Phase‑Slope Echo Detection: Theory and Insights¶

This guide delves into the theory of phase‑slope echo detection, examines how front‑end AGC and in‑band LTE signals affect group‑delay estimates, and presents a multi‑resolution scanning strategy to pinpoint disturbances.

1. Fundamental Principle¶

A simple two‑path channel (direct path + reflection) has the frequency response:

\[ H(f) = H_0 + H_1\,e^{-j2\pi f\,\tau_{rt}}, \]

where:

$H_0$ is the direct‑path complex gain.
$H_1$ is the echo path gain.
$\tau_{rt}$ is the round‑trip delay of the echo.

Taking the phase and unwrapping across subcarriers gives:

\[ \varphi(f) = \arg H(f) \approx -2\pi f\,\tau_{rt} + \text{constant}. \]

A linear fit $\varphi(f) \approx a f + b$ yields slope:

\[ a = \frac{d\varphi}{df} \approx -2\pi\,\tau_{rt}, \quad \tau_{rt} = -\frac{a}{2\pi}. \]

Thus, the one‑way delay is

\[ \tau = \frac{|\tau_{rt}|}{2}, \]

and the distance to the reflector:

\[ d = v\,\tau, \]

with propagation velocity $v = c_0 \times \mathrm{prop\_speed\_frac}$.

2. Effects of AGC and In‑Band Signals¶

AGC dynamics: The Automatic Gain Control adjusts amplifier gain based on total in‑band power. A strong LTE signal (e.g., 40 MHz) within a wider OFDM band (e.g., 100 MHz) shifts the AGC operating point.
Phase ripple: Gain adjustments introduce frequency‑dependent phase shifts (group‑delay ripple) that corrupt linear phase assumptions.
Impact: The measured slope reflects both echo delay and AGC/equalizer transients until front‑end circuits re‑settle.

3. Group‑Delay Flatness Metric¶

Let $\tau_k$ be the one‑way delay estimated at subcarrier $f_k$. Define:

Global statistics:

$$ \mu = \frac{1}{K}\sum_{k=1}^K \tau_k, \quad \sigma_{\mathrm{tot}} = \sqrt{\frac{1}{K-1}\sum_{k=1}^K (\tau_k - \mu)^2}. $$

Local variability: Divide the occupied channel bandwidth $B$ into $N_b$ bins (e.g., 1 MHz each). For bin $j$ with indices $\mathcal{K}_j$:

$$ \mu_j = \frac{1}{|\mathcal{K}j|}\sumj} \tau_k, \quad \sigma_j = \sqrt{\frac{1}{|\mathcal{K}_j|-1}\sum. $$}_j}(\tau_k - \mu_j)^2

Anomaly metric:

$$ \Delta\sigma_j = |\sigma_j - \sigma_{\mathrm{tot}}|. $$

Flag bin $j$ as disturbed if $\Delta\sigma_j > T$, where $T$ is a threshold based on baseline ripple levels.

4. Multi‑Resolution Scanning Strategy¶

Coarse scan: Compute $\Delta\sigma_j$ over large bins (e.g., 1 MHz).
Bin selection: Mark bins where $\Delta\sigma_j > T$.
Refinement: Subdivide flagged bins into finer bins (e.g., 500 kHz, then 100 kHz), recompute metrics, and localize disturbances.
Repeat: Continue until desired frequency resolution is achieved.

This hierarchical method focuses computation on suspect regions, optimizing performance.

5. Practical Considerations¶

Phase unwrapping: Use robust algorithms (e.g., numpy.unwrap) to avoid 2π jumps.
Threshold tuning: Set $T$ as a multiple (e.g., 3×) of baseline $\sigma_{\mathrm{tot}}$.
AGC/EQ modeling: Consider digital filter group‑delay and DC‑offset compensation.
Extensions: Combine with PSD analysis or pilot-correlation to reduce false positives.

6. References¶

Delay Estimation via Phase Slope, DSPRelated.com
Multipath Channel Models and Rake Receivers, WirelessPi

Tip: Always verify AGC settling time and remove large in-band interferers before echo analysis.