How is a Morlet CWT different from an STFT spectrogram?

STFT uses one window length for all frequencies. The CWT changes the analysis duration with scale, so high-frequency features are sharper in time and low-frequency features occupy longer windows — a natural multiresolution view.

What does ω₀ control?

ω₀ sets how many oscillations sit under the Gaussian envelope. Larger ω₀ improves frequency selectivity of the wavelet but worsens time localization; smaller ω₀ does the reverse.

Why is there a cone of influence?

Near the signal edges the wavelet overhangs the data and must be padded or truncated, so those coefficients are less trustworthy. The cone highlights that boundary artifact rather than genuine interior structure.

Morlet Wavelet (CWT)

Continuous wavelet transform with the complex Morlet wavelet: scaleogram |W(s,t)|, log-frequency axis, cone of influence, adjustable ω₀ and scale range.

About this model

The continuous wavelet transform (CWT) correlates a signal with scaled and translated copies of a mother wavelet, producing a scaleogram |W(s,t)|. This page uses the complex Morlet wavelet — a complex exponential modulated by a Gaussian — with adjustable central frequency ω₀ and a selectable scale range on a log-frequency axis. Unlike a fixed-window STFT, longer scales probe low frequencies with wider time support, while short scales resolve fast transients. The cone of influence marks edge regions where padding artificially affects coefficients. The implementation is a didactic CWT visualizer, not a discrete wavelet packet or orthogonal filter bank, and it does not claim invertibility for reconstruction demos. Vary ω₀ and the scale span to trade time versus frequency localization.

Who it's for: Signal processing, applied math, geophysics, and scientific computing courses on multiresolution analysis.

Key terms

Morlet wavelet
Continuous wavelet transform
Scaleogram
Cone of influence
Mother wavelet
Scale

How it works

Continuous Wavelet Transform with the complex Morlet wavelet ψ(η) = π^(−1/4) e^{iω₀η} e^{−η²/2}. Each row of the scaleogram is the convolution of the signal with a stretched, complex sinusoid windowed by a Gaussian — small scales give sharp time + broad frequency, large scales the opposite (constant Q ≈ ω₀/√2). Compare with the STFT: the wavelet adapts its window length to the analyzed frequency, so it follows a chirp without any window-size trade-off, and its scaleogram has crisp edges around clicks.

Key equations

W(s, τ) = (1/√s) ∫ x(t) ψ*((t − τ)/s) dt

ψ(η) = π⁻¹ᐟ⁴ e^{iω₀η} e^{−η²/2}, f_pseudo ≈ ω₀ f_s / (2π s)

Frequently asked questions

How is a Morlet CWT different from an STFT spectrogram?: STFT uses one window length for all frequencies. The CWT changes the analysis duration with scale, so high-frequency features are sharper in time and low-frequency features occupy longer windows — a natural multiresolution view.
What does ω₀ control?: ω₀ sets how many oscillations sit under the Gaussian envelope. Larger ω₀ improves frequency selectivity of the wavelet but worsens time localization; smaller ω₀ does the reverse.
Why is there a cone of influence?: Near the signal edges the wavelet overhangs the data and must be padded or truncated, so those coefficients are less trustworthy. The cone highlights that boundary artifact rather than genuine interior structure.

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Morlet Wavelet (CWT)

About this model

How it works

Key equations

Frequently asked questions

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How it works

Key equations

Frequently asked questions