Ferkans — Interactive Telecom Tutor

Pilot Design as an Optimization

Sections 1 and 2 designed an embedded pilot and a detector. This section quantifies the resulting performance — channel-estimation MSE and pilot overhead — and compares it to OFDM-based schemes. The comparison is concrete and favorable: OTFS embedded pilots achieve lower MSE at a fraction of the overhead. The underlying reason is the $P$ -sparse structure of the DD channel, which means we are estimating $P$ parameters, not $MN$ .

Theorem: MSE of Embedded Pilot Channel Estimation

Under the embedded pilot scheme with pilot power $|\alpha|^2$ , noise variance $\sigma^2$ , and a $P$ -path integer-Doppler channel, the least-squares estimates satisfy $\mathrm{Var}(\hat{h}_i - h_i) \;=\; \frac{\sigma^2}{|\alpha|^2}, \qquad i = 1, \ldots, P.$ The total mean-squared error of the channel-vector estimate is $\mathrm{MSE}_{\text{total}} \;=\; P \cdot \frac{\sigma^2}{|\alpha|^2}.$ This scales with the number of paths, not with the grid size $MN$ .

Each path contributes one complex parameter estimated from one DD grid cell. With $P$ independent cells and $P$ parameters, the MSE is simply the sum of per-cell variances. No $MN$ dependence appears because the sparse structure is fully exploited.

Proof

Per-path LS estimator

At the guard-region cell corresponding to path $i$ , $Y_{DD}[\cdot] = \alpha h_i + W$ with $W \sim \mathcal{CN}(0, \sigma^2)$ . LS: $\hat{h}_i = Y_{DD}/\alpha$ .

Per-path variance

$\hat{h}_i - h_i = W/\alpha$ , so variance is $\mathrm{Var}(W)/|\alpha|^2 = \sigma^2/|\alpha|^2$ .

Total MSE

Sum over $P$ independent paths (detections at different cells): $\mathrm{MSE}_{\text{total}} = P \cdot \sigma^2/|\alpha|^2$ .

No $MN$ dependence

The grid size determines the guard region size and pilot overhead but not the per-path MSE. The MSE is fundamentally a function of the sparsity level, not the grid. $\blacksquare$

Key Takeaway

Sparsity gives $O(P)$ MSE. The channel-estimation error scales with the number of paths — typically $P \leq 20$ — not with the grid size $MN \sim 10^4$ . This is the structural estimation advantage of OTFS over OFDM: at the same pilot power, OTFS achieves roughly $MN/P \sim 1000$ -times better MSE on the $P$ -dimensional channel parameter space. Equivalently, OTFS delivers the same MSE with roughly $1/\sqrt{MN/P}$ times less pilot power.

Theorem: MSE Comparison: OTFS Embedded Pilot vs OFDM DMRS

With OFDM DMRS spaced at the coherence-cell Nyquist rate (one pilot per resolvable $(B_c, T_c)$ cell), the per-TF-cell MSE is $\sigma^2/|\alpha|^2$ , and the total MSE over the $MN$ -cell grid is $\mathrm{MSE}_{\text{OFDM}} \;=\; \tau_{\max}\,f_D \cdot MN \cdot \sigma^2/|\alpha|^2.$ With OTFS embedded pilots, the total MSE is $P\,\sigma^2/|\alpha|^2$ . The MSE ratio OFDM/OTFS is $\frac{\mathrm{MSE}_{\text{OFDM}}}{\mathrm{MSE}_{\text{OTFS}}} \;=\; \frac{\tau_{\max}\,f_D\,MN}{P}.$ For typical terrestrial parameters ( $\tau_{\max}\,f_D \approx 10^{-3}$ , $MN = 10^4$ , $P = 10$ ), the ratio is $\sim 1$ — OTFS matches OFDM MSE at equal pilot power. However, OTFS uses a much smaller fraction of the grid for pilots, giving a large spectral-efficiency advantage at comparable MSE.

OFDM DMRS estimates the channel at every coherence cell, giving a high-resolution but low-MSE-per-cell estimate. OTFS embedded pilots estimate $P$ parameters directly, giving a low-MSE-per-parameter estimate but over a small number of parameters. Both strategies are valid; the OTFS advantage is pilot density, not per-cell estimation quality.

Proof

OFDM MSE accounting

OFDM places one pilot per coherence cell $B_c T_c \approx 1/(\tau_{\max}f_D)$ . Number of coherence cells in the frame: $TW \cdot \tau_{\max} f_D = MN \cdot \tau_{\max} f_D$ . Each pilot estimates one TF-grid cell with MSE $\sigma^2/|\alpha|^2$ . Total MSE over the whole grid: sum over all $MN$ cells, where each cell is interpolated from the nearest pilot — MSE per cell is approximately $\sigma^2/|\alpha|^2$ .

OTFS MSE accounting

$P$ independent LS estimates, each with MSE $\sigma^2/|\alpha|^2$ . Total MSE: $P \cdot \sigma^2/|\alpha|^2$ .

Ratio

Dividing: $\mathrm{MSE}_{\text{OFDM}}/\mathrm{MSE}_{\text{OTFS}} = \tau_{\max}f_D MN/P$ . This is the operational comparison metric. $\blacksquare$

Channel-Estimation MSE: OTFS vs OFDM at Varying SNR

Plot MSE as a function of pilot SNR for both schemes at fixed channel parameters. At all SNRs, the $1/\mathrm{SNR}$ scaling holds. The OFDM MSE has an additional factor of $\tau_{\max}f_D\,MN/P$ relative to OTFS — in typical vehicular parameters, $\sim 0$ dB gap (OTFS and OFDM have similar MSE), but in high-mobility or rich-scattering environments, OTFS gains several dB.

Parameters

Number of paths

P

8

\tau_{\max}

(

\mu

s)5

f_D

(Hz)500

Bandwidth

W

(MHz)10

Frame duration

T

(ms)4

Example: MSE and Overhead for a Vehicular Deployment

A 5G NR OTFS deployment at $\Delta f = 30$ kHz, $M = 512$ , $N = 16$ , with a channel of $\tau_{\max} = 4\,\mu$ s, $f_D = 500$ Hz, $P = 8$ . Pilot SNR = 25 dB. Compute MSE (dB) and pilot overhead.

Solution

Pilot-noise ratio

$|\alpha|^2/\sigma^2 = 10^{2.5} = 316$ .

OTFS MSE

$\mathrm{MSE}_{\text{OTFS}} = P/316 = 8/316 = 0.0253$ ( $\approx -16$ dB). Excellent channel estimate.

OFDM MSE

$\tau_{\max}f_D = 4\times 10^{-6} \cdot 500 = 0.002$ . $MN = 8192$ . Ratio $\tau_{\max}f_D MN/P = 0.002 \cdot 8192/8 = 2.05$ . $\mathrm{MSE}_{\text{OFDM}} \approx 2.05 \cdot 0.0253 = 0.052$ ( $\approx -13$ dB). About 3 dB worse than OTFS.

Overhead

$l_{\max} = \lceil 4\mu s \cdot 10\,\text{MHz}\rceil$ : with $W = M \Delta f = 15.36$ MHz, $l_{\max} = \lceil 4\cdot 10^{-6}\cdot 15.36\cdot 10^6\rceil = 62$ . $k_{\max} = \lceil 500\cdot T\rceil$ with $T = N/\Delta f = 533\,\mu$ s, $k_{\max} = \lceil 500\cdot 5.33\cdot 10^{-4}\rceil = 1$ . Guard region: $63 \cdot 3 = 189$ cells. Plus pilot: 190. Overhead: $190/8192 = 2.3\%$ . Far below typical OFDM DMRS density ( $7$ – $10\%$ ).

Pilot Overhead: OTFS Embedded vs 5G NR DMRS

Scheme	Pilot pattern	Overhead	Typical MSE at 25 dB SNR
OFDM DMRS (Type 1)	2 OFDM symbols × 1/2 REs	~14%	$\sim$ -13 dB
OFDM DMRS (Type 2)	2 OFDM symbols × 1/4 REs	~7%	$\sim$ -11 dB
OTFS embedded pilot	Single pilot + guard	1–3%	$\sim$ -16 dB
OTFS superimposed pilot (§4)	Pilot overlaid on data	0%	$\sim$ -12 dB

⚠️Engineering Note

Pilot-Power / Data-Power Trade-off

Raising pilot power improves channel estimation MSE but reduces power available for data. Analytically: if $\rho_p$ is the pilot power fraction, $\mathrm{MSE}_{\text{pilot}} \propto 1/\rho_p$ , while $\text{data-SNR} \propto 1 - \rho_p$ . The optimal balance is determined by the downstream detector's sensitivity to channel-MSE error.

Empirically: for OTFS with MMSE or MP detection, $\rho_p = 0.05$ (5% of frame power) is near-optimal — pilot SNR sufficient for reliable path detection without starving data.

In practice, the frame-level power budget balances: pilot cell carries maybe 100× data-cell energy ( $\rho_p \approx 0.05$ for a single pilot in $\sim 10^4$ cells). Dynamic range is not a practical issue at modern ADC resolutions.

Practical Constraints

•
Optimal pilot fraction: 0.03–0.10 of frame power
•
Pilot boost too high $\Rightarrow$ PAPR increase
•
Pilot boost too low $\Rightarrow$ missed detections

Frame-by-Frame vs Tracked Channel

This chapter analyzes per-frame channel estimation — estimate the channel from scratch at every OTFS frame. For slowly-varying channel geometries (reflectors move slowly), the DD channel kernel is correlated across frames, and a tracking filter (Kalman or RLS) can reduce per-frame pilot overhead further. This is explored in Chapter 14 (sensing-assisted communication) and in the cell-free OTFS extension (Chapter 17).

For this chapter, we assume frame-by-frame estimation as the baseline — the "block fading" assumption of Chapter 4.

MSE and Pilot Overhead Analysis