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EchoDetector — FFT/IFFT-based Echo Analysis (Docsis OFDM)

Overview

EchoDetector converts frequency-domain channel estimates (H[k]) into a time-domain impulse response (h[n]) via IFFT, then locates the direct path and subsequent echo peaks. Echo timing is mapped to one-way distance using a cable-dependent propagation speed (v = c_0 \cdot \text{VF}).

  • Supports: single snapshot or multiple snapshots (coherent average), complex arrays or ((\mathrm{Re},\mathrm{Im})) pairs.
  • OFDM context: sample rate (f_s = N \cdot \Delta f) for (N) subcarriers and spacing (\Delta f) (25/50 kHz typical in DOCSIS 3.1/4.0).
  • Distance model: one-way (d = \dfrac{v \cdot \Delta t}{2}), where (\Delta t) is echo delay relative to direct path.

Inputs & Assumptions

  • Frequency data (freq_data):

  • Shape: ((N,)) complex, ((M,N)) complex, ((N,2)) re/imag pairs, or ((M,N,2)) re/imag pairs.

  • If multiple snapshots (M>1), a coherent average across snapshots is used for detection.
  • Subcarrier spacing (\Delta f) (Hz): commonly 25 kHz or 50 kHz.
  • IFFT size (n_{\text{fft}}): optional zero-padding (n_{\text{fft}} \ge N) to improve time resolution of peak indices; the time step remains (\Delta t = 1/f_s).
  • Cable type / velocity factor (VF): picks (v = c_0\cdot\text{VF}) (RG6/RG11: ~0.87, RG59: ~0.82 by default).

Core Math

Frequency → Time

Given (H[k]) for (k=0,\dots,N-1): [ h[n] = \text{IFFT}{H[k]}{n=0}^{n, \quad t[n] = \frac{n}{f_s}, \quad f_s = N\cdot\Delta f . ]}}-1

Magnitude: [ |h[n]| = \sqrt{ \Re{h[n]}^2 + \Im{h[n]}^2 }. ]

Direct Path & First Echo

  1. Direct path index (i_0 = \arg\max_n |h[n]|), amplitude (A_0 = |h[i_0]|).
  2. Threshold (\tau = \text{threshold_frac} \cdot A_0).
  3. Guard region: skip indices (i \le i_0 + \text{guard_bins}).
  4. Echo: first (i_e>i_0) with (|h[i_e]|\ge\tau) (optionally capped by max_delay_s).

Delay & distance: [ \Delta t = t[i_e] - t[i_0], \qquad d = \frac{v \cdot \Delta t}{2}, \quad v=c_0\cdot\text{VF}. ]

Multiple Echoes

  • Find local maxima in the search window with (|h[i]|\ge\tau).
  • Sort by magnitude (desc), then enforce min spacing in time (min_separation_s) before selecting up to max_peaks.

Tuning Knobs

  • threshold_frac (0-1): sensitivity; larger ⇒ fewer echoes.
  • guard_bins (≥0): avoids main-lobe skirt around the direct path.
  • max_delay_s (optional): caps search window (controls false positives far out).
  • min_separation_s (multi): enforces spacing between returned echoes.
  • max_peaks (multi): upper bound on echoes reported.
  • n_fft (≥N): zero-padding improves bin granularity for peak location.

Returned Models

EchoDatasetInfo

Field Type Description
subcarriers int Number of frequency bins (N)
snapshots int Number of snapshots (M)
subcarrier_spacing_hz float (\Delta f) in Hz
sample_rate_hz float (f_s = N\cdot\Delta f)

EchoPath

Field Type Description
bin_index int Peak index in ( h )
time_s float Peak time (t)
amplitude float ( h ) at peak
distance_m float One-way distance (m)
distance_ft float One-way distance (ft)

EchoReflection (first-echo)

Field Type Description
direct_index int Direct-path index
echo_index int Echo index
time_direct_s float Direct time (t[i_0])
time_echo_s float Echo time (t[i_e])
reflection_delay_s float (\Delta t)
reflection_distance_m float One-way distance (m)
amp_direct float ( h [i_0])
amp_echo float ( h [i_e])
amp_ratio float ( h [i_e]/ h [i_0])
threshold_frac float Threshold fraction used
guard_bins int Guard bins used
max_delay_s float None Optional search window

TimeResponse (optional iframe for plotting)

Field Type Description
n_fft int IFFT length used
time_axis_s list[float] (t[n]=n/f_s), length (n_{\text{fft}})
time_response list[(float,float)] (h[n]) as ((\Re,\Im)) pairs

EchoDetectorReport (multi-echo)

Field Type Description
channel_id ChannelId OFDM downstream channel ID
dataset EchoDatasetInfo Shape & sampling meta
cable_type "RG6", "RG59", "RG11" Cable used for VF
velocity_factor float VF actually used
prop_speed_mps float (v=c_0\cdot\text{VF})
direct_path EchoPath Direct-path sample
echoes list[EchoPath] Selected echo peaks
threshold_frac float Threshold used
guard_bins int Guard bins used
min_separation_s float Min echo spacing (s)
max_delay_s float None
max_peaks int Max echoes returned
time_response TimeResponse None

Typical Workflow

  1. Construct the detector with (H) and (\Delta f):

  2. Optional: set (n_{\text{fft}}) and cable_type (or custom VF).

  3. First echo:

  4. Call first_echo(threshold_frac, guard_bins, max_delay_s?) → EchoReflection.

  5. Multi-echo:

  6. Call multi_echo(threshold_frac, guard_bins, min_separation_s, max_delay_s?, max_peaks, include_time_response) → EchoDetectorReport.

  7. Plotting:

  8. Use time_response block if returned, or reconstruct via (h=\text{IFFT}(H)).

Practical Notes

  • Resolution trade-offs:

  • Time grid is fixed by (f_s=N\cdot\Delta f).

  • Zero-padding refines bin placement but not the fundamental (\Delta t).

  • Leakage & windowing:

  • If leakage is strong, consider pre-conditioning (H[k]) or smoothing; guard_bins mitigates main-lobe bleed-through.

  • Robustness:

  • Choose threshold_frac to balance sensitivity vs. false positives.

  • Use max_delay_s to limit distant artifacts.
  • Increase min_separation_s when echoes cluster.

Example (conceptual)

# H: (N,) complex or (N,2) pairs; df=25e3 or 50e3
det = EchoDetector(H, subcarrier_spacing_hz=50_000.0, n_fft=2048, cable_type="RG6")

# First echo
ref = det.first_echo(threshold_frac=0.2, guard_bins=2)
print(ref.reflection_distance_m, ref.amp_ratio)

# Multiple echoes
rep = det.multi_echo(threshold_frac=0.2, guard_bins=1,
                     min_separation_s=1e-6, max_peaks=5,
                     include_time_response=True, channel_id=42)
for e in rep.echoes:
    print(e.time_s, e.distance_m)

Units & Conventions

  • (c_0 = 299,792,458\ \mathrm{m/s})
  • (\text{VF} \in (0,1)), defaults (can be overridden): RG6/RG11 ~ 0.87, RG59 ~ 0.82
  • Distances are one-way, relative to the modem input.
  • Time axis always starts at 0 with step (1/f_s).

Limitations

  • Assumes single dominant direct path; deep fades at the true direct path can bias detection.
  • Strong frequency-selective channels may require pre-equalization or smoothing before IFFT.
  • Echoes closer than the effective time resolution or within guard_bins cannot be separated.