Time-Frequency Domain Analysis of Discrete Signals Using Continuous Wavelet Transform

Continuous Wavelet Transform (CWT) serves as a crucial tool for analyzing time-frequency characteristics of signals, particularly suitable for stability evaluation of non-stationary signals. The core concept involves convolving the signal with mother wavelets at different scales to decompose the energy distribution across various frequencies and time points. In code implementation, this typically requires defining a scale vector and applying wavelet convolution through iterative loops or built-in functions like cwt in MATLAB.

For CWT implementation on discrete signals, key steps include: first selecting appropriate wavelet basis functions (such as Morlet, Mexican Hat, etc.), then scanning different frequency bands by adjusting scale parameters. The transformation results in a 2D time-frequency matrix where rows correspond to scales (frequencies) and columns represent time points, with matrix values indicating the similarity between the signal and wavelet at specific positions. Programmatically, this can be achieved using wavelet.transform functions with scale arrays and signal vectors as inputs, often requiring normalization and complex conjugate operations.

For stability assessment, distribution characteristics of time-frequency energy can be identified: stable signals exhibit relatively uniform energy concentration regions in the time-frequency plane, while unstable signals (such as abrupt changes or frequency jumps) manifest as intense energy fluctuations or local clustering. Further analysis can extract statistical features from the time-frequency matrix (such as entropy values, energy variance, etc.) as quantitative indicators. Implementation-wise, this involves computing energy distribution across scales and time indices, followed by statistical calculations using functions like std, var, or entropy measurements.

This method overcomes the fixed-resolution limitation of Short-Time Fourier Transform and finds wide applications in mechanical fault diagnosis, bioelectrical signal analysis, and other fields. Code optimization may involve parallel processing for large-scale computations and visualization techniques using spectrogram plots or heatmaps.