## 3 Designing the UWAC Channel Emulator
We can then modify (22) in the delay domain to obtain  
\[S(\tau,\eta)=\mathcal{F}\bigg{\{}\sum_{q=1}^{Q}\mathrm{E}[\mathbf{M_{q}}(t) \mathbf{M}_{\mathbf{q}}^{*}(t+\Delta t)]\delta(\tau-\tau_{\mathbf{q}}(t)) \bigg{\}} \tag{23}\]  
and from (23), we can obtain \(S(\tau_{\mathbf{q}},\eta)\), the delay-domain scattering of the \(q\)th path with respect to the Doppler spectrum. Reflections due to the surface, bottom, surface-bottom, and surface-bottom-surface combination give rise to different sets of reflected paths with a range of Rician \(K\)-factors. Thus, we will use the SoS assumption to generate channel samples with amplitudes that are Rician-distributed. We consider the wide sense stationary uncorrelated scattering (WSSUS)assumption here; therefore, we can represent channel samples \(\mathbf{M}_{q}[i]=\mathbf{M}_{q}[iT_{s}]\) to formulate the SoS model:  
\[\mathbf{M}[i]=\frac{1}{\sqrt{L(1+K)}}\left[\sum_{l=1}^{L}e^{i(\phi_{l}+2\pi f_{d }iT_{S})}+\sqrt{K}e^{i(\phi_{0}+2\pi f_{d}iT_{S})}\right] \tag{24}\]  
The time \(t\) is sampled into intervals of \(T_{S}\), where \(L\) represents the number of path bundles for each tap. Within each tap, there are uniformly distributed random phases \(\phi_{l}\) over the interval \([0,2\pi]\), an initial phase \(\phi_{0}\), and \(L\) Doppler frequencies \(f_{d}\). In (24), \(K\) is the Rician \(K\)-factor, depending on the ratio of the signal power over the direct path to that over scattered paths and the direct line-of-sight (LoS) path acquires a Doppler frequency of \(f_{0}\). It is evident that \(L\) sets of Doppler frequencies \(f_{d}\) can be selected from [PERSON]'s spectrum, such that these \(f_{d}\)s, in turn, can be used to generate (24), and the probability density function (pdf) of the range of \(f_{d}\) can be expressed as,  
\[S(\eta)=\frac{1}{2\chi}e^{-\frac{|\eta|^{\xi}}{\lambda}}. \tag{25}\]  
The values of \(f_{d}\) can be formulated by applying the inverse transform sampling lemma [21] on (25). The inverse transform sampling lemma states that if the distribution function of input \(X\) is given by \(f(x)\), the cumulative distribution function (cdf), \(F(x)\) will be uniformly distributed over the range \([0,1]\). Consequently, input samples can be generated using \(x=F^{-1}(\
u)\) where \(\
u\) is uniformly distributed over the range \([0,1]\), where \(F\) denotes the distribution function. Using the above concept, the cdf of the Doppler frequencies \(f_{d}\) can be expressed as,  
\[f_{d} =F^{-1}\bigg{[}\int_{-\infty}^{f_{d}}\frac{1}{2\chi}e^{-\frac{| \eta|^{\xi}}{\chi}}\mathrm{d}\eta\bigg{]}=F^{-1}\bigg{[}\frac{1}{2\chi}\Gamma \bigg{[}(1/xi)-(-f_{d}/\chi)^{\xi}\bigg{]}\bigg{]}\] \[\mathbf{M}(\tau,t,f)=\sum_{q=1}^{Q}|\mathbf{M_{q}}|\gamma_{\mathbf{q}}(f,t) \delta(\tau-\tau_{\mathbf{q}}(t)) \tag{18}\]0  
where the obtained \(f_{d}\)s follows a stretched exponential distribution, mod refers to the modulo operation, and \(W\) represents the product-Log or Lambert-\(W\) function. It is worthmentioning that the'modulo' operation returns the remainder after a division of \(2u\) by 1, and we consider that \(u=F(f_{d})\). A range of histograms of the \(f_{d}\)s using (25) and (26) is plotted in \[\mathbf{M}(\tau,t,f)=\sum_{q=1}^{Q}|\mathbf{M_{q}}|\gamma_{\mathbf{q}}(f,t) \delta(\tau-\tau_{\mathbf{q}}(t)) \tag{18}\]1 Using the discrete frequency-selective channel model, the channel impulse response is given by  
\[\mathbf{M}(\tau,t,f)=\sum_{q=1}^{Q}|\mathbf{M_{q}}|\gamma_{\mathbf{q}}(f,t) \delta(\tau-\tau_{\mathbf{q}}(t)) \tag{18}\]2  
where the term \(E_{Mq}(t)\) represent the fading coefficients corresponding to the originally formulated Milne energy associated with each path bundle, and \(\delta\) represents the delay function quantifying the delay experienced by the \(q\)th path. The delay is represented using the \(\delta\) function as we are trying to model the infinite bandwidth system. In terms of system implementation, an infinite bandwidth system needs to transmit and receive filters for effective functioning.