## 3 Transmission Parameters of Shallow-Water Channel
Then, it is possible to calculate the 2-dimensional scattering function:  
\[S(\
u,\tau)=\int_{\Delta t}\int_{\Delta f}R_{h}(\Delta t,\Delta f)e^{-j2\pi( \
u\Delta t-\tau\Delta f)}d\Delta td\Delta f \tag{4}\]  
where \(\Delta t\) and \(\Delta f\) are time and frequency differences, respectively, \(\
u\) is the Doppler spread, \(\tau\) is the delay, and \(R_{h}(\Delta t,\Delta f)\) is the STFCF of the channel. An example module of TVIR and corresponding scattering function are shown in Figure 4. The probable reflections of the transmitted signal are seen as the multipath components of TVIR. Figure 3: Sound speed profile measured during inland water experiment. Figure 2: Localisation of inland water experiment and hardware setup. The scattering function \(S(\
u,\tau)\) is the basis for the calculation of the transmission parameters: delay spread \(\tau_{M}\), Doppler spread \(\
u_{M}\), coherence time \(T_{c}\), and coherence bandwidth \(B_{c}\), which are used for designing the physical layer of the UAC data transmission system [8,12]. The delay spread is calculated on the basis of Power Delay Profile (PDP) \(P(\tau)\), which is obtained as the integral of \(S(\
u,\tau)\) over the Doppler shift \(\
u\) domain. It describes the average signal power reaching the receiver as a function of delay \(\tau\) and it characterizes the time dispersion of the channel. The time dispersion can be assessed as a maximum delay spread \(\tau_{M}\) measured at a given threshold level \(T_{r}\) of \(P(\tau)\), or as an rms value \(\tau_{rms}\):  
\[\tau_{rms}=\sqrt{\tau^{2}-\tau_{m}^{2}} \tag{5}\]  
where:  
\[\tau^{2}=\frac{\sum_{k}P(\tau_{k})\tau_{k}^{2}}{\sum_{k}P(\tau_{k})},\quad \tau_{m}^{2}=\frac{\sum_{k}P(\tau_{k})\tau_{k}}{\sum_{k}P(\tau_{k})}\]  
The values of delay spread averaged over the results of the analysis of the 16 measured impulse responses are as follows. The maximum delay spread was measured as a time duration between the first and last multipath component of a value higher than a threshold level of 0.1 or 0.01 of a maximum value of \(P(\tau)\)), which corresponds to a decrease of 10 dB or 20 dB from the maximum value, respectively. In the case of a threshold level of 0.1, the maximum delay spread was equal to 8.67 ms, and in the case of a threshold level of 0.01, it was equal to 27.13 ms. The rms value of the delay spread was equal to 17.12 ms. These are values typical for shallow-water channels with multipath propagation. The integral of \(S(\
u,\tau)\) over the delay \(\tau\) domain gives the Doppler Power Spectral Density (DPSD) \(P(\
u)\). The Doppler spread of the transmitted signal is calculated similarly to the delay spread, that is, as a maximum Doppler spread \(\
u_{M}\) or rms Doppler spread \(\
u_{rms}\). The maximum Doppler spread measured as a width of \(P(\
u)\) at a threshold level of 0.1 of its maximum value was equal to 2.23 Hz. Changing the threshold level to 0.01 of the maximum value of \(P(\
u)\) results in \(\
u_{M}\) equal to 3.5 Hz. The rms value of the Doppler spread was equal to 0.83 Hz. This is a small value of Doppler spread caused by the slow drift of the boat with the transmitting transducer. Assuming a carrier frequency of the transmitted signal equal to 30 kHz, a Doppler shift of 3.5 Hz corresponds to a motion of 0.17 m/s. The exemplary PDP and DPSD, calculated on the basis of the impulse response presented in Figure 4, are shown in Figure 5.  
After averaging \(R(\Delta t,\Delta f)\) over the \(\Delta t\) and \(\Delta f\) domain, Space-Frequency Correlation Function (SFCF) \(R(\Delta f)\) and Space-Time Correlation Function (STCF) \(R(\Delta t)\) are obtained (Figure 6). The coherence bandwidth \(B_{c}\), in which the signal amplitude spectrum is flat and its phase characteristic is linear, is calculated as the width of \(R(\Delta f)\) at a given threshold level \(T_{r}\). The threshold level is  
Figure 4: Module of TVIR (**a**) measured at a distance of 1035 m and corresponding scattering function (**b**). usually equal to 0.5 or 0.9 in the case of OFDM radiocommunication systems [8].