The chirplet transform: physical considerations. Mann, S. & Haykin, S. IEEE Trans. Signal Process., 43(11):2745--2761, November, 1995.
The chirplet transform: physical considerations [link]Paper  doi  abstract   bibtex   
We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call q-chirps for short), giving rise to a parameter space that includes both the time-frequency plane and the time-scale plane as 2-D subspaces. The parameter space contains a “time-frequency-scale volume” and thus encompasses both the short-time Fourier transform (as a slice along the time and frequency axes) and the wavelet transform (as a slice along the time and scale axes). In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shear in time (obtained through convolution with a q-chirp) and shear in frequency (obtained through multiplication by a q-chirp). Signals in this multidimensional space can be obtained by a new transform, which we call the “q-chirplet transform” or simply the “chirplet transform”. The proposed chirplets are generalizations of wavelets related to each other by 2-D affine coordinate transformations (translations, dilations, rotations, and shears) in the time-frequency plane, as opposed to wavelets, which are related to each other by 1-D affine coordinate transformations (translations and dilations) in the time domain only
@article{ mann_chirplet_1995,
  title = {The chirplet transform: physical considerations},
  volume = {43},
  url = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=482123},
  doi = {10.1109/78.482123},
  abstract = {We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call q-chirps for short), giving rise to a parameter space that includes both the time-frequency plane and the time-scale plane as 2-D subspaces. The parameter space contains a “time-frequency-scale volume” and thus encompasses both the short-time Fourier transform (as a slice along the time and frequency axes) and the wavelet transform (as a slice along the time and scale axes). In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shear in time (obtained through convolution with a q-chirp) and shear in frequency (obtained through multiplication by a q-chirp). Signals in this multidimensional space can be obtained by a new transform, which we call the “q-chirplet transform” or simply the “chirplet transform”. The proposed chirplets are generalizations of wavelets related to each other by 2-D affine coordinate transformations (translations, dilations, rotations, and shears) in the time-frequency plane, as opposed to wavelets, which are related to each other by 1-D affine coordinate transformations (translations and dilations) in the time domain only},
  number = {11},
  journal = {{IEEE} Trans. Signal Process.},
  author = {Mann, S. and Haykin, S.},
  month = {November},
  year = {1995},
  keywords = {2-D affine coordinate transformations, 2-D subspaces, Signal processing, chirplet transform, convolution, dilations, multidimensional parameter space, multiplication, physical considerations, q-chirps, quadratic chirp functions, rotations, shear in frequency, shear in time, shears, short-time Fourier transform, signal analysis, time-frequency plane, time-frequency-scale volume, time-scale plane, translations, wavelet transform},
  pages = {2745--2761}
}

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