Continuous Wavelet Transform (CWT)
The CWT decomposes a signal using scaled and translated versions of a mother wavelet ψ.
Discrete Wavelet Transform (DWT)
The DWT uses discrete scales and translations based on powers of 2.
Haar Wavelet
The simplest wavelet, piecewise constant with compact support.
Morlet Wavelet
Complex exponential modulated by Gaussian, ideal for time-frequency analysis.
Inverse Wavelet Transform
Reconstructs the original signal from wavelet coefficients.
Time-Frequency Localization
Wavelets provide a compromise between time and frequency resolution. According to the Heisenberg uncertainty principle, we cannot have perfect resolution in both domains simultaneously. Wavelets adapt: good time resolution at high frequencies (small scales) and good frequency resolution at low frequencies (large scales).
Multi-Resolution Analysis (MRA)
MRA represents a signal at different scales. Approximation coefficients (A) capture low-frequency components, while detail coefficients (D) capture high-frequency components. Each level further divides the approximation, creating a tree structure.
Orthogonality & Energy Preservation
Orthogonal wavelets preserve energy: the sum of squared coefficients equals the signal energy. This makes them ideal for compression and perfect reconstruction.
Compact Support
Wavelets with compact support are non-zero only over a finite interval. This localization enables efficient computation and edge detection in signals and images.
Image Compression (JPEG2000)
Wavelet transform decomposes images into subbands. Small coefficients can be discarded or quantized heavily, achieving high compression ratios while maintaining quality.
Signal Denoising
Noise typically appears in small detail coefficients. Thresholding these coefficients removes noise while preserving important signal features.
Edge Detection
Wavelet detail coefficients are large at signal discontinuities, making them excellent for detecting edges in images and transients in signals.
ECG & Biomedical Analysis
Wavelets detect QRS complexes, arrhythmias, and other features in biomedical signals where time localization is critical.