# Commenting frequency spectrum of a sampled data

the question below asked in my latest signal processing exam. I have no idea about it and i could not find any solution in literature. Could you please help me finding out the answer. Actually i believe that i am quite good at signal processing until this question.

A voltage signal is sampled digitally in the laboratory at three different sampling frequencies, and FFTs are performed on the sampled data. In all cases, 512 data points are taken. Here is a record of the observations: ‒ Data sampled at 115 Hz: A peak appears in the frequency spectrum at 2 Hz, with amplitude close to 3.0 V. Another peak appears in the frequency spectrum at 45 Hz, with amplitude close to 1.0 V. ‒ Data sampled at 280 Hz: A peak appears in the frequency spectrum at 48 Hz, with amplitude close to 3.0 V. Another peak appears in the frequency spectrum at 95 Hz, with amplitude close to 1.0 V. -Data sampled at 500 Hz: A peak appears in the frequency spectrum at 185 Hz, with amplitude close to 1.0 V. Another peak appears in the frequency spectrum at 232 Hz, with amplitude close to 3.0 V.

Question: What are most likely the true frequencies (and their corresponding amplitudes) in the voltage signal.

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## migrated from stackoverflow.comDec 3 '12 at 13:44

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You say sampled at three different frequencies but you have only listed observations for two ? – Paul R Dec 3 '12 at 7:33
You r right. In the exam three samples were listed but i did not mention here about the third one because two of them are enough for me to understand the underlying idea. – user1871830 Dec 3 '12 at 7:51
Well you just need to understand aliassing and then solve a couple of simultaneous equations. The solution won't be unique though without the third set of observations. – Paul R Dec 3 '12 at 9:52
I have just add the third one. Could you please run over an eye again. Regards – user1871830 Dec 3 '12 at 10:02
You can either do it formally with simultaneous equations or perhaps more easily just draw a frequency axis from say 0 to 1 kHz, mark all the components and all their aliasses and then see where they coincide. – Paul R Dec 3 '12 at 10:10

Consider the 3V component first (note that aliassing only affects frequency, not amplitude, so the 3V component will have the same magnitude at whatever frequencies it is aliassed to):

For Fs = 115 Hz the 3V component shows up at 2 Hz, so it's real frequency could be any value which satisfies f = 115*k +/- 2, i.e.

  2 Hz
113 Hz
117 Hz
228 Hz
232 Hz
...


Do the same thing for Fs = 280 Hz, where we have f = 280*k +/- 48:

 48 Hz
232 Hz
328 Hz
512 Hz
608 Hz
...


And then again for Fs = 500 Hz:

232 Hz
268 Hz
732 Hz
...


We can see that the only frequency which corresponds in all 3 cases is 232 Hz, so it looks like we have a 3V component at f = 232 Hz.

You can now apply the same approach to identify the frequency of the 1V component.

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