Matlab Analyze Time Frequency Content

Multiresolution Analysis in MATLAB

Decompose signals into additive time-domain components at different scales or equivalently different center frequencies using wavelet-based or data-adaptive methods.

Method Selection

Deterministic Methods: MODWT and MODWPT

MODWT + MODWTMRA Pipeline

modwt computes wavelet coefficients. modwtmra converts them to additive time-domain components. These are sequential steps, not alternatives.

% Decompose into additive components
wt = modwt(x, "sym4", 6);
mra = modwtmra(wt, "sym4");

% Components sum to original (perfect reconstruction)
maxErr = max(abs(x - sum(mra, 1)));  % ~1e-12

% Selective reconstruction: remove trend (approximation at last level)
detrended = x - mra(end, :);

% Keep only levels 3-5 (specific frequency band)
bandFiltered = sum(mra(3:5, :), 1);

When to use modwt alone (without modwtmra):

• Wavelet variance analysis: modwtvar

• Wavelet correlation by scale: modwtcorr, modwtxcorr

• Energy partitioning — the L2 norm of the coefficients equals the L2 norm of the signal

When to add modwtmra:

• Need time-domain signal components (for plotting, listening, reconstruction)

• Need to remove or keep specific frequency bands

• Need components that sum pointwise back to the original

MODWPT and modwptdetails

modwptdetails computes uniform-bandwidth signal components directly. It takes the signal as input — it is NOT a post-processor for modwpt output.

% Uniform-bandwidth decomposition — takes signal directly
[details, packetlevs, cfreq] = modwptdetails(x, "sym4", 4);

% Components sum to original (perfect reconstruction)
maxErr = max(abs(x - sum(details, 1)));  % ~1e-12

% Center frequencies in Hz (cfreq is normalized 0-0.5)
centerFreqsHz = cfreq * fs;

When to use modwpt alone (without modwptdetails):

• Energy or relative energy by frequency band — modwpt returns these directly:

[wpt, packetlevs, cfreq, energy, relenergy] = modwpt(x, "fk18", 4);
% energy: energy of wavelet packet coefficients at each terminal node
% relenergy: fraction of total signal energy in each band

• Wavelet packet coefficients for classification or feature extraction

When to use modwptdetails:

• Need time-domain signal components in uniform-width frequency bands

• Need to separate closely-spaced components that fall in the same MODWT detail level

Choosing MODWT vs MODWPT

Frequency Band Computation

MODWT — dyadic bands:

• Detail level j: [fs/2^(j+1), fs/2^j] Hz

• Approximation at level L: [0, fs/2^(L+1)] Hz

MODWPT — uniform bands at terminal level L:

• Bandwidth per band: bw = (fs/2) / 2^L

• Band k (1-indexed): [(k-1)*bw, k*bw] Hz

• Center frequency of band k: (k - 0.5) * bw Hz

modwpt and modwptdetails return rows in sequency order (monotonically increasing center frequency). Row 1 is the lowest frequency band, row 2^L is the highest. No reordering is needed — indexing by row directly corresponds to frequency.

These are approximate ideal passbands. Actual selectivity depends on the wavelet filter length.

Wavelet Selection and Frequency Selectivity

Longer wavelet filters produce sharper frequency bands (closer to ideal rectangular passbands) at the cost of worse time localization:

The Fejér-Korovkin (fk) family is specifically optimized for frequency concentration at a given filter length. "fk18" is the default for modwpt/modwptdetails.

Properties

Input Support

For complex-valued wavelet filters (csym family), use wavedec + wrcoef. This is the DWT path — it is NOT shift-invariant and has length constraints.

Energy Analysis with wavedec

wavedec provides L2-norm-preserving energy partitioning only when all three conditions are met:

1. Signal length is a power of 2

2. Orthogonal wavelet is used

3. dwtmode("per") (periodic extension)

If any condition is violated, energy partitioning is not exact. Prefer modwt for energy/variance analysis — it preserves the L2 norm unconditionally.

Data-Adaptive Methods: EMD and VMD

EMD (Empirical Mode Decomposition)

[imf, residual, info] = emd(x);
% imf: matrix where each column is an IMF (highest to lowest frequency)
% residual: monotonic remainder
% info: diagnostics (number of extrema, iterations per IMF)

Key behavior: MaxNumIMF is an upper bound, not a target. The algorithm terminates when the residual has fewer than MaxNumExtrema extrema (default 1), regardless of MaxNumIMF.

% This may return fewer than 10 IMFs
[imf, residual] = emd(x, MaxNumIMF=10);
numReturned = size(imf, 2);  % could be 5, 7, etc.

VMD (Variational Mode Decomposition)

[imf, residual, info] = vmd(x, NumIMFs=3);
% Always returns exactly 3 modes
% info.CentralFrequencies: normalized center frequencies of each mode

Key behavior: NumIMFs is an exact count. The optimization always returns this many modes. Central frequencies are in info.CentralFrequencies (normalized), NOT in the second output (which is the residual).

% Convert central frequencies to Hz (normalized cycles/sample * fs)
centralFreqsHz = info.CentralFrequencies * fs;

PenaltyFactor (default 1000): controls mode bandwidth. Lower values allow wider bandwidth modes — useful for chirps or frequency-varying components.

EMD Decomposes into Modes, Not Frequencies

EMD extracts intrinsic mode functions (IMFs) — signals with well-defined instantaneous frequency — not sinusoidal components. An amplitude-modulated signal is already a valid IMF:

t = linspace(0, 1, 1000);
x = (1 + cos(2*pi*10*t)) .* cos(2*pi*100*t);
imf = emd(x);
size(imf, 2)  % Returns 1 — EMD sees one mode

The STFT sees the same signal as three separate tones (90, 100, 110 Hz), but EMD correctly identifies the underlying modulated carrier as a single entity with time-varying envelope.

This means EMD is not interchangeable with MODWPT or VMD for separating closely-spaced tones. It operates on a fundamentally different notion of "component" — one defined by oscillatory structure and instantaneous frequency, not by Fourier frequency content.

Choosing EMD vs VMD

Why Wavelet MRA is Preferred for Machine Learning

Deterministic methods produce the same decomposition structure for every signal:

• Fixed number of bands at fixed frequencies → consistent feature dimensionality

• Training and inference use the identical transform

• Reproducible and parallelizable (no iterative convergence)

EMD produces different numbers of IMFs per signal — incompatible with fixed-shape feature matrices without padding/alignment. VMD has fixed count but mode frequency assignments vary across signals ("mode 2" means different bands for different inputs).

Deep learning entry points:

% dlmodwt in a custom training loop — filters MUST be numeric vectors
% (not wavelet name) so gradient-descent updates to the filters persist
[Lo, Hi] = wfilters("sym4");
dlX = dlarray(reshape(x, 1, 1, []), "CBT");  % Channel-Batch-Time
[wt, mra] = dlmodwt(dlX, Lo, Hi, 4);
% Use mra directly as multi-channel features — no separate modwtmra call needed

Why filter vectors, not wavelet name: dlmodwt makes the filters optionally learnable via automatic differentiation. Passing a wavelet name string would re-read fixed coefficients each call, discarding any gradient updates. Always pass Lo/Hi vectors obtained from wfilters.

Hilbert-Huang Transform and Instantaneous Frequency

hht — Hilbert Spectrum

hht computes the Hilbert spectrum: a time-frequency representation built from the instantaneous frequency and energy of each IMF. Unlike STFT/CWT which spread energy across a kernel, the Hilbert spectrum places energy at a single frequency per IMF per time sample.

Input is IMFs, not the raw signal. The correct workflow:

imf = emd(x);               % or vmd(x, NumIMFs=K)
[hs, f, t] = hht(imf, fs);  % Hilbert spectrum (sparse matrix)

Outputs (up to 5):

Name-value options:

Key points:

• Zero-output call produces a plot

• The sparse matrix hs is the actual Hilbert spectrum — a legitimate TF representation, not just visualization data

• Default frequency resolution (100 bins) is coarse — increase with FrequencyResolution for finer analysis

• Multiple IMFs contribute additively: if two IMFs have the same instantaneous frequency at time t, their energies accumulate in the same bin

• Internally computes: analytic signal → |z|² (energy), d/dt(unwrap(angle(z))) × fs/(2π) (frequency)

instfreq — Instantaneous Frequency Estimation

instfreq estimates instantaneous frequency of a signal. Two methods exist:

% Method 1: Hilbert (default) — requires monocomponent or pre-decomposed signal
[ifq, t] = instfreq(x, fs);                        % single signal via Hilbert
[ifq, t] = instfreq(imf, fs);                      % IMF matrix — one IF curve per column

% Method 2: TF-moment — works on raw multicomponent signals (centroid of spectrogram)
[ifq, t] = instfreq(x, fs, Method="tfmoment");     % first spectral moment of pspectrum
[ifq, t] = instfreq(P, F, T);                      % from pre-computed TFD (power, freq, time)

Critical distinction:

• Hilbert method: fs/(2π) × diff(unwrap(angle(hilbert(x)))) — physically meaningful only for narrowband/monocomponent signals. For multicomponent signals, decompose first with emd/vmd, then pass IMFs.

• TF-moment method: Weighted mean frequency from pspectrum at each time step. Works on raw signals but gives only a single average frequency (not per-component).

Output is 1 sample shorter with Hilbert method (forward difference loses one sample).

instbw — Instantaneous Bandwidth

[ibw, t] = instbw(x, fs);                          % from signal (via pspectrum internally)
[ibw, t] = instbw(P, F, T);                        % from pre-computed TFD
[ibw, t] = instbw(x, fs, FrequencyLimits=[flo fhi]);

• Only uses the TF-moment method (2nd central spectral moment of pspectrum) — no Hilbert option

• Measures spectral spread at each time instant

• Useful for detecting transitions, broadband transients, or time-varying bandwidth

Workflow Summary

Raw signal
    │
    ├── instfreq(x, fs, Method="tfmoment") → average IF (single curve)
    ├── instbw(x, fs) → spectral spread over time
    │
    └── emd(x) or vmd(x, NumIMFs=K)
            │
            ├── instfreq(imf, fs) → per-IMF IF curves (Hilbert method)
            └── hht(imf, fs) → Hilbert spectrum (full TF representation)

Interactive Exploration

signalMultiresolutionAnalyzer — interactive app for comparing MRA methods side by side.

• Supports: MODWT, EMD, VMD, EWT, TQWT

• Does NOT support: MODWPT

• Filters out complex-valued signals (only MODWT handles complex among app methods)

• Recommend when: User is exploring which method works best for their signal

• Use programmatic workflow when: Production code, batch processing, multichannel, complex-valued data, or need MODWPT

Common Mistakes

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