Ferkans — Interactive Telecom Tutor

Removing the Guard Region

The embedded pilot achieves $\sim 1$ – $3\%$ overhead, but that overhead is not zero. At scale — cell-free networks with hundreds of access points, LEO satellite constellations with dozens of beams — even a few percent overhead aggregates into significant loss. The superimposed pilot design eliminates the guard region by transmitting the pilot on top of data, recovering the full grid for data.

The point is that the pilot and data contributions to each received cell can be separated if we know the pilot pattern. At high SNR the separation is good; at low SNR there is a pilot-data interference residual that is the price paid for the zero-overhead operation. The trade is: guard overhead for receiver complexity.

Definition:
Superimposed Pilot

In superimposed pilot OTFS, the transmitted DD-grid symbol at each cell is the sum of a data symbol and a known pilot pattern: $X_{DD}[\ell, k] \;=\; \sqrt{1 - \rho_p}\,X_{\text{data}}[\ell, k] \;+\; \sqrt{\rho_p}\,P[\ell, k],$ where $P[\ell, k]$ is a known pilot sequence of unit power and $\rho_p \in (0, 1)$ is the pilot power fraction.

Common pilot patterns: a single impulse at $(\ell_p, k_p)$ , a Zadoff-Chu sequence along a diagonal of the DD grid, or a random Gaussian pattern. The choice affects the detection complexity.

,

Theorem: Superimposed Pilot Channel Estimation

With a superimposed pilot of pattern $P[\ell, k]$ and power fraction $\rho_p$ , the receiver can estimate the channel via correlation: $\hat{h}_i \;=\; \frac{1}{MN}\sum_{\ell, k} Y_{DD}[\ell, k]\,P^*[\ell - \ell_i,\,k - k_i]\,(\sqrt{\rho_p})^{-1}.$ The estimate has variance $\mathrm{Var}(\hat{h}_i - h_i) \;=\; \frac{\sigma^2 + (1 - \rho_p)\,\mathbb{E}|X_{\text{data}}|^2}{MN\,\rho_p}.$ For large $MN$ , the data-interference term averages out and the variance scales as $1/(MN)$ — far better than embedded pilot's $1/|\alpha|^2$ when $MN \gg |\alpha|^2$ .

The superimposed pilot is a "spread" pilot: its energy is distributed across all $MN$ grid cells. The correlation at the receiver integrates the contributions, benefiting from the $MN$ coherent samples. Data acts as additional "noise" with variance $(1 - \rho_p)\mathbb{E}|X_{\text{data}}|^2$ per cell, but averaged over $MN$ cells it becomes negligible.

The price is that the pilot-correlation requires all $MN$ cells as input (no guard region to concentrate analysis) and the data interference term must be iteratively suppressed (typically integrated with detection — see Chapter 8).

Proof

Receive equation

$Y_{DD} = h \star\star X_{DD} + \mathbf{w} = h \star\star (\sqrt{1 - \rho_p} X_{\text{data}} + \sqrt{\rho_p} P) + \mathbf{w}$ .

Correlation with the pilot

$\sum_{\ell, k} Y_{DD}[\ell, k] P^*[\ell - \ell_i, k - k_i]$ has a deterministic contribution $\sqrt{\rho_p}\,h_i \cdot MN$ (from the pilot auto-correlation at lag $(\ell_i, k_i)$ ) plus interference from data and noise.

Variance of the interference

Data interference: $(1-\rho_p)\,\mathbb{E}|X_{\text{data}}|^2 \cdot MN$ (cross-correlation over $MN$ cells). Noise: $\sigma^2\cdot MN$ . Normalizing: estimator variance is $(\sigma^2 + (1-\rho_p)\,\mathbb{E}|X_{\text{data}}|^2)/(MN\,\rho_p)$ .

Comparison with embedded

Embedded MSE per path: $\sigma^2/|\alpha|^2$ with pilot power concentrated in one cell. Superimposed MSE per path: spread over $MN$ cells, $MN$ -fold integration gain — favorable when $MN \cdot \rho_p \gg |\alpha|^2$ . $\blacksquare$

Key Takeaway

Superimposed pilots buy zero overhead with extra detector work. No guard region, no reserved cells for pilots — the full DD grid carries data. The receiver jointly estimates the channel (by correlation with the known pilot pattern) and decodes data (subtracting the estimated pilot contribution). This is a concrete cell-free OTFS CommIT contribution: at modest $\rho_p \sim 0.1$ (10% of energy to pilot), the zero-overhead scheme outperforms embedded-pilot at the same data-energy level.

🎓CommIT Contribution(2023)

Superimposed Pilot for Cell-Free OTFS

M. Mohammadi, H. Q. Ngo, M. Matthaiou, G. Caire — IEEE Trans. Wireless Communications

In the 2023 cell-free OTFS paper, Mohammadi, Ngo, Matthaiou, and Caire present both embedded-pilot and superimposed-pilot designs and show — under cell-free massive MIMO deployment — that the superimposed scheme wins at scale. The reason is fundamentally a pilot-reuse argument: with $L$ access points (APs), each UE needs channel estimates to every AP. Embedded pilots require $L$ distinct guard regions (or pilot allocation across APs); superimposed pilots with different sequences per UE need no reserved grid cells — all APs can estimate all UEs' channels from the same data-carrying grid.

Quantitatively, at 120 km/h vehicular mobility with $L = 16$ APs and $K = 8$ UEs per cell, the superimposed-pilot design gives a $25\%$ gain in 95%-likely per-UE throughput over embedded pilots, and a $35\%$ gain over OFDM DMRS. The analysis is the one we developed here (single-link, single-AP), generalized to the multi-AP setting. The CommIT contribution is this generalization and the empirical confirmation via system-level simulation. See Chapter 17 for the full cell-free OTFS development.

commitcell-freepilot-design

MSE: Embedded vs Superimposed Pilots

Plot channel-estimation MSE as a function of SNR for both schemes. Embedded pilot: fixed pilot power, $P/|\alpha|^2$ MSE. Superimposed: pilot energy spread over grid, MSE decreases as $MN$ grows. At typical parameters, superimposed wins at high SNR; at low SNR, the data-interference term dominates and embedded-pilot leads. Adjust $\rho_p$ , $MN$ , $P$ to see the crossover.

Parameters

Grid size

MN

4096

P

6

Pilot fraction

\rho_p

0.1

Embedded pilot boost (dB)10

Joint Channel Estimation and Data Detection (Superimposed)

Complexity:

O(T_{\text{iter}} \cdot MN \log(MN))

Input:

Y_{DD}

received grid, pilot pattern

P[\ell, k]

, pilot fraction

\rho_p

,

max iterations

T_{\text{iter}}

Output: Estimated channel

\{\hat{h}_i, \hat{\ell}_i, \hat{k}_i\}

, data estimate

\hat{X}_{\text{data}}

1. Initialize:

\hat{X}_{\text{data}}^{(0)} = 0

(no data estimate yet)

2. for iteration

t = 1, \ldots, T_{\text{iter}}

do

3.

\quad

Subtract estimated data contribution:

R^{(t)} = Y_{DD} - h \star\star(\sqrt{1-\rho_p}\,\hat{X}_{\text{data}}^{(t-1)})

4.

\quad

Correlate with pilot:

\hat{h}_i^{(t)} = \frac{1}{MN}\sum R^{(t)} P^*[\cdot]

5.

\quad

Decode data:

\hat{X}_{\text{data}}^{(t)} = \text{detect}(Y_{DD} - \sqrt{\rho_p}\hat{h}^{(t)}\star\star P)

6. end for

7. return

\hat{h}, \hat{X}_{\text{data}}

The iteration alternates between channel estimation (via pilot correlation on the pilot-only residual) and data detection (on the data-only residual with known channel). Two or three iterations typically suffice. This is the concrete algorithmic form of the CommIT cell-free OTFS contribution.

Embedded vs Superimposed Pilot: Trade-offs

Property	Embedded	Superimposed
Overhead	1–3%	0%
Receiver complexity	Simple threshold detection	Joint iteration (2-3 passes)
Low-SNR performance	Best	Data interference dominates
High-SNR performance	Saturates at $P/\|\alpha\|^2$	Scales as $1/MN$
Pilot-aware detector needed	No	Yes (joint)
Compatible with 5G DMRS	Easier (reserved slots)	Harder (data-pilot overlap)
Cell-free scaling	Limited (one pilot per AP)	Good (different sequences per UE)
Recommended use	Single-link, low-mobility	Cell-free massive MIMO

🔧Engineering Note

Adoption in 6G: Which Design Wins?

Current 3GPP study items (TR 38.912) consider both designs. The practical trade-off is:

Single-link with existing 5G NR hardware: embedded pilot is easier — aligns with existing DMRS slot structure.
Cell-free massive MIMO deployments: superimposed pilot wins via pilot reuse and zero overhead.
LEO satellite constellation: superimposed with longer pilot sequences (Zadoff-Chu) is favored to handle the extreme Doppler spread.

The consensus in the research community is that both designs will likely find deployment, with the choice driven by the architecture: embedded for single-link O-RAN cells, superimposed for distributed cell-free networks and NTN (non-terrestrial network) deployments.

Practical Constraints

•
Embedded pilot: simpler, proven in 5G NR simulation studies
•
Superimposed pilot: required for cell-free scalability
•
3GPP 6G study will likely standardize both as configurable options

📋 Ref: 3GPP TR 38.912 (6G study)

Common Mistake: Pilot Sequence Matters for Superimposed

Mistake:

Using a random pilot pattern without validating its auto-correlation properties. A pilot with poor auto-correlation creates correlated interference across cells, biasing the channel estimate.

Correction:

Use a Zadoff-Chu sequence on the DD plane — constant modulus and perfect periodic auto-correlation. Alternatively, use an m-sequence or a Kasami code. The auto-correlation peak at lag $(0, 0)$ should be much larger than the sidelobes to avoid ghosts in the channel estimate. For cell-free deployments with multiple UEs, different Zadoff-Chu root indices give mutually orthogonal pilots across UEs.

Superimposed Pilot Estimation