Continuous Wavelet Transform

Fourier Analysis · Axiom Academy

LESSON Continuous Wavelet Transform Understanding how wavelets analyze signals across time and frequency simultaneously The Continuous Wavelet Transform measures how well a signal f(t) matches a wavelet at different positions and scales. The wavelet coefficient W(a,b) is computed as: Here, ψ* a,b (t) is the complex conjugate of the wavelet, scaled by a and translated by b. The transform computes the inner product between the signal and the wavelet at each position. 2. Scale Parameter (a): Controlling Frequency The scale parameter a controls the width (and thus frequency) of the wavelet: By varying the scale, we analyze different frequency components of the signal, similar to how the Fourier Transform uses different frequencies. 3. Translation Parameter (b): Time Position The translation parameter b shifts the wavelet along the time axis, allowing us to probe the signal at different moments: This sliding window approach is what gives wavelets their time-localization property - we know not just which frequencies are present, but when they occur. 4. The Scalogram: Visualizing Time-Frequency Energy The scalogram displays the energy distribution of the signal across time and scale (frequency). It is defined as the squared magnitude of the wavelet coefficients: The scalogram creates a 2D heatmap where: Horizontal axis: Time (translation b) Vertical axis: Scale/Frequency (scale a) Color intensity: Energy magnitude

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