Coupled TRC Phase Shift Model

• Coupled TRC phase shift models are a family of frameworks where phase responses are constrained by coupling between channels, pathways, or control variables.
• They manifest in diverse settings such as Fano–Feshbach resonances, passive STAR-RIS systems with quadrature laws, and PRF-based oscillator networks with memory effects.
• The models inform both theoretical analyses and practical designs by highlighting constraints that turn independent phase variables into coupled, hybrid continuous–discrete control systems.

Searching arXiv for the exact topic phrase and closely related variants to ground the terminology and identify relevant papers. “Coupled TRC phase shift model” is not a single standardized construct across the arXiv literature. In the supplied corpus, it refers to several distinct frameworks in which a phase variable is constrained by coupling to another channel, pathway, or control variable. The term labels a two-channel coupled-square-well description of Fano–Feshbach resonances with a time-domain phase fixed by the Fano asymmetry parameter (Yan et al., 2017); passive, lossless STAR-RIS models in which transmission and reflection coefficients satisfy exact amplitude and quadrature constraints (Liu et al., 2021); and a generalized phase-shift response framework, mapped explicitly to the Phase Response Function (PRF), for strong or frequent pulse interactions in coupled oscillators (Klinshov et al., 2017). This suggests that the phrase is best treated as a context-dependent family of coupled phase-shift formalisms rather than a unique model.

1. Terminological scope and recurring structure

In the STAR-RIS literature, TRC denotes the transmission and reflection coefficients of each element, and the central issue is that the transmission and reflection phase shifts cannot be adjusted independently for purely passive, lossless hardware (Liu et al., 2021). In the oscillator literature represented here, “TRC” is explicitly mapped to the PRF formalism, where the phase shift caused by the current pulse depends on the history of several preceding pulses (Klinshov et al., 2017). Other papers in the corpus make clear that the term itself is not universal: the desynchronization study states that “The term ‘TRC’ does not appear in this paper” and instead analyzes an order-parameter-dependent phase shift in a nonlinear extension of the Kuramoto model (Burylko et al., 2011).

Despite this terminological heterogeneity, the recurring mathematical pattern is a constrained phase response generated by coupling. In the Fano–Feshbach setting, the phase is set by discrete–continuum interference through $\phi = 2\,\mathrm{arg}(q-i)$ (Yan et al., 2017). In passive STAR-RIS models, the phase of transmission is a $\pm \pi/2$-shifted copy of the reflection phase under energy-splitting operation (Zhong et al., 2022). In PRF-based oscillator reductions, the phase increment is history-dependent because residual amplitude variables have not decayed before the next pulse arrives (Klinshov et al., 2017).

2. Coupled-square-well and Fano–Feshbach phase correspondence

In the coupled-square-well formulation, one considers two coupled s-wave channels: an open channel $o$ with threshold set to zero, and a closed channel $c$ with threshold $E_c^{th} > 0$. The coupled-channel Hamiltonian acts on the two-component wavefunction $\Psi(r) = (\psi_o(r), \psi_c(r))^T$ as

$$H = \begin{pmatrix} -\frac{\hbar^2}{2m}\nabla^2 + V_o(r) & W(r) \ W(r) & -\frac{\hbar^2}{2m}\nabla^2 + V_c(r) \end{pmatrix},$$

with square wells inside a radius $a$,

$$V_o(r) = -V_{o0}\,\Theta(a - r),\quad V_c(r) = -V_{c0}\,\Theta(a - r),\quad W(r) = W_0\,\Theta(a - r).$$

If the closed channel supports a bound state $|\alpha\rangle$ of energy near the open-channel threshold, coupling $\pm \pi/2$0 hybridizes that discrete level with the open-channel continuum $\pm \pi/2$1, producing a Feshbach resonance; in the frequency domain, such discrete–continuum interference yields a Fano line shape, and in the time domain, it yields a transient response with a characteristic phase (Yan et al., 2017).

Near a single, isolated resonance at energy $\pm \pi/2$2, the absorption or scattering line shape takes the Fano form

$\pm \pi/2$3

In the weak-coupling regime and with a flat background near $\pm \pi/2$4, Fano’s configuration interaction theory leads to

$\pm \pi/2$5

and in the coupled-square-well model with fixed well parameters this implies

$\pm \pi/2$6

After a short $\pm \pi/2$7-like pulse at $\pm \pi/2$8, the time-dependent dipole response is the shifted Fourier transform of the line shape,

$\pm \pi/2$9

and the resonant part takes the form

$o$0

The phase is determined by the complex factor $o$1,

$o$2

For real $o$3 this reduces to

$o$4

The mapping implies that $o$5 gives $o$6, $o$7 gives $o$8, and $o$9 yields $c$0 between $c$1 and $c$2 depending on magnitude. The paper states that this phase–$c$3 correspondence is general for any Fano resonance in the weak coupling regime, independent of the transition mechanism, provided there is a single isolated Fano resonance and a flat background (Yan et al., 2017).

Within this model, resonance energies and widths are obtained from the poles of the open-channel $c$4-matrix in the complex energy plane. Matching the short-range and long-range solutions produces Wigner’s $c$5-matrix,

$c$6

and poles $c$7 solve

$c$8

With $c$9, $E_c^{th} > 0$0, and $E_c^{th} > 0$1 in hand, the transient phase is fixed by

$E_c^{th} > 0$2

A common misconception is that this correspondence is specific to electric-dipole excitation; the supplied description states that it is independent of whether the transition is electric dipole, magnetic dipole, or another mechanism, and depends instead on isolation of the resonance and weak coupling (Yan et al., 2017).

3. Passive, lossless STAR-RIS coupled transmission–reflection models

In passive, lossless STAR-RIS models, the per-element complex transmission and reflection coefficients are written as

$E_c^{th} > 0$3

with amplitudes in $E_c^{th} > 0$4 and phases in $E_c^{th} > 0$5 or $E_c^{th} > 0$6 depending on convention. The central hardware constraints are amplitude coupling through passivity and phase coupling through a quadrature law. In the energy-splitting mode,

$E_c^{th} > 0$7

and

$E_c^{th} > 0$8

Element-wise, one phase is treated as continuous and the other as a discrete offset,

$E_c^{th} > 0$9

so amplitudes are continuously tunable in ES mode, while one side’s phase is determined via a binary decision (Zhong et al., 2022).

This passive-lossless constraint is also expressed as

$\Psi(r) = (\psi_o(r), \psi_c(r))^T$0

with the consequence that if $\Psi(r) = (\psi_o(r), \psi_c(r))^T$1 and $\Psi(r) = (\psi_o(r), \psi_c(r))^T$2, then

$\Psi(r) = (\psi_o(r), \psi_c(r))^T$3

The literature repeatedly contrasts this with the ideal independent model, where transmission and reflection amplitudes and phases can be set independently subject only to energy conservation and hardware bounds. The coupled model “markedly shrinks the feasible set” and turns one of the two phase controls into a binary choice tied to the other (Zhong et al., 2022).

This hardware realism directly affects system modeling. In a two-user downlink STAR-RIS setting, the effective channels are

$\Psi(r) = (\psi_o(r), \psi_c(r))^T$4

while in multi-user MISO form the received signal on each side includes the direct path plus a STAR-RIS-mediated path with diagonal matrices $\Psi(r) = (\psi_o(r), \psi_c(r))^T$5 and $\Psi(r) = (\psi_o(r), \psi_c(r))^T$6 (Xu et al., 2021). A common misconception is that independent transmission and reflection phase control is a generic passive capability; the supplied papers state that passive, lossless elements cannot realize arbitrary independent T/R phase responses, and that independent control would require active or lossy implementations (Liu et al., 2021).

The coupled model also changes diversity and outage behavior. In the two-user passive, lossless STAR-RIS analysis, the primary–secondary phase-shift configuration (PS-PSC), diversity preserving phase-shift configuration (DP-PSC), and T/R-group phase-shift configuration (TR-PSC) strategies were derived analytically. The reported asymptotic diversity orders are:

• PS-PSC: $\Psi(r) = (\psi_o(r), \psi_c(r))^T$7, $\Psi(r) = (\psi_o(r), \psi_c(r))^T$8.
• DP-PSC: $\Psi(r) = (\psi_o(r), \psi_c(r))^T$9.
• TR-PSC: $$H = \begin{pmatrix} -\frac{\hbar^2}{2m}\nabla^2 + V_o(r) & W(r) \ W(r) & -\frac{\hbar^2}{2m}\nabla^2 + V_c(r) \end{pmatrix},$$0, $$H = \begin{pmatrix} -\frac{\hbar^2}{2m}\nabla^2 + V_o(r) & W(r) \ W(r) & -\frac{\hbar^2}{2m}\nabla^2 + V_c(r) \end{pmatrix},$$1.
• Random PSC: $$H = \begin{pmatrix} -\frac{\hbar^2}{2m}\nabla^2 + V_o(r) & W(r) \ W(r) & -\frac{\hbar^2}{2m}\nabla^2 + V_c(r) \end{pmatrix},$$2. The same study reports that DP-PSC achieves full diversity order simultaneously for users located on both sides of the STAR-RIS and has a comparable power scaling law with only about $$H = \begin{pmatrix} -\frac{\hbar^2}{2m}\nabla^2 + V_o(r) & W(r) \ W(r) & -\frac{\hbar^2}{2m}\nabla^2 + V_c(r) \end{pmatrix},$$3 dB reduction in received power relative to the independent upper bound, quantified as

$$H = \begin{pmatrix} -\frac{\hbar^2}{2m}\nabla^2 + V_o(r) & W(r) \ W(r) & -\frac{\hbar^2}{2m}\nabla^2 + V_c(r) \end{pmatrix},$$4

(Xu et al., 2021).

4. Optimization, learning, and secure control under coupled STAR-RIS constraints

Once the coupled TRC law is imposed, optimization becomes intrinsically hybrid. In a multi-user downlink MISO system, one formulation minimizes long-term BS power subject to minimum-rate constraints and the per-element coupling condition

$$H = \begin{pmatrix} -\frac{\hbar^2}{2m}\nabla^2 + V_o(r) & W(r) \ W(r) & -\frac{\hbar^2}{2m}\nabla^2 + V_c(r) \end{pmatrix},$$5

which in practice reduces to $$H = \begin{pmatrix} -\frac{\hbar^2}{2m}\nabla^2 + V_o(r) & W(r) \ W(r) & -\frac{\hbar^2}{2m}\nabla^2 + V_c(r) \end{pmatrix},$$6 for non-degenerate amplitudes. This turns the beamforming problem from fully continuous to hybrid continuous–discrete. The corresponding hybrid DDPG algorithm maps a normalized output $$H = \begin{pmatrix} -\frac{\hbar^2}{2m}\nabla^2 + V_o(r) & W(r) \ W(r) & -\frac{\hbar^2}{2m}\nabla^2 + V_c(r) \end{pmatrix},$$7 to

$$H = \begin{pmatrix} -\frac{\hbar^2}{2m}\nabla^2 + V_o(r) & W(r) \ W(r) & -\frac{\hbar^2}{2m}\nabla^2 + V_c(r) \end{pmatrix},$$8

and

$$H = \begin{pmatrix} -\frac{\hbar^2}{2m}\nabla^2 + V_o(r) & W(r) \ W(r) & -\frac{\hbar^2}{2m}\nabla^2 + V_c(r) \end{pmatrix},$$9

while the joint DDPG–DQN method assigns the continuous variables to DDPG and the binary $a$0 offsets to DQN (Zhong et al., 2022).

The reported performance trends are specific. In that study, STAR-RIS outperforms “double spliced” opposite-facing RIS by about $a$1 in the reported reward metric, both proposed algorithms outperform a baseline DDPG, and joint DDPG–DQN delivers about $a$2–$a$3 higher reward than hybrid DDPG at increased computational complexity (Zhong et al., 2022). In the airborne STAR-RIS setting, the coupled law is preserved by construction through

$a$4

with $a$5. The Dual Actor DDPG then separates continuous actions from discrete offset choices. Simulations report that accumulated reward improves by $a$6 versus single-actor DDPG and by $a$7 versus DQN, that three-dimensional UAV trajectory optimization achieves $a$8 higher communication efficiency than two-dimensional and altitude optimization, and that the HFI-based reward lowers QoS denial rates by about $a$9 compared to JFI-based benchmarks (Rizvi et al., 30 Aug 2025).

A complementary line of work treats the coupled phase-shift model through deterministic optimization. The general optimization framework based on WMMSE, penalty dual decomposition, and block coordinate descent enforces

$$V_o(r) = -V_{o0}\,\Theta(a - r),\quad V_c(r) = -V_{c0}\,\Theta(a - r),\quad W(r) = W_0\,\Theta(a - r).$$0

and provides closed-form auxiliary phase and amplitude updates. The paper states that the framework converges to the Karush-Kuhn-Tucker optimal solution under some mild conditions and that throughput with coupled phase shifts is very close to the independent-phase model (Wang et al., 2022).

Security-oriented formulations retain the same coupled element law. In the fairness-oriented secrecy problem, each STAR-RIS element satisfies

$$V_o(r) = -V_{o0}\,\Theta(a - r),\quad V_c(r) = -V_{c0}\,\Theta(a - r),\quad W(r) = W_0\,\Theta(a - r).$$1

with

$$V_o(r) = -V_{o0}\,\Theta(a - r),\quad V_c(r) = -V_{c0}\,\Theta(a - r),\quad W(r) = W_0\,\Theta(a - r).$$2

The penalty-based secrecy beamforming algorithm alternates active beamforming, lifted coefficient updates, and closed-form per-element phase and amplitude steps. The same study reports that the proposed scheme achieves higher secrecy capacity than conventional RIS and that $$V_o(r) = -V_{o0}\,\Theta(a - r),\quad V_c(r) = -V_{c0}\,\Theta(a - r),\quad W(r) = W_0\,\Theta(a - r).$$3-bit discrete phase shifters are sufficient for secrecy guarantee (Zhang et al., 2022).

5. Oscillator-network formulations: PRF memory, RC phase shift, and switching couplings

In the oscillator literature represented here, the closest formal definition of a coupled TRC phase-shift model is the PRF-based construction. The classical PRC assumption is that the effect of the stimulus vanishes before the next one arrives. The PRF generalizes this by introducing memory: for a pulse train with arrivals at times $$V_o(r) = -V_{o0}\,\Theta(a - r),\quad V_c(r) = -V_{c0}\,\Theta(a - r),\quad W(r) = W_0\,\Theta(a - r).$$4 and attributes $$V_o(r) = -V_{o0}\,\Theta(a - r),\quad V_c(r) = -V_{c0}\,\Theta(a - r),\quad W(r) = W_0\,\Theta(a - r).$$5, the discrete phase update is

$$V_o(r) = -V_{o0}\,\Theta(a - r),\quad V_c(r) = -V_{c0}\,\Theta(a - r),\quad W(r) = W_0\,\Theta(a - r).$$6

and for coupled oscillators the event-driven model becomes

$$V_o(r) = -V_{o0}\,\Theta(a - r),\quad V_c(r) = -V_{c0}\,\Theta(a - r),\quad W(r) = W_0\,\Theta(a - r).$$7

For short pulses, the paper derives the approximation

$$V_o(r) = -V_{o0}\,\Theta(a - r),\quad V_c(r) = -V_{c0}\,\Theta(a - r),\quad W(r) = W_0\,\Theta(a - r).$$8

showing exponentially decaying memory governed by $$V_o(r) = -V_{o0}\,\Theta(a - r),\quad V_c(r) = -V_{c0}\,\Theta(a - r),\quad W(r) = W_0\,\Theta(a - r).$$9 with $|\alpha\rangle$0 (Klinshov et al., 2017). The key misconception addressed there is that PRC remains adequate under strong or frequent stimulation; the paper states that PRC fails because the phase shift caused by each pulse depends on the history of several previous pulses.

The same general theme appears in oscillator systems with explicitly structured phase-shift mechanisms. In the RC phase-shift network based Chua’s circuits, each RC stage is approximated by

$|\alpha\rangle$1

and the triple-RC chain produces upstream and downstream phase offsets that allow complete synchronization, approximate lag synchronization, and approximate anticipating synchronization without delay or parameter mismatch. The reported similarity-function minima are $|\alpha\rangle$2 at $|\alpha\rangle$3 for anticipating synchronization and $|\alpha\rangle$4 at $|\alpha\rangle$5 for lag synchronization (Srinivasan et al., 2012).

A different coupled phase-shift mechanism is order-parameter-dependent lag in nonlinear Kuramoto–Sakaguchi systems,

$|\alpha\rangle$6

with the example

$|\alpha\rangle$7

Here the full synchronous state loses stability when $|\alpha\rangle$8, and the transition to partial synchrony proceeds through stable cluster states for small ensembles or directly to periodic or quasiperiodic regimes for larger ensembles (Burylko et al., 2011). In a related but distinct model, periodically switching couplings produce a temporal response delay in the order parameter, and the paper reports a “phase-shift inversion” near the synchronization transition, where the order parameter at the minimum interaction density can even be larger than that at the maximum interaction density (Lee et al., 2011).

The most explicit phase-shift-symmetry-breaking generalization in this group is

$|\alpha\rangle$9

which breaks continuous phase-shift symmetry and yields incoherent, oscillatory synchronized, and non-oscillatory synchronized states, including a three-state coexistence region (Manoranjani et al., 2021).

6. Related formulations, misconceptions, and cross-domain significance

Several additional papers in the corpus treat coupled phase shifts without using the same TRC nomenclature. In strongly coupled cavity QED, the transmitted phase is

$\pm \pi/2$00

and near antiresonance,

$\pm \pi/2$01

so that the antiresonance occurs exactly at $\pm \pi/2$02 and its width is set solely by the atom (Sames et al., 2013). In trapped two-particle coupled-channel scattering, the integrated correlation function satisfies

$\pm \pi/2$03

so the relation depends explicitly on the phase shifts in both channels but not on the inelasticity (Guo et al., 2024). In practical RIS hardware, a different amplitude–phase coupling law appears,

$\pm \pi/2$04

with configuration-set selection and capacity maximization performed under discrete phase choices (Hashemi et al., 2024).

The corpus also identifies a terminological pitfall: some works are about phase-shift detection rather than phase-shift generation. The change-point paper analyzes instantaneous phase shifts extracted via Hilbert-transform-based complex demodulation and proposes CUSUM and phase-derivative estimators; it explicitly notes that “The paper does not use the term ‘TRC’” (Marshall et al., 2014). This distinction matters because detection of phase-shift events, coupled phase dynamics, and hardware-coupled transmission/reflection coefficients are mathematically different problems even when the same words recur.

Across these domains, the strongest common misconception is that “coupled phase shift” denotes a single transferable formalism. The supplied papers do not support that reading. What they do support is a narrower shared principle: coupling restricts phase freedom and converts a nominally independent phase variable into one determined by another channel, another pathway, a memory buffer, or a hardware law. In Fano physics that restriction appears as $\pm \pi/2$05 (Yan et al., 2017); in passive STAR-RIS design it appears as exact quadrature and amplitude conservation (Liu et al., 2021); in PRF reductions it appears as a finite-memory functional of recent pulses (Klinshov et al., 2017). This suggests that “coupled TRC phase shift model” functions best as a descriptive umbrella for phase-shift models in which coupling is constitutive rather than perturbative.