Ferkans — Interactive Telecom Tutor

The Convenient Fiction

Chapters 4-9 presented OTFS under the assumption that each path's Doppler shift $\nu_i$ and delay $\tau_i$ land exactly on a DD-grid point: $\nu_i T \in \mathbb{Z}$ and $\tau_i W \in \mathbb{Z}$ . The math is clean, the diversity theorem is $P$ , and the detectors of Chapter 8 achieve it.

The point is that this integer assumption is a convenient fiction. Real channels produce paths with arbitrary real-valued Dopplers; each $\nu_i T$ is almost surely non-integer. When $\nu_i T = k_i^{\text{int}} + \epsilon_i$ with $|\epsilon_i| < 1/2$ , the path's response in the DD grid is no longer a single point but a smeared response spread across neighboring Doppler bins — the inter-Doppler interference (IDI).

This section defines fractional offsets, examines how they arise physically, and motivates the treatment of Sections 2-5.

Definition:
Fractional Delay and Doppler Offsets

A physical path with continuous delay $\tau_i$ and Doppler $\nu_i$ decomposes into integer and fractional parts on the $(M, N)$ DD grid: $\tau_i W \;=\; \ell_i^{\text{int}} + \epsilon_i^{(\tau)}, \qquad \nu_i T \;=\; k_i^{\text{int}} + \epsilon_i.$ with $\ell_i^{\text{int}}, k_i^{\text{int}} \in \mathbb{Z}$ and $|\epsilon_i^{(\tau)}|, |\epsilon_i| < 1/2$ . The integer part is the nearest DD grid cell; the fractional part quantifies how far off-grid the path is.

We typically assume $\epsilon_i^{(\tau)} = 0$ (delay resolution $\Delta\tau = 1/W$ is fine enough at realistic bandwidths; see §1.2) and focus on fractional Doppler $\epsilon_i$ as the dominant effect.

,

Why Delay Is Usually Fine, Doppler Rarely Is

Two asymmetries make fractional Doppler the practical problem, not fractional delay:

Resolution scale: $\Delta\tau = 1/W$ (bandwidth-limited); $\Delta\nu = 1/T$ (frame-duration-limited). At $W = 10$ MHz, $\Delta\tau = 100$ ns — finer than typical path delay separations. At $T = 4$ ms, $\Delta\nu = 250$ Hz — comparable to or coarser than individual path Dopplers.
Parameter range: $\tau_i \in [0, \tau_{\max}] \approx [0, 10\,\mu\text{s}]$ gives $\ell_i \in \{0, \ldots, 100\}$ — plenty of integer bins to land on. $\nu_i \in [-f_D, f_D] \approx \pm 500$ Hz gives $k_i \in \{-2, \ldots, 2\}$ — only 5 bins to quantize across. A typical path has $\epsilon_i \in (0, 0.5)$ with high probability.

Consequence: fractional delay is usually a perturbation of order $< 10\%$ of the delay resolution; fractional Doppler is routinely 50% of the Doppler resolution. The treatment below focuses on fractional Doppler; analogous arguments handle fractional delay at lower-bandwidth systems.

Theorem: DD Response Under Fractional Doppler

Consider a single path with delay $\tau_i = \ell_i/W$ (integer delay) and Doppler $\nu_i = (k_i + \epsilon_i)/T$ with fractional offset $\epsilon_i$ . Let $X_{DD}[\ell_p, k_p] = 1$ be an embedded-pilot impulse. The received DD response is $Y_{DD}[\ell_p + \ell_i, k_p + k] \;=\; h_i\,K_{\text{IDI}}(k - k_i, \epsilon_i) \;+\; W[\cdot],$ where the inter-Doppler interference (IDI) kernel is $K_{\text{IDI}}(k, \epsilon) \;=\; \frac{1}{N}\,\frac{\sin(\pi(k - \epsilon))}{\sin(\pi(k - \epsilon)/N)}\,e^{-j\pi(N - 1)(k - \epsilon)/N}.$ The kernel magnitude has a peak of $|K_{\text{IDI}}(0, 0)| = 1$ at $k = 0, \epsilon = 0$ and falls off as $|1/(k - \epsilon)|$ for distant $k$ — a Dirichlet-kernel shape.

A fractional-Doppler path does not produce a clean single-cell response. Instead, its energy spreads across many Doppler bins according to the Dirichlet kernel. At $\epsilon = 0$ (integer Doppler), the kernel collapses to $\delta_{k, 0}$ — the familiar integer case. At $\epsilon = 0.5$ (worst case), the kernel is evenly split between $k = 0$ and $k = 1$ , with $\sim 40\%$ spillover to $k = -1, 2$ .

This smearing is the single most important practical correction to the clean Chapter 4 theory. The rest of this chapter develops methods to handle it.

Proof

Path contribution to transmit waveform

The single path in the continuous DD model contributes $y(t) = h_i\,x(t - \tau_i)\,e^{j 2\pi \nu_i t}$ . After OTFS demodulation (Wigner + SFFT), the received DD grid receives contributions at integer-delay position $\ell_i$ (assumed integer) and a complex exponential in the Doppler axis $e^{j 2\pi \nu_i T n/N}$ for OFDM symbol $n = 0, \ldots, N - 1$ .

Doppler-axis DFT

The SFFT applies an $N$ -point DFT along the Doppler axis: $Y_{DD}[\cdot, k] = \frac{1}{\sqrt{N}}\sum_n e^{j 2\pi \nu_i T n/N}\,e^{-j 2\pi k n/N} = \frac{1}{\sqrt{N}}\sum_n e^{j 2\pi (k_i + \epsilon_i - k) n/N}$ .

Evaluate the geometric sum

$\sum_{n = 0}^{N - 1} e^{j 2\pi (k_i + \epsilon_i - k) n/N} = \frac{1 - e^{j 2\pi(k_i + \epsilon_i - k)}}{1 - e^{j 2\pi(k_i + \epsilon_i - k)/N}}$ . After normalization, this is the Dirichlet kernel stated in the theorem — with argument $(k - k_i - \epsilon_i)$ .

Read off the kernel

$K_{\text{IDI}}(k, \epsilon) = \frac{1}{N}\frac{\sin(\pi(k - \epsilon))}{\sin(\pi(k - \epsilon)/N)}\,e^{-j\pi(N - 1)(k - \epsilon)/N}$ . At $\epsilon = 0$ : the denominator $\sin(\pi k/N)$ makes the kernel $\delta_{k, 0}$ (Dirichlet's peak at the origin). At $\epsilon \neq 0$ : the kernel has a shifted peak and non-trivial sidelobes. $\blacksquare$

Key Takeaway

Fractional Doppler smears path energy across Doppler bins according to a Dirichlet kernel. A single path no longer produces a single DD-grid response; it produces a sinc-shaped pattern centered at the nearest integer bin, with energy leaking to $\pm 1, \pm 2, \ldots$ neighbors. At $\epsilon = 0.5$ (worst case), the leakage is roughly $40\%$ to neighbors. This is the fundamental OTFS problem addressed throughout this chapter.

Fractional Doppler and the IDI Kernel

The IDI kernel

|K_{\text{IDI}}(k, \epsilon)|^2

as

\epsilon

sweeps from 0 to 0.5. At

\epsilon = 0

: main lobe captures all energy. As

\epsilon

grows: main lobe shrinks, sidelobes appear at

k = \pm 1, \pm 2

. This is the source of inter-Doppler interference and the motivation for fractional-aware detection.

Inter-Doppler Interference Kernel $|K_{\text{IDI}}(k, \epsilon)|^2$

Plot the IDI kernel magnitude as a function of Doppler offset $k$ for several fractional values $\epsilon \in \{0, 0.1, 0.25, 0.5\}$ . At $\epsilon = 0$ , perfect single-bin localization. As $\epsilon$ grows, the kernel spreads, and at $\epsilon = 0.5$ the energy is evenly split between $k = 0$ and $k = 1$ . Observe the sinc-like sidelobe structure that decays as $1/k^2$ .

Parameters

Fractional Doppler

\epsilon

0.25

Doppler bins

N

16

Neighbors shown5

Example: Fractional Doppler in a Vehicular Channel

A vehicle at $v = 60$ km/h at 3.5 GHz produces a single-LOS-path Doppler $\nu_1 = 195$ Hz. The OTFS frame has $T = 2$ ms. Compute $(\ell_1^{\text{int}}, k_1^{\text{int}}, \epsilon_1)$ , and estimate the IDI power fraction to neighboring bins.

Solution

Doppler normalization

$\nu_1 T = 195 \times 0.002 = 0.39$ . $k_1^{\text{int}} = 0$ , $\epsilon_1 = 0.39$ .

Main-lobe magnitude

$|K_{\text{IDI}}(0, 0.39)|^2 \approx \text{sinc}^2(0.39) = 0.625$ . $62.5\%$ of energy at nearest bin; $37.5\%$ leaked.

Neighbor-bin split

$|K_{\text{IDI}}(1, 0.39)|^2 \approx \text{sinc}^2(1 - 0.39) = 0.27$ (27%). $|K_{\text{IDI}}(-1, 0.39)|^2 \approx \text{sinc}^2(-1 - 0.39) = 0.05$ (5%). Residual: $\sim 5\%$ in $k = \pm 2$ bins.

Implication

At this Doppler, 62% of path energy lands in bin 0, 27% in bin 1. Integer-Doppler detectors assigning all energy to bin 0 lose 27% of the path gain — a significant penalty. Mitigations of Sections 2-4 recover this energy.

⚠️Engineering Note

When to Worry About Fractional Doppler

Fractional Doppler is consequential when:

$f_D T \gtrsim 1$ : the maximum Doppler index covers more than one bin. Very common at vehicular mobility (100 km/h at 3.5 GHz, $T = 2$ ms gives $f_D T \approx 0.65$ ).
Single-path dominant channel: when LOS accounts for $\geq 70\%$ of channel energy, fractional offsets in the single dominant path dominate detection error.
Low SNR operation: at SNR $< 10$ dB, the IDI sidelobes mask paths that would otherwise be detected. Threshold-based estimators fail silently.
Long frames: LEO satellite with $T = 10$ ms pushes $f_D T$ into the tens — deeply fractional regime.

Fractional Doppler is not critical when:

Short frames: $T < 0.5$ ms keeps $f_D T \ll 1$ for typical terrestrial channels; rounding to the nearest integer bin is adequate.
Rich multipath: when many paths have IDI, the leakage averages out across DD cells and the per-cell impact is modest.

The rule of thumb: integer-Doppler assumption is fine for single-OFDM-symbol frames ( $N = 1$ ), degrades at $N \in [4, 16]$ , becomes unusable at $N \geq 32$ .

Practical Constraints

•
Integer assumption valid when $f_D T \leq 0.2$
•
IDI becomes 30%+ of signal when $f_D T > 0.5$
•
Frame duration $T$ is the lever — larger $T$ means finer Doppler but worse IDI

Integer vs Fractional Doppler: Summary

Aspect	Integer Doppler	Fractional Doppler
Channel response at path	Single DD cell	Dirichlet kernel across ~5 cells
Detector complexity	Sparse sum over $P$ paths	Dense coupling across 5-10 cells per path
Diversity order (§9 result)	$P$	$\min(P, k_{\max})$
Pilot estimation	Single threshold test	Interpolation / super-resolution
Prevalence	Only under special geometry	Generic case
Practical impact	Ideal baseline	Real-world mitigation needed
Treatment	Chapters 4-9	This chapter

IDI Kernel in the Doppler Axis — The inter-Doppler interference kernel $|K_{\text{IDI}}(k, \epsilon)|^2$ as a function of Doppler bin offset $k$ for several fractional offsets. At $\epsilon = 0$ : single spike at $k = 0$ . At $\epsilon = 0.5$ : bimodal split between $k = 0$ and $k = 1$ with significant sidelobes. The kernel magnitude is bounded by $1$ at the main lobe and decays as $1/k^2$ at distant bins — a classic Dirichlet / sinc pattern.

Integer vs Fractional Doppler