A Unified Mathematical Framework for Time–Frequency Feature Learning in Physical Signal Classification

Signals produced by physical systems often exhibitcomplex time-varying structures that reflect the internaldynamics of engineering processes. Accurate interpretation ofsuch signals is essential for diagnostics, monitoring, and datadrivendecision making in applications ranging from biomedicalmonitoring to industrial systems and renewable energy platforms.Conventional spectral analysis methods provide valuablefrequency-domain information but are often insufficient for analyzingtransient and non-stationary signal behavior. This paperpresents a unified mathematical framework for time–frequencyfeature learning in physical signal classification. The proposedformulation establishes a systematic connection between signalrepresentations, feature construction mechanisms, and statisticallearning modelsthe transient and non-stationary characteristicsofsformed into time–frequency representations using operatorssuch as the Short-Time Fourier Transform and the ContinuousWavelet Transform. Informative signal descriptors are then derivedthrough energy-based measures and entropy-based featuresthat characterize the structural distribution of signal energyacross the time–frequency plane. The resulting descriptors areorganized into compact feature vectors that provide a structuredrepresentation suitable for machine learning algorithms.The framework formalizes the mapping between signal space,time–frequency representation space, and feature space, enablingconsistent integration of signal processing techniques with classificationmodels such as artificial neural networks. The proposedformulation provides a general analytical structure for extractingdiscriminative patterns from complex engineering signals andsupports systematic development of signal-based classificationsystems across multiple application domains.