Time scale and Fourier transform

Question

Consider the Fourier transform $F(omega)$ of the function $f(t)$. The magnitude of $F(omega)$ depends on $omega$ and thus also depends on the scale of the $t$-axis. For example, when $f_1(t)$ is a box function which is 1 between $[0,1]$ s and $f_2(t)$ is a box function which is 1 between $[0,1]$ ns = $[0,10^{-9}]$ s, then the amplitude of $F_1(omega)$ is $10^9$ times larger than the amplitude of $F_2(omega)$. This is because the time scale is reduced by a factor $10^9$ in the second case.
However, no difference would be obtained when you do a numerical Fourier transform. If you have 100 points of the function $f_1(t)$ equally spaced between $[0,10]$ and also 100 points of the function $f_2(t)$ equally spaced between $[0,10times10^{-9}]$, then numerically the amplitude of the Fourier transform of both functions is the same (which is not correct). Numerically, you only consider the Fourier transform of the "y-values" $f(t_i)$, so the information about the time scale is lost...
You can solve this problem by multiplying the result with $10^{-9}$ in the second case because the Fourier transform of the rectangular function is proportional with $1/omega$. However, in general, this proportionality is not present...  How to solve this problem for a general function $F(omega)$? For example, when the Fourier transform is proportional to $e^omega$ or proportional to $1/omega^2$ or ...? Is there a numerical approach to take the scale of the time axis into account?

Laurent Duval · Answer

Most of the classical linear transformations (even filtering) may have three main types of scalings:

natural or none: sum or integral does not have an explicit scaling factor (but the inverse may need one)
in amplitude: because of linearity, at least one reference "unit" signal should have a unit amplitude in the transformed domain, for an easy read of graphs. Typically a unit amplitude sine (which is not $L_1$-integrable) should have a one amplitude at its frequency in the Fourier domain
in energy: to have constant energy in the original or the transformed domain.

Each of these options (sometimes, two of them are equal) depends on the purpose. Fourier  has a specific relation to scale (Where in “Discrete Time Signal Processing” (Oppenheim et al.) can I find the scale-change theorem?):
$$  s(alpha t) mapsto  frac{1}{|alpha|} S left(  frac{f}{|alpha|}right) $$
Obtaining scale-, shift- or orientation-invariant features is a long-lasting issue in DSP, and is still a concern in machine intelligence or deep learing.
Along the Fourier line, you can look at the Fourier-Mellin transform or scale represensation (by L. Cohen), and the many feature representations like SIFT, SURF, ORB, BRIEF: Image Matching Using SIFT, SURF, BRIEF and
ORB: Performance Comparison for Distorted

Time scale and Fourier transform

One Answer

Add your own answers!

Ask a Question