# Can I recover the physical wavelengths present in a light source from a time-series measurement of its amplitude?

Consider a mixture of different wavelengths being emitted from the same point (ex: a star). This light consists of a mixture of wavelengths and intensities at each wavelength. When measuring the combined signal as a time-series of amplitude values, we can then use something like Fourier Transform to get a frequency vs. power plot. Would this plot be a "physically" accurate description of what physical wavelengths are in the light, or is it somehow mathematically decomposed further.

I think... the same question - posed differently: Is it possible for the same combined waveform be generated by 2 distinct signals sources - with each source having its own set of emitted frequencies/intensities?

Suppose you have 1 star emitting at frequencies A, B, and C. Then another star emitting at frequencies J, K, and L. Is is somehow possible that the combined waveform (A+B+C) is the same as (J + K + L).

• To get the frequency from a time series, you need to sample at a rate faster than that frequency. For visible light that is difficult. A much easier way to measure the wavelength spectrum is with the aid of a diffraction grating. Nov 19, 2023 at 5:10
• Thank you. I think I would be approaching this from a more theoretical perspective. As to whether the there is a 1:1 relationship between the measured waveform and the original physical wavelengths present in the physical phenomena. Nov 19, 2023 at 5:17

• Re, "If I Fourier transform a signal..." Fourier Transform is only defined for periodic functions. If you sample an aperiodic signal over some time interval, $P$, and then you perform an FFT, and then you use the output of the FFT to program a synthesizer; the synthesizer will emit a periodic signal with period $P$ which, if you sample it for exactly one period, should give you back the same sequence of samples that you got when you sampled the original. In other words, the synth will keep repeating that same original clip, over and over again forever. Nov 19, 2023 at 19:04
• @codecitrus I answered that above: If you have a signal, lets call it $A(t)$, then there is a single $\tilde{a}(f)$ that will come out after you Fourier transform it. So, If you ever measure $B(t)=A(t)$, it does not matter, you will always get $\tilde{a}(f)$. Its a mathematical operation like multiplying by 2: if my starting condition is the same, I will always get the same answer. In your case if A and B are the same, then there is only one corresponding transform. Nov 19, 2023 at 19:29