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Power Spectral Density divergence (PSD)

Key: psd · Family: spectral · Better: lower ↓ · Per-lead: yes · Uses: fs

PSD asks: is the power distributed over the same frequencies in the real and synthetic ECGs? Heart rate, QRS sharpness, noise, mains interference and baseline wander all leave fingerprints in the spectrum that are hard to see in the waveform.

Try it

Top: the first 4 s of one real (upper) and one synthetic (lower) recording. Bottom: the average normalised spectrum of each set, on a log scale.

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What to notice:

  • Mains hum adds a sharp peak at 50 Hz that is barely visible in the waveform.
  • High-frequency noise lifts the whole tail above 30 Hz, where real ECGs have little power.
  • Baseline wander moves power to below 1 Hz.
  • Faster rhythm shifts the harmonic peaks of the heart rate to the right.

How it works

For each lead:

  1. Compute the PSD of every recording: Welch's method (SciPy, Hann segments of nperseg = min(256, T) samples) or a plain FFT periodogram without SciPy.
  2. Average the PSDs over the real set and over the synthetic set.
  3. Normalise each average to sum to 1, so that it is a distribution over frequency.
  4. Compute the Jensen–Shannon divergence between the two distributions.

The score is the mean over leads. The mean PSDs are kept in extra and drawn in the HTML report as an overlay plot.

The math

\[ \mathrm{JS}(P\,\|\,Q) = \tfrac12\,\mathrm{KL}(P\,\|\,M) + \tfrac12\,\mathrm{KL}(Q\,\|\,M), \qquad M = \tfrac12 (P+Q), \qquad \mathrm{KL}(P\,\|\,M) = \sum_f P(f)\ln\frac{P(f)}{M(f)} \]

where \(P\) and \(Q\) are the normalised mean PSDs of the real and synthetic sets.

Reading the score

  • Range [0, ln 2 ≈ 0.693]. 0 means identical average spectra.
  • It compares set averages, not individual recordings. A set where half the recordings are too noisy and half too smooth could average out; Spectral entropy looks at per-recording spread instead.
  • Normalising removes overall power (amplitude). Compare amplitudes separately.
  • Pass the correct sampling rate (fs); the CLI uses the real data's rate automatically.

Use it with ECGEN-Eval

ecgen-eval eval --real real.npy --synthetic synth.npy -m psd --fs 500 --output report.html

from ecgen_eval import load_ecg_npy, AVAILABLE_METRICS

real = load_ecg_npy("real.npy", fs=500)
synth = load_ecg_npy("synth.npy", fs=500)

psd = AVAILABLE_METRICS["psd"](fs=real.fs)
result = psd.compute(real.data, synth.data, lead_names=real.lead_names)
print(result.score)
mean_psd_real = result.extra["per_lead_psd_real"]   # mean PSD curves per lead

Install SciPy (pip install scipy) to use Welch's method. Parameters are in the Metric Reference.

Learn more

  • Welch, The use of fast Fourier transform for the estimation of power spectra, IEEE Trans. Audio Electroacoust. 1967.
  • Lin, Divergence measures based on the Shannon entropy, IEEE Trans. Inf. Theory 1991.