Orthogonal Time Frequency Space (OTFS)

• OTFS is a modulation framework that processes symbols in the delay-Doppler domain, leveraging the sparsity of physical propagation paths for robust performance.
• Its mathematical foundation, using the Zak transform and symplectic Fourier transforms, enables efficient mapping between delay-Doppler and time-frequency domains for improved equalization.
• OTFS supports both communications and radar sensing by unifying delay and Doppler processing, facilitating integrated sensing-and-communication systems.

Searching arXiv for OTFS foundational, implementation, radar, and detection papers to ground the encyclopedia article. arxiv_search(query="Orthogonal Time Frequency Space OTFS delay-Doppler foundational Zak transform radar sensing detection", max_results=10) arxiv_search(query="(Mohammed et al., 2023) OTFS Mathematical Foundation for Communication and Radar Sensing in the Delay-Doppler Domain", max_results=5) Orthogonal Time Frequency Space (OTFS) is a communication and active sensing framework in which information symbols are processed natively in the delay-Doppler (DD) domain rather than directly on the time-frequency (TF) grid. Its central premise is that a doubly selective wireless channel is more naturally described by a small number of physical propagation paths, each characterized by a delay and a Doppler shift, than by rapidly fluctuating TF coefficients. OTFS therefore places symbols on a DD lattice and uses carrier waveforms matched to that representation, so that channel interaction becomes localized, predictable, and, in a suitable operating regime, non-fading; the same representation also yields a common mathematical language for radar sensing and communications (Mohammed et al., 2023, Hadani et al., 2018).

1. Delay-Doppler viewpoint and motivation

OTFS is motivated by the mismatch between conventional TF-domain modulation and high-mobility, doubly selective propagation. In the DD representation, the channel spreading function is modeled as

$$h(\tau,\nu)=\sum_i h_i\,\delta(\tau-\tau_i)\delta(\nu-\nu_i),$$

and the noiseless received signal is

$$y(t)=\sum_i h_i\,x(t-\tau_i)e^{j2\pi \nu_i (t-\tau_i)}.$$

Here, delay $\tau_i$ is tied to path length, and Doppler $\nu_i$ is tied to relative motion. Because real channels are usually determined by a small number of dominant reflectors or paths, the DD representation is compact and sparse, and its parameters evolve according to geometry and motion rather than according to rapidly changing TF fading patterns (Mohammed et al., 2023).

This DD interpretation is the principal conceptual departure from OFDM and related TF-domain schemes. OFDM is highly effective when the channel is approximately time-invariant over one symbol, but it becomes vulnerable to severe Doppler spread, inter-carrier interference, shrinking coherence time, and costly TF-domain channel tracking in high-mobility or high-carrier-frequency scenarios. OTFS instead uses the domain in which propagation is physically parameterized. In the overview literature, the DD-domain channel is described as more stable, compact, separable, and potentially sparse; a representative compact-support condition is

$4\tau_{\max}\nu_{\max}\le 1,$

which expresses the fact that realistic channels occupy a bounded region in the DD plane (Wei et al., 2020).

A recurrent claim across the OTFS literature is that DD-domain multiplexing lets each information symbol experience the full TF channel diversity of a frame rather than a single local TF coefficient. In that sense, OTFS is not merely a coordinate change. It is a modulation strategy aligned with the physical degrees of freedom of mobile multipath channels, and it is especially targeted at high user speed, large Doppler spread, large carrier frequency such as mmWave/THz, vehicular and UAV links, LEO-satellite settings, and other doubly selective environments (Mohammed et al., 2023, Wei et al., 2020).

2. Mathematical structure: Zak transform, quasi-periodicity, and DD pulses

A foundational mathematical treatment describes OTFS through the Zak transform. Choosing delay and Doppler periods $\tau_p$ and $\nu_p$ such that $\nu_p=1/\tau_p$, the time Zak transform is

$x_{\mathrm{dd}}(\tau,\nu)\Define \mathcal{Z}_t(x(t)) =\sqrt{\tau_p}\sum_{k=-\infty}^{\infty}x(\tau+k\tau_p)e^{-j2\pi \nu k\tau_p}.$

The resulting DD-domain function is quasi-periodic rather than strictly periodic, satisfying

$$x_{\mathrm{dd}}(\tau+n\tau_p,\nu+m\nu_p) =e^{j2\pi n\nu\tau_p}x_{\mathrm{dd}}(\tau,\nu), \qquad n,m\in\mathbb Z.$$

This quasi-periodic structure is the mechanism by which OTFS obtains effective joint delay-Doppler localization (Mohammed et al., 2023).

Within a fundamental DD cell, a pulse localized near $$y(t)=\sum_i h_i\,x(t-\tau_i)e^{j2\pi \nu_i (t-\tau_i)}.$$0 has characteristic width $$y(t)=\sum_i h_i\,x(t-\tau_i)e^{j2\pi \nu_i (t-\tau_i)}.$$1 along delay and $$y(t)=\sum_i h_i\,x(t-\tau_i)e^{j2\pi \nu_i (t-\tau_i)}.$$2 along Doppler. Its time-domain realization is a pulsone: a pulse train of duration $$y(t)=\sum_i h_i\,x(t-\tau_i)e^{j2\pi \nu_i (t-\tau_i)}.$$3, pulse spacing $$y(t)=\sum_i h_i\,x(t-\tau_i)e^{j2\pi \nu_i (t-\tau_i)}.$$4, pulse width $$y(t)=\sum_i h_i\,x(t-\tau_i)e^{j2\pi \nu_i (t-\tau_i)}.$$5, modulated by a tone of frequency $$y(t)=\sum_i h_i\,x(t-\tau_i)e^{j2\pi \nu_i (t-\tau_i)}.$$6. The corresponding DD pulse occupies area about

$$y(t)=\sum_i h_i\,x(t-\tau_i)e^{j2\pi \nu_i (t-\tau_i)}.$$7

while the fundamental DD cell has area $$y(t)=\sum_i h_i\,x(t-\tau_i)e^{j2\pi \nu_i (t-\tau_i)}.$$8, so one can place approximately $$y(t)=\sum_i h_i\,x(t-\tau_i)e^{j2\pi \nu_i (t-\tau_i)}.$$9 almost orthogonal DD pulses in the cell. This yields a geometric interpretation of the usual time-bandwidth product and of the Nyquist-rate degrees of freedom (Mohammed et al., 2023).

OTFS is also presented as a family interpolating between conventional modulations. As $\tau_i$0, the pulsone tends to a single time pulse, corresponding to TDM. As $\tau_i$1, equivalently $\tau_i$2, it tends to a single frequency pulse, corresponding to FDM. In this sense, OTFS generalizes time-localized and frequency-localized signaling by using carriers localized in delay and Doppler rather than only in one of the two canonical dimensions (Mohammed et al., 2023, Hadani et al., 2018).

A parallel discrete treatment derives OTFS from the discrete Zak transform (DZT), with the inverse DZT mapping DD symbols into the transmitted time-domain vector and the DZT mapping the received vector back into DD. In that view, for rectangular pulses,

$\tau_i$3

which makes explicit the unitary relation between DD and time domains and the DD-domain effective channel matrix $\tau_i$4 (Li, 2022, Aghda et al., 2022).

3. Modulation chain and implementation architectures

In its standard sampled form, OTFS places symbols

$\tau_i$5

on an $\tau_i$6 DD grid. A common implementation uses the inverse symplectic finite Fourier transform (ISFFT) to map DD symbols to TF symbols,

$\tau_i$7

followed by a Heisenberg transform to synthesize the time-domain waveform. At the receiver, a Wigner transform or matched filtering returns TF samples, and the SFFT maps them back to DD-domain observations $\tau_i$8 (Raviteja et al., 2017, Raviteja et al., 2019).

A foundational Zak-domain formulation expresses the full modulator-demodulator chain as DD symbols $\tau_i$9 DD pulse shaping by twisted convolution $\nu_i$0 inverse Zak transform to time $\nu_i$1 channel $\nu_i$2 Zak transform back to DD $\nu_i$3 DD matched filtering $\nu_i$4 DD symbol samples. The effective sampled DD channel is itself obtained by twisted convolution of the physical channel with transmit and receive DD windows. This is a distinctive feature: pulse shaping is native to the DD operator structure rather than borrowed from a TF interpretation (Mohammed et al., 2023).

Several implementation viewpoints coexist. OTFS is often described as an overlay on OFDM hardware, since one may insert the ISFFT/SFFT around an otherwise standard multicarrier front end. A discrete-time vectorized analysis of OFDM-based OTFS makes every block explicit—ISFFT, windowing, OFDM modulation/demodulation, CP insertion/removal, and the linear time-varying channel—and extends naturally to MIMO (RezazadehReyhani et al., 2017). A reduced-CP interpretation goes further and describes OTFS as block-OFDM with a single cyclic prefix and deterministic time interleaving; the exact sample relation

$\nu_i$5

is used to show that reduced-CP OTFS can be viewed as block OFDM plus sample reordering (Rangamgari et al., 2020).

The literature also distinguishes Zak-OTFS from MC-OTFS. In Zak-OTFS, DD signals are quasi-periodic and DD-time mapping is direct via the Zak transform. In MC-OTFS, the mapping is approximated through an inverse symplectic finite Fourier transform to TF and then a Heisenberg transform to time. The foundational treatment suggests that Zak-OTFS may offer performance and complexity advantages, although the practical comparison remains open (Mohammed et al., 2023).

4. Input-output law, equalization, and detection

A central OTFS result is that the DD-domain input-output relation is structured. In one form it is written as a twisted convolution,

$\nu_i$6

and in overview treatments it is described as a 2D circular convolution between the DD symbol array and the effective DD channel (Mohammed et al., 2023, Wei et al., 2020). The point is not that symbols decouple one-tap-wise, as in ideal OFDM over delay-only channels, but that their coupling is localized, structured, and tied to the bounded DD support of the channel.

The pathwise interpretation is especially important. A DD pulse transmitted at $\nu_i$7 is shifted by path $\nu_i$8 to $\nu_i$9 with complex gain $4\tau_{\max}\nu_{\max}\le 1,$0. If the DD periods satisfy the crystallization condition

$4\tau_{\max}\nu_{\max}\le 1,$1

or, with localized filters, $4\tau_{\max}\nu_{\max}\le 1,$2 and $4\tau_{\max}\nu_{\max}\le 1,$3, quasi-periodic replicas do not overlap. In this crystalline regime, OTFS is called localized, predictable, and non-fading in the specific sense that the amplitude distribution of the response does not depend on transmitted DD symbol location; phases still vary, but predictably. This is the basis for the claim that the post-equalization SNR remains constant across all information symbols in a packet (Mohammed et al., 2023).

Receiver design then becomes a problem of exploiting DD sparsity and structure. For channels with integer Doppler, each path causes a circular shift in the DD grid; with fractional Doppler, energy leaks into neighboring Doppler bins, producing inter-Doppler interference (IDI). A low-complexity message passing detector models each received DD symbol as depending only on a sparse neighborhood, keeps $4\tau_{\max}\nu_{\max}\le 1,$4 significant Doppler terms per path, and was reported to outperform OFDM by about $4\tau_{\max}\nu_{\max}\le 1,$5 dB at BER $4\tau_{\max}\nu_{\max}\le 1,$6 in the simulated settings, while performance saturated once roughly $4\tau_{\max}\nu_{\max}\le 1,$7–$4\tau_{\max}\nu_{\max}\le 1,$8 neighboring Doppler terms were retained (Raviteja et al., 2017).

When fractional Doppler destroys DD-domain sparsity, cross-domain iterative detection has been proposed. That detector performs time-domain L-MMSE estimation, where the effective channel remains sparse, and DD-domain symbolwise denoising, where the constellation constraint is enforced, and iteratively exchanges extrinsic information through the unitary transform between the two domains. The analysis shows that the performance gain due to iterations comes from the non-Gaussian constellation constraint in the DD domain, and the paper proves convergence to almost the same error performance as maximum-likelihood sequence detection even in the presence of fractional Doppler shifts (Li et al., 2021).

Channel estimation has developed along similar lines. A superimposed channel-estimation method adds pilots on top of all DD data symbols and uses orthogonal matching pursuit (OMP) to recover sparse channel support and gains, thereby eliminating dedicated pilot/guard overhead. In the reported $4\tau_{\max}\nu_{\max}\le 1,$9, 4-QAM experiments, it achieved BER close to embedded-pilot and MMSE-superimposed schemes while exhibiting much lower PAPR than embedded-pilot and single-pilot-superimposed schemes (Aghda et al., 2022). A different line of work replaces explicit CSI-based equalization with one-shot online learning: a reservoir-computing equalizer is trained within each OTFS frame using pilots from that same frame and was reported to yield lower BER than conventional equalization methods in the low-SNR regime under large Doppler spreads (Zhou et al., 2021).

5. Radar sensing and integrated sensing-and-communication

Because radar reflectors are parameterized by delay and Doppler exactly as wireless propagation paths are, OTFS is also a radar waveform. In OTFS-based radar, the scene is modeled as

$\tau_p$0

with

$\tau_p$1

The received waveform is then processed by the usual OTFS chain, and target estimation becomes DD-domain channel estimation (Raviteja et al., 2019).

A matched-filter-based OTFS radar formulation assumes integer delay and Doppler taps on the grid and estimates sparse target coefficients from the delay-Doppler observation array. Under i.i.d. QPSK illumination, the corresponding Gram matrix becomes diagonally dominant as $\tau_p$2, so the matched filter approximates the true channel with post-processing noise variance $\tau_p$3. The method is not exact—it remains limited by off-diagonal Gram terms and therefore exhibits high-SNR saturation—but it provides a direct DD-domain estimator for target range and velocity (Raviteja et al., 2019).

The radar literature emphasizes two concrete advantages over OFDM. First, OTFS requires only one cyclic prefix for all $\tau_p$4 symbols in a frame, whereas OFDM radar requires $\tau_p$5 CPs, saving $\tau_p$6 symbols of transmission time and enabling either longer range radar or faster target tracking. Second, OTFS is stated to be inter-carrier-interference-free in this context, allowing much larger Doppler estimation: the paper states that OFDM can exactly detect Doppler only up to about $\tau_p$7, whereas OTFS can detect Doppler up to $\tau_p$8 without interference (Raviteja et al., 2019).

The numerical radar example at $\tau_p$9, $\nu_p$0, $\nu_p$1, and $\nu_p$2 reports $\nu_p$3, $\nu_p$4, $\nu_p$5, and $\nu_p$6. For a single target at $\nu_p$7 and $\nu_p$8, OTFS estimated both range and velocity correctly, whereas OFDM exhibited a velocity error of about $\nu_p$9 because the Doppler corresponded to about $\nu_p=1/\tau_p$0 of $\nu_p=1/\tau_p$1 and created strong ICI (Raviteja et al., 2019).

More recent integrated sensing-and-communication work addresses a limitation of standard OTFS sensing: finite-frame ambiguity in delay and Doppler. A cross-frame method designs multiple OTFS subframes with co-prime numbers of subcarriers and/or time slots and fuses the modulo-wrapped estimates using the Chinese remainder theorem. The paper states that this enlarges the maximum unambiguous range and maximum tolerable velocity from order $\nu_p=1/\tau_p$2 to roughly $\nu_p=1/\tau_p$3 with $\nu_p=1/\tau_p$4 subframes, while using the same total time and frequency resources (Zhang et al., 7 Apr 2025). This reinforces a broader theme in OTFS-based sensing: the DD-domain operator view is not only physically meaningful but algorithmically extensible.

6. Practical significance, variants, and limitations

The practical significance of OTFS is consistently framed in terms of robustness rather than in terms of a generic increase in fundamental capacity. One discrete-time MIMO analysis proves that, with time-domain cyclic prefix and perfect CSI at the receiver, MIMO OTFS and MIMO OFDM have the same ergodic capacity. In that formulation, an OTFS block decomposes into $\nu_p=1/\tau_p$5 non-interfering OFDM transmissions, so the information-theoretic capacity is unchanged. The OTFS advantage is therefore located in receiver design, channel estimation, sparsity exploitation, and robustness in high-Doppler settings, not in a universal capacity theorem (RezazadehReyhani et al., 2017).

This point helps resolve a common misconception. OTFS does not mean that the physical channel ceases to vary. Rather, it means that, after DD-domain modulation and suitable equalization, all symbols interact with the same structured DD operator instead of with different local TF fades. In the more engineering-oriented literature, this is summarized as all QAM symbols experiencing nearly the same effective channel gain, with markedly reduced symbol-to-symbol SNR fluctuation (Monk et al., 2016, Hadani et al., 2018).

Waveform design beyond rectangular pulses has also become important. Rectangular-pulse OTFS has high out-of-band radiation, which is undesirable in multi-user settings. A circular pulse-shaping framework recasts OTFS as ISFFT-precoded GFDM and identifies the frequency-localized circulant Dirichlet pulse as one pulse having the desirable unitary property. In the reported experiments, it reduced out-of-band radiation by around $\nu_p=1/\tau_p$6 dB without BER loss and lowered PAPR by about $\nu_p=1/\tau_p$7 dB at CCDF $\nu_p=1/\tau_p$8 relative to conventional OTFS (Tiwari et al., 2019).

OTFS is not without tradeoffs. It is a block modulation scheme and therefore introduces higher latency than classic OFDM; detector complexity can also be significantly higher. This has motivated hybrid frame structures in which OTFS and OFDM are orthogonally multiplexed in time to support both diversity-preferred and latency-preferred tasks. In the reported $\nu_p=1/\tau_p$9, $x_{\mathrm{dd}}(\tau,\nu)\Define \mathcal{Z}_t(x(t)) =\sqrt{\tau_p}\sum_{k=-\infty}^{\infty}x(\tau+k\tau_p)e^{-j2\pi \nu k\tau_p}.$0, LDPC-coded experiments, the hybrid design preserved most of the standalone benefits of both waveforms, with a BER penalty of less than $x_{\mathrm{dd}}(\tau,\nu)\Define \mathcal{Z}_t(x(t)) =\sqrt{\tau_p}\sum_{k=-\infty}^{\infty}x(\tau+k\tau_p)e^{-j2\pi \nu k\tau_p}.$1 dB at the same BER relative to standalone OTFS or OFDM (Yuan et al., 2023).

The literature is equally explicit about open issues. Recurrent limitations include fractional delay and fractional Doppler, which reduce exact DD sparsity; pilot and guard overhead for practical estimation; low-complexity high-performance detection; multi-target and clutter robustness in radar; scalable multiple access; MIMO-OTFS design; and the practical comparison between Zak-OTFS and MC-OTFS (Mohammed et al., 2023, Raviteja et al., 2019, Wei et al., 2020). Recent variants such as OTFS-aided media-based modulation show that OTFS is also becoming a platform for hybrid schemes: OTFS-MBM combines DD-domain robustness with RF-mirror index modulation and reported 84.2% energy savings compared to OTFS and 63.2% compared to OTFS-SM in one setup (Ozden et al., 12 Apr 2025).

Taken together, these developments position OTFS as a DD-domain modulation and signal-processing framework rather than as a single fixed waveform. Its defining feature is the replacement of TF-localized symbol placement by DD-domain symbol multiplexing, so that communications, channel estimation, equalization, radar sensing, and integrated sensing-and-communication can all be formulated against the same sparse or compact delay-Doppler operator.