Why do we follow Shannon's Theorem in audio sampling?

My professor told me that in sampling a signal, is not usual to respect the Shannon theorem because the interpolation formula contains sines and fractions which are difficult to calculate with a microprocessor.

Why is audio sampled at 44.1-48 KHz (Nyquist frequency of audio signals)? Is there a dedicated hardware for accomplishing this? I can sample sound with my PC, so does this mean that I have this hardware in my PC?

• dunno who your professor is (don't tell us), but what you said didn't make much sense. using a windowed $\mathrm{sinc}()$ for interpolation is common, but probably what is more common is an optimized interpolation function that looks very similar to the windowed $\mathrm{sinc}()$ but is derived from an FIR optimization program like Parks-McClellan. this function is sampled and stored in memory and then likely linear interpolation is used between the samples. – robert bristow-johnson Jan 11 '14 at 20:54
• He meant that non-audio signals are more easily representable with sampling at 4-5 times of the Nyquist frequency, without using interpolation formula. Probably this can give a quite good resolution, saving microprocessor resources for other aims, but probably audio needs more fidelity than other usual signals... – Federico Simonetta Jan 12 '14 at 9:22
• The interpolation formula is probably used in analysis and in remastering step, so not in microprocessor. For high quality reproduction 48 KHz is better, because in the reproduction of digital signal is unuseful converting to analog since it is the single word which is written to the speaker port, no interpolation formula required... – Federico Simonetta Jan 12 '14 at 10:48
• sorry, @fortea, nothing you're saying makes much sense to me. unless your sample-rate conversion ratio is $\frac{1}{N}$ where $N$ is an integer, interpolation of some form is necessary. – robert bristow-johnson Jan 12 '14 at 15:24

I do not know about the sinus values in embedded processors, since pre-computed values can be used. The principal problem with the interpolation with the cardinal series is that it is highly, one could even say extremely, acausal. To correctly implement it, you have to go back to times before the big bang and forward to the times after the dissolution of our galaxy to obtain the samples required for an exact reconstruction, and who in these days has that time and money.

So you always use filters that are imperfect from the theoretical point of view. To account for that gap between ideal theory and real practice, one uses a sampling frequency that is a higher multiple of the highest frequency in the signal than just the Nyquist double. Humanly perceptible sound goes to 12kHz according to vinyl and to 15kHz according to CDs and mp3. The range from 41-48kHz provides a safe multiple of 3 2.5-4 of the frequency range.

Please note "according to vinyl ... and mp3", which means that recordings of widely acceptable quality are produced with filtering the sound down to the frequency band limit of 12kHz-14kHz and thus having an oversampling factor of 3-4 when encoding with 41-48kHz. People trained in listening to orchestral music may notice the missing harmonics, especially at solos of high-pitched instruments.

• Human perceptible frequency is about 20 KHz. For Example, when you listen to music you can hear a note at whose fundamental frequency is at 4 KHz, but the harmonics are at 8KHz, 12 KHz, 16KHz, 32 KHz... (In this case you will listen to the third harmonic but not the fourth). – Federico Simonetta Jan 12 '14 at 9:12
• The perceptible range for most adults is already rather weak at 16kHz, newborns may react to sounds of 20kHz. For communication, a cut-off at 3kHz is acceptable, as decades of telephone communication have shown. Sound records on vinyl have a technological band limit of 12kHz, mp3 encodings a built-in limitation to 15kHz (that is seldom fully used). For recordings of classical music, the dynamical range is far more restricting than the frequency range. And if the recording quality is good enough, the sound you hear is the sound you expect to hear, i.e., your brain supplies the missing parts. – Dr. Lutz Lehmann Jan 12 '14 at 10:00
• This could have a sense, but what is the maximum frequency represantable in CD Audio standards? Also, mp3 are made sampling at about 40 KHz, so a little more than the double of 15 KHz... – Federico Simonetta Jan 12 '14 at 10:04
• I do not think that it is a limit in the CD standards. To avoid the aliasing distortions at frequencies too close to the half of the sampling frequency, the recordings are intentionally filtered down to a limiting frequency in the mid-tens. To account for shortcomings (too short filters while upsampling) in the reproduction device, the band limit is sometimes kept at the very low range. – Dr. Lutz Lehmann Jan 12 '14 at 10:14
• so are you saying that in CD Audio, frequencies above ≃15KHz are not played? – Federico Simonetta Jan 12 '14 at 10:28

What your professor probably meant is that in most audio converters a small amount of aliasing is permitted (for example if you put 24.1khz into a 48khz converter you will see an output component at 23.9kHz that is not very heavily attenuated). The reason for this is that manufacturers like to use so- called half-band filters in the decimation chain because this saves a factor of 2 in the number of multiplications. Halfband filters impose a response symmetry around FS_out/2 so the attenuation at Nyquist is only 6db. Since this is above the audio band it is not audible. However if you attempt to use this shortcut at lower sample rates like 8khz, it can be quite audible so it is not a recommended technique for lower sample-rates.

Bob

• He meant that non-audio signals are more easily representable with sampling at 4-5 times of the Nyquist frequency, without using interpolation formula. Probably this can give a quite good resolution, saving microprocessor resources for other aims, but probably audio needs more fidelity than other usual signals... – Federico Simonetta Jan 12 '14 at 9:21