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Beyond Single-Pulse Designs

The preceding sections treated OTFS as a single-pulse waveform: one transmit pulse $g_{\mathrm{tx}}$ modulating the DD grid. An alternative, explored in filter-bank-based OFDM (FBMC) and its extension filter-bank OTFS, uses a bank of filters — each tuned to a specific time-frequency region. This provides more flexibility: different sensing and comms streams can use different pulse shapes, with controlled overlap. This section develops filter-bank OTFS and discusses its trade-offs vs standard OTFS.

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Definition:
Filter-Bank OTFS

Filter-bank OTFS (FB-OTFS) uses $K$ parallel filters $\{g_i(t)\}_{i=1}^{K}$ at the transmitter, each modulated separately onto a distinct DD-sub-grid: $x(t) \;=\; \sum_{i=1}^{K} \sum_{\ell, m} s_i[\ell, m]\, g_i(t - \ell T_s^{(i)})\, e^{j 2\pi m W_s^{(i)} t}.$ Different filters can have different time-frequency localizations:

Narrow-band filters: precise Doppler resolution, long duration. Used for sensing.
Wide-band filters: good time resolution, short duration. Used for data.

Advantage: flexibility. Each stream optimized for its purpose.

Disadvantage: increased complexity (parallel filter banks at Tx and Rx). Cross-filter interference if not carefully designed.

Comparison with standard OTFS: FB-OTFS is a special case where $K > 1$ filters replace the single $g_{\mathrm{tx}}$ .

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Theorem: FB-OTFS Orthogonality Conditions

For $K$ -filter bank OTFS to maintain orthogonal streams, each pair of filters $(g_i, g_j)$ , $i \neq j$ , must satisfy $\int g_i(t) g_j(t-\tau)^* e^{-j 2\pi \nu t} dt \;=\; 0 \quad \forall (\tau, \nu) \in \mathrm{grid}.$ Additionally, each filter pair must satisfy its own bi-orthogonality condition with its matched receive filter.

Practical realization: use filters with disjoint time-frequency supports (approximate) or filters from orthogonal families (e.g., Hermite basis, truncated prolate spheroidal).

Coexistence constraint: FB-OTFS requires $K \cdot T_s W_s \leq 1$ (critical density across all filters). Beyond this, inter- filter interference is unavoidable.

FB-OTFS generalizes single-pulse OTFS by allowing $K$ coexisting streams, each with its own pulse. The orthogonality extends: not just across the DD grid for a single pulse, but also across pulses. Practical designs use disjoint regions (e.g., sensing at low-frequency, data at high-frequency) or orthogonal function families.

Proof

Inter-filter inner product

For stream $i$ transmitted at DD $(\ell, m)$ and stream $j$ at $(\ell', m')$ : cross-term $= \int g_i(t - \ell T_s^{(i)}) \cdot g_j(t - \ell' T_s^{(j)})^* e^{j2\pi \nu_{ij} t} dt$ .

Orthogonality at grid

Setting cross-term to zero at grid samples: requires cross-ambiguity of filters to vanish.

Density constraint

Each filter occupies bandwidth $1/T_s W_s$ . Sum across $K$ filters: $K/(T_s W_s) \leq 1$ by Nyquist in the TF plane.

Construction

Orthogonal Hermite basis provides a concrete FB-OTFS design satisfying all conditions. $\blacksquare$

Definition:
Hermite Filter-Bank OTFS

Hermite FB-OTFS uses the Hermite-Gauss functions $\psi_n(t) \;=\; \frac{1}{\sqrt{2^n n! \sqrt{2\pi}\sigma}}\, H_n(t/\sigma)\, e^{-t^2/(4\sigma^2)},$ where $H_n$ are Hermite polynomials. These are orthogonal in $L^2(\mathbb{R})$ .

Key properties:

$\psi_n$ have compact support in both time and frequency (approximate): 2D Gaussian × Hermite.
Orthogonal as a family.
Different $n$ : different time-frequency localizations — low $n$ compact, high $n$ spread.

FB-OTFS with $K$ Hermites: filter bank with first $K$ Hermite functions. Provides $K$ orthogonal streams with controllable TF resolution.

Applications:

Stream 1 ( $\psi_0$ , Gaussian): compact, low-Doppler sensing.
Stream 2 ( $\psi_1$ ): wider, mid-Doppler.
Stream $K$ : wide-band, high-Doppler.

Example: FB-OTFS for ISAC

Design a FB-OTFS system for ISAC: 3 filters for (a) data, (b) sensing fine-Doppler, (c) sensing fine-delay. What Hermite orders?

Solution

Data filter

Wide-band, short-duration. Maximize spectral efficiency. Hermite $\psi_0$ (Gaussian). Good TF concentration.

Fine-Doppler sensing

Narrow-band, long-duration. $\psi_1$ or higher. Doppler resolution $1/T_s^{(1)}$ where $T_s^{(1)}$ is long.

Fine-delay sensing

Wide-band, short-duration. $\psi_0$ or a higher-order wideband Hermite.

Orthogonality check

Different Hermite orders: orthogonal by construction. No inter-filter interference.

Resource allocation

Total bandwidth split: data (60%), fine-Doppler (25%), fine-delay (15%). Flexible per-service.

Hermite Filter Bank TF Response

Plot the time-frequency response of first 4 Hermite functions. Shows how each filter occupies a distinct region in the TF plane. Sliders: pulse width $\sigma$ , number of filters $K$ .

Parameters

\sigma

(normalized)0.5

K

filters4

Theorem: FB-OTFS Capacity Gain

For a FB-OTFS system with $K$ filters, each with its own service quality, the aggregate capacity is $C_{\mathrm{FB}} \;=\; \sum_{i=1}^{K} W_i \log_2(1 + \mathrm{SINR}_i),$ where $W_i$ is per-filter bandwidth and $\mathrm{SINR}_i$ its effective SINR.

For typical ISAC use case ( $K = 3$ filters, balanced allocation): $C_{\mathrm{FB}} = 0.85 \cdot C_{\mathrm{single-pulse}}$ — slightly less than single-pulse OTFS, but with richer functionality.

Consequence: FB-OTFS accepts a 15% capacity penalty for the flexibility of multi-service simultaneous operation.

FB-OTFS is not free. Each filter consumes spectrum; the sum must fit within total bandwidth. The penalty is the sidelobe overhead between filters. The benefit is flexibility — each stream optimized for its task. For single-service (pure data or pure sensing), single-pulse OTFS wins. For multi-service (ISAC, multi-flow), FB-OTFS offers structural advantages.

Proof

Per-filter rate

$R_i = W_i \log_2(1 + \mathrm{SINR}_i)$ . Independent across filters (orthogonal construction).

Aggregate

$C_{\mathrm{FB}} = \sum R_i$ . Single-pulse: $C = W \log(1 + \mathrm{SINR})$ with all bandwidth $W$ used.

Overhead

Filter bank requires filter-transition regions (roll-off). ~15% bandwidth overhead.

Ratio

$C_{\mathrm{FB}}/C_{\mathrm{single}} \approx 0.85$ for $K = 3$ with RRC $\gamma = 0.25$ . $\blacksquare$

Definition:
FB-OTFS Applications

Key applications of FB-OTFS:

ISAC with multi-objective sensing: data on one filter; coarse radar on another; fine radar on third. Each optimized for its task. Compared to single-pulse OTFS-ISAC: $\sim 2$ - $3$ dB better CRB for sensing, 15% data penalty.

Hybrid URLLC + eMBB: URLLC on narrow-band Hermite (low latency), eMBB on wide-band. Separates services without inter- service interference. Enables strict SLA.

Multi-service IoT: NB-IoT-like services on different filters. Massive machine-type communications (mMTC) with differentiated QoS.

Cognitive radio: primary service on wide-band filter, secondary (opportunistic) on narrow-band side filter. Compatible with primary user via frequency-domain separation.

⚠️Engineering Note

FB-OTFS Implementation Cost

FB-OTFS implementation complexity:

Filter bank: $K$ parallel RRC or Hermite filters at Tx/Rx. Per-sample compute: $\sim K \cdot$ FIR-length ops. For $K = 3$ and 50-tap filters: $\sim 150$ ops/sample.
ISFFT/SFFT: standard OTFS operations, unchanged.
Detection: per-filter detection + combining. Compute: $\sim K \cdot$ single-pulse detection.

Total: ~ $K$ -fold complexity over single-pulse OTFS. For $K = 3$ : 3 $\times$ compute. Substantial but feasible on modern gNB/UE silicon.

Deployment status (2026):

FB-OTFS in academic research and specialized ISAC prototypes.
Not yet in 3GPP 6G consideration (complexity vs single-pulse gains).
Expected: post-Rel. 22 (2030+) for ISAC-native applications.

Single-pulse OTFS remains the baseline for 6G standardization. FB-OTFS is a research direction for specialized scenarios.

Practical Constraints

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FB-OTFS: $K$ × compute of single-pulse
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For $K = 3$ : 3× complexity, 15% capacity penalty
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Deployment: post-Rel. 22 (specialized ISAC)
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Standard OTFS is 6G baseline

Common Mistake: Filter Cross-Coupling Hurts Sensing

Mistake:

Designing FB-OTFS with unclean filter separation. Even small cross-coupling ( $\sim 0.1$ ) between sensing filter and data filter contaminates the sensing estimate.

Correction:

Use orthogonal filter families (Hermite, prolate spheroidal) for clean separation. Alternatively, use disjoint frequency bands at expense of spectral efficiency. At the design stage, verify cross-ambiguity is below $-60$ dB at all DD grid samples. Sensing applications require tighter cross-coupling than data-only.

Filter-Bank OTFS