B.5.MPC.R:5.2 - A correct spectrum calculation answers an ambiguous physical question

A measurement uses uniformly spaced samples at f_s=100 samples/s. The earlier physical account admits a single sinusoidal signal with frequency below 50 Hz. A spectrum calculation returns a 10 Hz component, interpreted under that restriction.

The apparatus is changed and the admitted frequency range becomes 0 to 120 Hz. The same interpretation is now in question. For ideal samples of a unit-amplitude cosine at times n/100 seconds:

cos(2π90n/100)=cos(2π10n/100)

for every integer n. This follows because 90/100=1−10/100 and cosine is periodic and even. A 110 Hz cosine gives the same samples as well.

Consequently, the sampled sequence fits physical frequencies 10, 90 and 110 Hz in the new range. The spectrum calculation can remain correct for its input. Increasing its arithmetic precision or merely collecting more samples at the same times leaves this ambiguity.

To determine which frequency is present, change a contribution that distinguishes those cases. For the stipulated single-tone model, sampling at 300 samples/s places the whole admitted range below half the sampling rate. A three-second record of a 90 Hz signal then contains 270 cycles; the spectrum can return 90 Hz under the stated ideal timing and signal assumptions.

For a physical measurement, obtain the needed sampling and input-conditioning behavior from the instrument account. Frequencies outside the admitted band, timing error and an unsuitable analog input path can reopen the interpretation. A filter that removes the signal of interest changes the measurement question rather than recovering its frequency.

The repair is a changed acquisition and interpretation, with a recomputation on the new data. The receiving user can now distinguish a physical-frequency estimate from an unresolved alias. If the old record is all that is available, return the remaining alternatives instead of selecting one without further grounds.