units on x axis after FFT

Question

My signal is a static 1D pattern detected by the linear photodiode array with N pixels and pitch p.

What units will I get along the X-axis after FFT to spectrum?

Answer 1

If you have a signal f(x) with unit U depending on variable x with unit V. Then

• the continuous Fourier transform of f has unit UV and depends on a variable with unit 1/V.
  • Example 1: f(x) is a Voltage with x being time. then the Fourier transform has unit Vs (or V/Hz) versus variable 1/s (or Hz).
  • Example 2: f(x) is a power with x being space. Then the FT has unit Wm and the x axis (which is then a wavenumber) unit 1/m (this is probably your case).
• the Discrete Fourier transform (or FFT) has unit U (same as original) and depends on a discrete variable, (which has with unit 1 by definition because it is just a counter).

So the units of the X-Axis of a FFT are 1 (because it is a counter).

I included the continuous Fourier transform, because I suspect that you just confused the FFT (which is just the name of an algorithm for the discrete Fourier transform by the way) with the ordinary (continuous) Fourier transform.

Answer 2

Let me clarify my above question because of the shortage of initial meaningful data.

The question was related with inverse FT of a spatial interferogram (a.k.a. fringe pattern) formed from the optical radiation by a static Fourier-transform spectrometer and detected with a linear photo diode array to reconstruct finally the optical spectrum.

Therefore, the mathematically formal answer "So the units of the X-Axis of a FFT are 1 (because it is a counter)" is absolutely right.