# Does sampling ~> Nyquist frequency capture harmonics?

In a debate on the value of 24b/192KHz consumer audio (using a good brick wall filter, minimal jitter), a friend posed the argument:

Nyquist never properly dealt with harmonics, which change sound pressures by riding a higher frequency on a lower frequency "carrier", increasing the propagation of the higher frequency and lending some extra punch to the lower frequency.

A post from ComputerAudioPhile.com poses the same:

So let's say you have a flute that produces a tone at 14kHz. It may then produce overtones or harmonics at 28kHz, 42kHz, etc. If say the 28kHz overtone is sufficiently strong, and you truncate at redbook sampling frequency (corresponding to a 22kHz signal), then you might get an artifact at 22kHz - (28-22kHz) = 16kHz, which, if you are young enough, might be audible.

My Nyquestions are: How does this actually work? Which papers should I read?

• The reason to sample at 96kHz or 192kHz is so that one does not need a (near) brick wall filter (as these usually cause distortion). Mar 18 '14 at 23:36

No. Sampling at double the Nyquist frequency won't capture usable data at that frequency in the real world (finite length signals imperfectly filtered).

Sampling above double the Nyquist frequency will not capture any of the very common harmonics or overtones of a pitch that are around or above half the sampling frequency, even if the pitch is well below half the sampling rate. If this high frequency spectrum is not filtered out, it will be aliased and thus distort the recorded data.

Thus audio has to be low-pass filtered before sampling, which removes any overtones or harmonics around or above half the sample rate. However, for a high enough sample rate, say 48 kHz, any spectrum around or above half this is thought to be inaudible to most adult humans. If these overtones are inaudible, then any effect that they have on the fundamental pitch waveform is also thought to be inaudible (although the topic is debated.)

Added: For part 2 of your question, the subject is called "aliasing" in sampling. If the anti-aliasing low-pass filter isn't very good (stop-band well above the noise floor), then there might remain spectral content above half the sample rate in frequency, which will be folded down in frequency (aliased) by sampling, and thus cause distortion. The greater the difference between the highest frequency from the mic and half the sample rate, the less will be left behind by a non-good filter, or a different and cheaper reasonably-good filter might be suitable.

• You can read "At Nyquist frequency" as "At or above." The context is in terms of how good a 44.1 or 48KHz recording can be. I'll edit the topic to indicate this. Mar 18 '14 at 23:43

This is called "aliasing" and it is simply a result of periodicity of sine.

$\sin ak$ = $\sin (a+2b\pi)k$ (for integers $k$,$b$ - $k$ being the sample index)

So, if you maybe have $a=1.3\pi$, you could also see it as a signal at $-0.7\pi$ (a signal of negative phase at $0.7\pi$), or rather: you will not see any difference in the samples.