Wavelet Energy Spectrum Analysis

Wavelet energy spectrum analysis is a multi-scale signal processing method particularly suitable for analyzing non-stationary signals. By implementing a three-level wavelet decomposition, we can break down the original signal into different frequency bands and calculate the energy distribution across each frequency component.

The process begins with a first-level decomposition, which separates the signal into high-frequency detail coefficients (d1) and low-frequency approximation coefficients (a1). The decomposition continues by applying a second-level decomposition to a1, yielding coefficients d2 and a2. Finally, a third-level decomposition is performed on a2 to obtain d3 and a3. Each decomposition level enables signal analysis at progressively finer scales, implemented algorithmically using wavelet transform functions like wavedec() in MATLAB.

The wavelet coefficients d1, d2, and d3 correspond to distinct frequency bands, while a3 represents the lowest-frequency approximation of the signal. The energy of each frequency band can be computed by calculating the sum of squares of the corresponding coefficients - a key step typically implemented using vectorized operations like sum(d1.^2) in code. The low-frequency approximation coefficients (a3) generally capture the signal's main trend, whereas the high-frequency detail coefficients (d1-d3) reveal local features and noise components.

This analytical approach finds extensive applications in fault diagnosis, biomedical signal processing, and other fields, effectively identifying energy distribution characteristics across different frequency components in signals through programmable wavelet transform implementations.