symlets

Symlets, short for symmetric wavelets, are a family of orthogonal wavelets developed as nearly symmetric variants of the Daubechies wavelets. They preserve the compact support and the same number of vanishing moments as their Daubechies counterparts while imposing symmetry in the scaling and wavelet functions. This design yields improved phase or boundary properties in discrete transforms, which helps reduce artifacts when processing finite-length signals.

Each member is designated symN, where N denotes the number of vanishing moments. The corresponding filter has length 2N, and the high-pass and low-pass reconstruction filters are chosen to enforce symmetry while maintaining orthogonality. Symlets offer a balance between smoothness, locality, and symmetry, making them suitable for applications that benefit from better phase information.

Commonly used symlets include sym2 through sym9, with sym4 and sym8 appearing frequently in practice. They are implemented in many wavelet toolkits and literature, often alongside Daubechies wavelets. Applications span signal denoising, compression, feature extraction, and time-frequency analysis in audio, image, and communications contexts. The symmetry property is particularly helpful for boundary handling and reconstruction accuracy in finite-length data sets.

Compared with Daubechies wavelets, Symlets trade some maximal theoretical regularity for improved symmetry and near-linear phase characteristics, while remaining compactly supported and orthogonal with the same vanishing moments. As the order increases, the wavelets become longer and the symmetry remains a defining feature, guiding their use where phase accuracy is important.