Ferkans — Interactive Telecom Tutor

What Did We Lose?

The ideal OTFS performance results of Chapter 9 — full $P$ -order diversity, $\sim 10$ dB gain over OFDM at $10^{-5}$ BER — were derived under integer Doppler. This section quantifies the residual degradation after applying the Section 2-4 mitigations, and compares against OFDM (which does not enjoy any DD-domain advantage). The headline: practical OTFS at $\epsilon \sim 0.3$ with BEM $Q = 3$ + windowing loses $\sim 1$ dB relative to the ideal $P$ -order diversity — still a decisive $5-8$ dB advantage over OFDM.

Theorem: Diversity Order Under Fractional Doppler

For an OTFS system with $P$ paths at fractional Doppler offsets $\{\epsilon_i\}$ , using BEM of order $Q_{\text{BEM}}$ and integer delay, the achievable diversity order is $d_{\text{frac}} \;=\; \min(P,\,(2 k_{\max} + 1)).$ where $k_{\max}$ is the maximum integer Doppler index. In particular:

If $Q_{\text{BEM}} \geq 2k_{\max} + 1$ : $d_{\text{frac}} = P$ (full diversity recovered).
If $Q_{\text{BEM}} < 2k_{\max} + 1$ : $d_{\text{frac}} < P$ (BEM undersampled; some diversity lost).

The diversity order is fundamentally limited by the effective number of independent fading dimensions — at most $P$ (the path count) and at most $2k_{\max} + 1$ (the span of integer Doppler).

Fractional Doppler does not fundamentally reduce diversity — it just distributes the channel's Doppler information across more cells. With adequate BEM order, the diversity of $\min(P, 2k_{\max} + 1)$ is recovered. For most vehicular channels with $k_{\max} = 1$ - $3$ , $2k_{\max} + 1 \leq 7$ , so $P$ dominates. For LEO with $k_{\max} = 500$ , diversity is fundamentally bounded by $\min(P, 1001)$ — always $P$ in practice.

Proof

BEM-expanded path count

With BEM $Q$ , each fractional path expands to $Q$ effective integer paths. Total paths: $P Q$ . If $Q = 1$ (integer only): loses diversity. If $Q$ matches the BEM's effective Doppler spread: retains diversity.

Upper bound from $k_{\max}$

The integer Doppler occupies bins $k \in [-k_{\max}, k_{\max}]$ . Any linear combination of channel paths lies in a $(2k_{\max}+1)$ -dimensional subspace. Diversity order cannot exceed the subspace dimension.

Lower bound from $P$

With $P$ physically independent paths, diversity cannot exceed $P$ .

Tight bound

$d_{\text{frac}} = \min(P, 2k_{\max}+1)$ when BEM order is matched. $\blacksquare$

Key Takeaway

Fractional Doppler does not destroy diversity — it just requires bigger detectors. With BEM order matched to the channel's Doppler span, OTFS retains full $P$ -order diversity even under realistic fractional offsets. The diversity theorem of Chapter 9 holds for practical channels, not just the idealized integer case. The practical cost is a few dB of SNR and $3\text{-}5\times$ more compute — still a major win over OFDM.

BER Under Fractional Doppler: Integer Assumption vs Full Treatment

Plot uncoded BER vs SNR for OTFS with $P = 4$ and $\epsilon_i \in [0, 0.5]$ : (a) ignore fractional (baseline), (b) windowing only, (c) BEM $Q = 3$ , (d) BEM + windowing + oversample. Observe the gap between (a) and (d) is $\sim 5$ dB at BER $= 10^{-5}$ .

Parameters

P

4

Max fractional

|\epsilon|

0.4

Min SNR (dB)5

Max SNR (dB)30

Example: SNR Budget for a Real Vehicular Deployment

A 5G V2X link at 5.9 GHz, $v = 100$ km/h, target BER $10^{-5}$ uncoded, $P = 4$ paths. Compute required SNR for (a) integer detection, (b) BEM $Q = 3$ + Hamming window, (c) full treatment (BEM + window + $\beta = 2$ ).

Solution

Doppler setting

$\nu_{\max} = (27.8/3 \times 10^8) \cdot 5.9 \times 10^9 = 547$ Hz. At $T = 2$ ms: $\nu_{\max} T = 1.09$ . Fractional offsets generic; $\epsilon \sim 0.5$ worst case.

Ideal $P = 4$ diversity

$\mathrm{BER} = \binom{7}{4}/(4\rho)^4 = 35/(4\rho)^4 = 10^{-5}$ $\Rightarrow \rho = (35 \cdot 10^5)^{1/4}/4 = 19$ (13 dB).

(a) Integer detection

Ignore IDI. Diversity degrades to 1 (single-path Rayleigh). BER = $1/(4\rho) = 10^{-5}$ $\Rightarrow \rho = 25{,}000$ (44 dB). 31 dB penalty.

(b) BEM + window

Recovers most diversity; residual 1-2 dB gap to ideal. Required SNR: $\sim 14$ - $15$ dB.

(c) Full treatment

Near-ideal: $\sim 13$ - $14$ dB.

Conclusion

Ignoring fractional: totally infeasible (44 dB). BEM + windowing: $\sim 15$ dB — well within typical link budget. The mitigations are necessary, not optional.

Fractional-Doppler Impact by Scenario

Scenario	$\epsilon$ range	Mitigation level	SNR penalty
Pedestrian (indoor)	$\leq 0.05$	None needed	$<0.1$ dB
Urban vehicular (3.5 GHz)	0.1–0.3	Hamming window	~0.5 dB
Highway vehicular (mmWave)	0.2–0.5	BEM $Q = 3$ + window	~1 dB
HST at mmWave	0.3–0.5	BEM $Q = 5$ + Blackman	~1.5 dB
LEO satellite (10 GHz)	Arbitrary	BEM $Q = 5$ + $\beta = 4$	~2 dB

🔧Engineering Note

Practical Compute Cost of Fractional Treatment

The compute budget for a fractional-aware OTFS receiver:

Baseline integer-Doppler MMSE: $1\times$ (reference).
Hamming windowing: $1\times$ (negligible overhead, 1 pre-multiply).
BEM $Q = 3$ : $3\times$ (3× the effective path count, 3× factor- graph connections for MP; 3× MMSE via expanded matrix).
BEM $Q = 5$ : $5\times$ .
$\beta = 2$ oversampling: $2\times$ grid, $2\log 2 = 2\times$ FFT.

Typical 5G URLLC: $15\times$ baseline compute with BEM + windowing + $\beta = 2$ . For $MN = 10^4$ : $\sim 10^6$ ops baseline, $1.5 \times 10^7$ ops fractional-aware. Still well within modern SoC capability.

The complexity adds latency: fractional-aware detection may run on a second clock cycle, giving total detection latency of 2-3 slot periods (still within 5G NR URLLC targets).

Practical Constraints

•
BEM + windowing: $3\times$ compute for full diversity
•
Oversampling $\beta = 2$ : additional $2\times$
•
All within 5G realtime budget ( $10^8$ ops/sec)

🎓CommIT Contribution(2020)

Diversity of OTFS Under Fractional Doppler

G. D. Surabhi, A. Chockalingam, G. Caire — IEEE Trans. Vehicular Technology

The CommIT extension of the Chapter 9 diversity theorem to fractional Doppler, establishing $d_{\text{frac}} = \min(P, 2k_{\max}+1)$ with BEM of order $Q_{\text{BEM}} \geq 2k_{\max}+1$ . This quantitative characterization of the practical diversity bound is the information-theoretic anchor of fractional-aware OTFS detection: it tells the detector designer how many BEM terms are needed to recover full diversity for a given channel.

The paper also shows that when $Q_{\text{BEM}} < 2k_{\max}+1$ , the lost diversity is proportional to the "BEM mismatch" — providing a graceful-degradation curve for undersampled BEM deployments. This informs the reference $Q_{\text{BEM}}$ choices in this chapter.

commitfractional-dopplerdiversity

Common Mistake: Don't Assume Worst-Case $\epsilon = 0.5$

Mistake:

Designing OTFS for all paths at $\epsilon = 0.5$ (worst case). This over-provisions BEM order and compute unnecessarily.

Correction:

In practice, fractional offsets are approximately uniformly distributed in $[-0.5, 0.5]$ . Average IDI is $\mathbb{E}[\epsilon^2] = 1/12$ , not the worst-case $0.25$ . Designing for the average case reduces compute requirement by $2-3\times$ compared to worst-case design. Use BEM $Q = 3$ for typical channels; bump to $Q = 5$ only for known high-mobility scenarios. Measure real channels (3GPP TR 38.901 statistics) before provisioning.

Why This Matters: Sensing Chapter 11 Embraces Fractional Doppler

For communications, fractional Doppler is a nuisance to be mitigated. For sensing (Chapter 11), it is exactly what we want: the fractional part $\epsilon_i$ encodes the target's fine velocity — the very quantity the radar system aims to estimate. The techniques of this chapter reverse role: BEM becomes super- resolution Doppler estimation; oversampling becomes fine velocity resolution. The mathematical framework is the same; only the objective shifts from "remove IDI" to "resolve the fractional offset precisely."

Performance Impact and Design Guidelines