Optical EngineeringAdaptive "chirplet" transform: an adaptive generalization of the wavelet transform
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The "chirplet" transform unifies manyof the disparate signal representation methods. In particular, the wide ra nge of time-frequency (TF) methods such as the Fourier transform, spectrogram, Wigner distribution, ambiguity function, wideband ambiguity function, and wavelet transform may each be shown to be a special case of the chirplet transform. The above-mentioned TF methods as well as many new ones may be derived by selecting appropriate 2-D manifolds from within the 8-D "chirplet space" (with appropriate smoothing kernel). Furthermore, the chirplet transform is a framework for deriving new signal representations. The chirplet transform is a mapping from a 1-D domain to an 8-D range (in contrast to the wavelet, for example, which is a 1-D to 2-D mapping). Display of the 8-D space is at best difficult. (Although it may be displayed by moving a mesh around in a 3-D virtual world, the whole space cannot be statically displayed in its entirety.) Computation of the 8-D range is also difficult. The adaptive chirplet transform attempts to alleviate some of these problems by selecting an optimal set of bases without the need to manually intervene. The adaptive chirplet, based on expectation maximization, may also form the basis for a classifier (such as a radial basis function neural network) in TF space.