ofdm-echo-detection.md

OFDM Echo Detector and Coax Cable Propagation Guide

This guide explains the OFDMEchoDetector class, which performs FFT/IFFT-based echo detection on OFDM channel estimates, and provides typical propagation delay values for common coaxial drop cables.

1. Class Overview

The OFDMEchoDetector class:

• Accepts a 1-D array of complex OFDM channel estimates (H[k]).
• Computes the time-domain impulse response via inverse FFT.
• Identifies the direct path and the first significant echo reflection.
• Calculates the echo delay and converts it into distance using a configurable propagation speed factor.

Key Methods

• compute_time_response() Computes:

  • time_axis: $t[n]$ values in seconds.
  • time_response: impulse response $h[n]$.

• detect_reflection(threshold_frac, min_separation) Detects peaks in time_response:

  • Returns a dict containing indices, times, one-way delay, and distance.

• compute_freq_response(time_data) Optionally applies FFT to reconstruct frequency response from time-domain data.

2. Mathematical Foundations

1. Inverse FFT

   $$ h[n] = \frac{1}{N} \sum_{k=0}^{N-1} H[k] ,e^{j2\pi k n / N} $$

2. Time Axis

   $$ t[n] = \frac{n}{F_s}, \quad n=0,1,\ldots,N-1 $$

3. Reflection Delay

   $$ \Delta t = t_{\mathrm{echo}} - t_{\mathrm{direct}} $$

4. One-Way Distance

   $$ d = \frac{c_0 \times \mathrm{prop_speed_frac} \times \Delta t}{2} $$

   • $c_0$: speed of light in vacuum.
   • prop_speed_frac: velocity factor relative to $c_0$.

3. Typical Usage

1. Instantiate detector

   detector = OFDMEchoDetector(
       H,                 # Complex channel estimates
       sample_rate=Fs,    # Sampling rate in Hz
       prop_speed_frac=0.87  # Velocity factor
   )

2. Compute time-domain response

   t, h = detector.compute_time_response()

3. Detect echo reflection

   result = detector.detect_reflection(
       threshold_frac=0.2,  # Fraction of peak direct path magnitude
       min_separation=1     # Minimum sample separation between direct and echo
   )
   print(result)

4. Coax Cable Propagation Delays

Cable Type	Velocity Factor ($% of (c_0$))	Delay (ns/ft)	Delay (ns/m)
RG-59A/U PE	65.9	1.52	4.99
RG-6/U Quad Shield	84.5	1.18	3.87
RG-11/U Quad Shield	86.0	1.16	3.81

• Delay (ns/ft) = $1/\mathrm{velocity_factor}$.
• Delay (ns/m) = Delay (ns/ft) × 3.28084.

5. References

• RG-59A/U PE velocity and delay (CableLabs)
• RG-6/U Quad Shield velocity factors (Manufacturer datasheets)
• RG-11/U NVP and delay (Technical specifications)

Tip: Use the compute_freq_response method to verify the FFT/IFFT round-trip, ensuring no windowing artifacts affect echo detection.