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Higher-Resolution DD Grid

BEM (Section 3) and windowing mitigate fractional Doppler at the detector level — the grid stays at $(M, N)$ ; only the detector gets smarter. An alternative attacks the problem at the grid level: use an oversampled DD grid $(M_{\text{os}}, N_{\text{os}})$ with finer resolution so that fractional offsets become smaller relative to the grid spacing. The trade-off: extra complexity from the larger grid, versus reduced IDI per cell.

The point is that oversampling is a simpler conceptual fix — "finer sampling" — and works cleanly when the additional complexity is affordable. In practice it is used in niche scenarios (sensing applications, Chapter 11-12) where fine Doppler resolution is intrinsic to the objective.

Definition:
Oversampled DD Grid

An oversampled DD grid has dimensions $(M_{\text{os}}, N_{\text{os}}) = (\alpha M, \beta N)$ with oversampling factors $\alpha, \beta \geq 1$ (integers). Grid resolutions become $\Delta\tau_{\text{os}} = 1/(\alpha W)$ and $\Delta\nu_{\text{os}} = 1/(\beta T)$ — finer by factor $\alpha, \beta$ .

The total number of DD cells scales as $\alpha\beta MN$ . For $\alpha = 1, \beta = 2$ (Doppler oversampling), $N_{\text{os}} = 2N$ — twice as many Doppler bins. Fractional offsets $\epsilon \in (-0.25, 0.25)$ in the oversampled grid (halved from the nominal $(-0.5, 0.5)$ range).

Theorem: IDI Reduction via Doppler Oversampling

With Doppler oversampling factor $\beta$ , the IDI kernel in the oversampled grid satisfies $|K_{\text{IDI}}^{(\beta)}(k, \epsilon^{(\beta)})|^2 \;=\; |K_{\text{IDI}}(k, \beta \epsilon)|^2,$ where $\epsilon^{(\beta)} \in (-0.5/\beta, 0.5/\beta)$ is the reduced fractional offset. The main-lobe energy increases with $\beta$ : $|K_{\text{IDI}}^{(\beta)}(0, \epsilon^{(\beta)})|^2 \approx 1 - (\pi \beta \epsilon)^2/3,$ so oversampling by factor $\beta$ quadratically reduces IDI.

Doubling $N$ halves the effective fractional offset (in the oversampled grid), which quadratically reduces IDI energy. This is the Doppler-domain analog of oversampling in sampled-data DSP.

The cost: the detector operates on a $\beta$ -times-larger grid. MMSE: $O(MN_{\text{os}}\log(MN_{\text{os}}))$ — a factor of $\beta\log\beta$ more. MP: proportional to factor graph size. For moderate $\beta \leq 4$ , the complexity increase is acceptable.

Proof

Oversampled grid

With $N_{\text{os}} = \beta N$ , Doppler bin spacing shrinks: $\Delta\nu_{\text{os}} = 1/(\beta T) = \Delta\nu/\beta$ . A physical Doppler $\nu_i$ with $\epsilon_i = \nu_i T - k_i^{\text{int}}$ becomes $\epsilon_i^{(\beta)} = \beta \nu_i T - \lfloor \beta \nu_i T\rfloor$ , i.e., the fractional part in the finer grid.

Reduced fractional

For generic $\nu_i$ , $|\epsilon_i^{(\beta)}| \leq 0.5/\beta$ . Oversampling by $\beta$ compresses the fractional range.

Quadratic IDI reduction

Using the small- $\epsilon$ expansion of the IDI kernel: $\rho_{\text{IDI}} \approx (\pi \beta \epsilon)^2/3$ . With $\epsilon^{(\beta)} = \epsilon/\beta$ : $\rho_{\text{IDI}}^{(\beta)} \approx (\pi\epsilon)^2/(3\beta)$ . $\beta$ -fold reduction in IDI power.

Cost: complexity

Grid size scales as $\beta$ . FFT operations scale as $\beta\log\beta$ . Detector memory scales as $\beta$ . For $\beta = 2$ : factor 2 complexity increase. For $\beta = 4$ : factor 4-5. Diminishing returns. $\blacksquare$

Key Takeaway

Oversampling trades complexity for IDI reduction. Doubling the Doppler grid ( $\beta = 2$ ) reduces IDI by 6 dB at the cost of 2× complexity. Quadrupling ( $\beta = 4$ ) gives 12 dB reduction at 4-5× complexity. Oversampling is typically applied together with BEM and windowing to reach near-integer-Doppler performance at 10-30× the nominal compute budget.

BER Improvement vs Oversampling Factor

For a fixed fractional-Doppler channel ( $P = 4$ , $\epsilon_i$ uniform in $[-0.5, 0.5]$ ), plot the uncoded BER at various SNRs against oversampling factor $\beta \in \{1, 2, 4, 8\}$ . The curve flattens at high $\beta$ : most gains are captured at $\beta = 2$ , with diminishing returns beyond.

Parameters

P

4

SNR (dB)15

Max

\beta

8

Example: Oversampling for Doppler-Velocity Sensing

An OTFS-based Doppler radar targets a pedestrian at 5 km/h at 5 GHz. The target's Doppler is $(v/c) f_0 = 23$ Hz. With frame $T = 10$ ms, $\nu T = 0.23$ . Using oversampling $\beta$ , what is the velocity resolution achievable?

Solution

Baseline $\beta = 1$

$\Delta\nu = 1/T = 100$ Hz. Velocity resolution: $\Delta v = c \Delta\nu/f_0 = 6$ m/s (22 km/h). Cannot resolve pedestrian (5 km/h below resolution).

Oversample $\beta = 4$

$\Delta\nu_{\text{os}} = 25$ Hz. Velocity resolution: $\Delta v = 1.5$ m/s (5.4 km/h). Pedestrian at 5 km/h is at the limit.

Oversample $\beta = 16$

$\Delta\nu_{\text{os}} = 6.25$ Hz. Velocity resolution: $\Delta v = 0.375$ m/s (1.35 km/h). Pedestrian well-resolved.

Implication

Oversampling is essential for fine-Doppler sensing (Chapter 11). At $\beta = 16$ , the grid has $MN_{\text{os}} = 16 MN$ cells — detection complexity scales accordingly. But for sensing, resolution is the primary goal.

Oversampling Is Not Free for Data Transmission

The oversampled DD grid has $\beta$ -times more cells but the data rate does not increase: only $MN$ data symbols fit in the nominal spectrum $[0, W] \times [0, T]$ . The extra cells in the oversampled grid are redundant — they replicate the information in the nominal grid at a finer sampling rate.

For data transmission, oversampling costs $\beta$ -times more compute with no rate gain. Only when the fractional-offset detection gain outweighs this cost is oversampling worthwhile for communications. Typically $\beta \leq 2$ is the practical limit for data.

For sensing, oversampling does improve the output: the target's Doppler is resolved at finer granularity. This is why OTFS radar (Chapter 11) commonly uses $\beta = 4$ - $16$ .

Fractional-Doppler Mitigation: Comparison

Technique	IDI reduction	Complexity cost	Best use
Ignore fractional	0 dB	None	Small $\epsilon$ only
Windowing (Hamming)	~15 dB sidelobes	Negligible	Moderate $\epsilon \leq 0.3$
BEM ( $Q = 3$ )	~6 dB effective	$3\times$ paths	General purpose (recommended)
Oversampling ( $\beta = 2$ )	6 dB quadratic	$2\times$ compute	Sensing applications
Oversampling ( $\beta = 4$ )	12 dB quadratic	$4\times$ compute	Fine Doppler (radar)
BEM + windowing + $\beta = 2$	$\sim 0$ -IDI effective	$\sim 6\times$ compute	6G URLLC

🔧Engineering Note

Recommended Configurations

Recommended fractional-Doppler configurations by deployment:

eMBB (broadband, moderate mobility): Windowing (Hamming) + integer-Doppler detector. Assumes most paths have $\epsilon \leq 0.3$ after windowing suppresses sidelobes. $\sim 1$ dB gap to ideal.
URLLC (reliability-critical): BEM $Q = 3$ + windowing. Full fractional-aware MMSE or MP. $< 0.5$ dB gap, predictable latency.
V2X mmWave: BEM $Q = 5$ + Blackman window. Extreme Doppler requires more BEM terms and tighter windowing.
Sensing/ISAC (Chapter 11-12): Oversample $\beta = 4$ – $16$ for fine velocity resolution; BEM for joint data detection.
LEO satellite (Chapter 18): $\beta = 4$ + BEM $Q = 5$ . Extreme fractional Doppler from orbital dynamics requires the full toolkit.

The integer-Doppler assumption of Chapters 4-9 is the clean theoretical baseline; this section's techniques are what make the theory work in practice.

Practical Constraints

•
Typical eMBB: windowing alone is adequate (1 dB gap)
•
BEM $Q = 3$ is the sweet spot for compute vs performance
•
Oversampling $\beta = 2$ standard for cell-free and mobile

Oversampled DD Grid