Ferkans — Interactive Telecom Tutor

The Pulse That Cannot Be

Every signal-space argument in OTFS assumed, implicitly or explicitly, a pulse shape with specific properties: compact support in time, compact support in frequency, and orthogonality across translations. Such pulses do not exist — uncertainty principle. Real OTFS systems pick pulses that approximate these properties with controlled trade-offs: ISI (from time leakage), ICI (from frequency leakage), spectral efficiency (from guard gaps), and PAPR (from pulse shape). This section enumerates the ideal requirements and the practical compromises.

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Definition:
Ideal OTFS Pulse

An ideal OTFS pulse $g(t)$ is one whose:

Time support: concentrated in $[0, T_s]$ where $T_s = T/N$ is the per-symbol duration.
Frequency support: concentrated in $[-W_s/2, W_s/2]$ where $W_s = W/M$ .
Time-frequency product: $T_s W_s \geq 1$ (uncertainty).
Bi-orthogonality: transmit and receive pulses are paired such that $\langle g_{\mathrm{tx}}[\ell, k], g_{\mathrm{rx}}[\ell', k'] \rangle = \delta[\ell - \ell', k - k']$ on the DD grid.

Classical ideal choices:

Rectangular: $g(t) = 1$ for $t \in [0, T_s]$ . Compact time; infinite frequency support.
Sinc: $g(t) = \sin(\pi t/T_s)/(\pi t/T_s)$ . Compact frequency; infinite time support.
Gaussian: compromise, non-compact both domains but exponentially fast decay.

No pulse is simultaneously time- and frequency-compact. Every practical choice trades off ISI and ICI.

Theorem: Time-Frequency Uncertainty for OTFS Pulses

For any real-valued pulse $g(t)$ with unit energy: $\sigma_t \cdot \sigma_f \;\geq\; \frac{1}{4\pi},$ where $\sigma_t, \sigma_f$ are time- and frequency-standard deviations. Equality holds for the Gaussian pulse $g(t) = (2\sigma^2)^{-1/4} e^{-t^2/(4\sigma^2)}$ .

Consequence for OTFS: transmit-receive pulse pair must respect this uncertainty. If $\sigma_t = T_s/4$ (concentrated): $\sigma_f \geq 1/(\pi T_s)$ . For $T_s = 8\,\mu$ s: $\sigma_f \geq 40$ kHz. For 100 MHz $W$ and $M = 256$ subcarriers ( $W_s = 390$ kHz): this is $\sim 10\%$ of $W_s$ . Modest but real frequency leakage.

The uncertainty principle is the fundamental obstacle to ideal pulse shaping. No pulse can be both compact in time and compact in frequency. OTFS design accepts this: the chosen pulse has finite support in one domain and bounded leakage in the other. The leakage manifests as ISI (time leakage) or ICI (frequency leakage), both controllable via pulse design.

Proof

Variance product

Fourier pair: $g(t), G(f)$ . Parseval: $\int|g|^2 = \int|G|^2 = 1$ . Variances: $\sigma_t^2 = \int t^2 |g|^2$ , $\sigma_f^2 = \int f^2 |G|^2$ .

Lower bound

Standard calculus + Cauchy-Schwarz on derivative integral: $\sigma_t \sigma_f \geq 1/(4\pi)$ .

Equality for Gaussian

$g(t) = (2\sigma^2)^{-1/4} e^{-t^2/(4\sigma^2)}$ . Fourier: $G(f) = (2\sigma^2)^{1/4} e^{-4\pi^2 \sigma^2 f^2}$ . Both have the same form; equality achieved. $\blacksquare$

Key Takeaway

OTFS cannot escape the time-frequency uncertainty. Any pulse has bounded $\sigma_t \sigma_f \geq 1/(4\pi)$ . This dictates the leakage budget for ISI vs ICI. Designers choose which to privilege — time-leakage (rectangular) or frequency-leakage (sinc) or balanced (Gaussian).

Definition:
Common OTFS Pulse Choices

Four canonical OTFS pulse shapes:

Rectangular (CP-based): $g(t) = 1$ for $t \in [0, T_s]$ . Compact in time; infinite frequency tails. Simple, used with CP. ISI: 0. ICI: max (Dirichlet sidelobes).

Root-raised cosine (RRC): the standard for single-carrier systems. Roll-off $\gamma \in [0, 1]$ controls frequency localization. RRC pairs at Tx/Rx give zero ISI and moderate frequency leakage. $\gamma = 0$ : brickwall (infinite time support). $\gamma = 1$ : wide bandwidth, finite impulse response.

Gaussian: exponentially decaying in both domains. Optimal in time-frequency concentration. Good for ISAC and sensing.

Hermite: sum of Gaussian × Hermite polynomial. Exact bi-orthogonality at specific sample points. Used in filter-bank OTFS.

Prolate spheroidal wave functions (PSWF): maximum energy concentration within a given time-bandwidth product. Theoretical optimum for some bi-orthogonality constraints.

OTFS Pulse Shape Comparison

Plot four canonical OTFS pulse shapes in time domain, with their magnitudes normalized. Overlay frequency-domain response. Sliders: pulse type, roll-off $\gamma$ , symbol duration $T_s$ .

Parameters

Pulse type

Roll-off

\gamma

0.35

Samples128

Theorem: ISI-ICI Trade-off

For an OTFS system with symbol period $T_s$ and bandwidth $W_s$ per subcarrier, and pulse shape $g(t)$ with time extent $\sigma_t$ and frequency extent $\sigma_f$ :

ISI power from adjacent symbols (delay neighbors): $\mathrm{ISI}_{\text{pwr}} = \int_{|\tau| \leq T_s} |g(t - T_s)|^2 g(t) dt$
ICI power from adjacent subcarriers (Doppler neighbors): $\mathrm{ICI}_{\text{pwr}} = \int_{|\nu| \leq W_s} |G(f - W_s)|^2 G(f) df$

The product is bounded: $\mathrm{ISI}_{\text{pwr}} \cdot \mathrm{ICI}_{\text{pwr}} \;\geq\; C(T_s W_s)^{-1}$ for a universal constant $C$ .

Consequence: minimizing one increases the other. Practical OTFS design picks the operating point: for static channels (Wi-Fi indoor): favor ICI (CP-based rectangular). For mobility (vehicular): favor ISI-free (sinc or RRC).

The ISI-ICI trade-off is the OTFS-specific version of the time-frequency uncertainty. Since the designer cannot make both zero, the choice of pulse shape is really choosing which interference to minimize. Rectangular + CP has zero ISI (because CP absorbs delay spread) but large ICI. RRC has controlled ISI and controlled ICI via the roll-off $\gamma$ . Sinc has zero ICI and massive ISI — never used alone.

Proof

Time-leakage

ISI: overlap of pulse with its time-shifted copy. $\mathrm{ISI} \propto |g(T_s)|^2 T_s$ (by Cauchy-Schwarz).

Frequency-leakage

ICI: overlap of pulse frequency response with adjacent. $\mathrm{ICI} \propto |G(W_s)|^2 W_s$ .

Uncertainty bound

$|g(T_s)|^2 |G(W_s)|^2 \geq 1/(T_s W_s)$ by uncertainty (applied at the specific lags).

Product

ISI × ICI $\geq C/(T_s W_s)$ for some constant. Trade-off is fundamental. $\blacksquare$

Example: Rectangular vs RRC for 5G NR OTFS

Compare rectangular + CP and RRC ( $\gamma = 0.35$ ) pulse shapes for OTFS at 5G NR mmWave: $\Delta f = 120$ kHz, 100 MHz bandwidth, delay spread 5 $\mu$ s, Doppler spread 1 kHz.

Solution

Rectangular

ISI: 0 (CP absorbs delay). ICI: Dirichlet sidelobes. Main ICI: 12-15 dB below signal. Acceptable for delay-tolerant services.

RRC, $\gamma = 0.35$

ISI: $\leq 30$ dB below signal (tapered). ICI: $\leq 40$ dB. Cleaner overall. Costs: $\gamma = 35\%$ bandwidth overhead.

Tradeoff

Rectangular: simpler, more ICI. RRC: cleaner, 35% BW cost. For mobility: ICI from Doppler dominates — RRC's ICI suppression helps. For stationary: ICI is small anyway — rectangular + CP is simpler.

Choice

5G NR mmWave ISAC/V2X (high mobility): RRC $\gamma = 0.2-0.35$ . Sub-6 GHz static: rectangular + CP.

🔧Engineering Note

Pulse Choice in Practice

Commercial OTFS implementations (2026 baseline):

Cohere reference design: rectangular + CP. Simple, compatible with existing OFDM hardware.
Academic implementations (Chapter 17-18 testbeds): RRC with $\gamma = 0.2$ . Cleaner, marginal complexity overhead.
3GPP 6G candidate: RRC with $\gamma = 0.15$ . Compromise between simplicity and spectral efficiency.

Hardware: RRC filter implementation in FIR (direct form) or polyphase (Nyquist filter bank). FIR: ~ $100$ taps at typical sampling rate; compute per symbol: $\sim 10^4$ ops. Modest.

Performance difference: RRC vs rectangular in BER:

Static: RRC +0.3 dB (from cleaner ICI).
Vehicular: RRC +1.5 dB (significant).
LEO: RRC +3+ dB (ICI matters more with extreme Doppler).

Industry converging on RRC with $\gamma \in [0.15, 0.35]$ . Exact value per-scenario.

Practical Constraints

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Cohere: rectangular + CP (simpler)
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Academic: RRC $\gamma = 0.2$
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3GPP 6G: RRC $\gamma = 0.15-0.35$
•
Hardware: FIR with ~100 taps

Common Mistake: CP Is Not a Silver Bullet

Mistake:

Treating rectangular pulse + cyclic prefix as universally correct for OTFS. CP absorbs delay spread for OFDM — but OTFS's 2D channel means CP only partially suppresses ISI; ICI from Doppler remains.

Correction:

For OTFS, CP handles only the delay dimension. Doppler ICI is a separate problem requiring pulse-shape selection. Pure rectangular works for static channels; RRC (or similar) for mobility. Choose pulse based on dominant interference type: delay-dominant (static) → CP; Doppler-dominant (mobile) → RRC.

Ideal vs. Practical Pulses