I have noticed that there is an identical pattern between two signals:
one is a signal of n identically spaced samples over a single period of a sinusoid
the other is a signal constructed of the real component of the second (i.e. first periodic) Fourier bin of n circular shifts of the signal, so pseudocode would be:
allocate a final result, size n loop from 1 to n newSignal = circularly shift signal one step to the left newValue = take the FFT, second (first periodic) bin, real component finalResult(n) = newValue
The only difference between these two is the scaling. If one's input is known to be a single cycle of a sinusoid, this is a nice way to remove all noise without knowing the phase of the input. Obviously it works in part because the second Fourier bin corresponds to the frequency of the input. But that's as far as my intuition gets me. In terms of theory, I don't understand why a signal composed of the real component of the second Fourier bins of the circular shift of each step of the signal looks the same as the original signal. Can anyone shed insight?