ExpFxx

ExpFxx is a family of exponential-type functions used in time-domain and time-frequency modeling. It is commonly defined by a decaying oscillatory kernel, f(t; alpha, beta) = exp(-alpha t) cos(beta t) for t ≥ 0, with alpha > 0 and beta real. A complex form, f(t; alpha, beta) = exp(-alpha t + i beta t), is also used. The suffix xx in ExpFxx is a naming convention indicating a two-parameter subfamily distinguished by the decay rate alpha and the angular frequency beta.

ExpFxx serves as a sparse dictionary for decomposing signals into a sum of decaying oscillatory components. It is applied in areas such as audio signal processing, vibration analysis, and biological signal modeling. In practice, ExpFxx components are recovered with methods like Prony’s method, the matrix pencil technique, or greedy algorithms such as matching pursuit, often within a dictionary-learning framework.

Mathematically, the family is not orthogonal in general, but it provides a flexible basis for representing non-stationary signals. For fixed alpha, the Fourier transform of exp(-alpha t) cos(beta t) yields a pair of peaks near ±beta with width related to alpha; the complex form corresponds to a shift in the complex plane. Parameters may be estimated from data via nonlinear least squares or sparse regression.

History and usage: ExpFxx appears in academic discussions as a descriptive label for exponential-atom dictionaries. It is not a single universal standard but a recurring motif in research on exponential fitting and non-stationary signal analysis. Some open-source toolkits implement ExpFxx-inspired dictionaries to promote spectral sparsity and interpretability in signal representations.