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Discrete Wavelet Transform

Multi-resolution analysis - see both time and frequency

Original Signal

Level 1 - Approximation (A1)

Level 1 - Detail (D1) [High Freq]

Level 2 - Approximation (A2)

Level 2 - Detail (D2) [Mid Freq]

Level 3 - Approximation (A3)

Level 3 - Detail (D3) [Low Freq]

Reconstructed Signal

Wavelet vs Fourier Transform

Fourier: Perfect frequency resolution, no time localization. A short burst looks the same as a continuous tone in the spectrum.

Wavelets: Trade-off between time and frequency resolution at each scale. Short events are localized in time AND frequency.

Decomposition: Low-pass filter → Approximation (A), High-pass filter → Detail (D). Repeat on A for multi-level analysis.

Applications: Image compression (JPEG2000), denoising, feature detection, ECG analysis.