Peruswaveletin

Peruswaveletin is a proposed class of wavelets used in time–frequency analysis of non-stationary signals. It is defined as a family of compactly supported wavelets generated by applying a controlled periodic envelope modulation to a mother wavelet and then applying a reversible lifting refinement to produce a dual set that satisfies biorthogonality. The modulation parameter controls time localization, enabling sharper capture of transient features while maintaining a prescribed number of vanishing moments.

Constructive approach begins with a base wavelet ψ with N vanishing moments and finite support. A real envelope e(t; α) is chosen to emphasize recent or rapid transients; the modulated wavelet is ψα(t) = e(t; α) ψ(t). A dual wavelet ψα*(t) is obtained through a lifting scheme to satisfy biorthogonality. The family is indexed by scale s and translation k, with α controlling the envelope shape; the pair (ψα, ψα*) forms a biorthogonal pair. These steps can be embedded in discrete or continuous transform frameworks.

Properties include improved time localization for high-frequency content and transient features, while preserving a specified number of vanishing moments. The method aims to maintain stability and reconstruction accuracy in wavelet transforms. However, stronger envelope modulation can reduce regularity and increase computational cost, so there are trade-offs between time concentration, smoothness, and efficiency.

Applications are mainly in audio and speech processing, seismic waveform analysis, and biomedical signal processing where sharp transients are informative. Peruswaveletin remains largely within experimental or niche contexts, with limited adoption in mainstream toolchains. The term appears in a small body of conference papers and speculative notes, and it is generally regarded as an option for datasets with pronounced transients rather than a replacement for established wavelet families.