## 3 Signal Demodulation
### PLS-Based Detection Method
#### 3.2.2 Demodulation with PLS Regression
We quantify the multicollinearity of the simulated dataset by using condition indices (CIs) [58]:

\[CI_{i}=\sqrt{\frac{\lambda_{max}}{\lambda_{i}}} \tag{23}\]

where \(\lambda_{max}\) is the maximum eigenvalue of the symbol set, and \(\lambda_{i}\) is its \(i\)-th eigenvalue. [PERSON] et al. [59] suggest that when the value of CI is higher than 10, data dependencies are starting to affect the regression estimates. Figure 11 plots the condition indices of a 0.3-s symbol set, in which 30.42% of the elements are collinear with the others. Figure 10: Accuracy comparison results with bit error ratios: \(fs=4000\) Hz. (**a**) \(T=0.3\) s, \(k=40\); (**b**) \(T=0.6\) s, \(k=55\); (**c**) \(T=1\) s, \(k=70\). Unlike the reference algorithms, PLS models the fundamental relations between the training and response matrices by modeling their covariance structure. More precisely, a PLS model will try to find the multidimensional direction in the space of training matrix that explains the maximum multidimensional variance direction in the space of response matrix. The PLS family of methods is, therefore, particular suited for multicollinearity problem, providing better accuracy performance than the others. As for the BP networks, the matched filters achieve better accuracy than when the SNR is low. In our opinion, this is caused by the fact that the power of the noise components is higher than the transmitted signals within low SNR environment (\(SNR<0\) dB), whereas the given training examples are too sparse, resulting that the BP networks are drowned by the strong ambient noise and can hardly capture the useful feature information even if they are well converged. Yet, if the signal-to-noise ratio exceeds certain thresholds (in the case of this evaluation they are \(-1\), \(0\) and \(-2\) dB for the three datasets), the networks will provide better accuracy than matched filter. Additionally, the PLS models are trained by using the same datasets with the BP networks; it demonstrates, therefore, that incorporating variable relationship analysis into the regression problem can improve the capacity of the regressors in terms of handling the sparse training datasets. Finally, the accuracy performance of the matched filters significantly rises with the increasing of the symbol time length \(T\). For example, the bit error rates of the matched filters are around 14%, 11% and 0% at \(SNR=0\) dB. As mentioned in Section 5.2, the dinsguishability of the symbols varies in function of its length due to the time gain. According to our further test, with the same experimental protocol, both matched filter and PLS regression achieve the 0% BER at \(SNR=-5\) dB when \(T=1.5\) s. However, it should be noted that the communication rate is inversely proportional to \(T\), so the method of this paper is more appropriate when requiring a high communication rate. ### Practicability Analysis

Collecting high-density qualified underwater acoustic data is technically difficult for today's acoustic engineering. In this subsection, the environment and sample quality compatibility is evaluated to investigate whether the training datasets could be sparse to a certain extent. The evaluations are performed by testing the well-trained PLS models using the datasets with strange underwater sound channels and different SNRs. To do this, we have 50% of the testing symbols simulated via strange channels (not included in the training channels). The time interval between the two channel measurements is around 5 minutes, and their normalized correlation is around 0.8. The evaluation result is shown in Table 2, in which the rows correspond to the training datasets and the columns to the testing datasets. Figure 11: Condition indices of the 0.3-s symbol set.