# Recording and playback of sound, frequencies question

If I record a sound at a rate of 8000 samples per second I know that the highest frequency that can be reproduced from the sound is 4000 Hz. What I want to know is;

If the sound contains frequencies above 4000 Hz when I play the sound back what would I hear and what frequencies would be produced at the points where the frequency of the original sound is greater than 4000Hz.

If I record a sound at a rate of 8000 samples per second I know that the highest frequency that can be reproduced from the sound is 4000 Hz.

False! You can only reproduce frequency content that is below half the sample rate, or in your case, strictly below 4000 Hz (not at 4000 Hz). Spectral content at exactly half the sample rate might be partially reproduced depending on the phase. Spectral content near half the sample rate in a finite length window has a similar problem, where it can be strongly attenuated, depending on the phase.

Content above half the sample rate is just folded below half the sample rate. A frequency above Fs/2 by dF that isn't removed by an anti-aliasing filter will be heard at a frequency of (Fs/2) - dF. e.g. A 5500 Hz (4000 + 1500) sinewave will be heard as 2500 Hz (4000 - 1500) sinewave.

One way to think about content at exactly half the sample rate is that it gets folded back on itself, and thus interferes with itself, either additively or destructively, depending on phase. Not good.

• That's my favourite exam question, getting students to figure out that a sinusoid of frequency $f_s/2$, when sampled, has an ambiguity to it based on phase. Many never figure it out for themselves. – Peter K. Apr 30 '13 at 18:19

If your sampling frequency is $f_s=8000$ Hz, your maximum signal frequency is indeed 4000 Hz ($=f_s/2$). If your signal contains frequencies above $=f_s/2$ you would hear the results of aliasing. This means that the original spectrum is folded back into the range $[0,4000]$ Hz. What actually happens is that by sampling with a sampling frequency $f_s$ your spectrum is made periodic with period $f_s$. So if the signal is not band-limited with maximum frequency $f_s/2$, the resulting periodic spectrum will have regions of overlap. E.g., if your signal has frequency components up to 5000 Hz, these frequencies would get folded back to the frequency band between 3000 and 4000 Hz, and consequently lead to distortions in this band. This is the reason why sampling is always preceded by lowpass filtering to attenuate all frequency components that might lead to aliasing (anti-aliasing filter). You can listen to some examples of aliasing here.