Wavelet Analysis

Mathematical Principles & Theorems

Investigates time-frequency localized signal analysis. Continuous Wavelet Transform (CWT): \(W(a, b) = \frac{1}{\sqrt{|a|}}\int_{-\infty}^\infty x(t) \psi^*\left(\frac{t-b}{a}\right)dt\) with scale \(a > 0\) and translation \(b\), overcoming Heisenberg-Gabor uncertainty limitations of the Short-Time Fourier Transform. Discrete Wavelet Transform (DWT) via Haar wavelet \(\psi(t) = \mathbf{1}_{[0, 1/2)}(t) - \mathbf{1}_{[1/2, 1)}(t)\) and Daubechies filter banks. Wavelet shrinkage denoising via soft/hard thresholding of detail coefficients.

Operating Instructions

• Select test signal profile (Doppler, Blocks, Bumps, Noisy Sinusoid).
• Set threshold \(\tau\) using the numeric value box and toggle Soft / Hard thresholding.
• Click Run Decomposition to compute Multiresolution Analysis (MRA) detail and approximation levels.
• Inspect the 2D CWT scalogram time-frequency heatmap and reconstructed denoised signal.