Time-Modulated Intelligent Reflecting Surface

• Time-Modulated Intelligent Reflecting Surface (TM-IRS) is a technology that varies per-element reflection coefficients over an OFDM symbol period to induce direction-dependent inter-subcarrier mixing.
• It employs coordinated on/off switching and programmable phase control to scramble signals in unintended directions while preserving signal integrity for authorized users.
• TM-IRS design methodologies span rule-based and learning-based approaches, optimizing secure multi-user communications, radar beampatterning, and physical-layer security.

Searching arXiv for the cited TM-IRS papers and closely related background. Searching for "Time-Modulated Intelligent Reflecting Surface" on arXiv. Time-modulated intelligent reflecting surface (TM-IRS) denotes an intelligent reflecting surface whose per-element reflection coefficients vary within an OFDM symbol period through coordinated on/off switching and phase control, rather than remaining static over the symbol. In the formulation developed for secure OFDM transmission, the switching period is aligned with the OFDM subcarrier spacing so that the induced harmonics create direction-dependent inter-subcarrier mixing: authorized directions can be made to observe near-undistorted constellations, whereas other directions observe scrambled mixtures of subcarriers (Xu et al., 2023). Subsequent work extended the concept from waveform security to secure multi-user directional modulation designed by generative flow networks (GFlowNets) (Tao et al., 17 Jun 2025), and then to integrated sensing, communication, and security in dual-function radar-communication systems, where TM-IRS configurations are learned to jointly optimize multi-user rate, radar beampatterning, and physical-layer security (Tao et al., 6 Sep 2025). A broader electromagnetic antecedent is the time-varying IRS literature that treats spatiotemporal modulation and delay-controlled sideband steering from an aperture perspective (Yurduseven et al., 2020).

1. Historical development and conceptual position

A conventional IRS applies static unit-modulus phase shifts $c_{mn}$ across a passive aperture to shape a reflected wavefront toward desired directions. TM-IRS augments this by introducing per-element time variation $U_{mn}(t)$, typically implemented through high-speed single-pole-single-throw switches together with programmable phase shifters, so that each element is periodically turned on and off during an OFDM symbol (Tao et al., 6 Sep 2025). The essential distinction is that the time dimension does not merely retune a beam; it creates harmonic components aligned with OFDM subcarrier spacing and thereby induces structured, direction-dependent inter-subcarrier interference.

From an electromagnetic viewpoint, IRS control was first framed as holographic wavefront synthesis, in which the reflected aperture field is matched to a desired near-field focus or far-field beam through a phase profile derived from the incident reference wave and the desired radiated field. That literature later introduced a time-varying IRS concept based on time-delay control across unit-cell lattices, producing sidebands at $f_c \pm m f_0$ with harmonic-specific spatial steering (Yurduseven et al., 2020). This established the spatiotemporal interpretation on which later communication-oriented TM-IRS work rests.

The OFDM waveform-security formulation made the communication mechanism explicit. It showed that periodic switching at frequency $f_s$, equal to the subcarrier spacing, aliases the harmonics onto the OFDM subcarrier grid, so that the reflected signal on any given subcarrier becomes a weighted sum of all transmitted subcarriers. By choosing the TM pattern appropriately, the desired direction retains the original symbol mapping while unintended directions experience deterministic scrambling (Xu et al., 2023).

The 2025 line of work reframed parameter design as learning over a vast discrete space. In the multi-user secure TM-IRS design, GFlowNets were used to sample parameter configurations with probability proportional to a non-negative reward based on authorized users’ sum rate under constellation-integrity constraints (Tao et al., 17 Jun 2025). In the ISAC extension, the same learning principle was generalized to a dual-function radar-communication setting with worst-case secrecy and radar beampattern constraints, thereby tying TM-IRS to physical-layer security and sensing rather than waveform scrambling alone (Tao et al., 6 Sep 2025).

2. Signal model and harmonic mixing mechanism

In the OFDM formulations, the base station or transmitter emits

$$e(t)=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1} d(k)e^{j2\pi(f_c+kf_s)t}, \qquad 0\le t<T_s,$$

where $K$ is the number of subcarriers, $f_s$ is the subcarrier spacing, $f_c$ is the carrier, and $d(k)$ are zero-mean unit-variance symbols such as QPSK (Tao et al., 6 Sep 2025). Each IRS element $(m,n)$ applies a unit-modulus phase $U_{mn}(t)$0 and a periodic square-wave switching function

$U_{mn}(t)$1

where $U_{mn}(t)$2 is the turn-on instant and $U_{mn}(t)$3 is the on-duration, both normalized to the OFDM symbol period (Xu et al., 2023, Tao et al., 6 Sep 2025).

The IRS array response enters through the steering factor. In the ISAC formulation, the IRS steering vector is

$U_{mn}(t)$4

with half-wavelength spacing along both axes (Tao et al., 6 Sep 2025). After substituting the OFDM waveform and the Fourier series of the switching function, the reflected field can be regrouped into harmonic components. The corresponding coefficient governing the $U_{mn}(t)$5-th harmonic is

$U_{mn}(t)$6

where $U_{mn}(t)$7 (Tao et al., 17 Jun 2025, Tao et al., 6 Sep 2025).

After OFDM demodulation, the received symbol on subcarrier $U_{mn}(t)$8 takes the form

$U_{mn}(t)$9

in the ISAC model, and an analogous form appears in the secure multi-user model without explicit channel factors (Tao et al., 6 Sep 2025, Tao et al., 17 Jun 2025). The implication is central: each demodulated subcarrier is a direction-dependent linear combination of all transmitted subcarriers. The authorized direction is therefore not privileged a priori; it must be engineered through $f_c \pm m f_0$0 so that the main term $f_c \pm m f_0$1 dominates and the off-diagonal terms $f_c \pm m f_0$2 are suppressed or otherwise rendered non-destructive.

This mechanism is closely related to the harmonic beampattern formulation of the waveform-security paper, which writes the IRS-assisted contribution on subcarrier $f_c \pm m f_0$3 as

$f_c \pm m f_0$4

where $f_c \pm m f_0$5 is the $f_c \pm m f_0$6-th harmonic beampattern coefficient (Xu et al., 2023). In that notation, $f_c \pm m f_0$7 preserves the original mapping, whereas $f_c \pm m f_0$8 generates deterministic inter-carrier interference. The electromagnetic sideband-steering literature suggests the same underlying structure from an aperture perspective: time variation produces sidebands, and the spatial phase associated with each sideband can be controlled independently through per-element delays (Yurduseven et al., 2020).

3. Security, communication, and sensing criteria

The communication metric used in the multi-user and ISAC formulations is a per-subcarrier SINR that treats the off-diagonal harmonic terms as interference. For a legitimate communication user $f_c \pm m f_0$9,

$f_s$0

with achievable rate

$f_s$1

To preserve constellation integrity, the desired harmonic is also phase-constrained:

$f_s$2

where $f_s$3 depends on the modulation format; for $f_s$4-PSK, the secure multi-user paper states $f_s$5 (Tao et al., 17 Jun 2025, Tao et al., 6 Sep 2025).

The ISAC extension adds a sensing requirement through the radar beampattern. The instantaneous power toward the target direction is

$f_s$6

and the average beampattern over one OFDM symbol is

$f_s$7

The resulting secrecy metric is worst-case over a discretized suspected eavesdropper region $f_s$8:

$f_s$9

The design then maximizes $$e(t)=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1} d(k)e^{j2\pi(f_c+kf_s)t}, \qquad 0\le t<T_s,$$0 subject to the sensing threshold $$e(t)=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1} d(k)e^{j2\pi(f_c+kf_s)t}, \qquad 0\le t<T_s,$$1 and the constellation-phase constraints (Tao et al., 6 Sep 2025).

A recurring misconception is that TM-IRS security is necessarily identical to information-theoretic secrecy. The waveform-security formulation explicitly distinguishes waveform security from secrecy-rate guarantees: the former relies on deterministic angle-dependent constellation scrambling, whereas secrecy-rate optimization introduces channel-based rate differences under explicit worst-case or statistical models (Xu et al., 2023). The ISAC formulation moves closer to physical-layer secrecy by optimizing worst-case secrecy rates over a directional uncertainty set, but it still exploits the same direction-dependent inter-subcarrier mixing mechanism rather than requiring precise eavesdropper CSI (Tao et al., 6 Sep 2025).

4. TM-IRS design methodologies

The earliest OFDM TM-IRS designs were rule-based and closed-form. In the waveform-security paper, the element phases are first chosen to equalize the legitimate direction,

$$e(t)=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1} d(k)e^{j2\pi(f_c+kf_s)t}, \qquad 0\le t<T_s,$$2

so that array-dependent phases cancel at the authorized receiver. With a common duty cycle $$e(t)=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1} d(k)e^{j2\pi(f_c+kf_s)t}, \qquad 0\le t<T_s,$$3, the received symbol in Bob’s direction reduces to a sum containing

$$e(t)=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1} d(k)e^{j2\pi(f_c+kf_s)t}, \qquad 0\le t<T_s,$$4

and the design enforces this phasor sum to vanish for all $$e(t)=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1} d(k)e^{j2\pi(f_c+kf_s)t}, \qquad 0\le t<T_s,$$5, thereby eliminating harmonics at Bob (Xu et al., 2023). Two implementation modes are defined. In the linear mode, common TM parameters are imposed per row or per column, reducing hardware complexity because one high-speed switch per row or column suffices. In the planar mode, each element receives its own timing parameter, which yields stronger sidelobe suppression at the cost of more switches and finer control (Xu et al., 2023).

The GFlowNet-based approach reformulates TM-IRS parameter selection as a deterministic Markov decision process. A state encodes a partial assignment of the quantized parameters $$e(t)=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1} d(k)e^{j2\pi(f_c+kf_s)t}, \qquad 0\le t<T_s,$$6, $$e(t)=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1} d(k)e^{j2\pi(f_c+kf_s)t}, \qquad 0\le t<T_s,$$7, and $$e(t)=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1} d(k)e^{j2\pi(f_c+kf_s)t}, \qquad 0\le t<T_s,$$8 across all IRS elements; an action selects one unassigned parameter and fixes it to a discrete value; after $$e(t)=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1} d(k)e^{j2\pi(f_c+kf_s)t}, \qquad 0\le t<T_s,$$9 actions a terminal state is reached that represents a complete configuration $K$0 (Tao et al., 17 Jun 2025, Tao et al., 6 Sep 2025). In the secure multi-user design, the terminal reward is

$K$1

so infeasible terminal states receive zero reward (Tao et al., 17 Jun 2025). In the ISAC extension, the reward becomes

$K$2

which incorporates worst-case secrecy and sensing simultaneously. The paper notes that this multiplicative form enforces hard constraints and avoids hyperparameter tuning of additive penalty weights (Tao et al., 6 Sep 2025).

Learning is performed through forward and backward policies parameterized by a feedforward DNN and trained by the trajectory balance objective

$K$3

for a root-to-terminal trajectory $K$4 ending at terminal state $K$5 (Tao et al., 17 Jun 2025, Tao et al., 6 Sep 2025). Sampling masks already assigned parameters, applies softmax with temperature annealing, and updates the network and $K$6 using Adam. The stated objective is not to identify a single optimum only, but to learn a stochastic policy that samples diverse high-reward configurations with probability proportional to reward.

5. Reported experimental behavior

The secure multi-user GFlowNet study uses a $K$7 IRS, an $K$8 antenna ULA transmitter, $K$9 OFDM subcarriers, QPSK, path loss set to $f_s$0, and SNR $f_s$1 dB; the single-user experiments analytically fix $f_s$2 and train on $f_s$3 trajectories with learning rate $f_s$4 for $f_s$5 steps and $f_s$6 for $f_s$7 (Tao et al., 17 Jun 2025). The ISAC study uses $f_s$8, $f_s$9, $f_c$0, $f_c$1 OFDM symbols, QPSK, SNR $f_c$2 dB unless otherwise stated, and a $f_c$3-hidden-layer network with $f_c$4 neurons per layer trained on $f_c$5 trajectories (Tao et al., 6 Sep 2025).

A central empirical observation in both GFlowNet papers is sample efficiency. The single-user parameter space with $f_c$6, $f_c$7 has size $f_c$8 when only $f_c$9 and $d(k)$0 are optimized, yet the method is reported to converge after training on fewer than $d(k)$1 of all possible configurations (Tao et al., 17 Jun 2025). The ISAC paper states that the GFlowNet learns from only $d(k)$2 sampled trajectories, which is also less than $d(k)$3 of the space, while generalizing the reward distribution to unvisited configurations (Tao et al., 6 Sep 2025).

For single-user security, both the rule-based TM-IRS design and the GFlowNet design achieve near-zero SER at the desired direction $d(k)$4 and high SER near the target direction $d(k)$5 in the ISAC experiments (Tao et al., 6 Sep 2025). The secure multi-user study similarly reports low SER in the authorized direction and high SER elsewhere, confirming secure directional modulation (Tao et al., 17 Jun 2025). Both papers also note a limitation of static high-reward patterns: some unintended directions may still exhibit low SER. The mitigation proposed in both cases is to exploit diversity in the learned stochastic policy by switching among multiple high-reward TM configurations every $d(k)$6 OFDM symbols, which preserves low SER for legitimate users while averaging out leakages in unintended directions (Tao et al., 17 Jun 2025, Tao et al., 6 Sep 2025).

For multi-user operation, the GFlowNet framework is reported to support two users at $d(k)$7 and $d(k)$8 with low SER in both directions (Tao et al., 17 Jun 2025). In the ISAC study, the achievable sum rate versus $d(k)$9 exhibits strong peaks at the two communication-user directions, while the SER remains near-zero there and the sensing constraint is satisfied (Tao et al., 6 Sep 2025). Under equal iteration budgets, the ISAC paper further reports that GFlowNet consistently attains higher sum rates than simulated annealing with initial temperature $(m,n)$0 and decay $(m,n)$1, as well as random sampling, once SNR leaves the noise-limited regime (Tao et al., 6 Sep 2025). At SNR $(m,n)$2 dB, the same study reports much lower SER at the intended direction than the rule-based TM-IRS design, attributing the improvement to SINR-aware optimization that increases $(m,n)$3 while suppressing off-diagonal $(m,n)$4 (Tao et al., 6 Sep 2025).

6. Implementation constraints, comparative interpretation, and open directions

Hardware feasibility in the reported TM-IRS designs rests on three ingredients: unit-modulus phase control $(m,n)$5, high-speed SPST switching, and precise timing of $(m,n)$6 and $(m,n)$7 within the OFDM symbol. The ISAC study states that quantization levels $(m,n)$8 for the time parameters are feasible for digital controllers and that $(m,n)$9 phase states are standard for IRS hardware (Tao et al., 6 Sep 2025). The waveform-security paper emphasizes the hardware trade-off between linear mode and planar mode: linear mode requires fewer switches and simpler control, whereas planar mode yields stronger sidelobe suppression (Xu et al., 2023).

Several limitations are explicit in the literature. Time modulation produces harmonics and sidebands, so sideband leakage, timing jitter, and phase noise can perturb the ideal coefficients $U_{mn}(t)$00 or $U_{mn}(t)$01 (Xu et al., 2023, Tao et al., 6 Sep 2025). The ISAC model assumes a strong adversary in which eavesdroppers and communication users can compensate their channels to the IRS, treats LOS to the target as the dominant radar return, and omits weak NLOS sensing contributions via the IRS (Tao et al., 6 Sep 2025). The secure multi-user formulation omits channel terms in order to isolate the TM-IRS effect and assumes legitimate receivers know their channels for compensation (Tao et al., 17 Jun 2025). These assumptions delimit the scope of current guarantees.

Comparatively, conventional IRS beamforming improves SNR through static phase alignment but cannot by itself create direction-dependent symbol scrambling. TM-IRS adds a time dimension that induces controlled harmonic mixing across OFDM subcarriers and thereby realizes directional modulation without relying on precise eavesdropper CSI (Xu et al., 2023, Tao et al., 6 Sep 2025). Relative to artificial noise, the waveform-security paper argues that TM-IRS does not add noise power that could degrade Bob’s SNR; instead it creates deterministic inter-carrier interference in unintended directions (Xu et al., 2023). Relative to time-modulated transmit arrays, the same paper argues that IRS aperture gain helps offset the loss associated with periodic deactivation of elements (Xu et al., 2023). From the electromagnetic side, time-delay-based spatiotemporal control connects TM-IRS to broader mechanisms of sideband steering and selective focusing or defocusing; in near-field demonstrations, focusing versus randomized phases produced about $U_{mn}(t)$02 dB amplitude contrast at the intended receiver depth (Yurduseven et al., 2020).

The open problems identified in the 2025 papers are largely methodological and systems-oriented. They include real-time adaptation and low-latency online sampling, robustness to CSI errors and dynamic environments, training under hardware nonidealities such as timing jitter and quantizer limitations, and scalability to larger IRSs, richer quantization, and more users (Tao et al., 17 Jun 2025, Tao et al., 6 Sep 2025). The ISAC study additionally suggests lighter architectures such as CNNs or GNNs and curriculum training as possible directions for reducing training cost in much larger discrete design spaces (Tao et al., 6 Sep 2025). A plausible implication is that TM-IRS research is moving from closed-form waveform-preserving constructions toward learned stochastic control over large combinatorial spaces, while retaining the same physical mechanism: spatiotemporal modulation of a passive aperture to shape who observes coherent OFDM symbols and who observes scrambled harmonic mixtures.