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Say I have a signal that is not x-indexed. That is, the x-axis of the signal is the distance traversed by car and the y-axis is the heading direction of the car at the corresponding distance.

Can I apply the Fourier Transform to this signal? If so, what is the physical meaning of this transformation? I believe that the horizontal axis is no longer frequency any more. What is it in this case?

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Yes you can. The unit of the "frequency" axis after the transform will be $m^{-1}$, and is known as spatial frequency.

For example, if there is a strong peak in the Fourier transform at $3.10^{-4} m^{-1}$, it means that your original curve exhibits a strong pattern that repeats at a scale of every $3.3km$, and from that you could infer that maybe the signal was recorded from a vehicle doing laps at the Monaco Grand Prix. The harmonics of this spatial frequency would contain a "signature" of the shape of the circuit.

A practical application of this is handwriting recognition - looking at shapes in the Fourier domain yields representations invariant to scaling, rotations, or more robust to deformations than the original data.

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  • $\begingroup$ OMG. The answer is so awesome and it really enlightens me! Thank you so so much! I am asking this because I wish to use FFT to find out the similarity between two such signals. Because the two signals are not aligned on x-axis, so I think FFT might help. Could you please further give some comments? Yes just as you said, ultimately I wish to infer the road shapes. $\endgroup$ – Sibbs Gambling Sep 19 '13 at 7:58
  • $\begingroup$ More information: stackoverflow.com/questions/18887200/… I really appreciate your kind help! $\endgroup$ – Sibbs Gambling Sep 19 '13 at 8:49

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