## 3 Designing the UWAC Channel Emulator
Here, we consider a combination of a pulse-shaping transmit filter with an impulse response \(g(t)\), the propagation channel \(|\mathbf{M}|(t)\), and a matched receive filter, to represent a communication system with complex transmit samples \(x_{i}\) and an output signal \(z(t)\). If \(z(t)\) is sampled at intervals that are multiples of \(T\), or the sampling time \(T_{s}\), the overall communication system will embody a discrete linear time-varying filter consisting of \(n=1,2,\ldots,N\) filter taps. Therefore, such a communication system can be exemplified as a tapped delay line filter model. The filter taps, \(h_{n}(tT_{s})\), can be generated using digital filtering that has a sampling rate capable of modelling a usable channel bandwidth for the channel model of (27). These filter taps can be characterised by the following equation,  
\[\mathbf{M}(\tau,t,f)=\sum_{q=1}^{Q}|\mathbf{M_{q}}|\gamma_{\mathbf{q}}(f,t) \delta(\tau-\tau_{\mathbf{q}}(t)) \tag{18}\]3 \[\mathbf{M}(\tau,t,f)=\sum_{q=1}^{Q}|\mathbf{M_{q}}|\gamma_{\mathbf{q}}(f,t) \delta(\tau-\tau_{\mathbf{q}}(t)) \tag{18}\]4  
where \(\tau_{0}\) is the initial delay and \(|\mathbf{M}_{q}|\) are the average power levels observed over \(Q\) path bundles. If we want to account for the drift velocity \(v\) between the transmit and the receive terminals in an underwater environment, (28) is modified to  
\[\mathbf{M}(\tau,t,f)=\sum_{q=1}^{Q}|\mathbf{M_{q}}|\gamma_{\mathbf{q}}(f,t) \delta(\tau-\tau_{\mathbf{q}}(t)) \tag{18}\]5 \[\mathbf{M}(\tau,t,f)=\sum_{q=1}^{Q}|\mathbf{M_{q}}|\gamma_{\mathbf{q}}(f,t) \delta(\tau-\tau_{\mathbf{q}}(t)) \tag{18}\]6  
where \(t+vt/c=(1+a)t\), \(c\sim 1500\) m/s, and \(a\) is the Mach number. The concept of the Mach number is introduced to represent the time-stretching or Doppler shift, i.e., the time period \(t\) expands to \((t+vt/c)\). Here, \(N\) is an important factor in the emulator design as it offers a trade-off between computational complexity and model stationarity. A sample set of trajectories generated over a 3-s interval and \(L=1000\), \(K=3\), \(\xi=1.2\) is presented in \[\mathbf{M}(\tau,t,f)=\sum_{q=1}^{Q}|\mathbf{M_{q}}|\gamma_{\mathbf{q}}(f,t) \delta(\tau-\tau_{\mathbf{q}}(t)) \tag{18}\]7 We also include the corresponding Doppler frequency distribution, Doppler power spectrum, and temporal coherence in \[\mathbf{M}(\tau,t,f)=\sum_{q=1}^{Q}|\mathbf{M_{q}}|\gamma_{\mathbf{q}}(f,t) \delta(\tau-\tau_{\mathbf{q}}(t)) \tag{18}\]8 Seldom do practical measurements from trials align with the WSSUS assumption for underwater propagation channels. However, in our work, we derive the amplitude of paths/echoes  
\[\mathbf{M}(\tau,t,f)=\sum_{q=1}^{Q}|\mathbf{M_{q}}|\gamma_{\mathbf{q}}(f,t) \delta(\tau-\tau_{\mathbf{q}}(t)) \tag{18}\]9 Histogram of Doppler frequencies generated using (25) and (26) for different stretching exponents, \(\xi\), and different values of the closeness of the fit parameter, \(\chi\). of travelling sound waves, the delays experienced, and other relevant parameters directly from fundamental principles of physics. These concepts are derived from a mathematical description of how acoustic waves propagate through water and how water interacts with and influences the flow of acoustic waves. Therefore, it is possible to identify specific time periods and bandwidths within which the WSSUS assumption remains valid [22]. Accordingly, it is possible to determine an observation time \(T_{d}\) in which the autocorrelation function of \(|\mathbf{M}|(t,\tau)\), \(\mathcal{R}_{\mathbf{M}}(t,\Delta\tau)\) is stationary with the constant mean value. Moreover, we can define an observation bandwidth \(B_{d}\), where the frequency autocorrelation function, \(\mathcal{R}_{\mathbf{M}}(f,\Delta t)\), is stationary.