# How does Nyquist's theory apply to digital recording of opera music?

by amylase
Tags: apply, digital, music, nyquist, opera, recording, theory
P: 132
 Quote by amylase Cool. Thank you very much (and everyone who replied!). So if I'm recording a natural, real world phenomenon, like a live concert, as long as I am sampling at >= 2f the concert's highest frequency f then I can reconstruct back to exact original. That's clear to me now.
"Reconstruct back to exact original" is a nice idea, but what does it mean? You're talking about recording an opera. In fact, every person in the audience will hear a different version of the performance: depending on your position in the auditorium, you'll hear more of the singers, or the orchestra, or more from the left, or the right, with different proportions of direct and reverberated sound... A professional opera recording will use a large number of microphones: there'll be some on stage, some in the auditorium, some in the orchestra pit, some wireless ones for individual singers and maybe some backstage ones for an offstage chorus. The inputs from all these mikes will be mixed into an end product that will be different from any version that a single person in the audience heard.

We need to make a choice as to what we want to achieve. Here's a possible one: we try to reproduce what one particular person in the audience would have heard (maybe sitting at "the best seat in the house"). We use a dummy head setup with two microphones. Even the very best microphone doesn't have a flat response curve, and the frequency response usually drops sharply above 20 kHz. Most frequency response charts for professional microphones don't go any further than 20 kHz.

Now is a good time to ask the question: what use is sampling the sound at 192 kHz or 384 kHz? The input to the sound card is not the original sound: all the information the sound card is getting is coming from the output of the microphone. If the sound card registers information in frequencies above 100 kHz, it is due to artefacts created by the microphone and has no relation to the original sound.

Now that we have a digital version of the recorded sound, we need a way to get it into the ears of the hearer: an amplifier and loudspeakers or headphones. Our ideal is to get the air next to the listeners ears to vibrate in exactly the same way as the air vibrated around the dummy head as we were recording. Suffice it to say that no headphone or loudspeaker can create a pressure wave in the air that has the exact form of the one that is fed to it.

So "reconstruct to exact original" is a myth. We can make something that approaches the original. Maybe we can do this well enough that a blindfolded person siting on a chair in a concert hall couldn't tell if the sound they are hearing is coming from a live orchestra or a recording of the same orchestra. That's what really matters: even if some instruments are producing sounds in the ultrasound range, nobody can hear them so they could add nothing to the quality of the recording.

To go back to the Nyquist-Shannon theory, this is what it states:

"If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart."

Note that the wave is completely determined when sampled at twice the highest frequency. Sampling at a higher frequency simply doesn't add any more information (here's an article that goes into some detail about this). If the listeners cannot hear frequencies above 20 kHz, there's no sense in sampling at more than twice this frequency. 48 kHz is easily sufficient, not just for the "low quality mp3", but also for a high end opera recording.
 Sci Advisor P: 3,136 Verrrry interesting, folks .... Thanks to all. Probably the ear wouldn't distinguish between a 20 khz sine vs square wave, http://hyperphysics.phy-astr.gsu.edu.../place.html#c1 and if it won't, that's that in my book. Further that's something i could try if can find where i packed my old headphones three moves ago.. 48K vs 192K could become another "Tubes vs Transistors" debate. old jim
P: 269
 Quote by Michael C Note that the wave is completely determined when sampled at twice the highest frequency. Sampling at a higher frequency simply doesn't add any more information (here's an article that goes into some detail about this). If the listeners cannot hear frequencies above 20 kHz, there's no sense in sampling at more than twice this frequency. 48 kHz is easily sufficient, not just for the "low quality mp3", but also for a high end opera recording.
True, but there are conditions to be met.
First, source signal must not contain any frequencies above f/2, otherwise you'll get aliasing. So you have to use low-pass filter. Perfect low-pass aka brick-wall filters are in the same category as massless springs and absolute rigid bodies, i.e. do not exist. Real-world filters are a lot more sloppy. Plus this filter has to be before ADC in analog domain, and these days everything analog is fiddly and expensive and everything digital is dirt cheap and abundant.
Second, you need to use sinc interpolation to re-create the original function, which is again fiddly and expensive etc etc. Raising sampling frequency is the easy way out.

As sophiecentaur said, this is a mixture of theory and practice. And marketing. Surely 192Khz sounds 4 times better than 48 :)
PF Gold
P: 11,341
 Quote by Delta Kilo Surely 192Khz sounds 4 times better than 48 :)
And we hear through our wallets!
 P: 1 Goodness, a lot of interesting theories about how recording engineers do their jobs. I just had to register to clear a few things up. I am not a physicist, but a classical recording engineer. A few things: 1) Sampling rates are usually between 44.1 and 192kHz. 2) If there is a high channel count, 44.1 or 48, is probably the most common. 3) DXD formats like 394kHz are not common and are used primarily for PCM to DSD format conversion, as most Digital Audio Workstations cannot work with DSD formats. 4) High quality converters are coveted for their clean analog circuity and dynamic range, not the sample rate. 5) Most propaganda about the superiority of high sample rates sounding better are sales pitches and have no basis in reality. In truth, some inferior converters will sound better at 2x sample rates (88.2 and 96kHz) because of the stability of the clocking and performance of the converter, not because of the higher frequencies. 6) Forget any wave other than sin waves. Those are artificial waves that are not generated by any known acoustic instrument. We are talking about opera right? 7) Any limitation of the Nyquist theorem is due to the need for analog circuity in audio recording. 16bit audio works alright, but the theoretical limit of 144dB of dynamic range for 24 PCM recording has yet to be achieved due to analog limitations. Therefore a converter cannot output exactly what it records and any frequency rate, even though the converter is trying to produce an identical waveform from the original, (post decimation filter), analog signal.
P: 2,265
 Quote by AlephZero 96Khz is a fairly low sample rate for pro recording... Pros tend to use 384 Khz.
that's news to me. i haven't heard of any audio recording system that uses 384 kHz sampling rate.
P: 2,265
 Quote by Michael C To go back to the Nyquist-Shannon theory, this is what it states: "If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart." Note that the wave is completely determined when sampled at twice the highest frequency.
strictly speaking, the sampling frequency must be slightly higher than 2B. you will have a phase or amplitude ambiguity if you have a sinusoid at exactly B Hz and you sample it at exactly 2B.

even though Nyquist-Shannon didn't say this back in 1945 or whatever year it was, it has since been commonly expressed in the literature that you must sample at a frequency greater than 2B, if B is the highest possible frequency.
P: 2,265
 Quote by f95toli If you have a signal where the highest frequency component is f, and you sample at 2f then you can reproduce the original signal EXACTLY.

okay, for any $-\pi/2 < \theta < +\pi/2$, the following sampled at 2f will result in the same samples of +1, -1, +1, -1...

$$x(t) = \frac{1}{\cos(\theta)} \cos(2 \pi f t + \theta )$$

$$x(nT) = (-1)^n$$

$$T \equiv \frac{1}{2f}$$

So how are you going to reconstruct the highest component if the samples don't give you any hint as to what $\theta$ is?
PF Gold
P: 11,341
 Quote by rbj okay, for any $-\pi/2 < \theta < +\pi/2$, the following sampled at 2f will result in the same samples of +1, -1, +1, -1... $$x(t) = \frac{1}{\cos(\theta)} \cos(2 \pi f t + \theta )$$ $$x(nT) = (-1)^n$$ $$T \equiv \frac{1}{f}$$ So how are you going to reconstruct the highest component if the samples don't give you any hint as to what $\theta$ is?
That's so obvious. I'm sure he's just forgotten the 'greater than' sign.
P: 2,265
 Quote by sophiecentaur That's so obvious. I'm sure he's just forgotten the 'greater than' sign.
nyquist or shannon did too.
 Sci Advisor PF Gold P: 11,341 To be fair to them, they never built one which would have made it clear to them by producing a string of equal value pulses. It's easy to be wise after the event ;-)
 P: 5,634 great discussion!!! A practical insight into the above 'sampling' discussions can be found at this wiki article: http://en.wikipedia.org/wiki/T-carrier [T carrier is an older version of what is usually referred to as DS-1 today.] QUOTE]A more detailed understanding of how the rate of 1.544 Mbit/s was divided into channels is as follows. (This explanation glosses over T1 voice communications, and deals mainly with the numbers involved.) Given that the telephone system nominal voiceband (including guardband) is 4,000 Hz, the required digital sampling rate is 8,000 Hz (see Nyquist rate). Since each T1 frame contains 1 byte of voice data for each of the 24 channels, that system needs then 8,000 frames per second to maintain those 24 simultaneous voice channels. Because each frame of a T1 is 193 bits in length (24 channels × 8 bits per channel + 1 framing bit = 193 bits), 8,000 frames per second is multiplied by 193 bits to yield a transfer rate of 1.544 Mbit/s (8,000 × 193 = 1,544,000).[/QUOTE] A voice conversation, or music on a telephone line, gives you a bit of a subjective feel for what 'voice communications' sounds like when sampled at twice the nominal 4,000 hz analog bandwidth the former AT&T alloted for voice telephone calls here in the US. [As I recall, that analog bandwidth was based on subjective tests of human hearing by Bell Labs and what was needed to communicate human emotion and nuance and tones in typical voice conversations.] That digital sample rate resulted in subjectively 'higher quality' digital voice communications because background noise was noticeably reduced. One could tell early on if a conversation was on digital or analog facilities.... [These digital coding schemes required precise clock timing throughout the country via distribution of timing signals from the National Institute of Standards and Technology (NIST), formerly the National Bureau of Standards (NBS). That's not a problem in local audio systems. Loss or impairment of that timing signal was a cause for near 'panic' in testrooms around the country!!!!] But after reading the above posts I AM wondering if any error correction schemes are employed in modern digital audio systems???? [The digital scheme employed by AT&T utilized AMI which was replaced aby B8ZS error correction schemes.] And for those interested in some of the effects of analog filters, http://en.wikipedia.org/wiki/Bandwid...nal_processing)
P: 561
 Quote by Michael C Now is a good time to ask the question: what use is sampling the sound at 192 kHz or 384 kHz?
One of the biggest problems with recreating a digitally recorded audio signal with extreme high fidelity is the low pass or anti-aliasing filter that has to be implemented to ensure that a very minimum amount of energy ever gets above the Nyquist frequency.

The low pass filter in the case of 44 KHz or even 88 KHz and 96 KHz needs to be very, very steep. That introduces some severe phase distortion that varies sharply with frequency. Golden eared audiophiles with high quality analog equipment can definitely hear that, especially with sounds that are percussive such as triangles and cymbals. It's not just the individual sounds from each instrument or voice that matter but the spatial and timing relationships between them and sharp filters do some strange things with those. (I have experience as both a physicist and professional recording and production engineer) The higher sample rates reduce the requirements for the filter and the result should be much reduced phase distortion in the higher audio frequencies.

One of the smarter things that can be done is to upsample the signal to 192 KHz or 384 KHz and then apply the filter though that's not quite as refined as having the original signal there already.
 P: 3 Howdy folks. First post here. I should qualify it by saying that I am a recording engineer, not a scientist. From a practical standpoint, sampling rate depends as much on delivery format and track count as anything else when it comes to recording audio. Most adult humans can't hear much above 16 kHz, but we go with the assumption that the average human can hear up to 20 kHz. Can you perceive anything above that? Hard to say, but most evidence points to the fact that higher sampling rates sound better, because the slope of the anti-aliasing filter can be relaxed so that there are no ripples or resonances in the audio band, and the cutoff frequency can be above 20 kHz, rather than in the audio band. 44.1 kHz and 48 kHz were decided upon 30 years ago, since the first commercially available digital audio recorders were adapted video decks, and we seem to be stuck on multiples of these for PCM encoding. These sampling rates fulfilled two criteria: 1 - they were fast enough to recreate almost the entire bandwidth of human hearing (with rather sharply sloped anti-aliasing filters); 2 - they could function with the frame rates of the video recorders that were being used; A lot of classical recordings, if they are using PCM, will use a sample rate of 96 kHz. Few (if any) of my colleagues are specifying 192 kHz or above. If track counts are high, chances are it will be recorded at 44.1 kHz for CD release, sometimes 48 kHz. Nyquist/Shannon works. If you have listened to a CD in the past 30 years and heard all of the instruments, then you can be a witness. Are there some overtones or ultrasonic components missing? Maybe. But it must be said that most microphones and audio processors don't even pass audio signal much beyond 20 kHz. Even the best of the best might have a 50 kHz bandwidth at most. Your speakers sure don't reproduce anything that high. Which leaves only the quality of the ADC and DAC. High-quality analog components; stable, low-jitter clocks, temperature stability, and such all make far more a difference in the quality of reproduction than the sampling rate. I'll take a Genex or Mytek converter at 44.1 over a "soundcard" at 384 kHz any day, and if you heard the difference, you would too.
P: 132
 Quote by RobAnderson A lot of classical recordings, if they are using PCM, will use a sample rate of 96 kHz. Few (if any) of my colleagues are specifying 192 kHz or above. If track counts are high, chances are it will be recorded at 44.1 kHz for CD release, sometimes 48 kHz.
A few days ago I participated in a classical concert being recorded for the German radio network. I had a chat with the chief recording engineer, who told me that 48 kHz at a depth of 24 bit is still the standard for radio recordings here. He doesn't see any reason for this standard to change, since the microphones don't have any appreciable output above 24 kHz. He said that some colleagues use 96 kHz for classical recordings, but he would challenge anybody to hear the difference. Higher sampling rates, according to him, only make sense if you are using a lot of digital effects, which isn't the case in a classical recording where you are trying to reproduce the original as faithfully as possible.
P: 3
 Higher sampling rates, according to him, only make sense if you are using a lot of digital effects, which isn't the case in a classical recording where you are trying to reproduce the original as faithfully as possible.
Yep - 48 kHz is typically used for broadcast or visual media. Bit depth is an important part of this discussion, and probably makes more of a difference than sampling rate - especially in terms of reproducing the dynamic range found in classical music. 24-bit resolution vs 16-bit makes a much bigger difference than 48 vs. 44 kHz sampling rate.

Although his argument is counter to the way many folks on this side of the pond seem to think. For classical recordings, the lower track counts, the desire to maintain the utmost fidelity, and the reluctance to do much in the way of digital processing often leads folks to go with the 88 or 96 k sampling frequencies, while the pop and rock productions often go with the lower sampling rates because of the need for extreme amounts of processing on many tracks.
 P: 269 Well, you can always run ADC at 384KHz with 8x oversampling, followed by digital anti-aliasing filter and downsampling to 48KHz for transmission/storage. Same thing on the playback side - digitally upsample to 384KHz using proper sinc interpolation before sending it to the DAC. This way analog filters are a lot less critical, noise and jitter are much reduced, basically you get a couple of extra bits of resolution.
 P: 3 Here's a great article on the subject: http://people.xiph.org/~xiphmont/demo/neil-young.html Enjoy!

 Related Discussions Electrical Engineering 2 General Math 11 Classical Physics 2