Step-Like Time Modulation in Dynamic Systems

• Step-like time modulation is the use of piecewise-constant temporal profiles to abruptly change system parameters with controlled boundary matching.
• It underpins diverse applications such as harmonic generation, non-reciprocal scattering, Floquet dynamics, and entanglement initialization across various physical systems.
• By engineering temporal discontinuities, the method enables precise control over resonance, waveform synthesis, and spectral characteristics in both classical and quantum regimes.

Searching arXiv for the cited papers and closely related work on step-like time modulation. I’m checking whether the arXiv search capability is available in this session. Step-like time modulation denotes the use of piecewise-constant temporal profiles—single Heaviside jumps, rectangular pulses, multi-step profiles, or finite sequences of constant-duration steps—to vary a system parameter in time. Across current research, the modulated quantity may be an interface stiffness $K(t)=K_{1}+\Delta K\,H(t-t_{0})$, a plasma frequency $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$, a tunneling rate $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$, a coupling constant $g(t)$, or a periodically repeated two-step Hamiltonian with total period $T=T_{1}+T_{2}$ (Darche et al., 29 Mar 2025, Rizza et al., 2024, Fernandes et al., 13 Mar 2025, Lizunova et al., 2020, Liang et al., 2020). The same formal idea also appears in time-modulated arrays and in continuous-time realizations of AFDM, where the waveform is made piecewise constant in instantaneous frequency over each sampling interval while preserving the discrete AFDM samples (Maneiro-Catoira et al., 2024, Gao et al., 2020, Cao et al., 12 May 2026). In this broad sense, step-like time modulation is a unifying construction for harmonic generation, non-reciprocity, temporal reflection and transmission control, Floquet engineering, entanglement initialization, and spectrally cleaner waveform synthesis.

1. Canonical mathematical forms

A defining feature of step-like time modulation is that the governing parameter is constant on subintervals and discontinuous only at prescribed times. In the simplest single-jump case, the modulation is represented by a Heaviside function. Examples include the interface stiffness law

$K(t)=K_{1}+\Delta K\,H(t-t_{0}),$

the moving-index step

$n(z,t)=n_{1}+(n_{2}-n_{1})\,H(z-v_{\rm m}t),$

and the quench protocol

$$g(t)= \begin{cases} g_{1}, & t<0,\ g_{2}, & t>0. \end{cases}$$

These forms appear, respectively, in transient scattering at modulated interfaces, moving space-time discontinuities, and topological-field-theory quenches (Darche et al., 29 Mar 2025, Li et al., 2022, Lizunova et al., 2020).

A second archetype is the finite rectangular pulse. In interacting charge qubits, the lead-dot tunneling rate is switched on for a finite duration,

$$\Gamma(t)\equiv \Gamma_{0}\ \text{for}\ 0<t<\sigma,\qquad \Gamma(t)=0\ \text{otherwise},$$

or equivalently $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$. The pulse width is also written in dimensionless form as $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$0, where $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$1 is the fraction of the two-level oscillation period (Fernandes et al., 13 Mar 2025).

A third archetype is a periodic multi-step drive. In the driven transverse-field Ising model, one period consists of two constant Hamiltonians,

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$2

with quasiperiodic couplings $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$3 entering only in the second step (Liang et al., 2020). In purely time-modulated media, an $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$4-step piecewise-constant $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$5 is assigned to intervals $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$6, yielding a temporal slab in an effective-medium limit (Sini et al., 2023).

These formulations share a common algebraic consequence: exact solutions are built by matching across temporal boundaries. Depending on context, the matching variables are fields and their time derivatives, scattering coefficients, density matrices, or discrete-time Floquet operators. This suggests that the principal analytic burden of step-like time modulation is not the evolution within an interval, which is constant-parameter dynamics, but the interfacial constraints at the switching times.

2. Scattering at time-modulated interfaces and moving steps

In one-dimensional transient scattering with a modulated interface at $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$7, the bulk equations are

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$8

and the imperfect interface is governed by

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$9

For a step in interface stiffness, $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$0, the displacement remains continuous and the stress jump follows the time-varying stiffness. In frequency space, a single-frequency input no longer stays single-frequency: the abrupt change couples all frequencies and produces a continuum of temporal sidebands. The associated energy balance is

$\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$1

so a positive jump $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$2 injects energy $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$3, while a negative jump extracts it. The same framework identifies non-reciprocity, because waves arriving from opposite directions at the same absolute time generally see different instantaneous interface states (Darche et al., 29 Mar 2025).

A related but distinct problem is scattering from a moving space-time modulation step. In the laboratory frame,

$\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$4

Lorentz-frame hopping yields generalized phase matching and the critical-angle formula

$\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$5

For codirectional modulation ($\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$6), $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$7 is smaller than the static critical angle; for contradirectional modulation ($\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$8), it is larger and may exceed $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$9. In the codirectional case, $g(t)$0 as $g(t)$1, beyond which the incident wave cannot catch up with the interface (Li et al., 2022).

The space-time Fresnel prism reformulates the moving-step problem into a finite structure that repeatedly re-injects the same abrupt modulation. For the ideal moving interface, the transmitted and reflected waves each acquire a single shifted frequency,

$g(t)$2

with closed-form amplitudes

$g(t)$3

Two finite realizations are given. Prism I has useful-energy fraction $g(t)$4, while Prism II yields

$g(t)$5

with $g(t)$6 in a typical blue-shift design where $g(t)$7 and $g(t)$8 (Li et al., 2023).

3. Pure-time media, temporal slabs, and natural resonances

Step-like time modulation can also act in a spatially homogeneous medium. In a Lorentz-type dispersive interface of zero thickness, the plasma frequency is switched at $g(t)$9 from $T=T_{1}+T_{2}$0 to $T=T_{1}+T_{2}$1 according to

$T=T_{1}+T_{2}$2

For TE waves, the sheet conditions are

$T=T_{1}+T_{2}$3

and the surface polarization obeys

$T=T_{1}+T_{2}$4

Across the temporal boundary, $T=T_{1}+T_{2}$5, $T=T_{1}+T_{2}$6, $T=T_{1}+T_{2}$7, and $T=T_{1}+T_{2}$8 remain continuous. After the jump, the response consists of a new forced steady term plus free oscillations

$T=T_{1}+T_{2}$9

The generated spectral lines occur at the interface’s own resonances $K(t)=K_{1}+\Delta K\,H(t-t_{0}),$0, rather than at $K(t)=K_{1}+\Delta K\,H(t-t_{0}),$1. When $K(t)=K_{1}+\Delta K\,H(t-t_{0}),$2 lies below the light line, the process couples propagating waves to evanescent ones and directly excites surface-wave modes without spatial gratings or prisms (Rizza et al., 2024).

A complementary effective-medium treatment uses an $K(t)=K_{1}+\Delta K\,H(t-t_{0}),$3-step pure-time profile in the plasma-frequency-squared,

$K(t)=K_{1}+\Delta K\,H(t-t_{0}),$4

For a plane-wave ansatz $K(t)=K_{1}+\Delta K\,H(t-t_{0}),$5, the scalar temporal equation is

$K(t)=K_{1}+\Delta K\,H(t-t_{0}),$6

In the regime $K(t)=K_{1}+\Delta K\,H(t-t_{0}),$7, an effective constant plasma frequency $K(t)=K_{1}+\Delta K\,H(t-t_{0}),$8 emerges on $K(t)=K_{1}+\Delta K\,H(t-t_{0}),$9, producing a temporal slab with

$n(z,t)=n_{1}+(n_{2}-n_{1})\,H(z-v_{\rm m}t),$0

Matching at $n(z,t)=n_{1}+(n_{2}-n_{1})\,H(z-v_{\rm m}t),$1 and $n(z,t)=n_{1}+(n_{2}-n_{1})\,H(z-v_{\rm m}t),$2 gives the reflection and transmission amplitudes

$n(z,t)=n_{1}+(n_{2}-n_{1})\,H(z-v_{\rm m}t),$3

If $n(z,t)=n_{1}+(n_{2}-n_{1})\,H(z-v_{\rm m}t),$4, then $n(z,t)=n_{1}+(n_{2}-n_{1})\,H(z-v_{\rm m}t),$5 and full transmission occurs; otherwise, as $n(z,t)=n_{1}+(n_{2}-n_{1})\,H(z-v_{\rm m}t),$6, one has $n(z,t)=n_{1}+(n_{2}-n_{1})\,H(z-v_{\rm m}t),$7 and $n(z,t)=n_{1}+(n_{2}-n_{1})\,H(z-v_{\rm m}t),$8. The terminology of a temporal “wall” and temporal “well” is used to describe these two regimes (Sini et al., 2023).

Taken together, these results show that abrupt temporal changes do not merely broaden spectra. They can select discrete resonant channels, induce full reflection or full transmission, or launch bound surface waves, depending on whether the modulation acts as a jump condition, a pure-time slab, or a dispersive temporal boundary.

4. Floquet drives, gate pulses, and temporal quenches in quantum systems

In periodically driven quantum many-body systems, step-like time modulation is a natural Floquet construction. For the transverse-field Ising chain, the one-period evolution operator is

$n(z,t)=n_{1}+(n_{2}-n_{1})\,H(z-v_{\rm m}t),$9

with

$$g(t)= \begin{cases} g_{1}, & t<0,\ g_{2}, & t>0. \end{cases}$$0

In Majorana variables, the drive factorizes into two dimer unitaries, enabling analytical conditions for Majorana zero modes and $$g(t)= \begin{cases} g_{1}, & t<0,\ g_{2}, & t>0. \end{cases}$$1 modes localized at an open boundary. In the regime where all single-particle excitations are localized by strong quasiperiodic modulation, the system exhibits eigenstate order, exact $$g(t)= \begin{cases} g_{1}, & t<0,\ g_{2}, & t>0. \end{cases}$$2-spectral pairing in the thermodynamic limit, and stable period-doubling: the autocorrelation $$g(t)= \begin{cases} g_{1}, & t<0,\ g_{2}, & t>0. \end{cases}$$3 oscillates with period $$g(t)= \begin{cases} g_{1}, & t<0,\ g_{2}, & t>0. \end{cases}$$4, with a lifetime growing exponentially in system size under open boundary conditions (Liang et al., 2020).

A finite-duration step-like pulse appears in the entanglement dynamics of two interacting charge qubits. The system density matrix obeys a Lindblad equation with a time-dependent tunneling rate $$g(t)= \begin{cases} g_{1}, & t<0,\ g_{2}, & t>0. \end{cases}$$5, and entanglement is characterized by fidelity, linear entropy,

$$g(t)= \begin{cases} g_{1}, & t<0,\ g_{2}, & t>0. \end{cases}$$6

and negativity,

$$g(t)= \begin{cases} g_{1}, & t<0,\ g_{2}, & t>0. \end{cases}$$7

Scanning the pulse parameters $$g(t)= \begin{cases} g_{1}, & t<0,\ g_{2}, & t>0. \end{cases}$$8 identifies a “sweet spot” at

$$g(t)= \begin{cases} g_{1}, & t<0,\ g_{2}, & t>0. \end{cases}$$9

where the post-pulse fidelity to $$\Gamma(t)\equiv \Gamma_{0}\ \text{for}\ 0<t<\sigma,\qquad \Gamma(t)=0\ \text{otherwise},$$0 reaches $$\Gamma(t)\equiv \Gamma_{0}\ \text{for}\ 0<t<\sigma,\qquad \Gamma(t)=0\ \text{otherwise},$$1 and the subsequent negativity peaks at values $$\Gamma(t)\equiv \Gamma_{0}\ \text{for}\ 0<t<\sigma,\qquad \Gamma(t)=0\ \text{otherwise},$$2. Small dephasing hardly affects the first entanglement peak, moderate dephasing mildly suppresses it, and large dephasing kills it almost entirely (Fernandes et al., 13 Mar 2025).

In relativistic field theories with topological defects, a time-step in the coupling constant acts as a quench,

$$\Gamma(t)\equiv \Gamma_{0}\ \text{for}\ 0<t<\sigma,\qquad \Gamma(t)=0\ \text{otherwise},$$3

The field and its first time derivative are matched continuously at $$\Gamma(t)\equiv \Gamma_{0}\ \text{for}\ 0<t<\sigma,\qquad \Gamma(t)=0\ \text{otherwise},$$4, and the kink remains a rigid object because the topological charge

$$\Gamma(t)\equiv \Gamma_{0}\ \text{for}\ 0<t<\sigma,\qquad \Gamma(t)=0\ \text{otherwise},$$5

stays constant. Energy or momentum conservation yields the “hyperbolic” Snell’s law

$$\Gamma(t)\equiv \Gamma_{0}\ \text{for}\ 0<t<\sigma,\qquad \Gamma(t)=0\ \text{otherwise},$$6

which fixes the outgoing velocity after the quench. Radiation loss is reported to be $$\Gamma(t)\equiv \Gamma_{0}\ \text{for}\ 0<t<\sigma,\qquad \Gamma(t)=0\ \text{otherwise},$$7 in the numerical confirmation (Lizunova et al., 2020).

These examples show that step-like time modulation in quantum and nonlinear systems is not tied to a single objective. The same piecewise-constant logic can generate Floquet topology, initialize entangled charge states, or renormalize soliton kinematics through conserved quantities.

5. Stair-step pulses and phase-stepped modulation in arrays

In RF and microwave beamforming, step-like time modulation is often implemented through periodic stair-step or phase-step sequences. One representative construction uses a four-state phase sequence over one modulation period $$\Gamma(t)\equiv \Gamma_{0}\ \text{for}\ 0<t<\sigma,\qquad \Gamma(t)=0\ \text{otherwise},$$8,

$$\Gamma(t)\equiv \Gamma_{0}\ \text{for}\ 0<t<\sigma,\qquad \Gamma(t)=0\ \text{otherwise},$$9

The modulation factor $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$0 has Fourier coefficients

$\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$1

because the discrete phasor sum vanishes unless $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$2. Accordingly, the fundamental $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$3, all even harmonics, and all $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$4 harmonics are suppressed. The experimental implementation uses 2-bit phase shifters realized by four delay lines $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$5 and an FPGA-controlled switch network. Reported measurements at $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$6 GHz and $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$7 MHz show suppression $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$8 dB for the fundamental and even harmonics, an undesired $\Gamma(t)=\Gamma_{0}[\Theta(t)-\Theta(t-\sigma)]$9 sideband below $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$00 dBc, and an 80$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$01 beam-scanning range; the ideal insertion loss of the time-modulated module is $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$02 dB (Gao et al., 2020).

A different four-level periodic stair-step waveform is used for time-modulated arrays: $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$03 each lasting a quarter of the modulation period $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$04. For element $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$05,

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$06

with nonzero harmonics only on $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$07 and

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$08

At the first positive harmonic,

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$09

and beam steering to $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$10 is achieved by the linear delay law

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$11

A second SPST switch per element imposes a duty cycle $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$12, enabling sidelobe-level reconfiguration. In the reported simulations, scanning over $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$13 is obtained. For a 30-element array with $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$14 and uniform $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$15, the desired $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$16 sidelobe level is $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$17 dB, the largest unwanted harmonic is at $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$18 dB, and the total efficiency is $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$19. In a beamformer mode optimized for $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$20 dB sidelobe level, all other harmonics are below $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$21 dB and $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$22 (Maneiro-Catoira et al., 2024).

These array implementations demonstrate a characteristic advantage of step-like time modulation: harmonic content is engineered by finite phasor sets and switch timing rather than by continuous analog phase profiles. The resulting structures are described as small, cost-effective, and invariant in performance with carrier frequency in the reported TMA architecture (Maneiro-Catoira et al., 2024).

6. Stepped continuous-time AFDM waveforms

In communications, step-like time modulation appears in the continuous-time realization of AFDM. The inverse discrete affine Fourier transform produces Nyquist-rate samples

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$23

Canonical piecewise continuous AFDM (PC-AFDM) exhibits high out-of-band emission because frequency wrapping introduces internal envelope jumps between AFDM sampling instants. The mechanism is analytically separated from ordinary block truncation by the asymptotic subcarrier spectrum

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$24

where $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$25 is common to any finite-duration block and $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$26 are the internal complex-envelope jumps (Cao et al., 12 May 2026).

Stepped frequency division multiplexing replaces the wrapped chirp over each sampling interval $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$27 by a constant instantaneous frequency equal to the midpoint value,

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$28

while phase is accumulated continuously: $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$29 The transmit waveform is

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$30

and satisfies

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$31

The midpoint choice is proved to be the unique sample-preserving choice for arbitrary chirp-rate parameter under continuous phase accumulation and without additional phase correction (Cao et al., 12 May 2026).

The spectral consequence is that the jump term vanishes in SFDM. For the one-sided tail of the average ESD,

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$32

whereas PC-AFDM has the additional term

$$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$33

Thus SFDM enforces $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$34 and removes the extra $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$35 tail generated by wrapping-induced discontinuities. Because SFDM and PC-AFDM coincide at the samples $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$36, the sampled modulation matrix $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$37, guard-interval structure, and receiver processing remain unchanged; under integer-multiple delays the sampled channel matrix is identical, and under fractional delays SFDM suppresses the mismatch that becomes large near PC-AFDM envelope jumps (Cao et al., 12 May 2026).

The reported numerical results use $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$38, $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$39 Hz, $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$40, and SNR $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$41 dB. The full OOBE ratio outside $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$42 is $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$43 for PC-AFDM and $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$44 for SFDM, corresponding to about $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$45 dB OOBE reduction. In a mismatched single-path fractional-delay LMMSE receiver, PC-AFDM shows EVM spikes up to $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$46, while SFDM remains smooth with peak EVM $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$47; the worst-case EVM reduction is $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$48 dB and the 95th-percentile reduction is $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$49 dB. In random three-path channels with fractional-delay estimation error $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$50, the 99th-percentile LMMSE EVM improves from $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$51 to $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$52, or $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$53 dB in the high-percentile regime, while the median EVM is nearly identical (Cao et al., 12 May 2026).

7. Interpretive themes and recurrent misconceptions

A recurrent misconception is that step-like time modulation is synonymous with periodic Floquet driving. The literature shows a wider taxonomy: a single Heaviside jump in interface stiffness, a one-shot gate pulse, a moving space-time step, an $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$54-step pure-time medium, and a periodically repeated two-step Hamiltonian are all step-like temporal protocols (Darche et al., 29 Mar 2025, Fernandes et al., 13 Mar 2025, Li et al., 2022, Sini et al., 2023, Liang et al., 2020). Periodicity is therefore one special case rather than the general definition.

A second misconception is that abrupt temporal changes merely reproduce conventional spectral leakage. In AFDM, the high-frequency spectral tail of PC-AFDM is traced to internal envelope jumps caused by frequency wrapping, distinct from ordinary block truncation. SFDM eliminates this mechanism while preserving the standard digital AFDM structure (Cao et al., 12 May 2026). In array systems, by contrast, selected harmonics vanish exactly because a finite phasor sum closes, so abrupt switching can be used to suppress, rather than inevitably create, unwanted sidebands (Gao et al., 2020).

A third misconception is that the functionality must rely on nonlinearity. The transient-interface framework explicitly states that amplification, harmonic generation, impedance matching, and non-reciprocity can be obtained “without resorting to non-linear mechanisms,” because the control variable is time dependence itself (Darche et al., 29 Mar 2025). The Lorentz-interface study similarly identifies frequency generation at natural resonances under a linear time boundary, with no need for spatial gratings or prisms (Rizza et al., 2024).

A fourth misconception is that stronger temporal contrast always means stronger reflection. The pure-time temporal-slab analysis shows a discrete exception: when $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$55, full transmission occurs even though the effective slab is finite and time varying; away from those resonant values and as $$\omega_{p}^{2}(t)=A_{1}\omega_{0}^{2}+(A_{2}-A_{1})\omega_{0}^{2}U(t)$$56, the same medium approaches full reflection (Sini et al., 2023). This suggests that in step-like time modulation the magnitude of the jump is often less decisive than the matching conditions it imposes on the system’s natural phases, eigenfrequencies, or sample constraints.

Across these domains, the unifying principle is that a temporal discontinuity is an operator that redistributes energy, phase, or state population by boundary matching at switching times. The specific observable—critical angle, resonance line, autocorrelation period, negativity peak, sidelobe level, or OOBE tail—depends on the physical setting, but the controlling structure remains piecewise-constant evolution joined by exact temporal boundary conditions.