## 4 Results
### Simulation Tests  
The simulation tests were performed using the Watermark simulator. It is a freely available benchmark for physical-layer schemes for underwater acoustic communications [18]. Its core is a replay channel simulator driven by at-sea measurements of the time-varying impulse response. Three of the communication channels available in Watermark were selected, represented by impulse responses measured in: Norway-Oslofjord (NOF1), Norway-Continental Shelf (NCS1), and Brest Commercial Harbor (BCH1). The NOF1 channel was measured at a range of 750 m and the water depth was 10 m. The probe signal frequency band was 10-18 kHz. The NCS1 channel was measured at a range of 540 m and the water depth was 80 m. The probe signal frequency band was the same as in the case of NOF1 channel measurements. Finally, the BCH1 channel was measured at a range of 800 m, where the water depth was 20 m. The frequency band of the probe signal was 32.5-37.5 kHz. The transmitter was bottom-mounted in the case of NOF1 and NCS1, and suspended in the water column in BCH1. The receiver was a bottom-mounted hydrophone (NOF1, NCS1) or a vertically suspended hydrophone array (BCH1). A more detailed description of Watermark channels can be found in [18]. Figure 2 shows the absolute values of exemplary impulse responses for each channel. During the simulation tests performed, a PRBS signal was transmitted in the NOF1, NCS1, and BCH1 channels. For the construction of the PRBS signal, m-sequences of order 8 and 10 were used. The spectrum of the signal was in the 10-18 kHz band for the NOF1 and NCS1 channels, and 32.5-37.5 kHz for the BCH1 channel. The sampling frequency was equal to 200 kHz. Sixty transmission tests were carried out for each configuration of the signal and channel parameters. During a single transmission test, the probe sequence was repeated 32 times. On the basis of the received signals, after performing a matched filtration, TVIR estimates and the corresponding coefficients \(\hat{c}^{\mu}_{IQ}\) and \(\hat{c}^{\mu}_{IQQI}\) were determined. Next, the complex envelope \(y(t)\) of each of the received signal was divided into segments of a duration corresponding to the duration of a single PRBS sequence. The coefficients \(c_{IQ}\) and \(c_{IQQI}\) were calculated as mean values over all segments of the probe signal. The minimum value of the coefficient is 0, and the maximum is 1. It is assumed that a value of the cross-coefficient above 0.5 means a strong similarity, and below 0.5 a weak similarity. This is consistent with the assumption of a stochastic model of the wireless communication channel, that two received signals are similar if their correlation coefficient is equal or greater than 0.5. This rule applies, inter alia, when calculating coherence time and the coherence bandwidth of the channel as a time interval and a frequency shift, respectively, for which the correlation coefficient of such delayed or frequency-shifted signals is equal or greater than 0.5 [19]. It is reasonable to assume that if values of both cross-coefficients: \(c_{IQ}\) and \(c_{IQQI}\) are greater than 0.5, the channel can be described as a WSS process. Figure 2: The module of exemplary time-varying impulse responses of Watermark channels. In order to compare the coefficients obtained for the TVIR and for the complex envelope of the received signal, a euclidean distance measure \(d\) was calculated as:  
\[d=\sqrt{(\bar{c}_{IQ}^{h}-c_{IQ})^{2}+(\bar{c}_{IQQI}^{h}-c_{ IQQI})^{2}} \tag{10}\]  
It has a meaning of error of estimation of \(c_{IQ}\) and \(c_{IQQI}\) parameters, assuming that \(\bar{c}_{IQ}^{h}\) and \(\bar{c}_{IQQI}^{h}\) are the reference values. The distance \(d\) was averaged over 60 transmission tests for each channel. Table 1 presents the values of the coefficients calculated on the basis of TVIR, the coefficients obtained on the basis of the envelope of the PRBS signal, and values of distance measure \(d\).