# Hilbert–Huang Transform (EMD) Laboratory

Interactive Hilbert–Huang Transform lab: empirical mode decomposition sifting, intrinsic mode functions, instantaneous frequency and HHT vs STFT time-frequency comparison.

> Canonical page: https://elysiatools.com/en/visualizations/hilbert-huang-transform

- **Category:** Signal Processing

## Overview

Interactive Hilbert-Huang Transform (HHT) and Empirical Mode Decomposition (EMD) laboratory, distinct from the existing wavelet-transform/wavelet-synthesizer (preset mother-wavelet bases) and spectrogram-analyzer (fixed-window STFT): here the basis is fully data-driven. Full sifting pipeline from Huang et al. (Proc. R. Soc. Lond. A 454, 903, 1998): interior extrema detection, mirror-extended cubic-spline upper/lower envelopes (Burden-Faires natural spline with endpoint mirroring, the classic endpoint-effect treatment), envelope-mean subtraction, and IMF conditions (#extrema vs #zero-crossings differ by at most one, zero envelope mean); sifting stops with the scale-free Rilling-Flandrin-Goncalves criterion (sigma(t) = |m|/a < theta1 on (1-alpha) of samples, IEEE SPL 10, 134, 2003) with Huang's SD recorded for display, over/under-sifting explored via the theta1 slider, and negligible-energy edge-artifact modes rejected. Decomposition: x(t) = sum of IMFs + monotone residual, exact reconstruction (verified < 1e-9), orthogonality index, up to 8 IMFs. Hilbert spectral analysis: radix-2 FFT analytic signal (positive frequencies doubled, negative zeroed), amplitude and phase separation, instantaneous frequency dtheta/dt via branch-unwrap-safe centered differences valid up to Nyquist, amplitude-gated and median-smoothed; Hilbert amplitude spectrum H(t, omega) = sum a_j(t) delta(omega - omega_j(t)) on a 160x80 time-frequency grid plus marginal spectrum h(omega). STFT comparison panel (Hann window, adjustable 32-256 samples, log-color spectrogram) sharing the time axis to show the fixed-window trade-off against window-free HHT instantaneous frequency. Six physics-anchored signal presets, each with its own sampling rate and time unit: linear chirp 2-28 Hz with AM (IMF1 instantaneous frequency tracks the ramp with R^2 = 0.997), switched 32->8 Hz tones (a single adaptive IMF whose IF switches within one time bin while STFT arrival error is about half a window), synthetic ECG with PQRST timing, 20 Hz QRS ringing and baseline wander (QRS lands in IMF1 at ~23 Hz, wander recovered as the last IMF with correlation 0.98), seismogram with P/S/surface-wave packets (IF median 13.0/5.5/2.8 Hz in the respective arrival windows), tide record (M2/S2/N2 semidiurnal + K1/O1 diurnal constituents + storm surge, hourly-scale separation), and a vowel-like glottal source with 6 Hz vibrato (formant-weighted harmonics, vibrato visible in the IF spread). Adjustable added noise (0-0.5 x RMS, seeded) demonstrates extra high-frequency IMFs appearing. Panels: input waveform with playhead, animated sifting lab showing successive iterates with envelopes and mean for IMF1, stacked IMF lanes with per-IMF frequency/iterations/energy, HHT heatmap with overlaid instantaneous-frequency traces and marginal spectrum, and the side-by-side STFT spectrogram. Diagnostics chips: IMF count, total sifting iterations, IMF1 mean frequency, valid IF fraction, orthogonality index, HHT peak frequency, residual energy, reconstruction exactness. Educational content: EMD and sifting with IMF conditions and endpoint effects, Hilbert transform/analytic signal/instantaneous frequency and why monocomponentity matters, HHT vs Fourier vs wavelet uncertainty trade-offs, applications (ECG, seismology, tides, speech, machinery) and honest caveats (mode mixing, noise sensitivity, EEMD/CEEMDAN variants). Multi-language support (zh, en, es, fr, de, ru, pt).

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