Zak-OTFS over CP-OFDM: Methods & Performance

• Zak-OTFS over CP-OFDM is a modulation scheme that integrates Zak transform processing with CP-OFDM to map delay-Doppler symbols onto a predictable lattice.
• It employs direct discrete Zak transforms and pulse-shaping constraints to reduce complexity and improve BER performance in high-Doppler scenarios compared to conventional OFDM.
• By embedding Zak pre/post-processing into existing CP-OFDM chains, the method provides a migration path that leverages delay-Doppler equalization while addressing pilot design and CP coexistence challenges.

Zak-OTFS over CP-OFDM denotes a class of architectures and comparisons in which Zak-transform-based orthogonal time frequency space modulation is related directly to cyclic-prefix orthogonal frequency division multiplexing. In the architectural sense, Zak-OTFS places information on a delay-Doppler lattice and then realizes modulation and demodulation by Zak-transform-based preprocessing and post-processing around a standard CP-OFDM modem; in the comparative sense, it frames Zak-OTFS and CP-OFDM as two different responses to doubly selective propagation, the former exploiting a predictable delay-Doppler input-output relation and the latter relying on approximate subcarrier orthogonality (Mohammed et al., 5 Aug 2025, Khan et al., 22 Jan 2026). Direct discrete-Zak implementations also describe Zak-OTFS as a DD-to-time modulation chain, rather than as conventional two-step OTFS via a time-frequency intermediate, and report BER gains over OFDM in high-Doppler settings (Yogesh et al., 2023).

1. Conceptual position within OTFS and OFDM

Zak-OTFS is a delay-Doppler-domain realization of OTFS in which symbols are carried by quasi-periodic DD pulses and processed through the Zak transform. This differs from the conventional “two-step” OTFS implementation, where DD symbols are first mapped to the time-frequency domain through ISFFT and then converted to time samples by Heisenberg or OFDM-style synthesis. In the direct DZT-OTFS formulation, the transmitter applies the inverse discrete Zak transform directly to DD symbols, inserts a cyclic prefix, pulse-shapes the block, and the receiver applies matched filtering, CP removal, and a DZT directly to the received time-domain block (Yogesh et al., 2023).

Relative to CP-OFDM, the literature repeatedly presents the distinction as architectural rather than merely notational. CP-OFDM attempts to choose numerology so that inter-carrier interference is negligible and equalization remains simple; Zak-OTFS instead accepts structured interference, but expresses it through a predictable DD-domain interaction that can be jointly equalized (Khan et al., 22 Jan 2026). This distinction is closely related to the broader OTFS interpretation of reduced-CP OTFS as “block OFDM + time interleaving + one block CP,” although that earlier formulation does not explicitly use the Zak transform and is best read as a Zak-adjacent antecedent rather than a strict Zak-OTFS construction (Rangamgari et al., 2020).

The most explicit bridge appears in the later “Zak-OTFS over CP-OFDM” formulation, which shows that Zak-OTFS over CP-OFDM is a family of modulations parameterized by the delay period. Within that family, CP-OFDM itself is recovered when the delay period takes its minimum possible value, equal to the inverse bandwidth, i.e. $\tau_p = 1/B$ (Mohammed et al., 5 Aug 2025).

2. Delay-Doppler signal model and operator structure

Zak-OTFS is built on a DD lattice whose sampling is tied to bandwidth $B$, duration $T$, and period parameters $\tau_p$ and $\nu_p$, with $\tau_p \nu_p = 1$. In the CP-OFDM-compatible formulation, $M = B\tau_p$, $N = T\nu_p$, and therefore $MN = BT$; the information symbols $x[k,l]$, $B$0, $B$1, are embedded into a quasi-periodic DD sequence $B$2 (Mohammed et al., 5 Aug 2025). The same finite-lattice viewpoint also appears in sampled finite-duration Zak-OTFS models, where the DD lattice is written as $B$3 and modulation is defined by inverse Zak transformation after DD-domain pulse shaping (Jayachandran et al., 2024).

The key end-to-end operator is the DD-domain twisted convolution. In the comparative 6G analysis, the discrete Zak-OTFS input-output law is

$B$4

with discrete twisted convolution

$B$5

In this representation, every DD symbol interacts through the same effective DD filter, and the support of that filter is governed by channel delay spread and Doppler spread (Khan et al., 22 Jan 2026).

By contrast, CP-OFDM is described by a frequency-domain relation of the form

$B$6

so the baseline diagonal model survives only when Doppler-induced off-diagonal terms are negligible (Khan et al., 22 Jan 2026). This contrast is the technical basis for the recurring claim that Zak-OTFS has a predictable DD-domain I/O relation whereas CP-OFDM becomes difficult to characterize once ICI is significant.

The direct DZT-OTFS study reaches a closely related conclusion from a different implementation starting point. There, DD symbols $B$7 on a $B$8 grid are mapped directly to a time block by IDZT, transmitted with a CP of length $B$9, and recovered by a DZT after matched filtering and CP removal. The paper derives a compact DD-domain matrix model

$T$0

valid for integer and fractional delay-Doppler channels (Yogesh et al., 2023).

3. Realizing Zak-OTFS with standard CP-OFDM blocks

The defining architectural result is that Zak-OTFS can be realized as a low-complexity precoder over standard CP-OFDM when pulse shaping is constrained to sinc filtering of bandwidth $T$1, followed by rectangular time-windowing over duration $T$2, and with the corresponding matched filter at the receiver (Mohammed et al., 5 Aug 2025). The transmitter computes an inverse discrete frequency Zak transform to obtain frequency-domain samples $T$3, and the standard CP-OFDM IDFT then produces the transmitted block. At the receiver, standard CP-OFDM demodulation yields $T$4, after which a discrete frequency Zak transform recovers the DD-domain observations. In this construction, the Zak periods are

$T$5

with $T$6, the CP-OFDM subcarrier count (Mohammed et al., 5 Aug 2025).

This realization is exact under the stated filtering and windowing constraints. It also makes the family structure explicit: choosing $T$7 collapses the IDFZT/DFZT layers, so CP-OFDM is recovered as the minimum-delay-period member of the Zak-OTFS-over-CP-OFDM family (Mohammed et al., 5 Aug 2025).

A related construction, called Zak-OFDM, uses the DD-domain effective channel estimate to reconstruct the complete frequency-domain channel response and then jointly equalizes all subcarriers. In that formulation, naïve DD-domain equalization has complexity $T$8, whereas the reconstructed FD channel is banded, enabling $T$9 joint equalization for fixed channel spread (Mohammed et al., 29 Jun 2025). The same paper motivates the architecture by noting that direct CP-OFDM acquisition of the complete FD subcarrier-coupling matrix is very challenging in doubly spread channels, whereas DD-domain channel acquisition is predictable (Mohammed et al., 29 Jun 2025).

The direct DZT-OTFS paper makes a narrower complexity statement. It shows that the DZT/IDZT stage has complexity $\tau_p$0, compared with $\tau_p$1 for the two-step OTFS realization through the TF domain. It does not provide a direct complexity benchmark against CP-OFDM, and it explicitly warns against concluding from that paper alone that Zak-OTFS is simpler than CP-OFDM (Yogesh et al., 2023).

4. Predictability, crystallization, and pilot design

A central Zak-OTFS notion is the crystallization condition: the delay period must exceed the channel delay spread and the Doppler period must exceed the channel Doppler spread. In the comprehensive Zak-OTFS versus CP-OFDM comparison, this is written as

$\tau_p$2

and under this condition the response to a single pilot pulsone reveals the complete effective DD filter $\tau_p$3 without overlap from its quasi-periodic aliases (Khan et al., 22 Jan 2026). The same basic condition appears in Zak-OFDM, which states that when the channel delay spread is less than the delay period and the channel Doppler spread is less than the Doppler period, the response to a single Zak-OTFS carrier provides an image of the scattering environment and can be used to predict the effective channel at all other carriers (Mohammed et al., 29 Jun 2025).

Later pilot-design work extends this regime of predictable operation by interleaving pilots in the DD domain. For one pilot, predictable operation requires $\tau_p$4. With $\tau_p$5 interleaved pilots, the paper derives an effective crystallization condition equivalent to

$\tau_p$6

so the effective periods become $\tau_p$7 and $\tau_p$8. The same work states that linear reconstruction then has complexity $\tau_p$9, compared with $\nu_p$0 for cross-ambiguity or ML estimation (Jayachandran et al., 2024).

Predictability also motivates later receiver designs that reuse detected data as pilots. In the differential Zak-OTFS scheme, the paper argues that the cross-ambiguity between received data and transmitted or detected data is approximately a scaled DD-channel estimate, so previously detected data can be used as pilots for the next frame. In the reported simulations, one pilot frame is transmitted every 30 frame transmissions to curb error propagation, and the method is described as achieving full spectral efficiency and better BER than spread-pilot Zak-OTFS at lower complexity in the tested settings (Mattu et al., 16 Jul 2025).

5. Comparative performance against OFDM and CP-OFDM

The empirical record is consistent in one respect and qualified in another. It is consistent in reporting strong Zak-OTFS gains in doubly selective and especially high-Doppler channels; it is qualified in not claiming universal superiority over CP-OFDM.

The earliest direct BER study of DZT-OTFS reports that both DZT-OTFS and two-step OTFS outperform OFDM in doubly selective channels, and that DZT-OTFS slightly outperforms two-step OTFS. Under the main $\nu_p$1 setting with $\nu_p$2, fractional delay and Doppler, BPSK, and MMSE detection, the reported gain at BER $\nu_p$3 is about $\nu_p$4 dB over two-step OTFS and about $\nu_p$5 dB over OFDM. In a separate sweep with $\nu_p$6 varied from $\nu_p$7 Hz to $\nu_p$8 kHz, both OTFS variants are much more robust than OFDM, and DZT-OTFS remains better than two-step OTFS over a wide range of Doppler (Yogesh et al., 2023).

The broader Zak-OTFS-versus-CP-OFDM studies organize the comparison by cell size and mobility. In low-mobility, small-cell conditions, optimized Zak-OTFS and CP-OFDM are reported to have similar spectral efficiency; one representative point is $\nu_p$9 bps/Hz for Zak-OTFS versus $\tau_p \nu_p = 1$0 bps/Hz for CP-OFDM at $\tau_p \nu_p = 1$1 Hz and $\tau_p \nu_p = 1$2. In high-mobility, small-cell conditions, Zak-OTFS is described as typically $\tau_p \nu_p = 1$3 more spectrally efficient. In low-mobility, large-cell conditions, the summary states about $\tau_p \nu_p = 1$4 advantage, with an example of $\tau_p \nu_p = 1$5 versus $\tau_p \nu_p = 1$6 bps/Hz at $\tau_p \nu_p = 1$7 Hz and $\tau_p \nu_p = 1$8. In high-mobility, large-cell conditions, Zak-OTFS is the clear winner; at $\tau_p \nu_p = 1$9 Hz and $M = B\tau_p$0, the reported values are $M = B\tau_p$1 bps/Hz for Zak-OTFS and $M = B\tau_p$2 bps/Hz for CP-OFDM, i.e. about $M = B\tau_p$3 better (Khan et al., 22 Jan 2026).

The explicit “Zak-OTFS over CP-OFDM” implementation paper reports similar trends after imposing CP-OFDM-compatible pulse-shaping constraints. In a TDL-C scenario at $M = B\tau_p$4 GHz and $M = B\tau_p$5 km/h, Zak-OTFS over CP-OFDM achieves about $M = B\tau_p$6 higher effective spectral efficiency than CP-OFDM, while unconstrained Zak-OTFS is about $M = B\tau_p$7 higher still. At $M = B\tau_p$8 km/h in the same channel family, the reported improvement over CP-OFDM rises to about $M = B\tau_p$9. In a TDL-D scenario at $N = T\nu_p$0 GHz and $N = T\nu_p$1 km/h, the reported spectral efficiencies are $N = T\nu_p$2 bits/s/Hz for CP-OFDM, $N = T\nu_p$3 bits/s/Hz for Zak-OTFS over CP-OFDM, and $N = T\nu_p$4 bits/s/Hz for unconstrained Zak-OTFS (Mohammed et al., 5 Aug 2025).

In MIMO, the picture becomes explicitly crossover-based. The first MIMO Zak-OTFS paper states that CP-OFDM performs slightly better at low SNR and low Doppler, while Zak-OTFS excels at higher SNR or under severe Doppler dispersion, and that the SNR and Doppler crossover points shift inversely to each other. Under imperfect CSI, the reported throughput-versus-Doppler behavior shows CP-OFDM better at very low Doppler because of more efficient pilot utilization, whereas Zak-OTFS remains stable up to about $N = T\nu_p$5 Hz in the plotted $N = T\nu_p$6 setting before pilot-pulsone interference begins to hurt performance (Barati et al., 24 Jun 2026).

Representative setting	Reported outcome	Source
BER $N = T\nu_p$7 in DZT-OTFS study	about $N = T\nu_p$8 dB over two-step OTFS and about $N = T\nu_p$9 dB over OFDM	(Yogesh et al., 2023)
High-mobility, large-cell example $MN = BT$0	$MN = BT$1 bps/Hz for Zak-OTFS vs $MN = BT$2 bps/Hz for CP-OFDM	(Khan et al., 22 Jan 2026)
Zak-OTFS over CP-OFDM, high-speed TDL-D example	$MN = BT$3 bits/s/Hz vs $MN = BT$4 for CP-OFDM; unconstrained Zak-OTFS $MN = BT$5	(Mohammed et al., 5 Aug 2025)

Taken together, these results support a narrow but consistent synthesis: Zak-OTFS is not presented as a universal CP-OFDM replacement, but it is repeatedly reported as superior when delay spread and Doppler spread are both large enough that CP-OFDM’s diagonal approximation, sparse pilot interpolation, and low-complexity per-subcarrier equalization cease to be adequate.

6. Practical complications, coexistence issues, and system-level extensions

Standards-compatible OTFS-over-OFDM realizations introduce complications that ideal DD models suppress. In coexistence analyses with multiple CPs and unequal CP lengths, OTFS implemented through an OFDM stack acquires an effective DD-domain channel with spreading along Doppler from the repeated CP structure and spreading along delay from edge-carrier unloading. The resulting effective DD channel is less sparse than the canonical ideal OTFS model, and the coefficients depend explicitly on the unequal CP lengths. The same papers report that ignoring unequal CP lengths during detection degrades BER, and they develop pilot-aided or interference-cancellation-based channel estimation that uses the derived CP-aware DD kernels (Shafie et al., 2024, Shafie et al., 2023).

Despite these complications, over-the-air work has shown that Zak-OTFS is not merely a theoretical construction. A mmWave demonstration at $MN = BT$6 GHz on the COSMOS testbed uses root-raised-cosine filtering, a short $MN = BT$7-sample Zadoff-Chu preamble, higher-order modulations up to $MN = BT$8-QAM, and MMSE equalization in the DD domain. The same work argues that CFO and timing offsets can be absorbed into the effective DD-domain channel through shifted effective delay and Doppler parameters, and reports better uncoded BER with RRC filtering than with sinc filtering under motion (Ramachandran et al., 10 Nov 2025).

The architectural consequences extend beyond point-to-point links. In uplink grant-free coded random access for mMTC, Zak-OTFS-based CRA is compared against a CP-OFDM-based CRA baseline. The claimed system-level advantage is not blanket link superiority but better inter-slot channel reuse and therefore better SIC under high mobility. In the reported full-frame high-mobility case with Gaussian pulse shaping, Zak-OTFS supports about $MN = BT$9 active users while keeping packet loss rate below $x[k,l]$0, whereas OFDM SIC is described as ineffective in that regime (Mirri et al., 29 Jul 2025).

The cumulative picture is therefore dual. On one side, Zak-OTFS over CP-OFDM offers a concrete migration path: reuse CP-OFDM hardware, add Zak-domain pre/post transforms, and exploit DD predictability in precisely the regimes where OFDM becomes ICI-limited. On the other side, that migration is constrained by pulse-shaping assumptions, DD equalizer complexity, CP-aware coexistence effects, and the fact that low-Doppler or low-SNR operation may still favor CP-OFDM. Within the literature surveyed here, Zak-OTFS over CP-OFDM emerges not as a synonym for OTFS in general, but as a specific implementation doctrine: preserve the DD-domain structure of Zak-OTFS while embedding it into the installed CP-OFDM signal chain wherever the propagation regime justifies the additional structure and receiver burden.