## 3 Signal Demodulation
### PLS-Based Detection Method
#### 3.2.2 Demodulation with PLS Regression
This criterion has been widely used to evaluate the classification performance in pattern recognition field. Figure 9 plots the recognition rate curves of the three dataset versions over \(k\). As expected, the recognition rate raises with the increasing of the \(k\) linearly at the beginning then slows down. It can be seen that the curves of \(T=0.3\) and \(T=0.6\) trend towards to stability when \(k=40\), whereas \(T=1\) when \(k=65\). Meanwhile, the optimal recognition rates are obtained when \(k\) is 42, 55 and 70 for the three dataset versions, respectively. That is, the optimal recognition rates are 90.36%, 92.58% and 99.74% when \(T\) is equal to 0.3, 0.6 and 1 s with \(SNR=-8\) dB. Additionally, it is found also that the dataset with larger symbol time length leads to better accuracy. This is because the larger \(T\) is, the more time gains can be achieved in the slow-time-varying environment. This experiment demonstrates that the symbols with the period of 1 s lead to an accuracy of 99.74%, to nearly 100%, in a low signal-to-noise ratio environment (\(SNR=-8\) dB). Figure 9: _Cont._

### Demodulation Accuracy Evaluation

The experiment of this subsection evaluates the accuracy performance of the proposed method by comparing it with two reference algorithms, including traditional matched filter [7] and a three-layer back propagation (BP) neural network [56]. Bit error rate is used as the criterion of accuracy performance. Both of the reference algorithms can be directly inserted into the scheme described in Section 3.1. We compare the three demodulators with all of the three versions of the datasets: three of the six sessions for training, and the rests for testing. That is, for each data version, either the training or testing dataset has 256 symbol candidates \(\times\) three sessions \(=768\) simulated symbols. For the purpose of unbiased evaluation, the simulated symbols are randomly divided into two groups. The average bit error rate of three repetitions with different training and testing data is used as the evaluation criterion of the demodulation accuracy performance. Figure 10 compares the accuracy performance between the proposed method and the reference algorithms with the three versions of datasets. First of all, we can see that PLS regression always possesses the lowest bit error rates. For the three dataset versions, when SNR is around \(-2\), \(-6\) and \(-6.5\) dB, the BERs of PLS reach 1.0%, whereas the matched filter and BP network are 22.24%, 36.05%, 6.44%, and 16.79%, 40.33%, 39.87%, respectively. Additionally, PLS achieves an error-free demodulation results with the three datasets at \(SNR=0\), \(-5\) and \(-5\) dB, out of the available range of the y-axis with logarithmic scale. This demonstrates that the proposed method has the best accuracy performance with the experiment protocol of this paper. Figure 10: _Cont._

Figure 9: Recognition rate over the number of projection directions: \(SNR=-8\) dB and \(fs=4000\) Hz. (**a**) \(T=0.3\) s; (**b**) \(T=0.6\) s; (**c**) \(T=1\) sIn our case, the communication symbols are modulated with multi-segment piecewise linear mapping, which are essentially a set of pseudo-random sequences, implying that the predictors (elements of the symbol vector) are somehow correlated, resulting in a multicollinearity problem. The multicollinearity problem is a phenomenon where one predictor variable in a multiple regression model can be linearly predicted from the others with a substantial degree of accuracy, resulting in model distortions [57]. Mathematically, a set of variables is perfectly multicollinear if one or more exact linear relationships exist among some of the variables:

\[\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\cdots+\beta_{i}x_{i}=0 \tag{22}\]

where \(\beta_{i}\) are constants and \(x_{i}\) is the \(i\)-th element of the symbol vector \(\mathbf{u}_{n}^{\prime}\).