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 comfort with the analytic-signal and instantaneous-frequency ideas from Chapter 3, and the leakage and windowing tradeoffs from Section 7.1. If the Hilbert transform is unfamiliar, the analytic-signal construction later in this section, where each IMF becomes \(z_i(t) = a_i(t)e^{j\theta_i(t)}\), is self-contained enough to follow.

Intrinsic mode functions: letting the signal pick its own basis

The Hilbert transform assigns an instantaneous frequency to any signal, but that number is meaningful only when the signal is locally symmetric about zero with exactly one oscillation per zero crossing. Huang and colleagues formalized this as the intrinsic mode function (IMF): extrema and zero crossings differ in count by at most one, and the upper envelope (through the maxima) and lower envelope (through the minima) average to zero at every point. An IMF is a well-behaved oscillation with slowly varying amplitude and frequency, defined without ever naming a carrier frequency in advance.

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

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.

Cubic splines are the standard envelope interpolant: lowest-order with continuous curvature, so no spurious kink masquerades as an oscillation. The cost is at the edges: a natural spline has no data past the first and last extremum to constrain its curvature, so it can overshoot and drag the whole envelope, and every IMF pulled from it, off the true signal. Implementations mirror the outermost extrema before fitting, then truncate back to the original length, but the effect never fully disappears; an instantaneous frequency read near an edge, including in the code below, deserves less trust.

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) against 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 earns its axis from local phase rather than global sinusoidal projection. Where a spectrogram (Section 7.2) smears energy across a window's worth of time and a bin's worth of frequency, the Hilbert spectrum can place a sharp ridge where the instantaneous frequency is, trustworthy only when the IMFs are clean (the callout below explains why).

Doing it in practice, and where the shortcut lives

Writing a correct sifting loop, with spline envelopes, boundary mirroring, 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 first IMF's instantaneous frequency; the same eight lines 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 HHT when the signal is genuinely nonlinear or nonstationary and needs instantaneous behavior a fixed basis cannot express: swallowing and gait segments in a wearable inertial stream (Chapter 2 explains why), motor-unit bursts in EMG, speed-varying machine vibration, or the seismic and oceanographic records where HHT was born. Prefer a wavelet transform for a stable, orthogonal, invertible decomposition with an error bound, and the STFT for a fast, well-understood spectrogram of a stationary signal. EMD's weaknesses are real: no complete convergence theory, interpolation- and stoppage-dependent output, mode mixing that can scramble physical meaning, and compute-hungry ensembles. Use it where adaptivity buys something a fixed basis cannot, then validate every IMF against the physics.

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.