Empirical Mode Decomposition — Control Systems/Time Frequency

Empirical Mode Decomposition

Control Systems / Time Frequency

Sifts the last L samples into intrinsic mode functions and a residual – MATLAB's emd – and reports each mode's instantaneous frequency and energy, which is what hht builds its Hilbert spectrum from. A sift is: find the local maxima and minima, run an envelope through each set, subtract the mean of the two envelopes, and repeat until

‖r_prev − r‖² ÷ ‖r_prev‖² < Sift Relative Tolerance

or the sift count runs out. What is left is one mode; it is subtracted and the next is sifted out of the remainder. Unlike a filter bank, the modes come from the signal's own extrema, so a mode can change frequency along the frame.

Ports

• u – the sampled signal, scalar: one channel and its own frame.
• IMF – the modes, [L, Max Number of IMFs]: column j is mode j over the frame, oldest sample first. Columns past the number actually found are zero, because a wire has one size where MATLAB returns a narrower matrix.
• r – the residual, [L, 1]: the frame minus every mode found.
• n – [1, 1], how many modes were found (how many columns of IMF carry one).
• insf – instantaneous frequency, [L, Max Number of IMFs], in hertz: f_s/(2π) times the centred difference of each mode's unwrapped analytic phase. Zero in the columns with no mode.
• inse – instantaneous energy, [L, Max Number of IMFs]: the squared magnitude of each mode's analytic signal. Zero in the columns with no mode.

Parameters

• Frame Length – L, the samples one decomposition is made from: a whole number from 8 to 256. The three HDL targets hold one register per sample.
• Hop Size – the samples from one decomposition to the next, 1 to L. Between them all five outputs hold.
• Max Number of IMFs – MATLAB's MaxNumIMF, 1 to 16: the ceiling on the decomposition loop and the width of the three matrix outputs. The default 10 is MATLAB's.
• Sift Max Iterations – MATLAB's SiftMaxIterations, 1 to 400: the ceiling on the sift loop. The default 100 is MATLAB's, and a frame of a few dozen samples typically stops after 1 to 5 sifts.
• Sift Relative Tolerance – MATLAB's SiftRelativeTolerance, greater than 0 and less than 1; default 0.2. The first test is made against a tolerance of exactly 1, so every mode is sifted at least once.
• Max Number of Extrema – MATLAB's MaxNumExtrema, 0 or more; default 1. The decomposition stops when the residue has fewer extrema than this.
• Max Energy Ratio (dB) – MATLAB's MaxEnergyRatio; default 20. The decomposition stops when 10·log₁₀(‖x‖÷‖residue‖) passes it, i.e. when the residue is that many decibels below the frame.
• Interpolation – how the envelope runs through the extrema:
  • spline – the not-a-knot cubic, MATLAB's default and its interp1(..., 'spline').
  • pchip – MATLAB's shape-preserving cubic, which cannot overshoot between two extrema and usually yields one more mode.
  The two are different code, not a retuning, so each is exported and verified on its own.
• Sample Rate (Hz) – f_s, read only by insf. MATLAB's hht(imf) with no rate is this block at 2π, which puts the instantaneous frequency in radians per sample.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Frames and timing

The frame is the last L samples, oldest first. The first decomposition is made when the register fills – at sample L − 1, counting from 0 – and then every Hop Size samples; between them the outputs hold, and before the first one they are all zero. The decomposition is of the WHOLE frame, so the newest sample is at its edge, where the envelope is an extrapolation rather than a fit; that is the same edge effect MATLAB's emd has, and it is the reason a frame is decomposed rather than a single sample published.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. The sift loop, the decomposition loop and the envelope solve are emitted as loops whose bounds are the parameters above, so the generated file does not grow with the frame length and the worst-case cost of a frame is fixed – a decomposition MATLAB describes as "until converged" becomes a bounded program.

The three HDL targets carry the whole decomposition in real arithmetic and are therefore simulation-only: a tridiagonal solve, a division by an envelope difference and an arctangent do not belong in a Q16.16 datapath. The frame itself is held in the port's fixed-point format, so quantization happens only at the port boundary.

⚠ Sifting makes DISCRETE decisions – which samples are extrema, when the tolerance is met, how many modes are found – so a frame in which one of those decisions is within a quantum of going the other way decomposes differently in fixed point than in double precision, and the modes then differ by much more than the quantum. Measured on this block's own reference over 40 runs of 30 frames each with the input quantized to Q16.16: about 2.5% of runs contain such a frame. That is arithmetic, not a code-generation defect, and a re-run is the first thing to try when one of the three HDL columns is red.

Simulink bridge

None (Support::None). emd and hht are functions of the Signal Processing and Wavelet toolboxes and neither ships a Simulink library block for them, so there is no path a diagram could name. The bridge reports this block rather than dropping it silently, and it therefore has no parity testbench; code export verification still covers it across all ten languages. Sampling Time (s) → SampleTime, as on every block, has no counterpart here for the same reason.

Notes

• Stateful, and discrete by nature: a register of L samples, a frame counter and the held decomposition.
• Verified against R2026a: 44 frames over eight option sets, worst 1.6e-15 on the modes, 7.2e-16 on the residual, with the same mode count and sift count on every frame (the source banner carries what was checked).
• It is a frame decomposition, not a filter. Each frame is decomposed on its own, so two overlapping frames may split a signal differently; that is a property of the method, and it is why the hop size is worth choosing deliberately.
• No state space. Sifting is not linear in the frame, so model reduction correctly declines to merge it.