Three-Level Wavelet Transform of Signal f(x) with Frequency Band Energy Analysis

Performing a three-level wavelet transform on signal f(x) yields wavelet coefficients d1, d2, d3 and approximation coefficient a3. The objective is to calculate the energy distribution across different frequency bands, where energy is defined as the sum of squared wavelet coefficients. This represents a highly practical application for signal analysis. In this implementation, we can determine the energy distribution of the signal across various frequency bands by computing the sum of squared wavelet coefficients. This approach provides valuable insights into the signal's characteristics and properties. The three-level wavelet decomposition generates multiple sets of wavelet coefficients and approximation coefficients, significantly enhancing our capability to analyze and understand signal behavior. From a coding perspective, this typically involves using wavelet transform functions (such as wavedec in MATLAB or pywt.wavedec in Python's PyWavelets library) to perform the decomposition, followed by energy calculation using element-wise squaring and summation operations. The algorithm efficiently captures frequency-domain information while maintaining time localization advantages of wavelet analysis. The design and implementation of this application hold significant importance for research and practical applications in signal processing and data analysis fields, particularly for feature extraction, denoising, and compression tasks.