## 3 Signal Demodulation
### PLS-Based Detection Method
#### 3.2.2 Demodulation with PLS Regression
Comparing the diagonal of Table 2 with the simulation results of Figure 10b which trains and tests with the same sound channel, we can see that a strange channel somehow reduces the accuracy performance of the demodulator, but this impact is tiny, and the demodulator reaches the BER of \(0\%\) at \(SNR=-4\) dB, which is still much better than either BP networks or matched filters. This demonstrates that the proposed method has the ability to resist the channel variant. Meanwhile, both of the SNRs of the training and testing datasets positively affect the accuracy of the demodulator. Poor training data do not necessarily result in high BERs, and vice versa. Hence, the performance of our demodulator is not seriously affected by the gap between the training and testing datasets, implying that the training data can be collected with few constraints. ### Temporal Efficiency Evaluation

This subsection evaluates the temporal efficiency performance of the proposed demodulator by comparing it with the matched filter and BP network. This paper theoretically analyzes the efficiency performance of all the algorithms with asymptotic time complexity (also known as time complexity) [60]:

Figure 8:0

where \(\mathbf{n}\) is the scale of the problem to be solved and \(f(\mathbf{n})\) is a function having the same order of magnitude with \(\mathcal{T}(\mathbf{n})\), which makes their ratio \(\eta\) is a non-zero constant. The pseudocode of the demodulation scheme shown in Figure 3 is given in Algorithm 2, in which Line 7 is replaced by using the matched filter or BP network to perform the comparisons. The time complexity of this demodulation scheme \(\mathcal{T}_{total}\) can be expressed as:

Figure 8:1

with

Figure 8:2 Figure 8:3 (27) Figure 8:4

where \(\mathcal{T}_{FIR}\), \(\mathcal{T}_{norm}\) and \(\mathcal{T}_{Dce}\) are the time complexities of FIR filter, normalization operation and decision function, \(\mathcal{T}_{*}\) with \(*=``PLS\"\), \(``MF\"\) or \(``BP\"\) is the time complexity of the selected demodulator, \(n_{ord}\) is the order of the FIR filter, \(m\) is the symbol number of the input message, \(n=256\) is the number of symbol candidates, and \(f_{s}=4000\) Hz is the sampling rate. \begin{table}
\begin{tabular}{r r r r r r r r r r r r} \hline \hline
**BER** & **SNR** & \(-10\) & \(-9\) & \(-8\) & \(-7\) & \(-6\) & \(-5\) & \(-4\) & \(-3\) & \(-2\) & \(-1\) & \(0\) \\ \hline \(-10\) & 44.61 & 33.57 & 22.77 & 21.21 & 13.78 & 8.98 & 6.91 & 4.52 & 4.52 & 3.98 & 2.76 \\ \(-9\) & 30.78 & 25.09 & 10.32 & 9.87 & 6.31 & 5.92 & 3.33 & 2.51 & 0.91 & 0.69 & 0.54 \\ \(-8\) & 27.14 & 14.12 & 9.75 & 5.01 & 3.82 & 2.01 & 1.21 & 0.82 & 0.51 & 0.42 & 0.21 \\ \(-7\) & 15.34 & 7.82 & 4.98 & 4.57 & 1.27 & 0.98 & 0.39 & 0.26 & 0.13 & 0.00 & 0.00 \\ \(-6\) & 10.52 & 4.21 & 2.61 & 1.55 & 0.26 & 0.13 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \(-5\) & 6.45 & 3.13 & 1.27 & 0.52 & 0.26 & 0.13 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \(-4\) & 4.33 & 1.01 & 0.13 & 0.13 & 0.13 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \(-3\) & 3.78 & 0.26 & 0.13 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \(-2\) & 2.21 & 0.51 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \(-1\) & 1.88 & 0.31 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \(0\) & 0.91 & 0.26 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: BER (%) between the training and testing datasets with different underwater sound channel response and SNRs (dB): \(T=0.6\) s, \(k=55\) and \(f_{s}=4000\) Hz, the rows correspond to the training datasets and the columns to the testing datasets. For each symbol, according to the algorithms presented in Section 3, the PLS model runs in polynomial time (An algorithm is said to be of polynomial time if its running time is upper-bound by a polynomial expression in the size of the input for the algorithm, i.e., \(mathellT(\approx\sim\approx\propto\propto n)=O(\approx\sim\propto n^{c})\) for some positive constant \(\kappa\)[(6,61)].