Mallat Algorithm for Signal Decomposition and Reconstruction

The Mallat algorithm is a widely used signal processing method for decomposition and reconstruction. This implementation employs the db4 wavelet (Daubechies 4 wavelet) to decompose signals into different frequency bands and subsequently reconstruct them by merging these bands back together. The algorithm operates through a multi-resolution analysis approach, where signal decomposition involves successive high-pass and low-pass filtering followed by downsampling, while reconstruction performs upsampling followed by complementary filtering. In code implementation, key functions typically include: - Wavelet filter coefficient initialization for db4 wavelet - Convolution operations with decomposition filters - Downsampling/upsampling procedures - Recursive application for multiple decomposition levels This algorithm finds extensive applications in signal processing domains such as audio processing (e.g., noise reduction, compression) and image processing (e.g., edge detection, image compression). The db4 wavelet's four vanishing moments provide good balance between time-frequency localization and computational efficiency, making it suitable for practical implementations where both precision and performance are required.