## 2 Mathematical Model of Ship Radiation Noise Modulation Spectrum  
The ship radiated noise signal obtained by the passive sonar system can be approximated as  
\[s(t)=r(t)+q(t), \tag{1}\]  
where \(s(t)\) is the ship radiated noise signal, \(r(t)\) is the propeller noise with envelope modulation characteristics, and \(q(t)\) is the non-modulated component of the propeller noise as well as the radiated noise and environmental noise of other vibration systems of the ship. The signal-to-noise ratio can be defined as:  
\[SNR=20\log\frac{\int|r(t)|dt}{\int|q(t)|dt}\,\mathrm{dB}, \tag{2}\]  
For a ship with the \(M\)-blade propeller, one cycle in its propeller noise time-domain signal consists of \(M\) pulses, so it is divided into a group.  
The modulated signal envelope is assumed to be modeled as an impulsive stochastic process with a grouped structure, the same shape, the same repetition period, and a random amplitude [22; 23]. For a ship with the m-blade propeller, one cycle of propeller noise time-domain signal includes \(M\) pulses, so it is divided into a group. Assuming that the shaft frequency period of the ship is \(T\), the pulse repetition period is \(T/M\). The envelope function consisting of \(2N+1\) groups of pulses can be expressed as  
\[g(t)=\sum_{n=-N}^{N}\sum_{m=0}^{M-1}a_{nm}u(t-\frac{mT}{M}-nT), \tag{3}\]  
where \(a_{nm}\) is the amplitude of the \(m\)th pulse of the nth group, and \(u(t)\) is the expression of the pulse waveform. Assuming that the single pulse waveform is Gaussian [24,25], \(u(t)\) can be expressed as  
\[u(t)=\frac{1}{\sqrt{2\pi}}e^{\frac{-t^{2}}{2\pi^{2}}}, \tag{4}\]  
where \(\sigma\) is a parameter that affects the pulse width. \(a\) is a mutually independent random variable subject to a certain probability distribution. In the specific calculation, take the average value of the amplitude \(\overline{a}\), and \(a\) satisfies  
\[a\sim U(\frac{\overline{a}}{2},\frac{3\overline{a}}{2}), \tag{5}\]  
The propeller noise is a kind of cavitation noise, which is mainly generated by the propeller blades periodically cutting water resulting in the fragmentation of bubble groups. The signal is a non-stationary process [26], which can be regarded as the product of envelope modulation signal and stationary signal, which is  
\[r(t)=g(t)\cdot c(t), \tag{6}\]  
In the simulation of ship radiated noise, the bubble cluster noise is approximated as Gaussian noise [27]. The power spectrum analysis (Fourier transform) of propeller cavitation noise signal is performed with  
\[R(f_{n})=FFT(r(t)), \tag{7}\]  
where \(f_{n}=nf_{s}/N,n=0,1,\ldots,N-1\), \(f_{s}\) is the sampling frequency, \(N\) is the number of sampling points of a signal. Approximating the modulation spectrum of the ship radiated noise as the power spectral density of the propeller noise signal, then  
\[S(f_{n})=20\mathrm{lg}|R(f_{n})|-20\mathrm{lg}N, \tag{8}\]  
The modulation spectrum of ship radiation noise consists of a line spectrum and a continuous spectrum [28], which is different from the power spectrum in that the line spectrum in the modulation spectrum has the harmonic relationship and group structure, in which the frequency corresponds to the fundamental line spectrum is the ship shaft frequency.