The affine group The Cohen's class, as presented in the previous section, is based on the properties of covariance by shifts in time and in frequency. One important element of this class is the Wigner-Ville distribution, noteworthy for its numerous properties. In order to favor a time-scale approach of the signal, one can also choose to put forward, among these desirable properties, the covariance by translation in time and dilation. The corresponding group of transforms, counterpart of the Weyl-Heisenberg group (see section 3.1.1), is the affine group, noted $A$, already introduced in the context of wavelet transform (see section 3.2.2). Its action induced on a signal $x(t)$ is given by $$x(t) \ \rightarrow\ x_{a',b'}(t)={1\over\sqrt{\vert a'\vert}}\ x\left({t-b'\over a'}\right),$$ and on its Fourier transform by $$X(\nu) \ \rightarrow\ X_{a',b'}(\nu)=\sqrt{\vert a'\vert}\ e^{-j2\pi\nu b'} X(a'\nu).$$
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