\section{I. Introduction}

The complicated scattering of electromagnetic (EM) fields can be found in many applications \cite{Alt1998, Dietz2006, Kaya2009, Tait2011, Ma2019}.
Examples include the propagation of WiFi signals in an office space, and the propagation of high-power microwave signals in interconnected cabins in ships and aircraft.
When the enclosure dimension is much larger than the operating wavelength, the propagation of waves can be thought of in terms of ray trajectories.
In this limit, the ray dynamics are extremely sensitive to geometrical details, as well as to initial conditions.
If the trajectories exhibit chaos, that is, nearby trajectories diverge from one and other exponentially, such enclosures are labeled as ray-chaotic.
Returning to the short wavelength wave description, the enclosures are said to be wave chaotic \cite{Gradoni2014}.
The complicated dynamics of EM waves pose significant challenges for the exact solution of the EM fields.
With powerful computers, accurate modeling of wave propagation can be simulated, but only when the exact geometrical details of the scattering system are known \cite{Taflove1988, Yang1997, Davies2009}.
However, the precise geometrical details of such large systems are usually unknown in practice.
Moreover, the results computed from deterministic methods will be rendered invalid when perturbations are applied to the system, for example, by a simple change of system boundary.
Under these circumstances, statistical approaches are more suitable for studying wave-chaotic systems.

Random Matrix Theory (RMT) is widely applied to describe the statistics of spectral properties of wave chaotic systems.
Researchers originally developed RMT to model the spectra of the complicated Hamiltonians of large nuclei \cite{Wigner1955,Dyson1962,Mehta2004}.
Later,  Bohigas, Giannoni, and Schmit (BGS) proposed the applicability of RMT to simple wave propagation in domains with complicated boundaries.
It is now surmised that all systems that show ray-chaos in the classical regime will have wave properties that share universal statistical properties \cite{Bohigas1984}.
Based on the BGS conjecture, RMT has found broad application in various physical systems (atoms, quantum dots, quantum wires, microwave enclosures, acoustic resonators, and others) \cite{Haq1982,stockmann2000quantum,Alhassid2001,haake2010,Auregan2016}.
Depending on certain basic symmetries, the spectral properties of a specific chaotic system can be categorized into one of three groups \cite{Mehta2004,haake2010}.
The first two groups are the Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE) of random matrices for a system with and without time-reversal symmetry, respectively.
The third group is the so-called Gaussian Symplectic Ensemble (GSE) of random matrices for a system with half-integer spins and time-reversal symmetry.

The Random Coupling Model (RCM) has found great success in characterizing the statistical properties of various frequency-domain quantities, for example, the full scattering (S -) matrix and the impedance (Z -) matrix of wave chaotic systems \cite{Hemmady2005,Zheng2006,Zheng2006a,Hemmady2012,Gradoni2015}.
To use the RCM one must first identify the ports of entry and exit for waves in the enclosure.
The RCM then provides a statistical expression for the impedance matrix that linearly relates the port currents and voltages.

For a single chaotic cavity, RCM models the fluctuating impedance matrix as
\begin{equation}
\underline{\underline{Z}}_{cav}=iIm[\underline{\underline{Z}}_{avg}] + Re[\underline{\underline{Z}}_{avg}]^{1/2}\cdot \underline{\underline{\xi}}_{RCM}\cdot  Re[\underline{\underline{Z}}_{avg}]^{1/2}.
\end{equation}
Here $\underline{\underline{Z}}_{avg}$ is a frequency or realization averaged impedance matrix and $\underline{\underline{\xi}}_{RCM}$ is a universal, fluctuating matrix that imparts the statistical variations to the model $\underline{\underline{Z}}_{cav}$.
As mentioned, the quantity $\underline{\underline{Z}}_{avg}$ is the ensemble or frequency averaged system impedance, and it describes the early time dynamics of fields in the enclosure.
This includes radiation from the port into the enclosure as well as the propagation of radiation from one port to another in the enclosure over short paths.
The length of the paths that are retained depends on how the average is taken \cite{Yeh, Zhou2017, Hemmady2005, Xiao2018, Chen2022}.
If the average is over a window of frequencies, then only paths with travel times shorter than the inverse of the frequency window survive the averaging process.
If the average is taken by varying the orientation of a mode stirrer, then only paths that do not intersect the stirrer survive the averaging process.
The quantity $\underline{\underline{\xi}}_{RCM}$ is defined as
\begin{equation}
\underline{\underline{\xi}}_{RCM} (\omega_0)=-\frac{i}{\pi} \sum_n^N \frac{\underline{c}_{n}\, \underline{c}_{n}^T}{(\omega_0^2-\omega_n^2)/\Delta \omega_n^2+i\alpha}
\end{equation}
where the sum over $n$ represents a sum over the modes of the closed system.
It is through $\underline{\underline{\xi}}_{RCM}$ that the long-time behavior of the fields is modeled.
Here the vector $\underline{c}$ consists of $N$ independent, zero mean, unit variance Gaussian random variables.
Each of the variables represents the coupling between a port and the $n$th cavity mode of the enclosure.
This random choice of mode-port coupling strength is based on the Random Plane Wave hypothesis, which models the chaotic EM field as a superposition of many plane waves with random phases, amplitudes, and directions \cite{Berry1981, Wu1998, Chung2000}.

The quantities $\omega_0$ and $\omega_n$ are the operating frequency and the $n^{th}$ system resonant frequency, respectively.
Random Matrix Theory is utilized to generate a list of cavity mode frequencies which model those of the actual enclosure $\omega_n$ \cite{Hemmady2005, Lin2020}.
More details of the cavity mode generation in RCM are included in Appendix A.
The dimensionless loss parameter $\alpha$ characterizes the universal fluctuating impedance behavior of the system.
It is defined as
\begin{equation}
\alpha = \omega_0 \omega_n/(Q\,\Delta \omega_n^2)
\end{equation}
where $Q$ is the average quality factor of the modes and $\Delta\omega_n^2/(2\omega_n)$ is the mean-mode spacing near the operating frequency $\omega_0$ \cite{Gradoni2014}.
With the loss parameter $\alpha$ and the averaged impedance $\underline{\underline{Z}}_{avg}$ known, one can apply the RCM to generate a statistical ensemble of system impedance matrices $\underline{\underline{Z}}_{cav}$ via Monte-Carlo simulations.
We have recently extended the RCM from single enclosures to the treatment of multiple coupled enclosure systems \cite{Ma2020, Ma2020b}.

In this paper, we aim to further expand the RCM from the frequency domain to the time domain.
Previous work has directly applied the frequency-domain RCM to compute the system time-domain quantities (reflected voltage at the input port) by Fourier transforms \cite{Hart2009}.
However, this method can not be used to describe nonlinear devices in the scattering system.
The proposed time domain RCM (TD-RCM) method is a true time-domain calculation, which is based on the solution of a system of coupled ordinary differential equations.
Each equation describes the evolution of a cavity mode driven by currents in the ports.
Because the mode amplitudes are evolved in time, one can treat various types of port loads including both linear and nonlinear, passive and active devices, and time-dependent loads, such as those associated with time-dependent logic states of electronics \cite{Chen2022}.
The TD-RCM also explicitly includes non-universal effects through the incorporation of the port radiation impedances and the RCM loss parameter.
The short orbit effects are included explicitly in a natural way.

As introduced above, we focus on the treatment of time-domain aspects of short-wavelength excitations in complex scattering systems.
Researchers have developed several theoretical approaches to study the time-domain wave dynamics in complex electromagnetic resonators \cite{tomsovic1993long, Castaldi2012}.
Time-domain studies of reverberation chambers have resulted in several theoretical approaches to their dynamics.
One approach is to calculate the time dependence of the ensemble or spatially-averaged energy density in one or more coupled reverberation chambers \cite{Tait2011}.
Another approach is to consider the diffusion of wave amplitude through complex structures \cite{weaver2006transport, tanner2009dynamical}.
Other statistical time-domain methods are based on the Baum-Liu-Tesche equations \cite{Tesche2007}, the channel model \cite{saleh1987statistical,rappapor1989}, and the power balance method \cite{Tait2011}.
In acoustic systems, the time-domain wave dynamics in acoustically-large enclosures have also been studied \cite{young1959sabine, dunens1977impulsive}.
The present paper is unique in that all of the modes of the closed system evolve in time, and their interactions through the linear and nonlinear ports are included.
Our treatment also explicitly includes the effects of geometrical short-orbits, thus capturing important system-specific features in the time dynamics.
We also note that time-dependent coupled-mode theory (CMT) is broadly applied in many wave-systems \cite{Yariv1973, haus1984waves, Hardy1986, Suh2004, Liu2013}.
CMT usually treats the energy exchange between a few modes which belong to different resonant systems put in close proximity.
However, our approach differs from CMT in that the TD-RCM deals with the dynamics of many modes in one resonant system, where the modes do not directly couple to one another except through a discrete set of ports that connect the system to the outside world.
Moreover, the TD-RCM focuses on the detailed time-domain evolution of the modes, while CMT usually treats steady-state conditions.

The outline of the paper is as follows.
We first introduce the fundamental equations of the TD-RCM method in section II.
Section III introduces the short-pulse experimental verification of TD-RCM theory, followed by a comparison between the experimental results and the TD-RCM simulated results in sections IV and V.
We summarize the paper in section VI.

\section{II. TD-RCM formulation}

In this section, we discuss the detailed formulation of the TD-RCM method.
We first present the core mechanism of TD-RCM in Section II.A.
We next describe the treatment of system port loads (both linear and nonlinear loads) in Section II.B.
In Section II.C, we present a method to include short-orbits (SO) in TD-RCM, thus incorporating the non-statistical aspects of the early-time transmission inside the cavity.

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{fig1new4.png}
\caption{\label{fig:outline}
Schematic summary of the TD-RCM method.
The injected waveform $V_{in}(t)$ is sent into a wave chaotic enclosure through port 1.
The enclosure has $N$ eigenmodes excited.
The TD-RCM model computes the voltage and current on all $M$ ports, as well as all $N$ cavity mode voltages.
The system-specific quantities (cavity Q-factor, mean-mode-spacing $\Delta \omega^2$, the radiation resistance $R_{rad}(\omega)$ of the ports, and the load impedances attached to the ports) are color-coded in red, while the RMT generated quantities (cavity mode frequencies) are colored in green.
The result is predictions for the voltage waveforms appearing on all ports as a function of time, $V_i(t), i = 1, \dots, M$.
For example, a sketch of the voltage wave prediction at port $M$ is shown as $V_M(t)$ in the figure.
}
\end{figure}

\subsection{II.A TD-RCM Equation of Motion}

The time-domain Random Coupling Model (TD-RCM) models an enclosure supporting $N$ eigenmodes in the operating frequency bandwidth.
The enclosure can have $M$ ports, where each of the ports can serve as a transmitting (TX) port or a receiving (RX) port, or both.
We use Fig. \ref{fig:outline} to summarize the variables and parameters of the TD-RCM method.
The TD-RCM computes the temporal evolution of all cavity modes and ports.
The type of load connected at each of the ports can be different, and the degree of lossyness for each of the modes can also be specified.
The radiation resistance of the port $R_{rad}$ characterizes the free-space radiation property of the port antenna \cite{Hemmady2005, Zheng2006, Zheng2006a}.
The cavity modes ($U_n, n = 1, 2, \cdots, N$) are represented by the real voltages $U_n$, where $n$ is the mode index, and represents the amplitude of excitation of the $n^{th}$ eigenmode of the closed system.
The signal at port $j$ is represented by the port voltage $V_j$ and port current $I_j (j = 1, \dots, M)$.

Each cavity mode is described by a driven damped harmonic oscillator equation,
\begin{equation} \label{eq:modeV}
    \frac{d^2}{dt^2} U_n + v_n \frac{d}{dt} U_n + \omega_n^2 U_n = \omega_n \sum_{j=1}^M c_{nj} K_{nj} \frac{d}{dt} I_j,
\end{equation}
where port subscript $j$ runs through all $M$ system ports.
The port voltages ($V_j, j = 1, 2, \cdots, M$) are modeled by a linear summation over all cavity modes,
\begin{equation} \label{eq:portV}
    V_j = \sum_{n=1}^N c_{jn} \sqrt{ \frac{\epsilon}{\mu}} K_{nj} U_n - V_{jc}.
\end{equation}
As such, the ports act to couple all the modes to each other.
The quantity $V_{jc}$ represents the electrostatic mode contribution of port $j$, defined as $C_p \frac{d V_{jc}}{dt} = -I_j$.
The term $C_p$ is a port capacitance that is important at low frequencies for dipole-like ports.

In Eq. \ref{eq:modeV}, the damping term for a system mode is defined as $v_n = \alpha \Delta \omega_n^2 /\omega_n =\omega_n/Q$, where $\alpha$ is the RCM loss parameter.
The quantities $\omega_n$ and $\Delta \omega_n^2$ are the eigenmode frequency and mean-mode-spacing near $\omega_n$.
As in the frequency-domain RCM, the quantity $c_{jn}$ is a normalized, fluctuating coupling coefficient between mode $n$ and port $j$ (assumed to be Gaussian random variables).
The factor
\begin{equation}
K_{nj} = - \left( \frac{\mu}{\epsilon} \right )^{\frac{1}{4}} \left[ \frac{2R_{rad,j}(\omega_n) \, \Delta\omega_n}{\pi\omega_n } \right]^{\frac{1}{2}}
\end{equation}
is a dimensional factor having units of Ohms that give the strength of the coupling (see the derivation in Appendix B).
The quantities $\mu$ and $\epsilon$ are the (assumed uniform) permeability and permittivity of the enclosure volume material.
The quantity $R_{rad,j}(\omega_n)$ is the frequency domain radiation resistance of port $j$ evaluated at frequency $\omega_n$.
In Eq. \ref{eq:portV}, the random coupling coefficient $c_{jn}$ and mode voltages $U_n$ are the same quantities as in Eq. \ref{eq:modeV}.

For an enclosure with $N$ modes and $M$ ports, we have $N$ versions of Eqs. \ref{eq:modeV} and $M$ versions of Eqs. \ref{eq:portV}.
Note that these equations re-express in the time domain the same information contained in the frequency-dependent expression for the cavity impedance $\underline{\underline{Z}}_{cav}$.
The unknown quantities that need to be solved are the $N$ $U_n$s, the $M$ currents $I_j$, and the $M$ voltages $V_j$, all of which are functions of time.
Usually we set the initial condition ($t = 0$) as $U_n = dU_n/dt = V_j = 0$ for all $n$ and $j$, representing a quiet cavity.
When the input pulses are turned on, the port voltage values at the transmitting (TX) ports will have nonzero values as well as nonzero time derivatives.
As shown in the RHS of Eq. \ref{eq:modeV}, the change of port signals will then drive cavity modes to oscillate.
The oscillating cavity modes will further contribute to the nonzero signals at all ports (the RHS of Eq. \ref{eq:portV}).
Besides Eqs. \ref{eq:modeV} and \ref{eq:portV}, one still needs the explicit relationship between the port voltage and current signals to solve for all system mode ($U_n$) and port quantities ($I_j, V_j$).

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{portload3.png}
\caption{\label{fig:portload}
Diagrams of the TD-RCM port load modeling.
\textbf{a.} The linear load at port $j$ is represented with an impedance $Z_j$.
The dashed arrows mark the direction of waves flowing into and out of the cavity.
\textbf{b.} Schematic of a diode-loaded port with the resistor, diode, and capacitor components labeled.
}
\end{figure}

\subsection{II.B System Port Treatments}

In this subsection, we discuss the treatment of two generic types of loads that are present at ports: a linear impedance load (Fig. \ref{fig:portload} (a)) and a nonlinear load (Fig. \ref{fig:portload} (b)), where we utilize the specific example of a diode-loaded port.
We first show the treatment of ports with a linear load.

The voltage and current at the port $j$ can be expressed by
\begin{equation} \label{eq:linearload}
\begin{aligned}
    V_j &= V_{in} + V_{out}, \\
    I_j &= (V_{in} - V_{out})/Z_j,\\
\end{aligned}
\end{equation}
where $V_{in}$ ($V_{out}$) represents the incident (reflected) voltage wave coming into (out of) the cavity.
The direction of the waves is shown by the dashed arrows in Fig. \ref{fig:portload}, so that ``in'' corresponds to waves going into the cavity, and ``out'' represents the waves that leave the cavity.
The value of the load impedance $Z_j$ (often 50 $\Omega$) is represented by the load connected between the cavity and the ground in Fig. \ref{fig:portload} (a).
Note that the TD-RCM allows the cavity to have multiple ports, shown schematically in Fig. \ref{fig:portload} with the lines connected to the left side of the cavity.
For a signal input port, we have $I_j = (V_{in} - V_{out})/Z_j = (2V_{in} - V_{j})/Z_j$ where $V_{in}$ is the waveform injected into the cavity at that port.
For an output port, we have $I_j = -V_j/Z_j$ since $V_{in} = 0$.

One may express the I-V relation of both input and output ports by the following vector expression
\begin{equation} \label{eq:linearload2}
    \underline{I} = \underline{\underline{Y}} \, \underline{V} + \underline{D}.
\end{equation}
Vector quantities are used here for simplicity: $\underline{I}$ and $\underline{V}$ are two M-by-1 column vectors for all port currents and voltages, whose elements are $I_j$ and $V_j$ $(j = 1, \dots, M)$.
Here $\underline{\underline{Y}}$ is a diagonal M-by-M matrix where $Y_{jj} = 1/Z_{j}$ and $\underline{D}$ is a time-dependent vector with $D_j = 2V_{in, j}/Z_j$.
With Eqs. \ref{eq:modeV}, \ref{eq:portV} and \ref{eq:linearload2}, we have now obtained enough equations to solve for the port voltages.
Note that the values of port load impedance $(Z_j)$ and the injected voltage signals $(V_{in, j})$ values are treated as known quantities.
In a finite difference implementation, assuming $U_n$ and $dU_n/dt$ are known at time $t$, then $V_j$ and $dV_j/dt$ are known through Eq. \ref{eq:portV}.
Using Eq. \ref{eq:linearload2} then gives $dI_j/dt$ and consequently Eq. \ref{eq:modeV} can be used to update $dU_n/dt$ and $U_n$.

We next study the case with nonlinear output ports and use a generic diode-loaded port as a representative example, as illustrated in Fig. \ref{fig:portload} (b).
(Note that the TD-RCM can also treat nonlinear input ports utilizing this same formalism.)
As a true time-domain model, one of the strengths of TD-RCM is the ability to treat ports loaded with nonlinear devices.
As shown in Fig. \ref{fig:portload} (b), the diode-loaded port is modeled with a series resistance ($R$) and a shunt capacitance ($C$). The diode-port is modeled by the following equations:
\begin{equation} \label{eq:diode}
\begin{aligned}
    V_j &= V_D + R \, I_D, \quad I_j = -(I_C + I_D) \\
    I_C &= C \frac{d}{dt} V_j, \quad I_D = I_0[exp(\frac{qV_D}{k_BT})-1].
\end{aligned}
\end{equation}
The quantities $V_D, I_D$, and $I_C$ are the voltage drop at the diode, the currents at the diode path, and the shunt capacitor path (Fig. \ref{fig:portload} (b)).
The diode parameters $I_0, q, k_B, T$ are the reverse current, elementary charge, Boltzmann constant, and the temperature in Kelvin.
The subscripts $D$ and $C$ represent the diode and capacitor. For convenience, we set $I_D = 0$ for negative voltages. Eq. \ref{eq:diode} can be simplified to
\begin{equation} \label{eq:diode2}
    C \frac{dV_j}{dt} + \frac{V_j}{R} + I_j = -\frac{V_D}{R},
\end{equation}
\begin{equation} \label{eq:diode3}
    V_D = \frac{k_BT}{q} ln(1 + \frac{V_j - V_D}{R \, I_0}).
\end{equation}
The appearance of $dV_j/dt$ in Eq. \ref{eq:diode2} now adds a complication to the finite difference solution of our system of equations.
In particular, if Eq. \ref{eq:diode2} is differentiated in time to express $dI_j/dt$, and $dI_j/dt$ is inserted in Eq. \ref{eq:modeV}, then the second order derivative of $V_j$ appears on the right side of Eq. \ref{eq:modeV} and the second derivative of $U_n$ appears on the left side.
These must be solved for consistently with the relation Eq. \ref{eq:portV}.
This can be done by introducing a predictor-corrector method.
The voltage across the diode $V_D$ is computed by solving the transcendental equation \ref{eq:diode3}.
We examine the effectiveness of TD-RCM modeling of nonlinear elements in Appendix C, and provide an example of the TD-RCM modeling of a reservoir computer based on a microwave cavity containing diode-loaded output ports in Appendix D.

\subsection{II.C Inclusion of System-Specific Short-Orbits}

A limitation of the TD-RCM as outlined so far is the incomplete characterization of the early-time wave dynamics of a finite-size complex enclosure.
Specifically, we have represented the fields as a superposition of modes.
While this is, in principle, an exact representation if one uses the exact modes of the enclosure, we have replaced the exact modes with wave chaotic modes that are superpositions of random plane waves.
In doing this the ability to describe the propagation of wave energy directly from port to port, or indirectly with only a few reflections from the enclosure's walls has been lost.
The TD-RCM as formulated describes the evolution of fields on a longer time scale associated with the build-up of mode energy or the decay of mode energy due to losses \cite{tomsovic1993long, Tesche2007, Hart2009, Castaldi2012}.
However, before the setup of the reverberant fields, the receiver (RX) port-induced voltage signal will be mainly influenced by the short-orbit (SO) responses between the TX and RX ports \cite{Yeh, Xiao2016}.
Short orbits are ray-like trajectories that link ports by means of either ballistic, or few bounce, orbits.
This early-time RX signal rise-up comes from the non-statistical and system-specific prompt-response at the port location, and is present in many realizations of the cavity ensemble \cite{Yeh}.

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{addSO2.png}
\caption{\label{fig:addSO}
\textbf{a.} shows the computed time-domain Short-Orbit (SO) signal at the gigabox RX port.
The injected pulse is a 5 GHz, 5 ns single frequency sine wave. The inset is the enlarged view of the $V_{SO}$ curve from 30 to 32 ns window (black dashed box).
The $V_{SO}$ is computed with FT methods using the injected pulse and the measured ensemble averaged S-parameter (see details in the main text).
\textbf{b.} consists of the TD-RCM simulated RX port signals $V_{port}$ from 20 realizations of the model system. Here the SO contribution $V_{SO}$ is added onto the TD-RCM computed $V_{port}$.
\textbf{c.} shows the RMS ensemble averaged RX port signal  compiled from the port signals shown in \textbf{b}. }
\end{figure}

Here we propose a ``fix'' to TD-RCM which can incorporate the realistic SO response.
We consider the early-time transmission between a TX port and an RX port.
The general idea of the proposed SO treatment is similar to that of Ref. \cite{Hart2009}, where the frequency-domain quantities are used to compute time-domain responses.
The SO treatment utilizes the two frequency-domain system-specific quantities to compute the time-domain SO signal at the RX port.
The first quantity is the ensemble-averaged S-parameter linking the two ports, $\langle S_{RX,TX}(f) \rangle$ which describes the signal transmission from a TX to an RX port through the direct antenna radiation and SO trajectories.
The second quantity is the frequency spectrum of the injected time-domain pulse $\Tilde{V}_{in}(f)$.
With these quantities, one can compute the received signal at the RX port as $\Tilde{V}_{SO}(f) = \langle S_{RX,TX}(f) \rangle \cdot \Tilde{V}_{in}(f)$.
The corresponding time-domain signal ${V}_{SO}(t)$ is computed by Fourier transforming the computed $\Tilde{V}_{SO}(f)$.

Figure \ref{fig:addSO} (a) shows one example of the computed SO signal ${V}_{SO}(t)$.
Several measured quantities are used here for the computation of ${V}_{SO}(t)$.
The $\langle S_{RX,TX}(f) \rangle$ is measured in a $1$ $m^3$ enclosure (to be presented later) using a microwave network analyzer.
The injected waveform $V_{in}(t)$ is a modulated sine wave generated by the signal generators (to be explained).
The waveform $V_{in}(t)$ is then Fourier transformed to its frequency domain version $\Tilde{V}_{in}(f)$.
The frequency domain $\Tilde{V}_{SO}(f)$ is first computed and then transformed back to the time domain to obtain $V_{SO}(t)$.

With the time-domain SO signal ${V}_{SO}(t)$ computed, one will finally update the TD-RCM computed RX port signal $V_{port}(t)$ as $V_{port}(t) + {V}_{SO}(t)$, which is a coherent sum of oscillating voltages.
In Fig. \ref{fig:addSO} (b), we show the RX port signal $V_{port}(t)$ curves of 20 different system configurations, computed with the SO + TD-RCM treatment.
It is clear that the computed RX port responses for each realization have dramatically different time-dependent dynamics.
Figure \ref{fig:addSO} (c) shows the root-mean-square (RMS) average computation of the $V_{port}(t)$ data from 30 different system configurations (called RMS $V_{port}$).
In Fig. \ref{fig:addSO} (b), the SO signal $V_{SO}$ is difficult to discern from the single-realization RX port signals.
However, we find a signature of SO contributions after the RMS ensemble averaging process: a clear rise-up near 12 ns is observed in RMS $V_{port}(t)$, which demonstrates the effect of the SO signal.
The proposed method augments the TD-RCM model with a realistic characterization of the early-time dynamics.
The SO treatment only needs two quantities (the system-specific $\langle S_{RX,TX}\rangle$ and the waveform of interest $V_{in}(t)$).
However, this method of accounting for short orbits only applies to linear wave propagation environments because it is based on Fourier transforming frequency-domain quantities.
Note that we only present one possible method of incorporating realistic SO effects.
Alternative methods include time-gating signals deduced from Fourier transforming frequency domain data \cite{ADDISSIE2019123}, and another is the direct calculation of short orbits from the known geometry of the enclosure \cite{Xiao2016}.
To simulate the SO signal in a multi-port environment, one may compute the $V_{SO}(t)$ for each pair of TX and RX port by measuring the $\langle S_{RX,TX}\rangle$ for each pair.
Such treatment might seem complicated at first glance, however, we point out that a series of $\langle S_{RX,TX}\rangle$ determinations are just as common as the 'through connection' step in a vector network analyzer calibration.

\section{III. Short-pulse experiments}

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{fig0.png}
\caption{\label{fig:fig0}
Schematic of the short pulse time-domain experiment setup.
The injected waveform (left inset, a 5 ns pulse) is generated using the arbitrary waveform generator (AWG) and the programmable signal generator (PSG).
The short pulse is broadcast into the enclosure through the transmitting (TX) port (star).
The data ensemble is created by rotating a motorized mode-stirrer inside the enclosure.
The receiver (RX) port (star) induced voltage signal (right inset) is measured by the oscilloscope.
A lab computer is used for instrument control and data transmission/collection.
}
\end{figure}

To test the correctness and applicability of the TD-RCM method, we conducted short pulse injection experiments in the gigabox enclosure (shown schematically in Fig. \ref{fig:fig0}).
The gigabox is a metallic enclosure with a volume of $\sim 1$ $m^3$.
The round-trip time for short ray trajectories in the gigabox is $T_{rt} = 5$ ns.
A mode-stirrer, consisting of two thin metallic panels, is located inside the gigabox to randomize the EM fields.
One can rotate the angle of the mode-stirrer through many fixed values to obtain different scattering details, thus creating an ensemble of data.
To automate the measurement process, we installed a motor outside the gigabox to control the rotation angle of the mode stirrer.
Note that we only conduct 2-port pulse injection experiments.
One of the locations serves as the input port, where a single carrier frequency $f_c$ is modulated by a 20 ns short pulse envelope and is injected into the gigabox.
To generate the input pulse, we use the Tektronix AWG7052 5 GS/s arbitrary waveform generator (AWG) to produce a square pulse and then use the Agilent programmable signal generator (PSG) E8267D to act as a modulated waveform source with a fixed center frequency $f_c$.
We use a series of different $f_c$ values ranging from $5$ to $10$ GHz in the experiment.
A large number of system modes will be excited by these pulses.
For example, there exists about $2 \times 10^4$ modes from $4.5 \sim 5.5$ GHz, and about $9 \times 10^4$ modes from $9.5 \sim 10.5$ GHz inside the gigabox.
A non-zero induced voltage signal will be measured at the RX port.
A 50 $\Omega$ load is connected to the RX port.
Typical input and output waveforms are shown as insets in Fig. \ref{fig:fig0} where one should note the difference in time scales between the TX and RX waveforms.
We rotate the mode-stirrer to 200 fixed locations for each choice of the center frequency to create an ensemble of systems and measure the resulting received time-domain waveforms at the RX port using an Agilent DSO91304A 40 GS/s oscilloscope.
Statistical analysis is then done on this ensemble of time-domain waveforms.

\section{IV. Late-time Statistical Tests}

As introduced in the previous section, we have conducted short-pulse injection experiments in an electrically-large ray-chaotic cavity to test the performance of the TD-RCM model.
Here we study the late-time (beginning at $10T_{rt} \sim 50$ ns after the pulse injection) quantities where the short orbit effects are not present and a reverberant field structure has been established.
Using realistic information for the experimental setup, we simulate the experimental structure with the TD-RCM method.
More specifically, the information required are the gigabox loss parameter $\alpha$ \cite{Gradoni2014} at frequency $f_c$, the injected waveform $V_{in}(t)$, the mean mode-spacing, the antenna radiation resistance $R_{rad}$ over the bandwidth of the pulse for each port, and the value of the linear load impedance.
The gigabox TD-RCM simulation is run 1000 times where new lists of system modes $\omega_n$ and all random port-mode coupling coefficients $c_{jn}$ are refreshed each time.

We first study the statistics of the maximum receiver (RX) port voltage ($max\left[ V_{port} \right]$) in the time domain.
This quantity is defined as the maximum induced voltage signal at the RX port after the input pulse is injected.
We study the RCM simulation with or without the SO addition treatment.
We compare the statistics of $max\left[ V_{port} \right]$ obtained from the experiment and the TD-RCM simulations.
For each center frequency, we measure 200 different system configurations by rotating the mode stirrer to new angles inside the enclosure.
As summarized in Fig. \ref{fig:tdfig2}, we have found a relatively good agreement for max$[V_{port}]$ between the TD-RCM model prediction (red and yellow) and experimental data (blue).
Six different experiments are considered here, where we varied the center frequency of the injected sine wave from 5 GHz to 10 GHz in steps of 1 GHz.
Note that the value of the RX port signal decreases as the center frequency increases.
This can be explained by the loss parameter of the gigabox, which increases from $\alpha \sim 3$ (5 GHz) to $\alpha \sim 12$ (10 GHz).
Note that, as shown in Fig. \ref{fig:tdfig2}, the data computed from both the RCM model and the RCM model with the SO addition show relatively good agreement with the experimental data.
Two observations can be made.
First, the proposed SO ``fix'' does not significantly alter the long-term statistics of the original TD-RCM predictions.
Second, the maximum value of the RX port signal is a quantity that is not strongly affected by the early-time SO contributions.
This is a consequence of the high value of the quality factor for the experimental cavity.
The build-up time of the cavity modes is longer than the round-trip time $T_{rt}$ for short orbits.

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{newfig2o.png}
\caption{\label{fig:tdfig2}
Mean values of the receiver port maximum voltage data as a function of the center frequency of the input pulse.
The data measured experimentally are shown in solid blue, and the data computed by TD-RCM and the TD-RCM with the SO addition are shown in solid red and dashed yellow, respectively.
The input pulse is a single-frequency 20 ns modulated sine pulse. The center frequency $f_c$ is varied from 5 GHz to 10 GHz.
Note that the input pulse has a finite rise/fall time ($1 \sim 2$ ns, see Fig. \ref{fig:fig0} left inset).
The error bar is one standard deviation.
}
\end{figure}

\section{V. Early-time Statistical Tests}

\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{VrxPDF.png}
\caption{\label{fig:VrxPDF}
Comparison of RX received port voltage statistics in different time intervals for the gigabox experiment and related simulations.
The statistics of the distribution of the RX port voltages are shown from 3 data sets: the experimental data, the TD-RCM simulation neglecting SO, and the TD-RCM simulation with the SO addition.
A 5 GHz, 5 ns modulated single frequency sine pulse is adopted as the injected waveform into the gigabox for all three tests.
We show the RX port voltage statistics in a series of 5 ns windows. The window start time is swept from 0 to 25 ns in 5 ns steps, shown in (a) to (f), respectively.
}
\end{figure*}

In Section II we have introduced a method to include the direct short-orbit (SO) contribution to the TD-RCM.
The proposed method computes the time-domain SO contribution from a TX port to a RX port ($V_{SO}$), and directly adds $V_{SO}$ to the TD-RCM computed RX port voltages ($V_{port} \rightarrow V_{port} + V_{SO}$).
Here we study the effectiveness of the SO addition method by studying the detailed distribution of $V_{port}$ in the time domain.
We focus on the early-time ($< 8 \, T_{rt} = 40$ ns after the pulse injection) dynamics when the SO contribution dominates the induced voltage at the RX port.
We compare $V_{port}$ histogram data obtained from three different tests: the gigabox short-pulse experiments (Section III), the corresponding TD-RCM simulations that neglect SO, and the TD-RCM simulation that includes the SO contribution.
The distributions of the $V_{port}(t)$ values for different time intervals are shown in Fig. \ref{fig:VrxPDF}.
Because the injected pulse duration is 5 ns, we set the time window as 5 ns and look at the relatively earlier times (before $6 T_{rt} = 30$ ns) in $V_{port}(t)$.
The experimental and the RCM + SO histograms consistently show good agreement throughout the time evolution of the waveform.
However, the RCM-only results show a clear deviation from the experimental results during the 5 to 15 ns (1-3 times $T_{rt}$) time interval.
We believe the RCM-only results show smaller values of $V_{port}(t)$ due to a lack of SO-delivered energy.
As shown in Figs. \ref{fig:addSO} (a) and (c), the time-domain SO contribution shows the strongest amplitude contribution from 10 to 15 ns.
Inside the same time window, we also observe the most significant deviation between the RCM-only $V_{port}$ statistics and the other two data sets (Fig. \ref{fig:VrxPDF} (c)).
At a later time (15 ns after the pulse injection), we find that the $V_{port}$ statistics from the three data sets show a relatively good agreement (Figs. \ref{fig:VrxPDF} (d-f)).
We find that the late-time $V_{port}$ distribution fits nicely with a Gaussian distribution (not shown here).
On the other hand, a substantial deviation from Gaussian distribution in $V_{port}$ statistics marks the existence of substantial SO contributions in the system.
Examples of strong SO effects are the experimental and the RCM + SO datasets in Figs. \ref{fig:VrxPDF} (b) and (c).

\section{VI. Discussion and Summary}

The TD-RCM is a statistical treatment of time-dependent induced voltage signals in electrically-large complex enclosures.
It eschews geometrical details and represents the modes of the cavity using the random plane wave hypothesis.
A significant limitation of TD-RCM is the neglect of the nonzero time for signals to propagate through the cavity.
In making the random plane wave approximation, the model neglects the correlation between modes that carry the early-time information about the location of the ports.
The inclusion of the short orbit time domain response restores, to a degree, this information.

Another practical limitation of TD-RCM is the trade-off between computing speed and system mode density.
The TD-RCM models each and every one of the $N$ system eigenmodes that are excited by an input pulse.
We can estimate the required number of eigenmodes $N=BW/(\Delta f|_{f_c})$, where $BW$ is the bandwidth of the injected pulse, and $\Delta f|_{f_c}$ represents the system mean-mode-spacing at the pulse center frequency $f_c$.
Here is an estimate of the scale of $N$ in our gigabox short-pulse experiment.
For a 5 ns pulse at $f_c = 5$ GHz, there are $N \sim 8500$ modes excited.
Hence, the computation speed of the TD-RCM model is limited by the total number of modes that must be modeled.
A method of systematically reducing the total number of degrees of freedom to simulate is desired.

To sum up, the TD-RCM model computes the temporal evolution of the cavity modes and port signals.
The workflow of TD-RCM starts by generating a list of cavity modes using RMT.
Each of the cavity modes is characterized by a driven damped harmonic oscillator equation.
The port signal is computed by a random summation over all excited cavity modes.
A TD-RCM user will need to input the basic enclosure information (mode-density and RCM loss parameter $\alpha$), port information ($R_{rad}$ and voltage-current relationship for all nonlinear port loads), and the injected waveform.
An ensemble can be created by changing the list of eigenmodes and the mode-port random coupling coefficients.
One can then study many statistical properties of the time-dependent port voltages and system-mode dynamics, only a few of which have been discussed here.

To validate the TD-RCM, we have conducted short-pulse injection experiments into a large complex enclosure and studied the statistics of the peculiar/rare incidents (maximum receiver (RX) port voltage) in the time domain.
A relatively good agreement between the experimental data and the TD-RCM simulation is observed.
We have included the effect of SO by directly adding the SO signal obtained from the ensemble-averaged frequency domain scattering properties to the TD-RCM simulated RX port voltages.
We have also worked out the formalism for including nonlinear port loads, and demonstrated its use in Appendices C and D.

Concerning applications, one can use TD-RCM to simulate both the linear and nonlinear time-reversed wave focusing process between two ports in a complex scattering environment \cite{lerosey2004time, lerosey2006time, anlage2007new, taddese2010sensing, Frazier2013, hong2013focusing, Frazier2013a, hong2014nonlinear, Hong2015, Xiao2016, Frisco2019, 9130060}.
TD-RCM can also be applied to coda wave interferometry applications \cite{snieder2002coda, lobkis2003coda, Kuhl2005, snieder2006theory}, and the analysis of scattering fidelity studies \cite{schafer2005fidelity, gorin2006dynamics, taddese2009sensor}.
The reverberant wave-based reservoir computing hardware can also be studied with the nonlinear TD-RCM treatment (see Appendix D for an overview of the TD-RCM model of a resonant microwave cavity realization of a reservoir computer \cite{Ma2022}).
A TD-RCM user can set up a collection of SO signals to study the coherent energy delivery between the ports when the cavity geometry is known \cite{Xiao2016}.
For future directions, one may study the nonlinear properties of diode-loaded port with both TD-RCM and experimental methods \cite{Ma2020,Ma2020b}.
Another future direction is the inclusion of large multi-mode apertures between complex enclosures in TD-RCM.
Finally, TD-RCM may also be applied to simulate the linear and nonlinear dynamics for other systems which RMT can model.
Examples range from optical pulses in multi-mode fibers \cite{Xiong2016} to financial market fluctuations \cite{Fenn2011}.

\section{Acknowledgements}

We acknowledge discussions with Bisrat Addissie and Zachary Drikas of the US Naval Research Laboratory.
This work was supported by ONR under Grant No. N000141912481, DARPA Grant HR00112120021, and the Maryland Quantum Materials Center.

\section{Appendix}

\subsection{A. Generation of System Eigenmodes with RMT}

One will need to provide a list of eigenmodes of the closed system to initiate TD-RCM.
It is possible to obtain all cavity eigenmodes accurately using numerical simulation tools (CST, HFSS, COMSOL).
However, such a process would be time-consuming for an electrically large system and impossible when the accurate geometrical details are unknown.
Here we utilize the property of a chaotic system to simulate a list of system modes by only knowing the macroscopic system information (i.e., volume, area) and the operating frequency \cite{Gradoni2014}.

As introduced in the main text, a list of system eigenmodes $\omega_n$ can be generated using Random Matrix Theory (RMT).
We start by computing the eigenvalues $\lambda_{rmt}$ of large random matrices.
The matrix size equals the total number of system modes that we want to model, and the matrix elements are drawn from a Gaussian distribution controlled by the symmetry group of the system \cite{Dyson1962}.
The main symmetry classes of concern are the Gaussian Orthogonal and Gaussian Unitary ensembles of random matrices.
The system eigenmodes $\omega_n$ are computed from the relation $\lambda_{rmt} = \frac{\omega_c^2 - \omega_n^2}{\Delta \omega^2}$, where the quantity $\omega_c$ is the center frequency of the band that we want to model, and $\Delta \omega^2$ is the mean-mode-spacing near $\omega_c$.
To sum up, one first uses the total number of modes ($N$) to generate $N$ random matrix eigenvalues $\lambda_{rmt}$, and then uses the center frequency and mean-mode-spacing to compute $N$ system modes $\omega_n$ from $\lambda_{rmt}$.

\section{B. Derivation of Time Domain Random Coupling Model}

In this section, we outline the derivation of TD-RCM.
We start by representing the EM field inside the closed cavity in a basis of modes, $\mathbf{e}_n, \mathbf{h}_n, \phi_{n'}$, where
\begin{equation} \label{eq:eq1}
\begin{aligned}
    & \nabla \times \mathbf{e}_n = k_n \mathbf{h}_n\\
    & \nabla \times \mathbf{h}_n = k_n \mathbf{e}_n\\
    & \nabla^2 \phi_{n'} + \beta_n^2 \phi_{n'} = 0 \\
\end{aligned} \quad.
\end{equation}
The quantities $\mathbf{e}_n$ and $\mathbf{h}_n$ are the electric and magnetic fields for the $n^{th}$ cavity mode, $\phi_{n'}$ is potential of the electrostatic modes, and $\beta$ and $k$ are the corresponding wavenumbers.
For boundary conditions, we have the tangential components of $\mathbf{e}_n = 0$ at the boundary and the normal component of $\mathbf{h}_n = 0$.
For electrostatic modes, we have $\phi_{n'} = 0$ on the boundary.
Between two different cavity modes (mode $m$ and mode $n$, $m \neq n$), we have the orthogonal relationship
\begin{equation} \label{eq:ortho}
\begin{aligned}
    & \int_V d^3x \, \mathbf{e}_m \cdot \mathbf{e}_n = 0 \\
    & \int_V d^3x \, \mathbf{h}_m \cdot \mathbf{h}_n = 0 \\
\end{aligned} \quad.
\end{equation}
The quantity $V$ is the cavity volume.
We define a normalization process,
\begin{equation} \label{eq:normal}
    \int_V d^3x \, \mathbf{e}_n \cdot \mathbf{e}_n = k_n^{-1} = \int_V d^3x \, \mathbf{h}_n \cdot \mathbf{h}_n. \\
\end{equation}
Similarly, for the electrostatic modes, we have
\begin{equation} \label{eq:ortho2}
\begin{aligned}
    & \int_V d^3x \, \nabla \phi_{m'} \cdot \phi_{n'} = 0, \quad m' \neq n', \\
    & \int_V d^3x \, \nabla \phi_{n'} \cdot \phi_{n'} = \beta_{n'}^{-1}. \\
\end{aligned}
\end{equation}

We next expand the total EM field inside the cavity $\mathbf{E}$ and $\mathbf{H}$ in the basis of cavity modes,
\begin{equation} \label{eq:modes}
\begin{aligned}
    & \mathbf{E} = \sum_n U_n \mathbf{e}_n + \sum_{n'} U_{n'} \nabla \phi_{n'} \\
    & \mathbf{H} = \sum_n I_n \mathbf{h}_n \\
\end{aligned} \quad.
\end{equation}
Here we use $U_n$ and $I_n$ to represent the voltage and current of mode $n$.
Then we plug Eq. \ref{eq:modes} to the Maxwell's equations.
We dot Faraday's law with $\mathbf{h}_n$ and integrate over volume,
\begin{widetext}
    \begin{equation} \label{eq:max1}
    -\mu \frac{\partial}{\partial t} \int_V d^3x \, \mathbf{h}_n \cdot \mathbf{H} = -\frac{\mu}{k_n} \frac{\partial}{\partial t} I_n = \int_V d^3x \, \mathbf{h}_n \cdot \nabla \times \mathbf{E} = \int_V d^3x \, \mathbf{E} \cdot \nabla \times \mathbf{h}_n = U_n.
    \end{equation}
\end{widetext}
We dot Ampere's law with $\mathbf{e}_n$,
\begin{widetext}
\begin{equation} \label{eq:max2}
    \epsilon \frac{\partial}{\partial t} \int_V d^3x \, \mathbf{e}_n \cdot \mathbf{E} = \frac{\epsilon}{k_n} \frac{\partial}{\partial t} U_n = -\int_V d^3x \, \mathbf{e}_n \cdot \mathbf{J} + k_n\int_V d^3x \, \mathbf{h}_n \cdot \mathbf{H}.
\end{equation}
\end{widetext}
The quantities $\epsilon$ and $\mu$ are the permittivity and permeability of the system environment.
The current density $\mathbf{J}$ above has two components: the leakage current due to bulk conductivity representing cavity losses, and the driving current due to port current $I_j$ ($j$ is the index of the port),
\begin{equation} \label{eq:current}
    \mathbf{J} = \sigma \mathbf{E} + \sum_{ports} I_j \mathbf{u}_j.
\end{equation}
Here $\mathbf{u}_j$ is a profile function for the port $j$ current density.
It is normalized with unit magnitude has units $L^{-2}$.
The quantity $\sigma$ is the conductivity of the cavity wall.
Then the first term in Eq. \ref{eq:max2} becomes
\begin{widetext}
\begin{equation} \label{eq:e7}
    \int_V d^3x \, \mathbf{e}_n \cdot \mathbf{J} = \int_V d^3x \, \mathbf{e}_n \cdot \left( \sigma \mathbf{E} + \sum_{ports} I_j \mathbf{u}_j \right) = \frac{\sigma}{k_n} U_n + \sum_{ports} I_j \hat{c}_{nj}.
\end{equation}
\end{widetext}
Here the dimensionless coupling coefficient is defined $\hat{c}_{nj} = \int_V d^3x \, \mathbf{e}_n \cdot \mathbf{u}_j$, and the Ampere's law (Eq. \ref{eq:max2}) becomes
\begin{equation} \label{eq:amp2}
    \frac{\epsilon}{k_n} \frac{\partial}{\partial t} U_n + \frac{\sigma}{k_n} U_n = - \sum_{ports} I_j \hat{c}_{nj} + I_n.
\end{equation}
The port current $I_j$ will also excite an electrostatic field component,
\begin{equation} \label{eq:amp3}
    \epsilon \frac{\partial}{\partial t} \int_V d^3x \, \nabla \phi_{n'} \cdot \mathbf{E} = \frac{\epsilon}{\beta_{n'}} \frac{\partial}{\partial t} U_{n'} = - \frac{\sigma}{\beta_{n'}} U_{n'} - \sum_{ports} V_j \hat{c}'_{n'j}.
\end{equation}
Similarly we introduce a coupling coefficient $\hat{c}'_{n'j} = \int_V d^3x \, \nabla \phi_{n'} \cdot \mathbf{u}_j$.
Finally, we have the induced electric field at the port $j$ written as
\begin{equation} \label{eq:portVV}
    V_j = - \int_V d^3x \, \mathbf{u}_j \cdot \mathbf{E} = - \sum_{em-modes} U_n \hat{c}_{nj} - \sum_{es-modes} U_{n'} \hat{c}'_{n'j}.
\end{equation}

We write the complete time domain equations here
\begin{equation} \label{eq:all1}
\begin{aligned}
    -\mu \frac{\partial}{\partial t} I_n &= k_n U_n \\
    \frac{\epsilon}{k_n} \frac{\partial}{\partial t} U_n + \frac{\sigma}{k_n} U_n &= - \sum_{ports} \hat{c}_{nj} I_j + I_n \\
    \frac{\epsilon}{\beta_{n'}} \frac{\partial}{\partial t} U_{n'} + \frac{\sigma}{\beta_{n'}} U_{n'} &= - \sum_{ports} \hat{c}'_{n'j} I_j \\
\end{aligned} \quad.
\end{equation}
The coupling coefficients are $\hat{c}_{nj} = \int_V d^3x \, \mathbf{e}_n \cdot \mathbf{u}_j$ and $\hat{c}'_{n'j} = \int_V d^3x \, \nabla \phi_{n'} \cdot \mathbf{u}_j$.
We rewrite Eq. \ref{eq:all1} by eliminating the mode currents ($I_n$),
\begin{equation} \label{eq:all2}
\begin{aligned}
    & \frac{\partial^2}{\partial t^2} U_n + \frac{\sigma}{\epsilon} \frac{\partial}{\partial t} U_n + \frac{k_n^2}{\epsilon\mu} U_n = - \sqrt{\frac{\mu}{\epsilon}} \frac{k_n}{\sqrt{\mu \epsilon}} \sum_{ports} \hat{c}_{nj} \frac{\partial}{\partial t} I_j\\
    & \frac{\partial}{\partial t} U_{n'} + \frac{\sigma}{\epsilon} U_{n'} = - \frac{\beta_{n'}}{\epsilon} \sum_{ports} \hat{c}'_{n'j} I_j. \\
    & V_j = - \sum_{em-modes} U_n \hat{c}_{nj} - \sum_{es-modes} U_{n'} \hat{c}'_{n'j}.
\end{aligned}
\end{equation}

Here we study the port-mode coefficients $\hat{c}_{nj}$ in Eq. \ref{eq:all2} by switching to the frequency domain version.
We have
\begin{equation} \label{eq:f1}
    (\omega_n^2 - i\omega \gamma - \omega^2) U_n = i \sqrt{\frac{\mu}{\epsilon}} \omega_n \omega \sum_{ports} \hat{c}_{nj} I_j
\end{equation}
where $\gamma = \frac{\sigma}{\epsilon}$ and $\frac{k_n}{\sqrt{\mu \epsilon}} = \omega_n$.
The electrostatic modes are characterized by $(\gamma - i\omega) U_{n'} = - \frac{\beta_{n'}}{\epsilon} \sum_{ports} \hat{c}_{n'j} I_{n'}$.
The port voltage is written as
\begin{widetext}
\begin{equation} \label{eq:f2}
    V_j = -i \omega \sqrt{\frac{\mu}{\epsilon}} \sum_{em-modes} \sum_{ports} \frac{\omega_n \hat{c}_{nj'} \hat{c}_{nj}}{(\omega_n^2 - i\omega \gamma - \omega^2)} I_{j'} + \sum_{es-modes} \sum_{ports} \frac{\beta_{n'}}{\epsilon} \frac{\hat{c}_{n'j'} \hat{c}_{n'j}}{\gamma - i \omega} I_{j'}
\end{equation}
\end{widetext}
where $j$ and $j'$ are the indices of different ports.
The impedance matrix writes as $Z_{jj'} = -i \omega \sqrt{\frac{\mu}{\epsilon}} \sum_{em-modes} \frac{\omega_n \hat{c}_{nj'} \hat{c}_{nj}}{(\omega_n^2 - i\omega \gamma - \omega^2)}$

We next relate elements of the impedance matrix to the radiation resistance of the ports.
We first introduce the mode density.
We use $N(\omega)$ as the number of cavity modes with $\omega < \omega_n$.
Because we work with cavities with a high mode density, we have $\sum_{em-modes} \rightarrow \int d\omega_n N'(\omega_n), \quad N'(\omega_n) = dN(\omega)/d\omega |_{\omega = \omega_n}$.
The real part of the impedance $Re(Z_{jj'}) = -i \omega \sqrt{\frac{\mu}{\epsilon}} \int d\omega_n N' \frac{\omega_n \hat{c}_{nj'} \hat{c}_{nj}}{(\omega_n^2 - i\omega \gamma - \omega^2)} \approx  \omega \sqrt{\frac{\mu}{\epsilon}} \frac{\pi}{2} N' \hat{c}_{nj'} \hat{c}_{nj}$.

We redefine the coupling coefficient as $\sqrt{\frac{\mu}{\epsilon}} \hat{c}_{nj'} \hat{c}_{nj} = \left[ R_{rad}(\omega_n) / \left( \frac{\pi}{2} \omega_n N' \right) \right] c_{nj}^2$ where $c_{nj}$ is a zero mean, unit variance Gaussian random variable.
The quantity $R_{rad,j}$ is the radiation resistance of port $j$.
Thus we have defined the mode-port coupling coefficient as $c_{nj} = \left[ \sqrt{\frac{\epsilon}{\mu}} R_{rad, j}(\omega_n) / \left( \frac{\pi}{2} \omega_n N' \right) \right]^{1/2} \hat{c}_{nj}$.
We also have the system impedance matrix as
\begin{equation} \label{eq:f3}
    Z_{jj'} = -i \omega \sum_{em-modes} \frac{2(R_{rad,j}R_{rad,j'})^{1/2}}{\pi N'} \frac{c_{nj'} c_{nj}}{(\omega_n^2 - i\omega \gamma - \omega^2)}.
\end{equation}
Note that Eq. \ref{eq:f3} has the same form as the frequency domain single cavity RCM \cite{Gradoni2014}.

The final TD-RCM mode equations are
\begin{widetext}
\begin{equation} \label{eq:all4}
\begin{aligned}
    & \frac{\partial^2}{\partial t^2} U_n + \gamma_n \frac{\partial}{\partial t} U_n + \omega_n^2 U_n = - \left( \frac{\mu}{\epsilon} \right)^{1/4} \omega_n \sum_{ports} \left[ \frac{2R_{rad, j}(\omega_n)}{\pi \omega_n N'} \right]^{1/2} c_{nj} \frac{\partial}{\partial t} I_j\\
    & \gamma_n = \omega_n / Q_n \\
    & C \frac{\partial}{\partial t} V_{jc} = - I_j \\
\end{aligned} \quad.
\end{equation}
\end{widetext}
Here $N'(\omega_n)= 1/\Delta \omega_n$ is the inverse of the mode spacing at $\omega_n$, and $V_{jc}$ describes the electrostatic contribution.
The quantity $C$ is the port radiation reactance at lower frequencies as in the main text.
The TD-RCM port voltage equation is
\begin{equation} \label{eq:all5}
    V_j = - \left( \frac{\epsilon}{\mu} \right)^{1/4} \sum_{em-modes} \left[ \frac{2R_{rad, j}(\omega_n)}{\pi \omega_n N'} \right]^{1/2} c_{jn} U_n - V_{jc}.
\end{equation}
With Eqs. \ref{eq:all4} and \ref{eq:all5}, we have the complete equations of motion of TD-RCM.
The coefficient $K_{nj} = - \left( \frac{\mu}{\epsilon} \right )^{\frac{1}{4}} \left[ \frac{2R_{rad,j}(\omega_n) \, \Delta\omega_n}{\pi\omega_n } \right]^{\frac{1}{2}}$ is the combined coefficient on the RHS of the mode equation Eq. \ref{eq:all4}, and $K'_{jn}$ is the corresponding coefficient in Eq. \ref{eq:all5}.

\subsection{C. TD-RCM Treatment of Linear and Non-Linear Loads - Spectral Response}

\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{nlinload3.png}
\caption{\label{fig:nlin}
The TD-RCM simulated RX port signals. (a) and (b) show the time-domain RX port signal $V_{port}$ for a 50 $\Omega$ linear load and a diode load, respectively. The Fourier transform of the time-domain waveforms in (a) and (b) are shown in (c) and (d).
}
\end{figure*}

To expand the applicability of the TD-RCM model, we have presented the treatment of nonlinear port loads in Section III.B.
Here we test the nonlinear capability of TD-RCM with a simple single-tone stimulation of a cavity with a nonlinear output port.
In this numerical exercise, we inject a single frequency 5 GHz sine-wave for 50 ns through a linear TX port into a model cavity with $M=2$ ports, and study the induced voltage signal at a second RX port in the cavity.
The loss parameter of the system is set at $\alpha = 1$, and $N=1000$ system modes are used.
The mean-mode-spacing is set as $\Delta k^2 = 4\pi/A = 109.3 m^{-2}$ where $A = 0.115 m^2$ is the two-dimensional billiard area.
For convenience, the frequency-independent radiation resistance of both ports is set to $(18 + 50 i)\Omega$ (measured at 5 GHz).
Because we are only testing the port load responses, we have fixed the system spectrum ($\omega_n$ in Eq. \ref{eq:modeV}) and the port-mode coupling coefficients  ($c_{jn}$ in Eqs. \ref{eq:modeV} and \ref{eq:portV}).
For the linear system test, the load connected at the RX port is a 50 $\Omega$ linear impedance (Fig. \ref{fig:portload} (a)).
For the nonlinear system test, the RX port is a diode-loaded port, as shown in Fig. \ref{fig:portload} (b).
The following parameters are used for the diode-loaded port ($R = 50\Omega, C = 1.5e^{-20}F, I_0 = 1e^{-7}A$, and $T = 300K$).
The amplitude of the CW input sine-wave is 5V.
The computed time-domain linear and nonlinear RX port results are shown in Figs. \ref{fig:nlin} (a) and (b), respectively.
One can already see hints of nonlinearity from Fig. \ref{fig:nlin} (b) by eye.
We next show the spectrum of the linear and nonlinear time-domain waveforms in Figs. \ref{fig:nlin} (c) and (d), respectively.
Here, the linear-port spectrum is dominated by one peak at the fundamental frequency (5 GHz), while the spectrum of the nonlinear-port shows a clear peak at the second and several higher harmonics.
Thus the TD-RCM is capable of simulating nonlinear elements and emulating proper nonlinear behaviors.
Note that the speed of TD-RCM computation will slow down when nonlinear elements are included.

\subsection{D. TD-RCM Application: Reverberant Wave Reservoir Computing Hardware}

\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{RCtask.png}
\caption{\label{fig:Rossler}
The Rossler-observer task testing set performance.
The trained RC is given a continuous stream of the x(t) data generated from the coupled nonlinear Rossler equations and has to deduce the evolution of the y(t) and z(t) time series based only on training data.
(a) and (b) show the Rossler-y and Rossler-z series data, respectively.
The TD-RCM simulated reservoir computing data are shown in red dotted lines. The actual states are shown in solid blue lines.
The x-axis shows the index of the testing, ranging from the $32000^{th}$ to the $40000^{th}$ data points.
Each data index corresponds to a time bin of duration 250 ps.
}
\end{figure*}

Here we aim to demonstrate one successful application of the TD-RCM method.
Recently in Ref. \cite{Ma2022}, we have experimentally realized a physical system, namely reverberant wave reservoir computing (RC) hardware, which can execute multiple machine learning tasks on time-domain signals.
The RC hardware is based on a wave chaotic enclosure with multiple diode-loaded ports.
To operate the wave-based RC, one injects the input data stream into a wave chaotic enclosure through a linear port and then measures the induced voltage signal at nonlinear ports at several discrete locations.
These voltage signals will serve as the `neuron' responses in the traditional software RC.
After the training, the desired output signal is obtained by properly summing up the RC neuron responses \cite{Lu2016, Pathak2018}.
A RC realization requires a system with a complex and diverse set of modes, and requires a nonlinear evolution of the system state.
A ray-chaotic microwave cavity with diode-loaded output ports is very well suited for this task.
We use TD-RCM with nonlinear ports to simulate the microwave realization of an RC.

Here we utilized the nonlinear version of TD-RCM to simulate the dynamics of the wave-based RC hardware.
The diode-loaded ports are modeled as in Section III.B.
We have put a single linear input port and 100 diode-loaded output ports in our TD-RCM cavity model computation.
All diodes have the same characteristics ($R = 50\Omega, C = 1.5e^{-12}F, I_0 = 1e^{-7}A$, and $T = 300K$).
The specific type of machine learning (ML) task we studied here is the so-called Rossler observer task \cite{Lu2016}.
The Rossler system is governed by these equations,
\begin{equation} \label{eq:Rossler}
\begin{aligned}
    \Dot{x} &= -y-z,\\
    \Dot{y} &= x+ay,\\
    \Dot{z} &= b+z(x-c),\\
    \end{aligned}
\end{equation}
where $a=0.5, b=2.1, c=3.5$ and the over-dot denotes derivative with respect to time.
In this task, the ML model is expected to predict the Rossler attractor $y$- and $z$-component data when only the $x$-component information is supplied.
More details of the Rossler observer task setup can be found in Refs. \cite{Lu2016, Ma2022}.
The training of the wave-based RC is completed with a so-called out coupling matrix $W_{out}$.
The process of finding the optimized $W_{out}$ matrix is completed by running the Ridge regression algorithm on the digital computer \cite{Lu2016}.
Here, the training phase used $80\%$ of the entire dataset (80000 data points uniformly sampled in $\sim 200$ oscillation periods).
With optimized cavity parameters ($\alpha = 300$ and a 35 V input amplitude), the testing set results are shown in Fig. \ref{fig:Rossler}.
In this case, the trained RC is fed the $x(t)$ time domain waveform generated by an arbitrary waveform generator with a step size of 250 ps.
The task for the RC is to continuously generate the $y(t)$ and $z(t)$ time domain signals without knowledge of the underlying equations of motion for the Rossler attractor.
The performance of the RC prediction can be evaluated using the normalized mean square error, e.g., defined for the $y$-variable as NMSE $= \sum_n[s_n - s_n']^2/\sum_n s_n^2$, where $s'$ and $s$ denote true and RC-inferred values for the Rossler time series, respectively, and $n$ is the time index of the testing set data.
The $s'$ true signal is denoted as "from data", and the RC-inferred signal $s$ is denoted "from RC" in Fig. \ref{fig:Rossler}.
The NMSE is computed to be 0.055 and 0.083 for the results shown in Fig. \ref{fig:Rossler} (a) and (b), respectively, showing that the TD-RCM simulated RC can perform this ML task with high accuracy.
We note that NMSE generally decreases with an increasing number of output ports.

With TD-RCM, one can study the dynamics of the reverberant wave RC with great fidelity and detail because all system parameters can be easily controlled.
It is also possible to study the cavity realization of the RC with electromagnetic numerical simulation software \cite{Ma2022}.
However, the computing speed of TD-RCM is much faster than the numerical EM simulation because no accurate CAD drawing and the corresponding spatial-domain mesh-solving are required.

\end{spacing}

\section{I. Introduction}

The complicated scattering of electromagnetic (EM) fields can be found in many applications \cite{Alt1998, Dietz2006, Kaya2009, Tait2011, Ma2019}.
Examples include the propagation of WiFi signals in an office space, and the propagation of high-power microwave signals in interconnected cabins in ships and aircraft.
When the enclosure dimension is much larger than the operating wavelength, the propagation of waves can be thought of in terms of ray trajectories.
In this limit, the ray dynamics are extremely sensitive to geometrical details, as well as to initial conditions.
If the trajectories exhibit chaos, that is, nearby trajectories diverge from one and other exponentially, such enclosures are labeled as ray-chaotic.
Returning to the short wavelength wave description, the enclosures are said to be wave chaotic \cite{Gradoni2014}.
The complicated dynamics of EM waves pose significant challenges for the exact solution of the EM fields.
With powerful computers, accurate modeling of wave propagation can be simulated, but only when the exact geometrical details of the scattering system are known \cite{Taflove1988, Yang1997, Davies2009}.
However, the precise geometrical details of such large systems are usually unknown in practice.
Moreover, the results computed from deterministic methods will be rendered invalid when perturbations are applied to the system, for example, by a simple change of system boundary.
Under these circumstances, statistical approaches are more suitable for studying wave-chaotic systems.

Random Matrix Theory (RMT) is widely applied to describe the statistics of spectral properties of wave chaotic systems.
Researchers originally developed RMT to model the spectra of the complicated Hamiltonians of large nuclei \cite{Wigner1955,Dyson1962,Mehta2004}.
Later,  Bohigas, Giannoni, and Schmit (BGS) proposed the applicability of RMT to simple wave propagation in domains with complicated boundaries.
It is now surmised that all systems that show ray-chaos in the classical regime will have wave properties that share universal statistical properties \cite{Bohigas1984}.
Based on the BGS conjecture, RMT has found broad application in various physical systems (atoms, quantum dots, quantum wires, microwave enclosures, acoustic resonators, and others) \cite{Haq1982,stockmann2000quantum,Alhassid2001,haake2010,Auregan2016}.
Depending on certain basic symmetries, the spectral properties of a specific chaotic system can be categorized into one of three groups \cite{Mehta2004,haake2010}.
The first two groups are the Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE) of random matrices for a system with and without time-reversal symmetry, respectively.
The third group is the so-called Gaussian Symplectic Ensemble (GSE) of random matrices for a system with half-integer spins and time-reversal symmetry.

The Random Coupling Model (RCM) has found great success in characterizing the statistical properties of various frequency-domain quantities, for example, the full scattering (S -) matrix and the impedance (Z -) matrix of wave chaotic systems \cite{Hemmady2005,Zheng2006,Zheng2006a,Hemmady2012,Gradoni2015}.
To use the RCM one must first identify the ports of entry and exit for waves in the enclosure.
The RCM then provides a statistical expression for the impedance matrix that linearly relates the port currents and voltages.

For a single chaotic cavity, RCM models the fluctuating impedance matrix as
\begin{equation}
\underline{\underline{Z}}_{cav}=iIm[\underline{\underline{Z}}_{avg}] + Re[\underline{\underline{Z}}_{avg}]^{1/2}\cdot \underline{\underline{\xi}}_{RCM}\cdot  Re[\underline{\underline{Z}}_{avg}]^{1/2}.
\end{equation}
Here $\underline{\underline{Z}}_{avg}$ is a frequency or realization averaged impedance matrix and $\underline{\underline{\xi}}_{RCM}$ is a universal, fluctuating matrix that imparts the statistical variations to the model $\underline{\underline{Z}}_{cav}$.
As mentioned, the quantity $\underline{\underline{Z}}_{avg}$ is the ensemble or frequency averaged system impedance, and it describes the early time dynamics of fields in the enclosure.
This includes radiation from the port into the enclosure as well as the propagation of radiation from one port to another in the enclosure over short paths.
The length of the paths that are retained depends on how the average is taken \cite{Yeh, Zhou2017, Hemmady2005, Xiao2018, Chen2022}.
If the average is over a window of frequencies, then only paths with travel times shorter than the inverse of the frequency window survive the averaging process.
If the average is taken by varying the orientation of a mode stirrer, then only paths that do not intersect the stirrer survive the averaging process.
The quantity $\underline{\underline{\xi}}_{RCM}$ is defined as
\begin{equation}
\underline{\underline{\xi}}_{RCM} (\omega_0)=-\frac{i}{\pi} \sum_n^N \frac{\underline{c}_{n}\, \underline{c}_{n}^T}{(\omega_0^2-\omega_n^2)/\Delta \omega_n^2+i\alpha}
\end{equation}
where the sum over $n$ represents a sum over the modes of the closed system.
It is through $\underline{\underline{\xi}}_{RCM}$ that the long-time behavior of the fields is modeled.
Here the vector $\underline{c}$ consists of $N$ independent, zero mean, unit variance Gaussian random variables.
Each of the variables represents the coupling between a port and the $n$th cavity mode of the enclosure.
This random choice of mode-port coupling strength is based on the Random Plane Wave hypothesis, which models the chaotic EM field as a superposition of many plane waves with random phases, amplitudes, and directions \cite{Berry1981, Wu1998, Chung2000}.

The quantities $\omega_0$ and $\omega_n$ are the operating frequency and the $n^{th}$ system resonant frequency, respectively.
Random Matrix Theory is utilized to generate a list of cavity mode frequencies which model those of the actual enclosure $\omega_n$ \cite{Hemmady2005, Lin2020}.
More details of the cavity mode generation in RCM are included in Appendix A.
The dimensionless loss parameter $\alpha$ characterizes the universal fluctuating impedance behavior of the system.
It is defined as
\begin{equation}
\alpha = \omega_0 \omega_n/(Q\,\Delta \omega_n^2)
\end{equation}
where $Q$ is the average quality factor of the modes and $\Delta\omega_n^2/(2\omega_n)$ is the mean-mode spacing near the operating frequency $\omega_0$ \cite{Gradoni2014}.
With the loss parameter $\alpha$ and the averaged impedance $\underline{\underline{Z}}_{avg}$ known, one can apply the RCM to generate a statistical ensemble of system impedance matrices $\underline{\underline{Z}}_{cav}$ via Monte-Carlo simulations.
We have recently extended the RCM from single enclosures to the treatment of multiple coupled enclosure systems \cite{Ma2020, Ma2020b}.

In this paper, we aim to further expand the RCM from the frequency domain to the time domain.
Previous work has directly applied the frequency-domain RCM to compute the system time-domain quantities (reflected voltage at the input port) by Fourier transforms \cite{Hart2009}.
However, this method can not be used to describe nonlinear devices in the scattering system.
The proposed time domain RCM (TD-RCM) method is a true time-domain calculation, which is based on the solution of a system of coupled ordinary differential equations.
Each equation describes the evolution of a cavity mode driven by currents in the ports.
Because the mode amplitudes are evolved in time, one can treat various types of port loads including both linear and nonlinear, passive and active devices, and time-dependent loads, such as those associated with time-dependent logic states of electronics \cite{Chen2022}.
The TD-RCM also explicitly includes non-universal effects through the incorporation of the port radiation impedances and the RCM loss parameter.
The short orbit effects are included explicitly in a natural way.

As introduced above, we focus on the treatment of time-domain aspects of short-wavelength excitations in complex scattering systems.
Researchers have developed several theoretical approaches to study the time-domain wave dynamics in complex electromagnetic resonators \cite{tomsovic1993long, Castaldi2012}.
Time-domain studies of reverberation chambers have resulted in several theoretical approaches to their dynamics.
One approach is to calculate the time dependence of the ensemble or spatially-averaged energy density in one or more coupled reverberation chambers \cite{Tait2011}.
Another approach is to consider the diffusion of wave amplitude through complex structures \cite{weaver2006transport, tanner2009dynamical}.
Other statistical time-domain methods are based on the Baum-Liu-Tesche equations \cite{Tesche2007}, the channel model \cite{saleh1987statistical,rappapor1989}, and the power balance method \cite{Tait2011}.
In acoustic systems, the time-domain wave dynamics in acoustically-large enclosures have also been studied \cite{young1959sabine, dunens1977impulsive}.
The present paper is unique in that all of the modes of the closed system evolve in time, and their interactions through the linear and nonlinear ports are included.
Our treatment also explicitly includes the effects of geometrical short-orbits, thus capturing important system-specific features in the time dynamics.
We also note that time-dependent coupled-mode theory (CMT) is broadly applied in many wave-systems \cite{Yariv1973, haus1984waves, Hardy1986, Suh2004, Liu2013}.
CMT usually treats the energy exchange between a few modes which belong to different resonant systems put in close proximity.
However, our approach differs from CMT in that the TD-RCM deals with the dynamics of many modes in one resonant system, where the modes do not directly couple to one another except through a discrete set of ports that connect the system to the outside world.
Moreover, the TD-RCM focuses on the detailed time-domain evolution of the modes, while CMT usually treats steady-state conditions.

The outline of the paper is as follows.
We first introduce the fundamental equations of the TD-RCM method in section II.
Section III introduces the short-pulse experimental verification of TD-RCM theory, followed by a comparison between the experimental results and the TD-RCM simulated results in sections IV and V.
We summarize the paper in section VI.

\section{II. TD-RCM formulation}

In this section, we discuss the detailed formulation of the TD-RCM method.
We first present the core mechanism of TD-RCM in Section II.A.
We next describe the treatment of system port loads (both linear and nonlinear loads) in Section II.B.
In Section II.C, we present a method to include short-orbits (SO) in TD-RCM, thus incorporating the non-statistical aspects of the early-time transmission inside the cavity.

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{fig1new4.png}
\caption{\label{fig:outline}
Schematic summary of the TD-RCM method.
The injected waveform $V_{in}(t)$ is sent into a wave chaotic enclosure through port 1.
The enclosure has $N$ eigenmodes excited.
The TD-RCM model computes the voltage and current on all $M$ ports, as well as all $N$ cavity mode voltages.
The system-specific quantities (cavity Q-factor, mean-mode-spacing $\Delta \omega^2$, the radiation resistance $R_{rad}(\omega)$ of the ports, and the load impedances attached to the ports) are color-coded in red, while the RMT generated quantities (cavity mode frequencies) are colored in green.
The result is predictions for the voltage waveforms appearing on all ports as a function of time, $V_i(t), i = 1, \dots, M$.
For example, a sketch of the voltage wave prediction at port $M$ is shown as $V_M(t)$ in the figure.
}
\end{figure}

\subsection{II.A TD-RCM Equation of Motion}

The time-domain Random Coupling Model (TD-RCM) models an enclosure supporting $N$ eigenmodes in the operating frequency bandwidth.
The enclosure can have $M$ ports, where each of the ports can serve as a transmitting (TX) port or a receiving (RX) port, or both.
We use Fig. \ref{fig:outline} to summarize the variables and parameters of the TD-RCM method.
The TD-RCM computes the temporal evolution of all cavity modes and ports.
The type of load connected at each of the ports can be different, and the degree of lossyness for each of the modes can also be specified.
The radiation resistance of the port $R_{rad}$ characterizes the free-space radiation property of the port antenna \cite{Hemmady2005, Zheng2006, Zheng2006a}.
The cavity modes ($U_n, n = 1, 2, \cdots, N$) are represented by the real voltages $U_n$, where $n$ is the mode index, and represents the amplitude of excitation of the $n^{th}$ eigenmode of the closed system.
The signal at port $j$ is represented by the port voltage $V_j$ and port current $I_j (j = 1, \dots, M)$.

Each cavity mode is described by a driven damped harmonic oscillator equation,
\begin{equation} \label{eq:modeV}
    \frac{d^2}{dt^2} U_n + v_n \frac{d}{dt} U_n + \omega_n^2 U_n = \omega_n \sum_{j=1}^M c_{nj} K_{nj} \frac{d}{dt} I_j,
\end{equation}
where port subscript $j$ runs through all $M$ system ports.
The port voltages ($V_j, j = 1, 2, \cdots, M$) are modeled by a linear summation over all cavity modes,
\begin{equation} \label{eq:portV}
    V_j = \sum_{n=1}^N c_{jn} \sqrt{ \frac{\epsilon}{\mu}} K_{nj} U_n - V_{jc}.
\end{equation}
As such, the ports act to couple all the modes to each other.
The quantity $V_{jc}$ represents the electrostatic mode contribution of port $j$, defined as $C_p \frac{d V_{jc}}{dt} = -I_j$.
The term $C_p$ is a port capacitance that is important at low frequencies for dipole-like ports.

In Eq. \ref{eq:modeV}, the damping term for a system mode is defined as $v_n = \alpha \Delta \omega_n^2 /\omega_n =\omega_n/Q$, where $\alpha$ is the RCM loss parameter.
The quantities $\omega_n$ and $\Delta \omega_n^2$ are the eigenmode frequency and mean-mode-spacing near $\omega_n$.
As in the frequency-domain RCM, the quantity $c_{jn}$ is a normalized, fluctuating coupling coefficient between mode $n$ and port $j$ (assumed to be Gaussian random variables).
The factor
\begin{equation}
K_{nj} = - \left( \frac{\mu}{\epsilon} \right )^{\frac{1}{4}} \left[ \frac{2R_{rad,j}(\omega_n) \, \Delta\omega_n}{\pi\omega_n } \right]^{\frac{1}{2}}
\end{equation}
is a dimensional factor having units of Ohms that give the strength of the coupling (see the derivation in Appendix B).
The quantities $\mu$ and $\epsilon$ are the (assumed uniform) permeability and permittivity of the enclosure volume material.
The quantity $R_{rad,j}(\omega_n)$ is the frequency domain radiation resistance of port $j$ evaluated at frequency $\omega_n$.
In Eq. \ref{eq:portV}, the random coupling coefficient $c_{jn}$ and mode voltages $U_n$ are the same quantities as in Eq. \ref{eq:modeV}.

For an enclosure with $N$ modes and $M$ ports, we have $N$ versions of Eqs. \ref{eq:modeV} and $M$ versions of Eqs. \ref{eq:portV}.
Note that these equations re-express in the time domain the same information contained in the frequency-dependent expression for the cavity impedance $\underline{\underline{Z}}_{cav}$.
The unknown quantities that need to be solved are the $N$ $U_n$s, the $M$ currents $I_j$, and the $M$ voltages $V_j$, all of which are functions of time.
Usually we set the initial condition ($t = 0$) as $U_n = dU_n/dt = V_j = 0$ for all $n$ and $j$, representing a quiet cavity.
When the input pulses are turned on, the port voltage values at the transmitting (TX) ports will have nonzero values as well as nonzero time derivatives.
As shown in the RHS of Eq. \ref{eq:modeV}, the change of port signals will then drive cavity modes to oscillate.
The oscillating cavity modes will further contribute to the nonzero signals at all ports (the RHS of Eq. \ref{eq:portV}).
Besides Eqs. \ref{eq:modeV} and \ref{eq:portV}, one still needs the explicit relationship between the port voltage and current signals to solve for all system mode ($U_n$) and port quantities ($I_j, V_j$).

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{portload3.png}
\caption{\label{fig:portload}
Diagrams of the TD-RCM port load modeling.
\textbf{a.} The linear load at port $j$ is represented with an impedance $Z_j$.
The dashed arrows mark the direction of waves flowing into and out of the cavity.
\textbf{b.} Schematic of a diode-loaded port with the resistor, diode, and capacitor components labeled.
}
\end{figure}

\subsection{II.B System Port Treatments}

In this subsection, we discuss the treatment of two generic types of loads that are present at ports: a linear impedance load (Fig. \ref{fig:portload} (a)) and a nonlinear load (Fig. \ref{fig:portload} (b)), where we utilize the specific example of a diode-loaded port.
We first show the treatment of ports with a linear load.

The voltage and current at the port $j$ can be expressed by
\begin{equation} \label{eq:linearload}
\begin{aligned}
    V_j &= V_{in} + V_{out}, \\
    I_j &= (V_{in} - V_{out})/Z_j,\\
\end{aligned}
\end{equation}
where $V_{in}$ ($V_{out}$) represents the incident (reflected) voltage wave coming into (out of) the cavity.
The direction of the waves is shown by the dashed arrows in Fig. \ref{fig:portload}, so that ``in'' corresponds to waves going into the cavity, and ``out'' represents the waves that leave the cavity.
The value of the load impedance $Z_j$ (often 50 $\Omega$) is represented by the load connected between the cavity and the ground in Fig. \ref{fig:portload} (a).
Note that the TD-RCM allows the cavity to have multiple ports, shown schematically in Fig. \ref{fig:portload} with the lines connected to the left side of the cavity.
For a signal input port, we have $I_j = (V_{in} - V_{out})/Z_j = (2V_{in} - V_{j})/Z_j$ where $V_{in}$ is the waveform injected into the cavity at that port.
For an output port, we have $I_j = -V_j/Z_j$ since $V_{in} = 0$.

One may express the I-V relation of both input and output ports by the following vector expression
\begin{equation} \label{eq:linearload2}
    \underline{I} = \underline{\underline{Y}} \, \underline{V} + \underline{D}.
\end{equation}
Vector quantities are used here for simplicity: $\underline{I}$ and $\underline{V}$ are two M-by-1 column vectors for all port currents and voltages, whose elements are $I_j$ and $V_j$ $(j = 1, \dots, M)$.
Here $\underline{\underline{Y}}$ is a diagonal M-by-M matrix where $Y_{jj} = 1/Z_{j}$ and $\underline{D}$ is a time-dependent vector with $D_j = 2V_{in, j}/Z_j$.
With Eqs. \ref{eq:modeV}, \ref{eq:portV} and \ref{eq:linearload2}, we have now obtained enough equations to solve for the port voltages.
Note that the values of port load impedance $(Z_j)$ and the injected voltage signals $(V_{in, j})$ values are treated as known quantities.
In a finite difference implementation, assuming $U_n$ and $dU_n/dt$ are known at time $t$, then $V_j$ and $dV_j/dt$ are known through Eq. \ref{eq:portV}.
Using Eq. \ref{eq:linearload2} then gives $dI_j/dt$ and consequently Eq. \ref{eq:modeV} can be used to update $dU_n/dt$ and $U_n$.

We next study the case with nonlinear output ports and use a generic diode-loaded port as a representative example, as illustrated in Fig. \ref{fig:portload} (b).
(Note that the TD-RCM can also treat nonlinear input ports utilizing this same formalism.)
As a true time-domain model, one of the strengths of TD-RCM is the ability to treat ports loaded with nonlinear devices.
As shown in Fig. \ref{fig:portload} (b), the diode-loaded port is modeled with a series resistance ($R$) and a shunt capacitance ($C$). The diode-port is modeled by the following equations:
\begin{equation} \label{eq:diode}
\begin{aligned}
    V_j &= V_D + R \, I_D, \quad I_j = -(I_C + I_D) \\
    I_C &= C \frac{d}{dt} V_j, \quad I_D = I_0[exp(\frac{qV_D}{k_BT})-1].
\end{aligned}
\end{equation}
The quantities $V_D, I_D$, and $I_C$ are the voltage drop at the diode, the currents at the diode path, and the shunt capacitor path (Fig. \ref{fig:portload} (b)).
The diode parameters $I_0, q, k_B, T$ are the reverse current, elementary charge, Boltzmann constant, and the temperature in Kelvin.
The subscripts $D$ and $C$ represent the diode and capacitor. For convenience, we set $I_D = 0$ for negative voltages. Eq. \ref{eq:diode} can be simplified to
\begin{equation} \label{eq:diode2}
    C \frac{dV_j}{dt} + \frac{V_j}{R} + I_j = -\frac{V_D}{R},
\end{equation}
\begin{equation} \label{eq:diode3}
    V_D = \frac{k_BT}{q} ln(1 + \frac{V_j - V_D}{R \, I_0}).
\end{equation}
The appearance of $dV_j/dt$ in Eq. \ref{eq:diode2} now adds a complication to the finite difference solution of our system of equations.
In particular, if Eq. \ref{eq:diode2} is differentiated in time to express $dI_j/dt$, and $dI_j/dt$ is inserted in Eq. \ref{eq:modeV}, then the second order derivative of $V_j$ appears on the right side of Eq. \ref{eq:modeV} and the second derivative of $U_n$ appears on the left side.
These must be solved for consistently with the relation Eq. \ref{eq:portV}.
This can be done by introducing a predictor-corrector method.
The voltage across the diode $V_D$ is computed by solving the transcendental equation \ref{eq:diode3}.
We examine the effectiveness of TD-RCM modeling of nonlinear elements in Appendix C, and provide an example of the TD-RCM modeling of a reservoir computer based on a microwave cavity containing diode-loaded output ports in Appendix D.

\subsection{II.C Inclusion of System-Specific Short-Orbits}

A limitation of the TD-RCM as outlined so far is the incomplete characterization of the early-time wave dynamics of a finite-size complex enclosure.
Specifically, we have represented the fields as a superposition of modes.
While this is, in principle, an exact representation if one uses the exact modes of the enclosure, we have replaced the exact modes with wave chaotic modes that are superpositions of random plane waves.
In doing this the ability to describe the propagation of wave energy directly from port to port, or indirectly with only a few reflections from the enclosure's walls has been lost.
The TD-RCM as formulated describes the evolution of fields on a longer time scale associated with the build-up of mode energy or the decay of mode energy due to losses \cite{tomsovic1993long, Tesche2007, Hart2009, Castaldi2012}.
However, before the setup of the reverberant fields, the receiver (RX) port-induced voltage signal will be mainly influenced by the short-orbit (SO) responses between the TX and RX ports \cite{Yeh, Xiao2016}.
Short orbits are ray-like trajectories that link ports by means of either ballistic, or few bounce, orbits.
This early-time RX signal rise-up comes from the non-statistical and system-specific prompt-response at the port location, and is present in many realizations of the cavity ensemble \cite{Yeh}.

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{addSO2.png}
\caption{\label{fig:addSO}
\textbf{a.} shows the computed time-domain Short-Orbit (SO) signal at the gigabox RX port.
The injected pulse is a 5 GHz, 5 ns single frequency sine wave. The inset is the enlarged view of the $V_{SO}$ curve from 30 to 32 ns window (black dashed box).
The $V_{SO}$ is computed with FT methods using the injected pulse and the measured ensemble averaged S-parameter (see details in the main text).
\textbf{b.} consists of the TD-RCM simulated RX port signals $V_{port}$ from 20 realizations of the model system. Here the SO contribution $V_{SO}$ is added onto the TD-RCM computed $V_{port}$.
\textbf{c.} shows the RMS ensemble averaged RX port signal  compiled from the port signals shown in \textbf{b}. }
\end{figure}

Here we propose a ``fix'' to TD-RCM which can incorporate the realistic SO response.
We consider the early-time transmission between a TX port and an RX port.
The general idea of the proposed SO treatment is similar to that of Ref. \cite{Hart2009}, where the frequency-domain quantities are used to compute time-domain responses.
The SO treatment utilizes the two frequency-domain system-specific quantities to compute the time-domain SO signal at the RX port.
The first quantity is the ensemble-averaged S-parameter linking the two ports, $\langle S_{RX,TX}(f) \rangle$ which describes the signal transmission from a TX to an RX port through the direct antenna radiation and SO trajectories.
The second quantity is the frequency spectrum of the injected time-domain pulse $\Tilde{V}_{in}(f)$.
With these quantities, one can compute the received signal at the RX port as $\Tilde{V}_{SO}(f) = \langle S_{RX,TX}(f) \rangle \cdot \Tilde{V}_{in}(f)$.
The corresponding time-domain signal ${V}_{SO}(t)$ is computed by Fourier transforming the computed $\Tilde{V}_{SO}(f)$.

Figure \ref{fig:addSO} (a) shows one example of the computed SO signal ${V}_{SO}(t)$.
Several measured quantities are used here for the computation of ${V}_{SO}(t)$.
The $\langle S_{RX,TX}(f) \rangle$ is measured in a $1$ $m^3$ enclosure (to be presented later) using a microwave network analyzer.
The injected waveform $V_{in}(t)$ is a modulated sine wave generated by the signal generators (to be explained).
The waveform $V_{in}(t)$ is then Fourier transformed to its frequency domain version $\Tilde{V}_{in}(f)$.
The frequency domain $\Tilde{V}_{SO}(f)$ is first computed and then transformed back to the time domain to obtain $V_{SO}(t)$.

With the time-domain SO signal ${V}_{SO}(t)$ computed, one will finally update the TD-RCM computed RX port signal $V_{port}(t)$ as $V_{port}(t) + {V}_{SO}(t)$, which is a coherent sum of oscillating voltages.
In Fig. \ref{fig:addSO} (b), we show the RX port signal $V_{port}(t)$ curves of 20 different system configurations, computed with the SO + TD-RCM treatment.
It is clear that the computed RX port responses for each realization have dramatically different time-dependent dynamics.
Figure \ref{fig:addSO} (c) shows the root-mean-square (RMS) average computation of the $V_{port}(t)$ data from 30 different system configurations (called RMS $V_{port}$).
In Fig. \ref{fig:addSO} (b), the SO signal $V_{SO}$ is difficult to discern from the single-realization RX port signals.
However, we find a signature of SO contributions after the RMS ensemble averaging process: a clear rise-up near 12 ns is observed in RMS $V_{port}(t)$, which demonstrates the effect of the SO signal.
The proposed method augments the TD-RCM model with a realistic characterization of the early-time dynamics.
The SO treatment only needs two quantities (the system-specific $\langle S_{RX,TX}\rangle$ and the waveform of interest $V_{in}(t)$).
However, this method of accounting for short orbits only applies to linear wave propagation environments because it is based on Fourier transforming frequency-domain quantities.
Note that we only present one possible method of incorporating realistic SO effects.
Alternative methods include time-gating signals deduced from Fourier transforming frequency domain data \cite{ADDISSIE2019123}, and another is the direct calculation of short orbits from the known geometry of the enclosure \cite{Xiao2016}.
To simulate the SO signal in a multi-port environment, one may compute the $V_{SO}(t)$ for each pair of TX and RX port by measuring the $\langle S_{RX,TX}\rangle$ for each pair.
Such treatment might seem complicated at first glance, however, we point out that a series of $\langle S_{RX,TX}\rangle$ determinations are just as common as the 'through connection' step in a vector network analyzer calibration.

\section{III. Short-pulse experiments}

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{fig0.png}
\caption{\label{fig:fig0}
Schematic of the short pulse time-domain experiment setup.
The injected waveform (left inset, a 5 ns pulse) is generated using the arbitrary waveform generator (AWG) and the programmable signal generator (PSG).
The short pulse is broadcast into the enclosure through the transmitting (TX) port (star).
The data ensemble is created by rotating a motorized mode-stirrer inside the enclosure.
The receiver (RX) port (star) induced voltage signal (right inset) is measured by the oscilloscope.
A lab computer is used for instrument control and data transmission/collection.
}
\end{figure}

To test the correctness and applicability of the TD-RCM method, we conducted short pulse injection experiments in the gigabox enclosure (shown schematically in Fig. \ref{fig:fig0}).
The gigabox is a metallic enclosure with a volume of $\sim 1$ $m^3$.
The round-trip time for short ray trajectories in the gigabox is $T_{rt} = 5$ ns.
A mode-stirrer, consisting of two thin metallic panels, is located inside the gigabox to randomize the EM fields.
One can rotate the angle of the mode-stirrer through many fixed values to obtain different scattering details, thus creating an ensemble of data.
To automate the measurement process, we installed a motor outside the gigabox to control the rotation angle of the mode stirrer.
Note that we only conduct 2-port pulse injection experiments.
One of the locations serves as the input port, where a single carrier frequency $f_c$ is modulated by a 20 ns short pulse envelope and is injected into the gigabox.
To generate the input pulse, we use the Tektronix AWG7052 5 GS/s arbitrary waveform generator (AWG) to produce a square pulse and then use the Agilent programmable signal generator (PSG) E8267D to act as a modulated waveform source with a fixed center frequency $f_c$.
We use a series of different $f_c$ values ranging from $5$ to $10$ GHz in the experiment.
A large number of system modes will be excited by these pulses.
For example, there exists about $2 \times 10^4$ modes from $4.5 \sim 5.5$ GHz, and about $9 \times 10^4$ modes from $9.5 \sim 10.5$ GHz inside the gigabox.
A non-zero induced voltage signal will be measured at the RX port.
A 50 $\Omega$ load is connected to the RX port.
Typical input and output waveforms are shown as insets in Fig. \ref{fig:fig0} where one should note the difference in time scales between the TX and RX waveforms.
We rotate the mode-stirrer to 200 fixed locations for each choice of the center frequency to create an ensemble of systems and measure the resulting received time-domain waveforms at the RX port using an Agilent DSO91304A 40 GS/s oscilloscope.
Statistical analysis is then done on this ensemble of time-domain waveforms.

\section{IV. Late-time Statistical Tests}

As introduced in the previous section, we have conducted short-pulse injection experiments in an electrically-large ray-chaotic cavity to test the performance of the TD-RCM model.
Here we study the late-time (beginning at $10T_{rt} \sim 50$ ns after the pulse injection) quantities where the short orbit effects are not present and a reverberant field structure has been established.
Using realistic information for the experimental setup, we simulate the experimental structure with the TD-RCM method.
More specifically, the information required are the gigabox loss parameter $\alpha$ \cite{Gradoni2014} at frequency $f_c$, the injected waveform $V_{in}(t)$, the mean mode-spacing, the antenna radiation resistance $R_{rad}$ over the bandwidth of the pulse for each port, and the value of the linear load impedance.
The gigabox TD-RCM simulation is run 1000 times where new lists of system modes $\omega_n$ and all random port-mode coupling coefficients $c_{jn}$ are refreshed each time.

We first study the statistics of the maximum receiver (RX) port voltage ($max\left[ V_{port} \right]$) in the time domain.
This quantity is defined as the maximum induced voltage signal at the RX port after the input pulse is injected.
We study the RCM simulation with or without the SO addition treatment.
We compare the statistics of $max\left[ V_{port} \right]$ obtained from the experiment and the TD-RCM simulations.
For each center frequency, we measure 200 different system configurations by rotating the mode stirrer to new angles inside the enclosure.
As summarized in Fig. \ref{fig:tdfig2}, we have found a relatively good agreement for max$[V_{port}]$ between the TD-RCM model prediction (red and yellow) and experimental data (blue).
Six different experiments are considered here, where we varied the center frequency of the injected sine wave from 5 GHz to 10 GHz in steps of 1 GHz.
Note that the value of the RX port signal decreases as the center frequency increases.
This can be explained by the loss parameter of the gigabox, which increases from $\alpha \sim 3$ (5 GHz) to $\alpha \sim 12$ (10 GHz).
Note that, as shown in Fig. \ref{fig:tdfig2}, the data computed from both the RCM model and the RCM model with the SO addition show relatively good agreement with the experimental data.
Two observations can be made.
First, the proposed SO ``fix'' does not significantly alter the long-term statistics of the original TD-RCM predictions.
Second, the maximum value of the RX port signal is a quantity that is not strongly affected by the early-time SO contributions.
This is a consequence of the high value of the quality factor for the experimental cavity.
The build-up time of the cavity modes is longer than the round-trip time $T_{rt}$ for short orbits.

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{newfig2o.png}
\caption{\label{fig:tdfig2}
Mean values of the receiver port maximum voltage data as a function of the center frequency of the input pulse.
The data measured experimentally are shown in solid blue, and the data computed by TD-RCM and the TD-RCM with the SO addition are shown in solid red and dashed yellow, respectively.
The input pulse is a single-frequency 20 ns modulated sine pulse. The center frequency $f_c$ is varied from 5 GHz to 10 GHz.
Note that the input pulse has a finite rise/fall time ($1 \sim 2$ ns, see Fig. \ref{fig:fig0} left inset).
The error bar is one standard deviation.
}
\end{figure}

\section{V. Early-time Statistical Tests}

\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{VrxPDF.png}
\caption{\label{fig:VrxPDF}
Comparison of RX received port voltage statistics in different time intervals for the gigabox experiment and related simulations.
The statistics of the distribution of the RX port voltages are shown from 3 data sets: the experimental data, the TD-RCM simulation neglecting SO, and the TD-RCM simulation with the SO addition.
A 5 GHz, 5 ns modulated single frequency sine pulse is adopted as the injected waveform into the gigabox for all three tests.
We show the RX port voltage statistics in a series of 5 ns windows. The window start time is swept from 0 to 25 ns in 5 ns steps, shown in (a) to (f), respectively.
}
\end{figure*}

In Section II we have introduced a method to include the direct short-orbit (SO) contribution to the TD-RCM.
The proposed method computes the time-domain SO contribution from a TX port to a RX port ($V_{SO}$), and directly adds $V_{SO}$ to the TD-RCM computed RX port voltages ($V_{port} \rightarrow V_{port} + V_{SO}$).
Here we study the effectiveness of the SO addition method by studying the detailed distribution of $V_{port}$ in the time domain.
We focus on the early-time ($< 8 \, T_{rt} = 40$ ns after the pulse injection) dynamics when the SO contribution dominates the induced voltage at the RX port.
We compare $V_{port}$ histogram data obtained from three different tests: the gigabox short-pulse experiments (Section III), the corresponding TD-RCM simulations that neglect SO, and the TD-RCM simulation that includes the SO contribution.
The distributions of the $V_{port}(t)$ values for different time intervals are shown in Fig. \ref{fig:VrxPDF}.
Because the injected pulse duration is 5 ns, we set the time window as 5 ns and look at the relatively earlier times (before $6 T_{rt} = 30$ ns) in $V_{port}(t)$.
The experimental and the RCM + SO histograms consistently show good agreement throughout the time evolution of the waveform.
However, the RCM-only results show a clear deviation from the experimental results during the 5 to 15 ns (1-3 times $T_{rt}$) time interval.
We believe the RCM-only results show smaller values of $V_{port}(t)$ due to a lack of SO-delivered energy.
As shown in Figs. \ref{fig:addSO} (a) and (c), the time-domain SO contribution shows the strongest amplitude contribution from 10 to 15 ns.
Inside the same time window, we also observe the most significant deviation between the RCM-only $V_{port}$ statistics and the other two data sets (Fig. \ref{fig:VrxPDF} (c)).
At a later time (15 ns after the pulse injection), we find that the $V_{port}$ statistics from the three data sets show a relatively good agreement (Figs. \ref{fig:VrxPDF} (d-f)).
We find that the late-time $V_{port}$ distribution fits nicely with a Gaussian distribution (not shown here).
On the other hand, a substantial deviation from Gaussian distribution in $V_{port}$ statistics marks the existence of substantial SO contributions in the system.
Examples of strong SO effects are the experimental and the RCM + SO datasets in Figs. \ref{fig:VrxPDF} (b) and (c).

\section{VI. Discussion and Summary}

The TD-RCM is a statistical treatment of time-dependent induced voltage signals in electrically-large complex enclosures.
It eschews geometrical details and represents the modes of the cavity using the random plane wave hypothesis.
A significant limitation of TD-RCM is the neglect of the nonzero time for signals to propagate through the cavity.
In making the random plane wave approximation, the model neglects the correlation between modes that carry the early-time information about the location of the ports.
The inclusion of the short orbit time domain response restores, to a degree, this information.

Another practical limitation of TD-RCM is the trade-off between computing speed and system mode density.
The TD-RCM models each and every one of the $N$ system eigenmodes that are excited by an input pulse.
We can estimate the required number of eigenmodes $N=BW/(\Delta f|_{f_c})$, where $BW$ is the bandwidth of the injected pulse, and $\Delta f|_{f_c}$ represents the system mean-mode-spacing at the pulse center frequency $f_c$.
Here is an estimate of the scale of $N$ in our gigabox short-pulse experiment.
For a 5 ns pulse at $f_c = 5$ GHz, there are $N \sim 8500$ modes excited.
Hence, the computation speed of the TD-RCM model is limited by the total number of modes that must be modeled.
A method of systematically reducing the total number of degrees of freedom to simulate is desired.

To sum up, the TD-RCM model computes the temporal evolution of the cavity modes and port signals.
The workflow of TD-RCM starts by generating a list of cavity modes using RMT.
Each of the cavity modes is characterized by a driven damped harmonic oscillator equation.
The port signal is computed by a random summation over all excited cavity modes.
A TD-RCM user will need to input the basic enclosure information (mode-density and RCM loss parameter $\alpha$), port information ($R_{rad}$ and voltage-current relationship for all nonlinear port loads), and the injected waveform.
An ensemble can be created by changing the list of eigenmodes and the mode-port random coupling coefficients.
One can then study many statistical properties of the time-dependent port voltages and system-mode dynamics, only a few of which have been discussed here.

To validate the TD-RCM, we have conducted short-pulse injection experiments into a large complex enclosure and studied the statistics of the peculiar/rare incidents (maximum receiver (RX) port voltage) in the time domain.
A relatively good agreement between the experimental data and the TD-RCM simulation is observed.
We have included the effect of SO by directly adding the SO signal obtained from the ensemble-averaged frequency domain scattering properties to the TD-RCM simulated RX port voltages.
We have also worked out the formalism for including nonlinear port loads, and demonstrated its use in Appendices C and D.

Concerning applications, one can use TD-RCM to simulate both the linear and nonlinear time-reversed wave focusing process between two ports in a complex scattering environment \cite{lerosey2004time, lerosey2006time, anlage2007new, taddese2010sensing, Frazier2013, hong2013focusing, Frazier2013a, hong2014nonlinear, Hong2015, Xiao2016, Frisco2019, 9130060}.
TD-RCM can also be applied to coda wave interferometry applications \cite{snieder2002coda, lobkis2003coda, Kuhl2005, snieder2006theory}, and the analysis of scattering fidelity studies \cite{schafer2005fidelity, gorin2006dynamics, taddese2009sensor}.
The reverberant wave-based reservoir computing hardware can also be studied with the nonlinear TD-RCM treatment (see Appendix D for an overview of the TD-RCM model of a resonant microwave cavity realization of a reservoir computer \cite{Ma2022}).
A TD-RCM user can set up a collection of SO signals to study the coherent energy delivery between the ports when the cavity geometry is known \cite{Xiao2016}.
For future directions, one may study the nonlinear properties of diode-loaded port with both TD-RCM and experimental methods \cite{Ma2020,Ma2020b}.
Another future direction is the inclusion of large multi-mode apertures between complex enclosures in TD-RCM.
Finally, TD-RCM may also be applied to simulate the linear and nonlinear dynamics for other systems which RMT can model.
Examples range from optical pulses in multi-mode fibers \cite{Xiong2016} to financial market fluctuations \cite{Fenn2011}.

\section{Acknowledgements}

We acknowledge discussions with Bisrat Addissie and Zachary Drikas of the US Naval Research Laboratory.
This work was supported by ONR under Grant No. N000141912481, DARPA Grant HR00112120021, and the Maryland Quantum Materials Center.

\section{Appendix}

\subsection{A. Generation of System Eigenmodes with RMT}

One will need to provide a list of eigenmodes of the closed system to initiate TD-RCM.
It is possible to obtain all cavity eigenmodes accurately using numerical simulation tools (CST, HFSS, COMSOL).
However, such a process would be time-consuming for an electrically large system and impossible when the accurate geometrical details are unknown.
Here we utilize the property of a chaotic system to simulate a list of system modes by only knowing the macroscopic system information (i.e., volume, area) and the operating frequency \cite{Gradoni2014}.

As introduced in the main text, a list of system eigenmodes $\omega_n$ can be generated using Random Matrix Theory (RMT).
We start by computing the eigenvalues $\lambda_{rmt}$ of large random matrices.
The matrix size equals the total number of system modes that we want to model, and the matrix elements are drawn from a Gaussian distribution controlled by the symmetry group of the system \cite{Dyson1962}.
The main symmetry classes of concern are the Gaussian Orthogonal and Gaussian Unitary ensembles of random matrices.
The system eigenmodes $\omega_n$ are computed from the relation $\lambda_{rmt} = \frac{\omega_c^2 - \omega_n^2}{\Delta \omega^2}$, where the quantity $\omega_c$ is the center frequency of the band that we want to model, and $\Delta \omega^2$ is the mean-mode-spacing near $\omega_c$.
To sum up, one first uses the total number of modes ($N$) to generate $N$ random matrix eigenvalues $\lambda_{rmt}$, and then uses the center frequency and mean-mode-spacing to compute $N$ system modes $\omega_n$ from $\lambda_{rmt}$.

\section{B. Derivation of Time Domain Random Coupling Model}

In this section, we outline the derivation of TD-RCM.
We start by representing the EM field inside the closed cavity in a basis of modes, $\mathbf{e}_n, \mathbf{h}_n, \phi_{n'}$, where
\begin{equation} \label{eq:eq1}
\begin{aligned}
    & \nabla \times \mathbf{e}_n = k_n \mathbf{h}_n\\
    & \nabla \times \mathbf{h}_n = k_n \mathbf{e}_n\\
    & \nabla^2 \phi_{n'} + \beta_n^2 \phi_{n'} = 0 \\
\end{aligned} \quad.
\end{equation}
The quantities $\mathbf{e}_n$ and $\mathbf{h}_n$ are the electric and magnetic fields for the $n^{th}$ cavity mode, $\phi_{n'}$ is potential of the electrostatic modes, and $\beta$ and $k$ are the corresponding wavenumbers.
For boundary conditions, we have the tangential components of $\mathbf{e}_n = 0$ at the boundary and the normal component of $\mathbf{h}_n = 0$.
For electrostatic modes, we have $\phi_{n'} = 0$ on the boundary.
Between two different cavity modes (mode $m$ and mode $n$, $m \neq n$), we have the orthogonal relationship
\begin{equation} \label{eq:ortho}
\begin{aligned}
    & \int_V d^3x \, \mathbf{e}_m \cdot \mathbf{e}_n = 0 \\
    & \int_V d^3x \, \mathbf{h}_m \cdot \mathbf{h}_n = 0 \\
\end{aligned} \quad.
\end{equation}
The quantity $V$ is the cavity volume.
We define a normalization process,
\begin{equation} \label{eq:normal}
    \int_V d^3x \, \mathbf{e}_n \cdot \mathbf{e}_n = k_n^{-1} = \int_V d^3x \, \mathbf{h}_n \cdot \mathbf{h}_n. \\
\end{equation}
Similarly, for the electrostatic modes, we have
\begin{equation} \label{eq:ortho2}
\begin{aligned}
    & \int_V d^3x \, \nabla \phi_{m'} \cdot \phi_{n'} = 0, \quad m' \neq n', \\
    & \int_V d^3x \, \nabla \phi_{n'} \cdot \phi_{n'} = \beta_{n'}^{-1}. \\
\end{aligned}
\end{equation}

We next expand the total EM field inside the cavity $\mathbf{E}$ and $\mathbf{H}$ in the basis of cavity modes,
\begin{equation} \label{eq:modes}
\begin{aligned}
    & \mathbf{E} = \sum_n U_n \mathbf{e}_n + \sum_{n'} U_{n'} \nabla \phi_{n'} \\
    & \mathbf{H} = \sum_n I_n \mathbf{h}_n \\
\end{aligned} \quad.
\end{equation}
Here we use $U_n$ and $I_n$ to represent the voltage and current of mode $n$.
Then we plug Eq. \ref{eq:modes} to the Maxwell's equations.
We dot Faraday's law with $\mathbf{h}_n$ and integrate over volume,
\begin{widetext}
    \begin{equation} \label{eq:max1}
    -\mu \frac{\partial}{\partial t} \int_V d^3x \, \mathbf{h}_n \cdot \mathbf{H} = -\frac{\mu}{k_n} \frac{\partial}{\partial t} I_n = \int_V d^3x \, \mathbf{h}_n \cdot \nabla \times \mathbf{E} = \int_V d^3x \, \mathbf{E} \cdot \nabla \times \mathbf{h}_n = U_n.
    \end{equation}
\end{widetext}
We dot Ampere's law with $\mathbf{e}_n$,
\begin{widetext}
\begin{equation} \label{eq:max2}
    \epsilon \frac{\partial}{\partial t} \int_V d^3x \, \mathbf{e}_n \cdot \mathbf{E} = \frac{\epsilon}{k_n} \frac{\partial}{\partial t} U_n = -\int_V d^3x \, \mathbf{e}_n \cdot \mathbf{J} + k_n\int_V d^3x \, \mathbf{h}_n \cdot \mathbf{H}.
\end{equation}
\end{widetext}
The quantities $\epsilon$ and $\mu$ are the permittivity and permeability of the system environment.
The current density $\mathbf{J}$ above has two components: the leakage current due to bulk conductivity representing cavity losses, and the driving current due to port current $I_j$ ($j$ is the index of the port),
\begin{equation} \label{eq:current}
    \mathbf{J} = \sigma \mathbf{E} + \sum_{ports} I_j \mathbf{u}_j.
\end{equation}
Here $\mathbf{u}_j$ is a profile function for the port $j$ current density.
It is normalized with unit magnitude has units $L^{-2}$.
The quantity $\sigma$ is the conductivity of the cavity wall.
Then the first term in Eq. \ref{eq:max2} becomes
\begin{widetext}
\begin{equation} \label{eq:e7}
    \int_V d^3x \, \mathbf{e}_n \cdot \mathbf{J} = \int_V d^3x \, \mathbf{e}_n \cdot \left( \sigma \mathbf{E} + \sum_{ports} I_j \mathbf{u}_j \right) = \frac{\sigma}{k_n} U_n + \sum_{ports} I_j \hat{c}_{nj}.
\end{equation}
\end{widetext}
Here the dimensionless coupling coefficient is defined $\hat{c}_{nj} = \int_V d^3x \, \mathbf{e}_n \cdot \mathbf{u}_j$, and the Ampere's law (Eq. \ref{eq:max2}) becomes
\begin{equation} \label{eq:amp2}
    \frac{\epsilon}{k_n} \frac{\partial}{\partial t} U_n + \frac{\sigma}{k_n} U_n = - \sum_{ports} I_j \hat{c}_{nj} + I_n.
\end{equation}
The port current $I_j$ will also excite an electrostatic field component,
\begin{equation} \label{eq:amp3}
    \epsilon \frac{\partial}{\partial t} \int_V d^3x \, \nabla \phi_{n'} \cdot \mathbf{E} = \frac{\epsilon}{\beta_{n'}} \frac{\partial}{\partial t} U_{n'} = - \frac{\sigma}{\beta_{n'}} U_{n'} - \sum_{ports} V_j \hat{c}'_{n'j}.
\end{equation}
Similarly we introduce a coupling coefficient $\hat{c}'_{n'j} = \int_V d^3x \, \nabla \phi_{n'} \cdot \mathbf{u}_j$.
Finally, we have the induced electric field at the port $j$ written as
\begin{equation} \label{eq:portVV}
    V_j = - \int_V d^3x \, \mathbf{u}_j \cdot \mathbf{E} = - \sum_{em-modes} U_n \hat{c}_{nj} - \sum_{es-modes} U_{n'} \hat{c}'_{n'j}.
\end{equation}

We write the complete time domain equations here
\begin{equation} \label{eq:all1}
\begin{aligned}
    -\mu \frac{\partial}{\partial t} I_n &= k_n U_n \\
    \frac{\epsilon}{k_n} \frac{\partial}{\partial t} U_n + \frac{\sigma}{k_n} U_n &= - \sum_{ports} \hat{c}_{nj} I_j + I_n \\
    \frac{\epsilon}{\beta_{n'}} \frac{\partial}{\partial t} U_{n'} + \frac{\sigma}{\beta_{n'}} U_{n'} &= - \sum_{ports} \hat{c}'_{n'j} I_j \\
\end{aligned} \quad.
\end{equation}
The coupling coefficients are $\hat{c}_{nj} = \int_V d^3x \, \mathbf{e}_n \cdot \mathbf{u}_j$ and $\hat{c}'_{n'j} = \int_V d^3x \, \nabla \phi_{n'} \cdot \mathbf{u}_j$.
We rewrite Eq. \ref{eq:all1} by eliminating the mode currents ($I_n$),
\begin{equation} \label{eq:all2}
\begin{aligned}
    & \frac{\partial^2}{\partial t^2} U_n + \frac{\sigma}{\epsilon} \frac{\partial}{\partial t} U_n + \frac{k_n^2}{\epsilon\mu} U_n = - \sqrt{\frac{\mu}{\epsilon}} \frac{k_n}{\sqrt{\mu \epsilon}} \sum_{ports} \hat{c}_{nj} \frac{\partial}{\partial t} I_j\\
    & \frac{\partial}{\partial t} U_{n'} + \frac{\sigma}{\epsilon} U_{n'} = - \frac{\beta_{n'}}{\epsilon} \sum_{ports} \hat{c}'_{n'j} I_j. \\
    & V_j = - \sum_{em-modes} U_n \hat{c}_{nj} - \sum_{es-modes} U_{n'} \hat{c}'_{n'j}.
\end{aligned}
\end{equation}

Here we study the port-mode coefficients $\hat{c}_{nj}$ in Eq. \ref{eq:all2} by switching to the frequency domain version.
We have
\begin{equation} \label{eq:f1}
    (\omega_n^2 - i\omega \gamma - \omega^2) U_n = i \sqrt{\frac{\mu}{\epsilon}} \omega_n \omega \sum_{ports} \hat{c}_{nj} I_j
\end{equation}
where $\gamma = \frac{\sigma}{\epsilon}$ and $\frac{k_n}{\sqrt{\mu \epsilon}} = \omega_n$.
The electrostatic modes are characterized by $(\gamma - i\omega) U_{n'} = - \frac{\beta_{n'}}{\epsilon} \sum_{ports} \hat{c}_{n'j} I_{n'}$.
The port voltage is written as
\begin{widetext}
\begin{equation} \label{eq:f2}
    V_j = -i \omega \sqrt{\frac{\mu}{\epsilon}} \sum_{em-modes} \sum_{ports} \frac{\omega_n \hat{c}_{nj'} \hat{c}_{nj}}{(\omega_n^2 - i\omega \gamma - \omega^2)} I_{j'} + \sum_{es-modes} \sum_{ports} \frac{\beta_{n'}}{\epsilon} \frac{\hat{c}_{n'j'} \hat{c}_{n'j}}{\gamma - i \omega} I_{j'}
\end{equation}
\end{widetext}
where $j$ and $j'$ are the indices of different ports.
The impedance matrix writes as $Z_{jj'} = -i \omega \sqrt{\frac{\mu}{\epsilon}} \sum_{em-modes} \frac{\omega_n \hat{c}_{nj'} \hat{c}_{nj}}{(\omega_n^2 - i\omega \gamma - \omega^2)}$

We next relate elements of the impedance matrix to the radiation resistance of the ports.
We first introduce the mode density.
We use $N(\omega)$ as the number of cavity modes with $\omega < \omega_n$.
Because we work with cavities with a high mode density, we have $\sum_{em-modes} \rightarrow \int d\omega_n N'(\omega_n), \quad N'(\omega_n) = dN(\omega)/d\omega |_{\omega = \omega_n}$.
The real part of the impedance $Re(Z_{jj'}) = -i \omega \sqrt{\frac{\mu}{\epsilon}} \int d\omega_n N' \frac{\omega_n \hat{c}_{nj'} \hat{c}_{nj}}{(\omega_n^2 - i\omega \gamma - \omega^2)} \approx  \omega \sqrt{\frac{\mu}{\epsilon}} \frac{\pi}{2} N' \hat{c}_{nj'} \hat{c}_{nj}$.

We redefine the coupling coefficient as $\sqrt{\frac{\mu}{\epsilon}} \hat{c}_{nj'} \hat{c}_{nj} = \left[ R_{rad}(\omega_n) / \left( \frac{\pi}{2} \omega_n N' \right) \right] c_{nj}^2$ where $c_{nj}$ is a zero mean, unit variance Gaussian random variable.
The quantity $R_{rad,j}$ is the radiation resistance of port $j$.
Thus we have defined the mode-port coupling coefficient as $c_{nj} = \left[ \sqrt{\frac{\epsilon}{\mu}} R_{rad, j}(\omega_n) / \left( \frac{\pi}{2} \omega_n N' \right) \right]^{1/2} \hat{c}_{nj}$.
We also have the system impedance matrix as
\begin{equation} \label{eq:f3}
    Z_{jj'} = -i \omega \sum_{em-modes} \frac{2(R_{rad,j}R_{rad,j'})^{1/2}}{\pi N'} \frac{c_{nj'} c_{nj}}{(\omega_n^2 - i\omega \gamma - \omega^2)}.
\end{equation}
Note that Eq. \ref{eq:f3} has the same form as the frequency domain single cavity RCM \cite{Gradoni2014}.

The final TD-RCM mode equations are
\begin{widetext}
\begin{equation} \label{eq:all4}
\begin{aligned}
    & \frac{\partial^2}{\partial t^2} U_n + \gamma_n \frac{\partial}{\partial t} U_n + \omega_n^2 U_n = - \left( \frac{\mu}{\epsilon} \right)^{1/4} \omega_n \sum_{ports} \left[ \frac{2R_{rad, j}(\omega_n)}{\pi \omega_n N'} \right]^{1/2} c_{nj} \frac{\partial}{\partial t} I_j\\
    & \gamma_n = \omega_n / Q_n \\
    & C \frac{\partial}{\partial t} V_{jc} = - I_j \\
\end{aligned} \quad.
\end{equation}
\end{widetext}
Here $N'(\omega_n)= 1/\Delta \omega_n$ is the inverse of the mode spacing at $\omega_n$, and $V_{jc}$ describes the electrostatic contribution.
The quantity $C$ is the port radiation reactance at lower frequencies as in the main text.
The TD-RCM port voltage equation is
\begin{equation} \label{eq:all5}
    V_j = - \left( \frac{\epsilon}{\mu} \right)^{1/4} \sum_{em-modes} \left[ \frac{2R_{rad, j}(\omega_n)}{\pi \omega_n N'} \right]^{1/2} c_{jn} U_n - V_{jc}.
\end{equation}
With Eqs. \ref{eq:all4} and \ref{eq:all5}, we have the complete equations of motion of TD-RCM.
The coefficient $K_{nj} = - \left( \frac{\mu}{\epsilon} \right )^{\frac{1}{4}} \left[ \frac{2R_{rad,j}(\omega_n) \, \Delta\omega_n}{\pi\omega_n } \right]^{\frac{1}{2}}$ is the combined coefficient on the RHS of the mode equation Eq. \ref{eq:all4}, and $K'_{jn}$ is the corresponding coefficient in Eq. \ref{eq:all5}.

\subsection{C. TD-RCM Treatment of Linear and Non-Linear Loads - Spectral Response}

\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{nlinload3.png}
\caption{\label{fig:nlin}
The TD-RCM simulated RX port signals. (a) and (b) show the time-domain RX port signal $V_{port}$ for a 50 $\Omega$ linear load and a diode load, respectively. The Fourier transform of the time-domain waveforms in (a) and (b) are shown in (c) and (d).
}
\end{figure*}

To expand the applicability of the TD-RCM model, we have presented the treatment of nonlinear port loads in Section III.B.
Here we test the nonlinear capability of TD-RCM with a simple single-tone stimulation of a cavity with a nonlinear output port.
In this numerical exercise, we inject a single frequency 5 GHz sine-wave for 50 ns through a linear TX port into a model cavity with $M=2$ ports, and study the induced voltage signal at a second RX port in the cavity.
The loss parameter of the system is set at $\alpha = 1$, and $N=1000$ system modes are used.
The mean-mode-spacing is set as $\Delta k^2 = 4\pi/A = 109.3 m^{-2}$ where $A = 0.115 m^2$ is the two-dimensional billiard area.
For convenience, the frequency-independent radiation resistance of both ports is set to $(18 + 50 i)\Omega$ (measured at 5 GHz).
Because we are only testing the port load responses, we have fixed the system spectrum ($\omega_n$ in Eq. \ref{eq:modeV}) and the port-mode coupling coefficients  ($c_{jn}$ in Eqs. \ref{eq:modeV} and \ref{eq:portV}).
For the linear system test, the load connected at the RX port is a 50 $\Omega$ linear impedance (Fig. \ref{fig:portload} (a)).
For the nonlinear system test, the RX port is a diode-loaded port, as shown in Fig. \ref{fig:portload} (b).
The following parameters are used for the diode-loaded port ($R = 50\Omega, C = 1.5e^{-20}F, I_0 = 1e^{-7}A$, and $T = 300K$).
The amplitude of the CW input sine-wave is 5V.
The computed time-domain linear and nonlinear RX port results are shown in Figs. \ref{fig:nlin} (a) and (b), respectively.
One can already see hints of nonlinearity from Fig. \ref{fig:nlin} (b) by eye.
We next show the spectrum of the linear and nonlinear time-domain waveforms in Figs. \ref{fig:nlin} (c) and (d), respectively.
Here, the linear-port spectrum is dominated by one peak at the fundamental frequency (5 GHz), while the spectrum of the nonlinear-port shows a clear peak at the second and several higher harmonics.
Thus the TD-RCM is capable of simulating nonlinear elements and emulating proper nonlinear behaviors.
Note that the speed of TD-RCM computation will slow down when nonlinear elements are included.

\subsection{D. TD-RCM Application: Reverberant Wave Reservoir Computing Hardware}

\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{RCtask.png}
\caption{\label{fig:Rossler}
The Rossler-observer task testing set performance.
The trained RC is given a continuous stream of the x(t) data generated from the coupled nonlinear Rossler equations and has to deduce the evolution of the y(t) and z(t) time series based only on training data.
(a) and (b) show the Rossler-y and Rossler-z series data, respectively.
The TD-RCM simulated reservoir computing data are shown in red dotted lines. The actual states are shown in solid blue lines.
The x-axis shows the index of the testing, ranging from the $32000^{th}$ to the $40000^{th}$ data points.
Each data index corresponds to a time bin of duration 250 ps.
}
\end{figure*}

Here we aim to demonstrate one successful application of the TD-RCM method.
Recently in Ref. \cite{Ma2022}, we have experimentally realized a physical system, namely reverberant wave reservoir computing (RC) hardware, which can execute multiple machine learning tasks on time-domain signals.
The RC hardware is based on a wave chaotic enclosure with multiple diode-loaded ports.
To operate the wave-based RC, one injects the input data stream into a wave chaotic enclosure through a linear port and then measures the induced voltage signal at nonlinear ports at several discrete locations.
These voltage signals will serve as the `neuron' responses in the traditional software RC.
After the training, the desired output signal is obtained by properly summing up the RC neuron responses \cite{Lu2016, Pathak2018}.
A RC realization requires a system with a complex and diverse set of modes, and requires a nonlinear evolution of the system state.
A ray-chaotic microwave cavity with diode-loaded output ports is very well suited for this task.
We use TD-RCM with nonlinear ports to simulate the microwave realization of an RC.

Here we utilized the nonlinear version of TD-RCM to simulate the dynamics of the wave-based RC hardware.
The diode-loaded ports are modeled as in Section III.B.
We have put a single linear input port and 100 diode-loaded output ports in our TD-RCM cavity model computation.
All diodes have the same characteristics ($R = 50\Omega, C = 1.5e^{-12}F, I_0 = 1e^{-7}A$, and $T = 300K$).
The specific type of machine learning (ML) task we studied here is the so-called Rossler observer task \cite{Lu2016}.
The Rossler system is governed by these equations,
\begin{equation} \label{eq:Rossler}
\begin{aligned}
    \Dot{x} &= -y-z,\\
    \Dot{y} &= x+ay,\\
    \Dot{z} &= b+z(x-c),\\
    \end{aligned}
\end{equation}
where $a=0.5, b=2.1, c=3.5$ and the over-dot denotes derivative with respect to time.
In this task, the ML model is expected to predict the Rossler attractor $y$- and $z$-component data when only the $x$-component information is supplied.
More details of the Rossler observer task setup can be found in Refs. \cite{Lu2016, Ma2022}.
The training of the wave-based RC is completed with a so-called out coupling matrix $W_{out}$.
The process of finding the optimized $W_{out}$ matrix is completed by running the Ridge regression algorithm on the digital computer \cite{Lu2016}.
Here, the training phase used $80\%$ of the entire dataset (80000 data points uniformly sampled in $\sim 200$ oscillation periods).
With optimized cavity parameters ($\alpha = 300$ and a 35 V input amplitude), the testing set results are shown in Fig. \ref{fig:Rossler}.
In this case, the trained RC is fed the $x(t)$ time domain waveform generated by an arbitrary waveform generator with a step size of 250 ps.
The task for the RC is to continuously generate the $y(t)$ and $z(t)$ time domain signals without knowledge of the underlying equations of motion for the Rossler attractor.
The performance of the RC prediction can be evaluated using the normalized mean square error, e.g., defined for the $y$-variable as NMSE $= \sum_n[s_n - s_n']^2/\sum_n s_n^2$, where $s'$ and $s$ denote true and RC-inferred values for the Rossler time series, respectively, and $n$ is the time index of the testing set data.
The $s'$ true signal is denoted as "from data", and the RC-inferred signal $s$ is denoted "from RC" in Fig. \ref{fig:Rossler}.
The NMSE is computed to be 0.055 and 0.083 for the results shown in Fig. \ref{fig:Rossler} (a) and (b), respectively, showing that the TD-RCM simulated RC can perform this ML task with high accuracy.
We note that NMSE generally decreases with an increasing number of output ports.

With TD-RCM, one can study the dynamics of the reverberant wave RC with great fidelity and detail because all system parameters can be easily controlled.
It is also possible to study the cavity realization of the RC with electromagnetic numerical simulation software \cite{Ma2022}.
However, the computing speed of TD-RCM is much faster than the numerical EM simulation because no accurate CAD drawing and the corresponding spatial-domain mesh-solving are required.

\end{spacing}