## 2 Communication System Model and Theory
### Correlation Cost Factor-Based Updates to Matched Filters Reception Method for M-SS (CCF-MSS)
#### 2.3.3 Theoretical Derivation of the Receiver Decoding Process  
The following presents the theoretical derivation of the decoding process for the \(i\)-th. The frequency offset value output after evaluating the correlation cost factor is denoted as \(f_{cs_{i}t,i}=f_{cs_{i}t,i-1}+k_{i}{}^{*}\cdot df\). Here, \(i\) denotes the \(i\)-th symbol (where \(i\geq 2\)), \(f_{cs_{i}t,i-1}=f_{c}{}^{*}+\sum_{i=2}^{i}k_{i-1}{}^{*}\cdot df\), \(f_{cs_{i}t,i-1}\) representing the CFO value transmitted from the \((i-1)\)-th, \(k_{i-1}^{*}\cdot df\) is the selected CFO value during the compensation process for the \(i\)-th symbol. Next, we employ a non-coherent demodulation method. First, two local carrier signals are generated at the receiver, \(y_{c}=\Re\big{\{}\ e^{j2\pi f_{c}t}\ \big{\}}\) and \(y_{s}=\Im\big{\{}\ e^{j2\pi f_{c}t}\ \big{\}}\), which are then used to down-convert the received signal, followed by passing through a low-pass filter (LPF):  
\[ \begin{split}\tilde{r}_{c}(t)&=LPF(r(t)\Re\Big{\{}e ^{j2\pi f_{cs_{i}t}t}\Big{\}})\\ &=\frac{1}{2}\Re\left\{\sum_{i=1}^{\infty}\sum_{q=0}^{Q-1}A_{q,i} (t)a_{i}[0]c_{i,h}e^{i\phi_{q}(t)}\right\}+w(t)\end{split}  \tag{13}\]  
Here, the carrier phase shift \(\phi_{q,i}(t)=2\pi(f_{cs_{i}t,i}t-f_{c}\tau_{q,i}(t))\). This includes the time-varying phase of the \(q\)-th path, where \(w(t)\) represents the baseband noise. When the dominant Doppler shift is caused by large-scale movement of the receiver, it can be assumed that for a single symbol, all paths have equal Doppler factors [29], and thus \(\tau_{q,i}(t)=\tau_{i}(t)\ \phi_{q,i}(t)=\phi_{i}(t)\ q=[0,Q-1]\). In this paper, considering the underwater acoustic communication process under the conditions of low SNR and significant time-varying Doppler shifts, the energy of the main path is significantly greater than that of other paths; therefore, the energy of other paths is treated as noise in this study, and after normalizing the impulse response amplitude of the main path, we have the following:  
\[ \begin{split}\tilde{r}_{c}(t)&=\Re\left\{\sum_{i= 1}^{\infty}a_{i}[0]c_{i,h_{i}}(t-\tau_{i}(t)-iT_{s})e^{i\phi_{i}(t)}\right\}+ w(t)\\ &=\sum_{i=1}^{\infty}a_{i}[0]c_{i,h_{i}}(t-\tau_{i}(t)-iT_{s}) cos(\phi_{i}(t))+w(t)\end{split}  \tag{14}\]Similarly, the signal from the other path is as follows:  
\[\tilde{r}_{s}(t)=\sum_{i=1}^{\infty}a_{i}[0]c_{i,b_{i}}(t-\tau_{i}(t)-iT_{s})sin( \phi_{i}(t))+w(t) \tag{15}\]  
where \(\phi_{i}(t)=2\pi(({f_{c}}^{*}+\sum_{i=2}^{i}{k_{i-1}}^{*}\cdot df)t-f_{c}\tau_{q,i}(t))\) represents the difference between the estimated carrier phase shift and the actual carrier phase shift, referred to as the residual carrier phase shift error. At this point, \(t\in[0,T_{s}]\). Since the symbol duration is typically very short, we assume that the carrier phase shift remains constant within a single symbol, and thus, we set \(t=T_{s}\) to control the time variable. If the Doppler tolerance is set to \(10^{-4}\), then in this paper, \(df\approx 0.7\) Hz. After constraining \(t\) and \(df\), we consider \({k_{i}}^{*}\) as a continuous variable, and thus, \(\phi_{i}(t)\) should be expressed as \(\phi_{i}(T_{s},k,df)\), setting \(\phi_{i}(T_{s},k,df)=0\); we obtain the following:  
\[\sum_{i=1}^{i}{k_{i}}^{*}=(\frac{f_{c}\tau(T_{s})}{T_{s}}-{f_{c}}^{*})\frac{1} {df} \tag{16}\]  
It is evident that the equality holds consistently only when the value of \({k_{i}}^{*}\) changes with \(\tau_{i}(T_{s})\). In this paper, we accurately update the value of \({k_{i}}^{*}\) based on the correlation cost factor, striving to maintain equality and minimize \(\phi_{i}(t)\). Assuming that \(\phi_{i}(t)\) has been minimized, it still exerts some influence on the despreading process. Therefore, we next employ a non-coherent down-conversion method to eliminate \(\phi_{i}(t)\), which will not result in significant energy loss for \(\phi_{i}(t)\)[30].