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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

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

  1. Time Axis

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

  1. Reflection Delay

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

  1. 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
)
  1. Compute time-domain response
t, h = detector.compute_time_response()
  1. 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.