## 3 Designing the UWAC Channel Emulator  
Linear time-varying (LTV) systems are primarily used to characterise UWAC channels owing to their time-varying nature and varying environmental factors affecting the flow of acoustic signals through the water. When an acoustic signal travels through water, the signal is reflected by the surface and bottom of the sea, resulting in multiple delayed versions of transmission rays or path bundles. If we have \(Q\) distinct reflected and refracted  
Figure 2: Visual representation of the acoustic signal path between a transmitter and a receiver located at different distances from the water body surface and different distances from each other. Scenario 1: Tx (transmitter) height = 10 m from the bottom and Rx (receiver) height = 50 m from the bottom, depth of water body = 100 m, the distance between Tx and Rx = 1 km. Scenario 2: Tx height = 50 m from the bottom and Rx height = 50 m from the bottom, depth of water body = 500 m, the distance between Tx and Rx = 2 km. Scenario 3: Tx height = 200 m from the bottom and Rx height = 100 m from the bottom, depth of water body = 500 m, the distance between Tx and Rx = 5 km. rays (echoes) with amplitudes represented by \(|\mathbf{M_{q}}|\) (where \(|\cdot|\) represents the determinant of a matrix), and delays are denoted as \(\tau_{\mathbf{q}}\), then the propagation channel with multiple information paths can be mathematically represented as follows:  
\[\mathbf{M}(\tau,t,f)=\sum_{q=1}^{Q}|\mathbf{M_{q}}|\gamma_{\mathbf{q}}(f,t) \delta(\tau-\tau_{\mathbf{q}}(t)) \tag{18}\]  
where \(\gamma_{\mathbf{q}}(f,t)\) is the fading parameter characterising small-scale variations with each path bundle. If \(\gamma_{\mathbf{q}}(f,t)\) is stationary in time, then we can use the time correlation function to derive the statistical description of \(\gamma_{\mathbf{q}}(f,t)\),  
\[\mathcal{R}_{\mathbf{q}}(f,\Delta t)=\mathrm{E}[\gamma_{\mathbf{q}}(f,t) \gamma_{\mathbf{q}}^{*}(f,t)] \tag{19}\]  
where \(\mathrm{E}[\cdot]\) denotes the expectation operation and \(*\) denotes the complex conjugate. In this case, the Fourier transform of \(\mathcal{R}_{\mathbf{q}}(f,\Delta t)\) yields the channel Doppler spectrum,  
\[\mathcal{F}\{\mathcal{R}_{\mathbf{q}}(f,\Delta t)\}=S_{\mathbf{q}}(f,\eta) \propto e^{-\frac{|\eta|}{\chi}\xi} \tag{20}\]  
which can be approximated by the stretched exponential distribution. In (20), \(\mathcal{F}\{\cdot\}\) denotes the discrete Fourier transform (DFT), \(\eta\) represents the Doppler frequency, \(\chi\) represents how closely the Doppler power spectrum matches the stretched exponential function, and \(\xi\) denotes the stretching exponent. The time-domain autoregressive memory process-based generation of the stretched exponential spectrum is neither practical nor accurate in representation. This is because \(\mathcal{R}_{\mathbf{q}}(f,\Delta t)\) has no closed form, and its Fourier transform is no longer exponentially distributed. Instead, in our developed platform, we generate random processes directly from the Doppler spectrum using the sum-of-sinusoids (SoS) model. The delay power spectrum of the channel and its frequency correlation function can be expressed as  
\[\mathcal{R}_{\mathbf{M}}(\tau)=\mathrm{E}[\mathbf{M}(t,\tau)\mathbf{M}^{*}(t,\tau^{\prime})]=\sum_{q=1}^{Q}\mathrm{E}[|\mathbf{M_{q}}(t)|^{2}]\delta(\tau -\tau_{\mathbf{q}}(t)) \tag{21}\]  
and  
\[\mathrm{E}[\mathcal{F}\{\mathbf{M}(t,\tau^{\prime})\}\mathcal{F}\{\mathbf{M}^{ *}(t,\tau)\}]=\frac{1}{2\pi}\sum_{q=1}^{Q}\mathrm{E}[|\mathbf{M_{q}}(t)|^{2}] e^{j2\pi(f-f^{\prime})\tau_{\mathbf{q}}} \tag{22}\]  
respectively, assuming stationarity and uncorrelatedness. The difference between the initial frequency \(f\) and the final frequency \(f^{\prime}\) defines the amount of the frequency correlation. Owing to the uncorrelated nature of \(\tau_{\mathbf{q}}\), it is possible to calculate each path delay separately.