Ferkans — Interactive Telecom Tutor

Reading Paths Out of the Guard Region

The embedded pilot creates a channel-response pattern in the guard region (Section 1). The receiver now faces a simple statistical problem: which of the $|\mathcal{G}|$ guard-region cells carry path energy, and what are the corresponding amplitudes? This is the threshold-based path detection problem — the computational workhorse of OTFS channel estimation.

The point is that path detection is a per-cell hypothesis test, and because we know the noise statistics (i.i.d. $\mathcal{CN}(0, \sigma^2)$ from Chapter 4), we can set the threshold to achieve a target false-alarm rate. This is the standard radar-detection problem adapted to the DD grid.

Threshold-Based Path Detection

Complexity:

O(|\mathcal{G}|) = O(l_{\max} k_{\max})

Input: Received guard-region samples

\{Y_{DD}[\ell_p + \ell, k_p + k]\}_{(\ell, k) \in \mathcal{G}}

,

pilot amplitude

\alpha

, noise variance

\sigma^2

, threshold

\gamma_{\text{th}}

Output: Estimated path parameters

\{(\hat{h}_i, \hat{\ell}_i, \hat{k}_i)\}_{i=1}^{\hat{P}}

1. for each

(\ell, k) \in \mathcal{G}

do

2.

\quad

Compute

z[\ell, k] \leftarrow Y_{DD}[\ell_p + \ell,\,k_p + k]/\alpha

(normalize by pilot)

3.

\quad

if

|z[\ell, k]|^2 \geq \gamma_{\text{th}}

then

4.

\quad\quad

Record path

(\hat{\ell}_i, \hat{k}_i) \leftarrow (\ell, k)

,

\hat{h}_i \leftarrow z[\ell, k]

5.

\quad

end if

6. end for

7. return the list of detected paths

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

The algorithm is a single pass over the guard region with a per-cell test. It scales as $O(|\mathcal{G}|)$ — trivial compared to the detection step (Chapter 8). The threshold $\gamma_{\text{th}}$ controls the false-alarm / missed-detection trade-off, designed next.

Theorem: Threshold for Target False-Alarm Rate

Let $\sigma^2$ be the noise variance of each DD-grid cell and $\alpha$ the pilot amplitude. Under the no-path hypothesis, the normalized cell $|z[\ell, k]|^2$ follows a scaled chi-squared distribution with 2 degrees of freedom (since $z \sim \mathcal{CN}(0, \sigma^2/|\alpha|^2)$ ). To achieve a target false-alarm rate $P_{\text{FA}}$ per cell, set $\gamma_{\text{th}} \;=\; -\frac{\sigma^2}{|\alpha|^2}\,\ln(P_{\text{FA}}).$ The probability of missing a true path with gain $h_i$ is $P_{\text{MD}}(h_i) \;=\; 1 - Q_1\!\left(\sqrt{\frac{2|h_i|^2|\alpha|^2}{\sigma^2}},\,\sqrt{\frac{2\gamma_{\text{th}}}{\sigma^2}}\right),$ where $Q_1$ is the Marcum Q-function of order 1.

Under noise only, the cell magnitude is Rayleigh-distributed, so $|z|^2$ is exponential with mean $\sigma^2/|\alpha|^2$ . The threshold $\gamma_{\text{th}} = -\sigma_z^2\ln(P_{\text{FA}})$ is the standard exponential-tail quantile. Choosing the right threshold is a classical detection-theory problem.

Proof

No-path distribution

Under $H_0$ (no path at this cell), $z[\ell, k] = W/\alpha$ with $W \sim \mathcal{CN}(0, \sigma^2)$ . Therefore $z \sim \mathcal{CN}(0, \sigma^2/|\alpha|^2)$ and $|z|^2 \sim$ Exponential $(|\alpha|^2/\sigma^2)$ — mean $\sigma^2/|\alpha|^2$ .

False-alarm probability

$P_{\text{FA}} = \Pr(|z|^2 \geq \gamma_{\text{th}} | H_0) = e^{-\gamma_{\text{th}} |\alpha|^2/\sigma^2}$ . Solving: $\gamma_{\text{th}} = -\sigma^2\ln(P_{\text{FA}})/|\alpha|^2$ .

Missed-detection probability

Key Takeaway

The detection threshold is set by the noise variance and pilot power. A target false-alarm rate $P_{\text{FA}} = 10^{-3}$ corresponds to $\gamma_{\text{th}} \approx 7 \sigma^2/|\alpha|^2$ . For a 25 dB pilot boost, the threshold is 30 dB below the pilot level — cleanly separating paths from noise in the guard region. The design is essentially identical to constant-false-alarm-rate (CFAR) detection in radar.

Example: Setting the Threshold for 25 dB Pilot SNR

The pilot SNR is $|\alpha|^2/\sigma^2 = 10^{2.5} \approx 316$ (25 dB). Target false-alarm rate per cell: $P_{\text{FA}} = 10^{-3}$ . Compute the detection threshold $\gamma_{\text{th}}$ relative to the pilot level, and the expected missed-detection probability for a path with $|h_i|^2 = 0.1$ (−10 dB relative to the pilot).

Solution

Threshold

$\gamma_{\text{th}} = -\sigma^2 \cdot \ln(10^{-3})/|\alpha|^2 = -\ln(10^{-3}) \cdot (1/316) \approx 6.91/316 \approx 0.022$ . Expressed in dB relative to pilot: $10\log_{10}(0.022) \approx -16.6$ dB.

Missed-detection

$|h_i|^2 \cdot|\alpha|^2/\sigma^2 = 0.1 \cdot 316 = 31.6$ (15 dB). With threshold at 16.6 dB below pilot and path peak at 15 dB below pilot, the path is above the threshold with high confidence — $P_{\text{MD}}$ is on the order of $10^{-3}$ or better.

Design rule

Set the pilot boost so that the weakest path (relative to the pilot) is at least 5 dB above the threshold. For a dynamic range of 15 dB between the strongest and weakest paths, a 25 dB pilot boost with $P_{\text{FA}} = 10^{-3}$ is typical.

ROC Curve: Detection vs False Alarm

Plot the detection probability $1 - P_{\text{MD}}$ against the false-alarm rate $P_{\text{FA}}$ for different pilot SNRs and path strengths. A receiver-operating-characteristic (ROC) curve: good pilot SNR gives a sharp knee (high detection with low false alarm); weak pilot SNR gives a nearly diagonal ROC (indistinguishable from guessing). Slide the pilot SNR and watch the ROC shift.

Parameters

Pilot SNR (dB)25

Path power rel. pilot (dB)-10

Per-Cell vs Frame-Level False-Alarm Rate

The threshold above gives per-cell false-alarm rate. In the full guard region of $|\mathcal{G}|$ cells, the expected number of false alarms is $|\mathcal{G}| \cdot P_{\text{FA}}$ . For a guard region of size $\sim 250$ cells and $P_{\text{FA}} = 10^{-3}$ , we expect roughly 0.25 spurious detections per frame — a mostly-clean output.

For aggressive false-alarm control, use a frame-level target: $P_{\text{FA,frame}} = P_{\text{FA,cell}} \cdot |\mathcal{G}|$ . Setting the cell threshold via the Bonferroni correction $P_{\text{FA,cell}} = 0.001 / |\mathcal{G}|$ ensures the frame-level false-alarm rate remains below $0.1\%$ .

Adaptive (CFAR) Threshold Detection

Complexity:

O(|\mathcal{G}|^2)

Input: Received guard-region samples, pilot amplitude

\alpha

, target

P_{\text{FA}}

Output: Detected paths

1. Compute cell magnitudes:

z[\ell, k] = |Y_{DD}[\ell_p + \ell,\,k_p + k]|^2/|\alpha|^2

.

2. for each

(\ell, k) \in \mathcal{G}

do

3.

\quad

Estimate local noise power

\hat{\sigma}^2_{\text{local}}

as the median of

\{z[\ell', k'] : (\ell', k')

in a ring around

(\ell, k)\}

.

4.

\quad

Set local threshold

\gamma[\ell, k] = -\hat{\sigma}^2_{\text{local}}\,\ln(P_{\text{FA}})

.

5.

\quad

Declare path at

(\ell, k)

if

z[\ell, k] \geq \gamma[\ell, k]

.

6. end for

7. return the detected paths.

CFAR (Constant False Alarm Rate) detection adapts the threshold to local noise statistics, which improves robustness in non-stationary noise environments. The median estimator is robust against strong nearby paths that would otherwise bias a mean-based estimate. For standard OTFS with well-calibrated noise, the fixed-threshold algorithm (Algorithm AThreshold-Based Path Detection) is adequate and cheaper.

⚠️Engineering Note

Practical Thresholds for 5G NR Deployment

Typical OTFS deployment thresholds:

Normal operation: $P_{\text{FA}} = 10^{-3}$ per cell, corresponding to $\gamma_{\text{th}} \approx 7\sigma^2/|\alpha|^2$ .
High-reliability: $P_{\text{FA}} = 10^{-5}$ per cell, $\gamma_{\text{th}} \approx 11\sigma^2/|\alpha|^2$ .
Low-SNR operation: use $P_{\text{FA}} = 10^{-2}$ with subsequent validation via data-aided refinement.

The detection threshold is decoupled from the channel's path strength distribution; the same threshold works across indoor, urban, and highway deployments. Only the pilot SNR boost needs to be adapted to the channel's worst-case dynamic range. 3GPP implementations typically set the pilot at 6 dB above data power and use $P_{\text{FA}} = 10^{-3}$ .

Practical Constraints

•
Fixed $P_{\text{FA}} = 10^{-3}$ works across URLLC, eMBB, mMTC traffic classes
•
Pilot boost 6–10 dB keeps 99% of path detections correct
•
CFAR variant is only needed for unknown noise environments (LEO uplink, unlicensed spectrum)

📋 Ref: ITU-R M.2410, §4.2

Common Mistake: Correlated Paths Break Per-Cell Detection

Mistake:

Assuming per-cell independence when the channel has paths at fractional DD positions. A fractional-Doppler path (Chapter 10) bleeds across multiple grid cells, creating correlated detections.

Correction:

For fractional Doppler, use a two-step detector: (i) locate the peak cell via threshold detection, (ii) refine the fractional offset by quadratic interpolation over the peak's neighbors. Alternatively, apply a matched-filter over the fractional-offset dictionary (compressed sensing / OMP, Chapter 10). For integer-only channels (the assumption of this chapter), per-cell detection is optimal.