# Underwater Target Recognition Based on Multi-Decision
## 2 System Model
### Signal Decomposition Algorithm Based on Resonance
#### 2.1.2 Adjustable Q-Factor Wavelet Transform  
TQWT is a discrete wavelet transform that can flexibly adjust the constant \(Q\) factor according to the resonance of the processed signal, which has an overcomplete basis and can be perfectly reconstructed [30]. This section uses the TQWT toolbox to complete simulation experiments and signal processing. The implementation framework consists of two filter banks which are analysis filter bank and integrated filter bank. They are shown in Figures 3 and 4. A filter bank refers to a group of filters. They have a common input, or a common output. The analysis filter bank has a common input to obtain multiple subband signals. In contrast, the integrated filter has a common output, combining multiple subband signals into a single signal to restore the original analyzed signal.  
The analysis filter bank of each layer is composed of high-pass filter \(H_{high}(w)\), low-pass filter \(H_{low}(w)\), and the corresponding scaling process, which are defined as follows:  
\[H_{high}(w)=\left\{\begin{array}{ll}0&|w|\leq(1-\beta)\pi\\ \theta(\frac{\alpha\pi-w}{\alpha+\beta-1})&(1-\beta)\pi\leq w\leq\alpha\pi\\ 1&\alpha\pi\leq|w|\leq\pi\end{array}\right., \tag{7}\]  
\[H_{low}(w)=\left\{\begin{array}{ll}1&|w|\leq(1-\beta)\pi\\ \theta(\frac{w+(\beta-1)\pi}{\alpha+\beta-1})&(1-\beta)\pi\leq w\leq\alpha\pi \\ 0&\alpha\pi\leq|w|\leq\pi\end{array}\right.. \tag{8}\]  
\(\theta(w)=0.5(1+cos(w))\sqrt{2-\cos(w)}\) is the Daubechies filter with second-order disappearing moment [27]. \(\alpha\), \(\beta\) (\(0<\alpha<1,0<\beta<1\)) are the scaling factors after the signal passes through the low-pass and high-pass filters, respectively. The scaling process of low-pass and high-pass are defined as:  
\[Y(w)=X(\alpha w),|w|\leq\pi, \tag{9}\]  
Figure 4: Integrated filter bank.  
Figure 3: Analysis filter bank.  
\[Y(w)=\left\{\begin{array}{ll}X(\beta w+(1-\beta)\pi)&0\leq w\leq\pi\\ X(\beta w-(1-\beta)\pi)&-\pi<w<0\end{array}\right.. \tag{10}\]  
The \(Q\) factor quantifies the degree of signal resonance, and its definition is \(f_{c}/BW\), where \(f_{c}\) represents the center frequency of the signal and \(BW\) represents the bandwidth.  
If the sampling frequency of the original input signal is \(f_{s}\), then the center frequency \(f_{c}\), the filter bank level \(j\) and \(\alpha\), \(\beta\)[31] can be expressed as:  
\[f_{c}=\alpha^{j}\frac{2-\beta}{4\alpha}f_{s}. \tag{11}\]  
Similarly, bandwidth \(BW\) can be expressed as:  
\[BW=0.5\beta\alpha^{j-1}\pi. \tag{12}\]  
Therefore, the \(Q\) factor is derived as:  
\[Q=\frac{2-\beta}{\beta}. \tag{13}\]  
After the original signal passes through the filter bank, the output of the low-pass channel is iteratively inputted to the deeper level filter bank until the preset level \(J\). At the same time, the wavelet basis functions \(\Phi_{h}\), \(\Phi_{l}\) are constructed by selecting the oversampling rate \(r\). The deepest level \(J_{\max}\) and the oversampling rate \(r\) are defined as follows:  
\[r=\frac{\beta}{\alpha+1}, \tag{14}\]  
\[J_{\max}=\bigg{|}\frac{\log(\beta N/8)}{\log(1/\alpha)}\bigg{|}. \tag{15}\]  
In summary, in the TQWT algorithm, \(Q\), \(r\), \(J\) can be calculated by selecting \(\alpha\), \(\beta\), and \(\alpha\), \(\beta\) selection is only determined by the inherent oscillation characteristics of the signal. Therefore, it can flexibly select \(\alpha\), \(\beta\) according to the specific requirements of \(Q\), \(r\), \(J\). For the input signal of ship radiated noise, we need to set \(Q_{h}\), \(r_{h}\), \(J_{h}\) to extract its high-resonance information and set \(Q_{l}\), \(r_{l}\), \(J_{l}\) to extract its low-resonance information.