Per-Cell Delay-Doppler Filters

• Per-cell delay-Doppler filters are specialized two-dimensional filter banks that precisely manage interference and channel tap estimation on discrete delay-Doppler grids.
• Design strategies such as Hermite-basis, DDOP, and tunable Gaussian pulses optimize key trade-offs between ISI suppression, localization, and channel capacity.
• Adaptive per-cell configuration enables dynamic filter selection to balance delay-Doppler trade-offs in high-mobility and nonstationary environments.

Per-cell delay-Doppler filters are specialized time-frequency structures enabling explicit interference control and channel tap estimation at each cell of a discrete delay–Doppler (DD) plane. Their design directly determines symbol orthogonality, intersymbol interference (ISI), and input–output (I/O) relationship in DD-domain multicarrier modulation schemes such as Zak-OTFS and ODDM. Approaches include Hermite-basis pulse design minimizing ISI, delay–Doppler plane orthogonal pulse (DDOP) construction enacting local orthogonality, and tunable parametric families such as the tunable Gaussian pulse (TGP). The filter selection and configuration define DD-to-TF localization tradeoffs, pilot-based channel estimation tractability, and robustness in high-mobility or nonstationary environments.

1. Fundamental Principles and Definitions

Per-cell DD filters operate over finite or periodic grids indexed by delay (Δτ) and Doppler (Δν) resolutions, forming an M×N lattice. A filter is assigned to each grid point; together, they form a two-dimensional filter bank—enabling precise demultiplexing and matched filtering per DD slot. Orthogonality at the cell level is achieved when the ambiguity function between filter pairs vanishes at all nonidentical integer grid offsets. This behavior is essential for independent detection in massive-multipath and high-Doppler conditions (Lin et al., 2023, Jesbin et al., 20 Oct 2025).

Designs must respect inherent constraints such as the Balian–Low Theorem, which governs the achievable localization–orthogonality tradeoff in $L^2(\mathbb{R})$. Sinc and Gaussian pulses represent theoretical extremes: the sinc pulse enforces strict orthogonality with poor localization, while the Gaussian pulse provides maximal localization but significant ISI. Composite and basis expansion approaches (e.g., Gaussian-sinc, Hermite sums) yield tunable intermediate solutions (Jesbin et al., 20 Oct 2025, Costa et al., 16 Dec 2025).

2. Hermite-Basis Design and ISI Minimization

A systematic mode of per-cell DD filter construction exploits the Hermite basis: for each pulse $p(t) = \sum_{n=0}^{N-1} c_n h_n(t)$, the $h_n(t)$ are orthonormal Hermite functions ($H_n$ polynomials times a Gaussian envelope). This sum can be generalized by introducing a width parameter $\sigma$ and forming scaled Hermites $\phi_n(\sigma;t)$ with $\|p\|^2 = \sum |c_n|^2 = 1$ ensuring unit energy (Jesbin et al., 20 Oct 2025).

Optimal coefficients $\{c_n\}$ are determined by explicitly minimizing total ISI energy at DD grid points, constrained by unit energy. The ISI metric is mapped to a quadratic form via a sampling matrix $\Phi$ whose $(m, n)$ element is $h_n(m/B)$. The minimization

$$\min_{\mathbf{c} \in \mathbb{C}^N} \ \mathbf{c}^H (\Phi^H\Phi)\mathbf{c}\quad \text{s.t.} \ \|\mathbf{c}\| = 1$$

is solved by computing the singular value decomposition of $\Phi$. The $\mathbf{c}_{\mathrm{opt}}$ is the last right singular vector of $\Phi$ (corresponding to the smallest singular value). ISI suppression and pulse localization trade off as the expansion order $N$ increases: higher $N$ enforces more off-diagonal nulls (orthogonality) but increases sidelobes (reduces localization).

3. Delay–Doppler Plane Orthogonal Pulses (DDOP) and Local Orthogonality

DDOPs leverage local orthogonality on finite DD grids by constructing the transmit pulse as a train of M root-Nyquist sub-pulses $a(t)$, each spaced by the delay period $T$. The sub-pulse $a(t)$ is chosen to satisfy orthogonality with respect to grid-shifts of length $\Delta \tau = T/M$; collectively,

$g(t) = \sum_{m=0}^{M-1} a(t-mT)$

where $g(t)$ is the DDOP. For non-integer support, a cyclic extension is added to preserve orthogonality at DD boundary cells (Lin et al., 2023).

Ambiguity analysis shows that the cross-ambiguity $A_{g_c,g}(k \Delta \tau,\ell \Delta \nu)$ is Kronecker-delta at $(k,\ell)$ up to grid bounds, with sharp nulls off the main diagonal—thus, each per-cell matched filter acts as an independent channel without inter-cell leakage. Frequency-domain and Zak-domain representations expose the dual concentration and orthogonality structure.

4. Tunable Gaussian Pulse and Parametric Tradeoffs

The tunable Gaussian pulse (TGP) encompasses a continuously-adjustable family of DD pulses. Its general expression is

$$x_{dd}(\tau, \nu) = \exp\left[-\pi \left( \frac{2 \tau^2}{\gamma T^2} + 2 \gamma \frac{\nu^2}{B^2} \right) \right] \cdot \exp\left[ j\pi \left( \alpha_c \frac{\tau^2}{T^2} + \beta_c \tau \nu \right) \right]$$

with tunables: aspect ratio $\gamma$, chirp rate $\alpha_c$, and bilinear phase coupling $\beta_c$ (Costa et al., 16 Dec 2025).

• $\gamma$ controls the envelope aspect ratio, trading spread between delay and Doppler.
• $\alpha_c$ determines quadratic phase coupling in delay, activating coupling terms in the Fisher information matrix for parameter estimation.
• $\beta_c$ encodes a bilinear delay-Doppler phase, reshaping the ambiguity function and interference covariance structure.

Closed-form Cramér–Rao lower bounds (CRLBs) for delay and Doppler parameter estimation, as well as explicit communication-domain channel covariances, can be derived as functions of the TGP tunables. Benchmarking shows TGP can trace a Pareto frontier between Sinc (max capacity/poor sensing) and root-raised cosine (max sensing/limited capacity), with: e.g., $97\%$ of max Sinc capacity at small sensing loss, or near-RRC CRLB with $90\%$ Sinc capacity.

5. Per-Cell Input–Output Relations and Channel Learning

The per-cell I/O relationship in Zak-OTFS incorporates the effective DD-domain channel response, sampled at grid points, producing a transfer function:

$$y[m, k] = \sum_{m', k'} H_{(m, k), (m', k')} x[m', k'] + w[m, k]$$

with

$$H_{(m, k), (m', k')} = h_{\text{eff}}\left( \frac{m-m'}{B}, \frac{k-k'}{T} \right) e^{j 2\pi \frac{m'}{M}(k-k')/N}.$$

The explicit form of $h_{\text{eff}}(\tau, \nu)$ depends on the prototype pulse design; for Hermite-based pulses, it is a sum over ambiguity functions of Hermite basis members, channel tap weights, and pulse coefficients.

Per-cell channel taps can be rapidly learned by embedding single-cell pilots in the DD domain. Under crystallization conditions ($\tau_{\max} < \tau_p$, $2\nu_{\max} < \nu_p$), the response to a pilot waveform directly yields $h_{\text{eff}}[k, \ell]$. For extended coverage, multiple interleaved pilots can be placed across the DD grid, and Vandermonde systems are solved per-cell to reconstruct multiple overlapping channel taps. Compared to cross-ambiguity exhaustive methods, this approach is computationally efficient, scaling as $O(Q^3 MN)$, where $Q$ is pilot count (Jayachandran et al., 2024).

6. Sidelobe, Interference, and Performance Benchmarks

The filter family enables explicit control over sidelobes and ISI nulls at integer-grid offsets. Orthogonality is assured via minimization or construction, producing deep zeros at nonzero DD shifts. Localization is regulated via pulse order ($N$ for Hermites, $\gamma$ for TGP) and scaling. Increased order nulls more ISI, but with higher sidelobe energy—this is quantified via heat-maps of $|h_{\mathrm{eff}}(m/B, k/T)|$ for various design choices (Jesbin et al., 20 Oct 2025).

Performance metrics include:

• Optimal Hermite pulse (N=9): $E_{\mathrm{ISI}} \approx -40\, \mathrm{dB}$
• NMSE of I/O estimation: Hermite and GS pulses within 1–2 dB of pure Gaussian; sinc is 5–10 dB worse.
• BER: Hermite ($N=9$) matches state-of-the-art GS, exceeding both sinc (estimation error-limited) and Gaussian (ISI-limited).
• TGP capacity / sensing: at $20$ dB SNR, TGP can achieve $97\%$ of Sinc capacity with higher precision, or $1.2\times$ RRC CRLB at $90\%$ of capacity.

7. Practical Filter Selection and Adaptation Procedures

Each cell (or sector) can adapt its DD filter in response to time-varying channel statistics such as delay and Doppler spread. For parameterized pulses, this entails:

• Monitoring per-cell spread.
• Optimizing parameters (e.g., $\gamma, \alpha_c, \beta_c$ for TGP) to jointly minimize estimation error (CRLB) and maximize ergodic capacity within operational constraints (Costa et al., 16 Dec 2025).
• In practice, discrete parameter sweeps, low-dimensional optimization (e.g., Nelder–Mead), or lookup tables are employed; Balian–Low bounds restrict the parameter ranges to avoid excessive ISI/ICI.
• For Hermite-basis and DDOPs, similar adaptation may involve adjusting order or sub-pulse support.

This enables networks to inject, remove, or tune per-cell filters dynamically, supporting diverse delay-Doppler profiles, tailored I/O estimation, and interference environments.

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References:

• "Delay-Doppler Pulse Shaping in Zak-OTFS Using Hermite Basis Functions" (Jesbin et al., 20 Oct 2025)
• "On Delay-Doppler Plane Orthogonal Pulse" (Lin et al., 2023)
• "Tunable Gaussian Pulse for Delay-Doppler ISAC" (Costa et al., 16 Dec 2025)
• "Zak-OTFS with Interleaved Pilots to Extend the Region of Predictable Operation" (Jayachandran et al., 2024)