Generic discrete wavelets

The work concentrates on orthogonal wavelets and documents a generic method for designing discrete wavelets which encompasses the well-documented Haar and Daubechies. Only wavelets up to 6 taps are considered, these being derived from seed equations capable of explicit solution; higher degrees than this require iterative solution. Much of the work describes the derivation of the generic method, leading to equations which although comprising simple algebra do become complicated, especially for the 6 tap case, which "reduces" to a single quartic equation of great complexity. The seed equations are the usual ones of orthonormality and moment conditions with the highest moment condition replaced by a generic parameterized "lock" condition which introduces a new degree of freedom not found in the Daubechies wavelets. The 4 and 6 tap wavelets exist as two classes, each encompassing a continuous space of wavelets of their respective lengths, mutable by use of a single controlling parameter that can be exploited to change the wavelet's shape, regularity and support. Applications of compression are examined, using the control parameter as a tuning mechanism to best match given signals. Additionally, application to transient detection is presented, again using the new wavelet's tuning capability to optimize the detection process