From the Fourier transform to the Mellin transform A second argument encourages one to find more specific tools to analyze broad-band signals: the eigenvectors of the Weyl-Heisenberg group are the familiar complex exponentials, on which the Fourier transform decomposes a signal, whereas for the affine group, the eigenvectors are hyperbolas. From a slightly different point of view, the Fourier transform is invariant in modulus to translations in frequency, but not to dilations. Therefore, the Fourier transform is no longer the appropriate transform to change the representation space of these signals. It has to be replaced by a new transform, the Mellin transform, which is invariant in modulus to dilations, and decomposes the signal on a basis of hyperbolic signals. This transform can be defined as: $$M_X(\beta)=\int_0^{+\infty} X(\nu)\ \nu^{j2\pi \beta-1}\ d\nu$$ where $X(\nu)$ is the Fourier transform of the analytic signal corresponding to $x(t)$. We can show easily that $$Y(\nu)=X(a\nu)\ \Rightarrow\ M_Y(\beta)=a^{-j2\pi \beta}\ M_X(\beta),$$ which demonstrates the invariance by dilation. The basic elements are waves of the form $\nu^{-j2\pi \beta}$, whose group delay is hyperbolic: $$t_x(\nu)={\beta\over\nu}.$$ Thus, the $\beta$-parameter can be interpreted as a hyperbolic modulation rate, and has no dimension; it is called the Mellin's scale. In the discrete case, the Mellin transform can be calculated rapidly using a fast Fourier transform. Its algorithm, called the fast Mellin transform, is computed thanks to the file fmt.m. For further details on this transform, see for example [Ova94]. This transform is often used in the Time-Frequency Toolbox to implement functions which are connected to the affine class.
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