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When Tx and Rx Pulses Disagree

OTFS transmission sends symbol $s[\ell, m]$ through transmit pulse $g_{\mathrm{tx}}$ and receives it via pulse $g_{\mathrm{rx}}$ . For the DD grid to be orthogonal in the receive direction — that is, for the transmitted symbol at DD cell $(\ell, m)$ to arrive cleanly at cell $(\ell, m)$ and not spread across other cells — the pair $(g_{\mathrm{tx}}, g_{\mathrm{rx}})$ must satisfy the bi-orthogonality condition. This is the formal counterpart of the Nyquist ISI criterion in single-carrier systems, lifted to the 2D DD plane.

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Definition:
Bi-Orthogonality Condition

The bi-orthogonality condition for OTFS transmit pulse $g_{\mathrm{tx}}$ and receive pulse $g_{\mathrm{rx}}$ is $A_{g_{\mathrm{rx}}, g_{\mathrm{tx}}}(\ell T_s, m W_s) \;=\; \delta[\ell, m], \qquad \ell, m \in \mathbb{Z},$ where $A_{g_{\mathrm{rx}}, g_{\mathrm{tx}}}(\tau, \nu)$ is the cross-ambiguity function: $A_{g_{\mathrm{rx}}, g_{\mathrm{tx}}}(\tau, \nu) \;=\; \int g_{\mathrm{rx}}(t)^* g_{\mathrm{tx}}(t - \tau) e^{j 2\pi \nu t} dt.$

Physical meaning: a symbol sent at DD cell $(\ell_0, m_0)$ appears at the receiver filter output at cell $(\ell_0, m_0)$ and nowhere else. No ISI, no ICI across the DD grid.

Same-pulse case: $g_{\mathrm{tx}} = g_{\mathrm{rx}} = g$ . Condition becomes $A_{g, g}(\ell T_s, m W_s) = \delta[\ell, m]$ — the auto-ambiguity is a Dirac comb.

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Theorem: Existence of Bi-Orthogonal Pulse Pairs

A bi-orthogonal pulse pair $(g_{\mathrm{tx}}, g_{\mathrm{rx}})$ with compact support in time and frequency exists if and only if the time-frequency product satisfies $T_s W_s = 1$ (Nyquist crammed).

Weyl-Heisenberg system: the collection $\{g_{\mathrm{tx}}(t - \ell T_s) e^{j 2\pi m W_s t}\}_{\ell, m}$ forms a Riesz basis of $L^2(\mathbb{R})$ iff $T_s W_s = 1$ (critical sampling).

For $T_s W_s > 1$ (over-sampled): pulses can have any shape; bi-orthogonality is trivial. But spectral efficiency drops.

For $T_s W_s < 1$ (under-sampled): bi-orthogonality is impossible. System is inevitably interfered.

Practical OTFS: operates at $T_s W_s = 1$ (or slightly above for safety). Uses Hermite, Gaussian, or RRC pulses that achieve near-bi-orthogonality.

The existence result formalizes why OTFS designers have so much freedom at critical sampling $T_s W_s = 1$ : they can pick any pulse pair that forms a Riesz basis. Below $T_s W_s = 1$ , no such pair exists, and OTFS necessarily leaks. Above, the pulses are "easy" but spectral efficiency suffers. This is the counterpart of Nyquist's theorem for 2D DD signaling.

Proof

Forward direction

Construct a bi-orthogonal pair via Zak transform. At $T_s W_s = 1$ , the Zak transform is well-defined and invertible (Chapter 2). Choose $g_{\mathrm{tx}}$ arbitrary; define $g_{\mathrm{rx}}$ via inverse Zak.

Converse

For $T_s W_s < 1$ , the 2D sampling is too dense: each pulse's Fourier transform aliases into its shifted copies. Orthogonality impossible.

Riesz basis

$T_s W_s = 1$ is the critical density for a Weyl-Heisenberg system to span $L^2(\mathbb{R})$ . At this density, unique bi-orthogonality exists. $\blacksquare$

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Key Takeaway

OTFS operates at critical TF sampling $T_s W_s = 1$ . This is the design sweet spot: ideal bi-orthogonality exists (with the right pulse choice), and spectral efficiency is maximized. Departures cost either bi-orthogonality (sub-critical) or spectral efficiency (super-critical).

Definition:
RRC Bi-Orthogonal Pair

Root-raised cosine pair: widely-used bi-orthogonal design. Transmit and receive are both root-raised-cosine (RRC) filters with the same roll-off $\gamma$ . Their cascade (Tx $\to$ channel $\to$ Rx) gives a raised-cosine filter — which has zero ISI at integer symbol times.

For OTFS with RRC:

Time support: $[-2T_s, 2T_s]$ (tapered).
Frequency: $|f| \leq (1 + \gamma) W_s/2$ .
Cross-ambiguity at DD grid samples: $\delta$ (bi-orthogonal).
Roll-off $\gamma = 0.2$ - $0.35$ typical.

Performance: ISI = 0 at perfect sync. ICI reduced by $\gamma$ (the "excess" bandwidth relaxes the sampling). Ideal for mobility.

Cost: $\gamma \cdot 100\%$ extra bandwidth for the roll-off transition. For $\gamma = 0.35$ : 35% bandwidth overhead vs brickwall pulses.

Theorem: RRC Bi-Orthogonality Proof

For two RRC filters with roll-off $\gamma$ , applied at Tx and Rx, the cascade is a raised-cosine (RC) filter. The RC filter satisfies the Nyquist ISI criterion: $\mathrm{RC}(\ell T_s) \;=\; \delta_\ell.$ Equivalently: the cross-ambiguity function of the RRC pair is zero at all non-zero DD grid samples.

Consequence: RRC pair is bi-orthogonal. At critical sampling $T_s W_s = 1 + \gamma$ , spectral efficiency is $1/(1 + \gamma)$ of brickwall.

The raised cosine's zero-crossings at integer sample times is the key to ISI-free single-carrier transmission. In OTFS, the same property extends to 2D: zero-crossings at all DD grid samples except origin. This is why RRC is the natural choice for OTFS — and the same reason it dominates DFT-s-OFDM uplink in 5G.

Proof

RRC-RRC cascade

Time-domain convolution of two RRC = RC filter (standard result from filter theory).

RC Nyquist property

$\mathrm{RC}(t)$ has zeros at $t = T_s, 2T_s, 3T_s, \ldots$ (by construction).

2D extension

OTFS 2D pulse = $g(t) \cdot e^{j 2\pi f t}$ separable at critical sampling. Each axis independently Nyquist.

Cross-ambiguity

$A_{g_{\mathrm{rx}}, g_{\mathrm{tx}}}(\ell T_s, m W_s) = 0$ for $(\ell, m) \neq 0$ by the Nyquist property. $\blacksquare$

Example: RRC Design for a 6G OTFS Deployment

Design RRC filters for 6G OTFS: 100 MHz total bandwidth, $M = 128$ delay bins ( $W_s = 780$ kHz per subcarrier), $T_s = 1.28 \mu$ s per symbol. Choose roll-off $\gamma$ for optimal ISI-ICI balance.

Solution

Critical sampling check

$T_s W_s = 1.28 \cdot 10^{-6} \cdot 780 \cdot 10^3 = 1.0$ . Exactly critical. Good starting point.

Roll-off choice

Mobility scenario: favor ICI suppression. $\gamma = 0.25$ gives 25% excess bandwidth but 12-15 dB ICI suppression.

Effective bandwidth

Bandwidth with $\gamma$ : $W_s (1 + \gamma) = 780 \cdot 1.25 = 975$ kHz per subcarrier. $M \cdot 975 = 125$ MHz — exceeds 100 MHz budget by 25%.

Adjustment

Reduce $M$ to 80 delay bins. Each subcarrier: 1.25 MHz. Total: 100 MHz. With $\gamma = 0.25$ , $T_s = 0.8 \mu$ s.

Performance

ISI $< -30$ dB, ICI $< -25$ dB at 120 km/h mobility. Acceptable for 6G target KPIs.

Cross-Ambiguity Function for RRC Pair

2D plot of $|A_{g_{\mathrm{rx}}, g_{\mathrm{tx}}}(\tau, \nu)|^2$ for an RRC pulse pair. DD grid samples overlaid in red. Shows bi-orthogonality at grid samples and controlled leakage elsewhere. Sliders: roll-off $\gamma$ , pulse type.

Parameters

Roll-off

\gamma

0.35

Pulse type

🔧Engineering Note

CommIT Recommended Pulse Choices

CommIT group's pulse shape recommendations by scenario:

Static (indoor, BS-BS backhaul): rectangular + CP. Simple. $\gamma = 0$ . ICI acceptable (no Doppler).

Low-mobility (pedestrian, $< 10$ km/h): RRC $\gamma = 0.15$ . Minor ICI suppression.

Vehicular (30-150 km/h, 5G NR mmWave): RRC $\gamma = 0.25$ . Balanced ISI/ICI.

High-mobility (HST, 200-500 km/h): RRC $\gamma = 0.35$ . Aggressive ICI suppression. Tolerable 35% bandwidth cost.

LEO satellite (>7000 km/h, extreme Doppler): Gaussian $\sigma = 0.3 T_s$ . Best time-frequency concentration. Bandwidth cost $\sim 40\%$ but essential for LEO performance.

ISAC (automotive radar + comms): same as vehicular, with tailored $\gamma = 0.2-0.35$ . Ambiguity function tuned for sensing (Chapter 11).

Deployment: CommIT reference designs use these per-scenario defaults. Adaptive pulse selection in 6G gNB (picks per UE based on profile).

Practical Constraints

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Static: rectangular + CP, $\gamma = 0$
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Vehicular: RRC $\gamma = 0.25$
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HST: RRC $\gamma = 0.35$
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LEO: Gaussian $\sigma = 0.3 T_s$

Common Mistake: Tx and Rx Pulses Must Be Designed Together

Mistake:

Picking transmit pulse optimally without considering the receive filter. Bi-orthogonality is a property of the pair, not the individual pulses.

Correction:

Design Tx and Rx pulses jointly. For RRC pair: both Tx and Rx are RRC with the same $\gamma$ . Gaussian pair: both Gaussian with matched $\sigma$ . Hermite pair: matched orders. Adjusting one without the other breaks bi-orthogonality and introduces interference. Commercial OTFS implementations fix both Tx and Rx to a single reference pulse (e.g., RRC $\gamma = 0.25$ ); changing one forces a retune of the other.

The Bi-Orthogonality Condition