The Importance of Sampling Rate in Audio Recording

What happens when you sample a sine wave at a rate of 20000 Hz with a sine wave frequency of 13000 Hz?

What will end up on your recording?

Answer:

The resulting recording will contain a 3000 Hz sine wave instead of the original 13000 Hz sine wave.

When sampling a sine wave at a rate of 20000 Hz, the Nyquist-Shannon sampling theorem states that the highest frequency that can be accurately represented is half of the sampling rate. In this case, the Nyquist frequency is 10000 Hz (20000 Hz / 2).

Since the sine wave you're sampling has a frequency of 13000 Hz, it is higher than the Nyquist frequency. According to the Nyquist-Shannon sampling theorem, frequencies above the Nyquist frequency will lead to aliasing.

Aliasing occurs when frequencies above the Nyquist frequency "fold back" and are mistakenly represented as lower frequencies. In this case, the 13000 Hz sine wave will be folded back and aliased as a lower frequency.

To determine the aliased frequency, you can subtract the Nyquist frequency (10000 Hz) from the original frequency (13000 Hz). In this case, 13000 Hz - 10000 Hz equals 3000 Hz.

Therefore, the resulting recording will contain a 3000 Hz sine wave instead of the original 13000 Hz sine wave.