# When sampling theory condition fails, what should I do?

I met with a problem: if I sample audio data at 11kHz (yes, that is slow!) for 10 seconds and apply a Fourier Transform to the data, at what frequency will a 15 kHz tone appear?

So the original signal is 15kHz? which is greater than half of sampling rate 5.5kHz, right? and thus alias appears. And then, in this case, how should I find the appears one?

Thank you!

Because you've sampled at 11kHz the Frequency spectrum becomes periodic with a period of 11 KHz. So you're 15kHz will not only appear at 15 kHz but also at every frequency that is a multiple of 11 kHz away from 15 kHz. So the frequencies at which the 15 KHz tone are given by $f= 15\times10^3 +N \cdot 11 \times10^3$, where $N$ is an integer.

By convention you are looking at the frequency bins from -5.5 kHz to 5.5 kHz. Therefore you're 15 kHz tone will appear at 4 KHz.

Note that if your original signal was real then there is also a freqency component at -15kHz. This also gets extended periodically every 11 kHz - so this will produce a corresponding peak at -4 kHz.