Haar Wavelet Decomposition and Reconstruction with MATLAB Implementation

This implementation demonstrates Haar wavelet decomposition and reconstruction, which is a fundamental signal processing technique. In MATLAB, this method can be efficiently implemented using built-in wavelet functions from the Wavelet Toolbox. The decomposition process involves breaking down signals into different frequency components through a pyramidal algorithm that uses averaging and differencing operations, while reconstruction reverses this process to reassemble the original signal from its wavelet coefficients. The key MATLAB functions used include: - wavedec: Performs multi-level 1D wavelet decomposition - waverec: Reconstructs the signal from the wavelet decomposition structure - dwt: Single-level discrete wavelet transform - idwt: Single-level inverse discrete wavelet transform This method is widely applied in image processing and compression, where it effectively extracts signal characteristics while reducing data redundancy through thresholding and coefficient quantization techniques. The MATLAB implementation allows straightforward processing of various signal types including audio, images, and sensor data, providing accurate results with computational efficiency through optimized matrix operations and filter bank implementations. The algorithm typically involves: 1. Applying high-pass and low-pass filters to separate detail and approximation coefficients 2. Downsampling to create multi-resolution representations 3. Thresholding coefficients for compression applications 4. Upsampling and filtering during reconstruction phase Using MATLAB's wavelet functions enables custom parameter configuration for decomposition levels, wavelet types, and boundary conditions, making it suitable for both educational purposes and practical engineering applications.