One of the most useful ways to characterize the channel's behavior is through its impulse and frequency response. As described, the signal reaches the receiver with a number of small delays that will be referred to as $\tau_i$. It is assumed that $\tau_i$ is constant in time. The channel impulse response can now be described as a sum of time-variant coefficients $a_i(t)$ multiplied with the time-invariant weights $w_i$, received with the delay $\tau_i$ [1]. The resulting equation $$ h(t, \tau) = \sum_{i}^{}w_i \cdot a_i(t)\cdot \delta(\tau - \tau_i) $$ has the form of a linear time-variant (LTV) filter. As such, the different delays $\tau_i$ are referred to as taps. Since the $a_i(t)$ are independent and wide-sense stationary, the resulting model is a wide-sense stationary uncorrelated scattering (WSSUS) model [3,8].
The coefficients $a_i(t)$ represent the Doppler-shifted signal parts arriving from the angles $\alpha_l$ with phase $\varphi_l$ and are given by the sum of individual shifts. The model used in this webdemo is baseband equivalent. The paper by Beaulieu, Xiao, and Zheng [3] describes the equation $$ a_i(t) = \sum_{l=0}^{L-1}\left[ \exp\left[\mathrm{j} 2\pi f_\mathrm{D}\cos(\alpha_l) \cdot t \right] \cdot \exp(\mathrm{j} \varphi_l) \right] + \sqrt{K}\exp\left[\mathrm{j}2\pi f_\mathrm{D} \cos(\theta_{\mathrm{LoS}}) \cdot t + \mathrm{j}\varphi_{\mathrm{LoS}} \right] $$ for the coefficients $a_i(t)$. The LoS component is represented by latter part of the equation. The Doppler frequency $f_\mathrm{D}$ is determined by the relative speed $v$ of the receiver and transmitter, as well as the carrier frequency $f_\mathrm{c}$. It can be calculated using $$ f_\mathrm{D} = \dfrac{\Delta v}{c} \cdot f_\mathrm{c}. $$ The angle $\theta_{\mathrm{LoS}}$ can be specified by the user. If a tap does not contain a LoS component, then $K$ is equal to zero. In this model, it is assumed that $\tau_0 = 0$. Additionaly, it is assumed that the LoS component has the shortest delay and hence, only the first ($i=0$) tap $\tau_0$ has a factor $K$ not equal to zero. The value of $K$ can be adjusted using the webdemo inputs.
The channel frequency response can be obtained from the impulse response via a Fourier transform. Since the timescale of $\tau_i$ is a lot smaller than that of the symbol time $T_\mathrm{S}$, the resulting frequency response given by $$ H(t, f) = \sum_{i}^{} w_i \cdot a_i(t)\cdot \exp[-\mathrm{j}2\pi f\tau_i] $$ is dependent on both time and frequency.
Both the impulse and frequency response can be written in vector notation as $$ h(t, \tau) = \mathbf{w} \odot \pmb{a}(t)\cdot \delta(\tau -\pmb{\tau}), \\ H(t, f) = \mathbf{w} \odot \pmb{a}(t)\cdot \exp[-\mathrm{j}2 \pi f \pmb{\tau}]. $$
The time-invariant weights $w_i$ and delays $\tau_i$ form a power delay profile (PDP). To control the magnitude of the delays a RMS delay spread $\tau_\mathrm{rms}$ can be specified. The user can also choose a PDP. This will control how the delays are distributed and how the taps are weighted. The options are "Default," which will assign uniform weights to uniformly spaced delays, "Decay," which will assign a series of exponentially decaying weights $\beta^i$ to uniformly spaced delays, and TDL A through E, which are predefined delay profiles given by 3GPP TR 38.901 [4].
The underlying system is an OFDM system with $N_\mathrm{sub}$ subcarriers and symbol duration $T_\mathrm{S}$. The values of $N_\mathrm{sub}$ and $T_\mathrm{S}$ can be specified by the user. If a single carrier system is desired instead the user may choose $N_\mathrm{sub} = 1$.
[1] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge: Cambridge University Press, 2005.
[3] N. C. Beaulieu, Chengshan Xiao and Yahong Rosa Zheng, “Novel Sum-of-Sinusoids Simulation Models for Rayleigh and Rician fading Channels,” IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, vol. 5, no. 12, pp. 3667–3679, 2006.
[4] 3GPP, “Study on channel model for frequencies from 0.5 to 100 GHz”, 3rd Generation Partnership Project (3GPP), Technical Report (TR) 38.901, Version 18.0.0 Release 18, Apr. 2024. [Online].
[8] A. F. Molisch, Wireless Communications, 2nd ed. Hoboken, NJ: Wiley-Blackwell, 2010.