \(A_{n}\) and \(B_{n}\) are the constant coefficients. The multiplication of Equations (4) and (5) will produce various harmonic components. After passing through the designed low-pass filter, only the modulated signal remains. Then, the spectrum analysis of the low-frequency modulation signal is carried out after a Fourier transform.  
According to the spectrum distribution, we extract the line spectrum components representing the characteristics of the ship for the next step of the ranging analysis.