Applying Low-pass and High-pass Filters to the Original Signal

The one-level wavelet decomposition process involves first applying low-pass and high-pass filters to the original signal separately, followed by binary downsampling for each filtered output. This yields two sets of coefficients: low-frequency (approximation) and high-frequency (detail) coefficients. Multi-level decomposition extends this process recursively by applying wavelet decomposition to the low-frequency coefficients obtained from the previous level. Through this hierarchical approach, increasingly detailed information can be extracted from the signal, resulting in more comprehensive analysis outcomes.

From an implementation perspective, this process typically utilizes filter banks with specific wavelet functions (like Daubechies or Haar wavelets). The filtering operation can be implemented using convolution between the signal and wavelet filters, while downsampling reduces the signal length by half by retaining every other sample. In programming terms, this decomposition can be efficiently computed using pyramid algorithms that maintain computational efficiency through iterative processing of approximation coefficients.