Empirical mode decomposition and the Hilbert-Huang transform

"Fourier hands me a basis and asks the signal to conform. EMD asks the signal what it is, then listens."

An Adaptive AI Agent

This section assumes you are comfortable with the analytic-signal and instantaneous-frequency ideas introduced alongside sampling in Chapter 3, and with the leakage and windowing tradeoffs from Section 7.1. If the Hilbert transform is unfamiliar, the analytic-signal box below is self-contained enough to follow.

Intrinsic mode functions: letting the signal pick its own basis

The Hilbert transform can assign an instantaneous frequency to any signal, but the number is only physically meaningful when the signal is locally symmetric about zero and contains exactly one oscillation per zero crossing. Huang and colleagues formalized this with the intrinsic mode function (IMF), a component satisfying two conditions: the number of extrema and the number of zero crossings differ by at most one, and at every point the mean of the upper envelope (through the maxima) and the lower envelope (through the minima) is zero. An IMF is what a "well-behaved oscillation with a slowly varying amplitude and frequency" looks like when you refuse to name a carrier frequency in advance.

Why insist on the zero-mean-envelope property? Because the Hilbert transform of a signal riding on a slow trend produces nonsensical negative or wildly swinging instantaneous frequencies. The IMF condition guarantees the analytic signal traces a clean, mostly-positive-radius spiral in the complex plane, so that differentiating its phase gives a frequency you can trust. This is the crucial difference from Fourier: an IMF may change both its amplitude and its frequency from one cycle to the next, which a fixed sinusoid can never do.

Sifting: the algorithm that manufactures an IMF

EMD extracts IMFs one at a time with an iterative procedure called sifting. Given a signal \(x(t)\), you find all local maxima and interpolate them with a cubic spline to form the upper envelope \(e_{\max}(t)\); do the same for the minima to get \(e_{\min}(t)\). Their mean is a local trend

and the candidate detail is \(h(t) = x(t) - m(t)\). If \(h\) already satisfies the IMF conditions you stop; otherwise you treat \(h\) as the new signal and repeat. Sifting is halted by a stoppage criterion, classically a Cauchy-type test on the normalized change between successive siftings,

with \(\varepsilon\) commonly near 0.2 to 0.3. Once an IMF \(c_1(t)\) emerges, you subtract it, \(r_1(t) = x(t) - c_1(t)\), and sift the residual for the next, lower-frequency IMF. The loop ends when the residual becomes monotone or has too few extrema to interpolate. The result is a fully reconstructing decomposition

where the IMFs \(c_i\) run from fastest to slowest oscillation and \(r_n\) is the residual trend. Unlike a filter bank, no band edges were chosen; the extrema of your data alone set the partition.

From IMFs to the Hilbert spectrum

Decomposition is only half of the Hilbert-Huang transform. For each IMF \(c_i(t)\) you form the analytic signal \(z_i(t) = c_i(t) + j\,\mathcal{H}\{c_i\}(t) = a_i(t)\,e^{j\theta_i(t)}\), where \(\mathcal{H}\) is the Hilbert transform. The instantaneous amplitude is \(a_i(t) = \lvert z_i(t)\rvert\) and the instantaneous frequency is

Plotting amplitude (or its square, energy) as a function of both time and this instantaneous frequency produces the Hilbert spectrum \(H(\omega, t)\). Integrating it over time gives the marginal spectrum, an energy-versus-frequency curve that looks like a Fourier power spectrum but earns its frequency axis from local phase rather than from global sinusoidal projection. The payoff is resolution: a spectrogram (Section 7.2) smears energy across a window's worth of time and a bin's worth of frequency, while the Hilbert spectrum can place a sharp ridge exactly where the instantaneous frequency is, unconstrained by the uncertainty-principle box of the short-time Fourier transform. The cost is that this sharp picture is only trustworthy when the IMFs are clean, which loops you straight back to mode mixing.

Doing it in practice, and where the shortcut lives

Writing a correct sifting loop with spline envelopes, boundary handling, and a defensible stoppage rule is a day of careful work and a longer day of debugging edge effects. In practice you reach for a maintained library. The code below decomposes a two-tone chirp-plus-trend signal with CEEMDAN and reads off the instantaneous frequency of the first IMF; the same eight lines would replace roughly 150 lines of hand-rolled sifting, envelope interpolation, and ensemble bookkeeping.

import numpy as np
from PyEMD import CEEMDAN            # pip install EMD-signal
from scipy.signal import hilbert

fs = 1000
t = np.arange(0, 2, 1 / fs)
# a rising chirp, a steady tone, and a slow drift, all superimposed
x = (np.sin(2*np.pi*(5 + 15*t)*t)   # 5 -> 35 Hz chirp
     + 0.6*np.sin(2*np.pi*40*t)     # steady 40 Hz tone
     + 0.3*t)                       # low-frequency trend

imfs = CEEMDAN().ceemdan(x, t)      # data-driven; no band edges chosen
phase = np.unwrap(np.angle(hilbert(imfs[0])))
inst_freq = np.diff(phase) / (2*np.pi) * fs   # instantaneous frequency, Hz

print(f"{imfs.shape[0]} IMFs; IMF-1 mean inst. freq "
      f"{inst_freq[fs//4:-fs//4].mean():.1f} Hz")

When to reach for HHT, and when not to

Choose the Hilbert-Huang transform when the signal is genuinely nonlinear or nonstationary and you need instantaneous behavior a fixed basis cannot express: swallowing and gait segments in a wearable inertial stream (Chapter 2 explains why those sensors produce such signals), motor-unit bursts in EMG, speed-varying machine vibration, or seismic and oceanographic records where HHT was born. Prefer a wavelet transform when you want a stable, orthogonal, invertible decomposition with a mathematical error bound, and prefer the STFT when you need a fast, interpretable, well-understood spectrogram for a roughly stationary signal. The honest weaknesses of EMD are real and worth stating plainly: it has no complete theory, its output depends on interpolation and stoppage choices, mode mixing can scramble physical meaning, and ensemble variants cost a great deal of compute. Use it where its adaptivity buys you something the fixed bases cannot, validate the IMFs against physics, and never treat an IMF as ground truth without checking it.

What's Next

In Section 7.5, we turn to cepstral and envelope analysis, the workhorse pair for machines with periodic structure. Where HHT chases instantaneous frequency, the cepstrum unwraps periodicities-of-periodicities to expose gear-mesh sidebands and harmonic families, and envelope demodulation isolates the impulsive signature of a spalling bearing. Together they complete the rotating-machinery toolkit that this section only opened.