--- title: 'Zak-OTFS over CP-OFDM: Methods & Performance' url: https://www.emergentmind.com/topics/zak-otfs-over-cp-ofdm type: topic --- # Zak-OTFS over CP-OFDM: Methods & Performance 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 [2508.03906][2601.15602]. 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 [2303.12496]. ## 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 [2303.12496]. 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 [2601.15602]. 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 [2001.02446]. 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. \(\tau_p = 1/B\) [2508.03906]. ## 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 \(x[k,l]\), \(k=0,\dots,M-1\), \(l=0,\dots,N-1\), are embedded into a quasi-periodic DD sequence \(x_{dd}[k,l]\) [2508.03906]. The same finite-lattice viewpoint also appears in sampled finite-duration Zak-OTFS models, where the DD lattice is written as \(\Lambda_{dd}=\{(k/B,l/T)\}\) and modulation is defined by inverse Zak transformation after DD-domain pulse shaping [2408.09379]. 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 \[ y_{dd}[k,l] = h_{\text{eff}}[k,l] *_\sigma x_{dd}[k,l] + n_{dd}[k,l], \] with discrete twisted convolution \[ h_{\text{eff}}[k,l] *_\sigma x_{dd}[k,l] = \sum_{k',l'\in\mathbb Z} h_{\text{eff}}[k',l']\,x_{dd}[k-k',\,l-l']\,e^{j2\pi l' \frac{k-k'}{MN}}. \] 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 [2601.15602]. By contrast, CP-OFDM is described by a frequency-domain relation of the form \[ Y[m] = X[m]\, H_{\text{ofdm}}[m,m] + \sum_{k\neq m} X[k]\, H_{\text{ofdm}}[m,k] + N[m], \] so the baseline diagonal model survives only when Doppler-induced off-diagonal terms are negligible [2601.15602]. 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 \(Z_x[k,l]\) on a \(K\times L\) grid are mapped directly to a time block by IDZT, transmitted with a CP of length \(N_{CP}\ge \lceil \tau_{\max}/T_s\rceil\), and recovered by a DZT after matched filtering and CP removal. The paper derives a compact DD-domain matrix model \[ \mathbf y_{DD} = \mathbf x_{DD}\mathbf H + \mathbf v_{DD}, \] valid for integer and fractional delay-Doppler channels [2303.12496]. ## 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 \(B\), followed by rectangular time-windowing over duration \(T+T_{cp}\), and with the corresponding matched filter at the receiver [2508.03906]. The transmitter computes an inverse discrete frequency Zak transform to obtain frequency-domain samples \(S[i]\), and the standard CP-OFDM IDFT then produces the transmitted block. At the receiver, standard CP-OFDM demodulation yields \(Y[i]\), after which a discrete frequency Zak transform recovers the DD-domain observations. In this construction, the Zak periods are \[ \nu_p=\frac{N}{T}=N\Delta f,\qquad \tau_p=\frac{1}{\nu_p}=\frac{M}{B}, \] with \(MN=K\), the CP-OFDM subcarrier count [2508.03906]. This realization is exact under the stated filtering and windowing constraints. It also makes the family structure explicit: choosing \(M=1\) collapses the IDFZT/DFZT layers, so CP-OFDM is recovered as the minimum-delay-period member of the Zak-OTFS-over-CP-OFDM family [2508.03906]. 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 \(\mathcal O(M^3N^3)\), whereas the reconstructed FD channel is banded, enabling \(\mathcal O(M^2N^2)\) joint equalization for fixed channel spread [2506.23045]. 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 [2506.23045]. The direct DZT-OTFS paper makes a narrower complexity statement. It shows that the DZT/IDZT stage has complexity \(O(KL\log_2 K)\), compared with \(O(KL\log_2 KL)\) 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 [2303.12496]. ## 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 \[ \tau_p > \tau_s,\qquad \nu_p > \nu_s, \] and under this condition the response to a single pilot pulsone reveals the complete effective DD filter \(h_{\text{eff}}[k,l]\) without overlap from its quasi-periodic aliases [2601.15602]. 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 [2506.23045]. Later pilot-design work extends this regime of predictable operation by interleaving pilots in the DD domain. For one pilot, predictable operation requires \(2\nu_{\max}<\nu_p\). With \(Q\) interleaved pilots, the paper derives an effective crystallization condition equivalent to \[ \tau_{\max}<\frac{\tau_p}{Q},\qquad 2\nu_{\max}