Matlab Analyze Time Frequency Content

CWT Reference Guide

Workflow

1. Understand the Signal and User Goals

Before computing the CWT, determine:

• Signal type: Real-valued or complex-valued?

• Feature type: Oscillatory components (frequency resolution matters) or impulsive events (time localization matters)?

• Sampling frequency: Known? If not, analysis uses normalized frequency (cycles/sample).

• Goal: Visualization only, numerical outputs, or both?

2. Run Pre-CWT Diagnostics (Recommended)

assessBoundary(x, Fs=1000)
illustrateBoundary()
assessSpectralEnergy(x, 1000, ThresholdDB=-20)
illustrateTransientLocalization()
plotAntiAnalyticScalogram(abs(cfs), f, t)

3. Choose the Calling Pattern

Critical rule: nargout == 0 triggers auto-plotting. Capturing ANY output suppresses the plot entirely.

4. Configure the CWT

Wavelet Selection (signal-feature-driven)

Key Parameters

FrequencyLimits Details

• Performance lever: Restricting the range reduces scales computed (fewer rows in output).

• Lower limit = 0: Gives lowest valid frequency from cwtfreqbounds (no guesswork).

• Upper limit = Fs/2: Forces highest wavelet to peak at Nyquist. Use when signal has significant near-Nyquist content. Default is conservative (wavelet decays by Nyquist).

• Use cwtfreqbounds to query achievable range for given signal length and wavelet.

Boundary Selection

Oscillations at boundaries: When the signal has active oscillations at an edge, "reflection" reverses the wave direction at that point, creating a cusp (instantaneous frequency doubling artifact). Prefer "zeropad" in this case — it attenuates to zero rather than creating a false continuation. Use "reflection" only when the signal is locally stationary (slowly varying) at its edges.

For large signals (>100K samples), "periodic" avoids doubling the FFT length. Validate with boundary assessment script first.

Why performance differs: The CWT computation is dominated by inverse FFT of the signal-wavelet product. "periodic" uses N-point FFT (DFT naturally assumes periodicity). "reflection" or "zeropad" extend the signal to ~2N points before transforming.

5. Compute and Interpret

Output Signatures

cwt():

[cfs, f, coi, fb, scalcfs] = cwt(signal, ...);

wt():

[cfs, f, coi, scalcfs] = wt(fb, signal);

• cfs — Wavelet coefficients. For complex signals: 3D array (scales x time x 2) where page 1 = positive frequencies (analytic) and page 2 = negative frequencies (anti-analytic).

• f — Frequencies in Hz (if SamplingFrequency set), cycles/sample (if no Fs), or periods as duration (if SamplingPeriod used).

• coi — Cone of influence values (one per time sample).

• fb — Filter bank object (from cwt only, 4th output).

• scalcfs — Scaling coefficients (lowpass filter capturing near-DC content wavelets cannot reach).

Interpreting Coefficient Values

MATLAB CWT uses L1 normalization: wavelets scaled by 1/s. This means:

• abs(cfs) ≈ amplitude of the signal component at each time-frequency point

• A unit-amplitude sinusoid yields abs(cfs) ≈ 1 at the matching scale

• Values are directly comparable across frequencies

• Plot abs(cfs), not abs(cfs).^2 or log(abs(cfs))

This differs from DWT (which uses L2 normalization for energy preservation).

Scaling Coefficients

Wavelets must have zero mean — cannot capture DC/near-DC content. The scaling filter is a lowpass filter covering frequencies below the lowest wavelet. Request scaling coefficients (last output) when:

• Low-frequency content matters to the analysis

• Planning to reconstruct via icwt (required for perfect reconstruction)

Cone of Influence

The COI marks regions potentially affected by edge effects. It is a caution zone, not an exclusion zone:

• Inside COI boundary: coefficients are reliable

• Outside (near edges): potentially affected by boundary extension — treat with increased skepticism

• A well-chosen boundary method reduces (but never eliminates) edge contamination

6. Visualize (if needed)

See Patterns section below.

Key Functions

cwtfilterbank Inspection Methods

Patterns

Basic Scalogram (Visualization Only)

% Auto-plots scalogram — do NOT capture outputs
cwt(signal, "Morse", Fs);

Scalogram into Specific Axes (App Designer)

% Plot into uiaxes or axes — no outputs
cwt(signal, "Morse", Fs, 'Parent', ax);

Efficient Repeated Analysis

fb = cwtfilterbank(SignalLength=length(signal), SamplingFrequency=Fs, ...
    Wavelet="Morse", TimeBandwidth=60, VoicesPerOctave=12);

% Option A: familiar cwt interface + auto-plot
cwt(signal1, 'FilterBank', fb);

% Option B: programmatic access
[cfs, f, coi] = wt(fb, signal2);

Manual Scalogram (Real-Valued Signal)

Use when you need both the data AND a plot.

[cfs, f, coi] = cwt(signal, "Morse", Fs);
t = (0:length(signal)-1) / Fs;

ax = newplot;
imagesc(ax, t, f, abs(cfs))
ax.YDir = "normal";
ax.YScale = "log";
xlabel(ax, "Time (s)")
ylabel(ax, "Frequency (Hz)")
title(ax, "Scalogram")
colorbar(ax)

% Optional: add COI
hold(ax, "on")
plot(ax, t, coi, 'w--', LineWidth=1.2)

Manual Scalogram (Complex-Valued Signal, Combined Plot)

For complex signals, cfs is a 3D array (scales x time x 2): page 1 contains positive-frequency (analytic) coefficients and page 2 contains negative-frequency (anti-analytic) coefficients. Use scripts/plotAntiAnalyticScalogram.m for a combined -Fs/2 to Fs/2 view:

fb = cwtfilterbank(SignalLength=length(signal), SamplingFrequency=Fs);
[cfs, f] = wt(fb, signal);
t = (0:length(signal)-1) / Fs;

plotAntiAnalyticScalogram(abs(cfs), f, t);

Why surf instead of imagesc for complex signals: The frequency axis spans negative to positive. Setting YScale = "log" would fail (log of negative numbers). surf handles the non-uniform logarithmic spacing correctly without requiring a log axis setting.

Transient Localization

When the goal is to precisely locate impulsive events or abrupt changes:

Wavelet choice for time localization (best to worst):

[cfs, f, coi] = cwt(signal, "Morse", Fs, TimeBandwidth=20);
t = (0:length(signal)-1) / Fs;

% Extract the highest-frequency scale (first row = finest scale)
finestScale = abs(cfs(1, :));

% Plot to identify transient locations
figure
plot(t, finestScale)
xlabel("Time (s)")
ylabel("|CWT| at finest scale")
title("Transient Detection")

Run scripts/illustrateTransientLocalization.m to compare all three wavelets on an impulse signal.

Constant-Q Property and Limitations

CWT is a constant-Q analysis: bandwidth is proportional to center frequency.

• High frequencies: broad bandwidth (less frequency resolution), short time support (better time resolution)

• Low frequencies: narrow bandwidth (better frequency resolution), long time support (less time resolution)

Limitation: At high frequencies, the wide bandwidth means CWT cannot resolve closely-spaced tones. If tone separation Δf is small relative to center frequency (Δf/fc < 1/Q), the components will be smeared together. Use STFT (constant bandwidth) instead.

Example: DTMF tones at 3600 and 3700 Hz (100 Hz apart) at fs=8000 Hz. CWT at 3650 Hz with bump wavelet (Q~20) has bandwidth ~182 Hz — cannot separate them. STFT with M=256 gives 31.25 Hz resolution — easily resolves both.

Scale-Frequency Relationship

Scale and frequency are inversely related: s = f_psi / f, where f_psi is the mother wavelet's peak frequency (in cycles/sample).

To convert to physical Hz: f_Hz = f_psi * Fs / s.

Key points:

• Scale is unitless regardless of Fourier transform convention

• Large scale → low frequency, small scale → high frequency

• The CWT uses L1 normalization, so dilating by s gives a filter centered at f_psi/s

• centerFrequencies(fb) returns the frequency at each scale for a given filter bank

Interactive Exploration

Recommend waveletTimeFrequencyAnalyzer when the user needs to explore parameters and export a script.

Common Mistakes

Low-Frequency Analysis

The lowest achievable frequency depends on signal length and wavelet time standard deviation:

• Constraint: 2σ_t of wavelet ≤ signal length at largest scale

• Higher TimeBandwidth → wider wavelets → lowest frequency further from DC

• Use cwtfreqbounds to check achievable range

• Request scaling coefficients (last output of cwt/wt) to capture near-DC content

Inverse CWT (icwt)

Reconstruction Methods

icwt has two fundamentally different inversion paths:

Exact Reconstruction (filter bank method)

Perfect reconstruction requires the analysis filter bank, which you obtain from cwtfilterbank:

% Forward CWT — must use periodic boundary
fb = cwtfilterbank(SignalLength=length(x), SamplingFrequency=Fs, Boundary="periodic");
[cfs, ~, ~, scalcfs] = wt(fb, x);

% Get analysis filters
psif = freqz(fb, FrequencyRange="twosided", IncludeLowpass=true);

% Exact inversion
xrec = icwt(cfs, [], ScalingCoefficients=scalcfs, AnalysisFilterBank=psif);
% norm(xrec' - x, Inf) ≈ 1e-16

Or equivalently using cwt (filter bank is the 4th output):

[cfs, f, coi, fb, scalcfs] = cwt(x, Fs, Boundary="periodic");
psif = freqz(fb, FrequencyRange="twosided", IncludeLowpass=true);
xrec = icwt(cfs, [], ScalingCoefficients=scalcfs, AnalysisFilterBank=psif);

Critical: AnalysisFilterBank cannot be combined with SignalMean, VoicesPerOctave, TimeBandwidth, or WaveletParameters. It is a separate inversion path.

Approximate Reconstruction (Morlet single-integral)

Uses the admissibility constant to sum CWT coefficients over scale. Misses DC/near-DC content that wavelets cannot capture.

% Option A: scalar mean (simplest)
[cfs, f] = cwt(x, Fs);
xrec = icwt(cfs, 'SignalMean', mean(x));

% Option B: time-varying trend (better low-frequency recovery)
trend = smoothdata(x, 'movmean', 100);
xrec = icwt(cfs, 'SignalMean', trend);

% Option C: scaling coefficients (best approximate method)
[cfs, ~, ~, ~, scalcfs] = cwt(x, Fs);
xrec = icwt(cfs, 'ScalingCoefficients', scalcfs);

Cannot specify both SignalMean and ScalingCoefficients — they serve the same role (DC recovery).

Bandpass-Filtered Reconstruction

Extract a frequency-localized approximation by specifying a frequency range:

[cfs, f] = cwt(x, Fs);
xrec = icwt(cfs, [], f, [flo fhi], 'SignalMean', mean(x));

This zeros out coefficients outside [flo, fhi] before applying the Morlet formula. Useful for isolating specific oscillatory components.

Wavelet Must Match

If you used a non-default wavelet in the forward CWT, pass it to icwt:

[cfs, ~, ~, ~, scalcfs] = cwt(x, 'bump');
xrec = icwt(cfs, 'bump', 'ScalingCoefficients', scalcfs);

Decision Tree

Need exact reconstruction?
├── Yes → Use AnalysisFilterBank + ScalingCoefficients
│         (requires Boundary="periodic" in forward CWT)
└── No
    ├── Need bandpass filtering? → icwt(cfs, [], f, [flo fhi], SignalMean=...)
    └── Just need approximate signal back?
        ├── Have scaling coefficients? → icwt(cfs, ScalingCoefficients=scalcfs)
        ├── Know the mean? → icwt(cfs, SignalMean=mean(x))
        └── Have a trend estimate? → icwt(cfs, SignalMean=trend)

Deep Learning Integration

Both use the exact filter-bank reconstruction path only (no Morlet approximation).

dlicwt — Custom Training Loops

% Forward: dlcwt outputs complex dlarray
fb = cwtfilterbank(SignalLength=N, Boundary="periodic");
[psif, indices] = cwtfilters2array(fb, 1e-8, IncludeLowpass=true);

% In training loop:
cfs = dlcwt(x_dl, psif, indices);          % complex output
xrec = dlicwt(cfs, psif, indices);          % accepts complex input

• Accepts complex dlarray — pairs naturally with dlcwt

• Requires psif (analysis filters as (1,1,:) dlarray) and indices (filter index matrix)

• Formatted/unformatted pattern: if input is formatted, output is formatted "CBT"; if DataFormat is specified explicitly, output is unformatted

icwtLayer — dlnetwork

% Construct layer
layer = icwtLayer(SignalLength=1024, VoicesPerOctave=12, ...
    IncludeLowpass=true, Wavelet="Morse", TimeBandwidth=60);

• Input must be real — expects real/imaginary parts concatenated along channel dimension (pairs with cwtLayer which outputs real)

• Channel dimension gets halved in output (real/imag → complex → real signal)

• Weights are learnable — initialized from cwtfilterbank analysis filters but can be fine-tuned. Default WeightLearnRateFactor=0 (frozen). Set > 0 to learn modified reconstruction filters.

• Always uses Boundary="periodic" internally

Deep Learning Gotchas

Common icwt Mistakes

Documentation References

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