Help understanding audio sampling

Started by Ritual April 14, 2007
Please forgive me if this is the wrong place to ask such a question as
this, but I really don't know offhand where else to inquire. I am a
musician who often works with WAV files and am trying to understand
how sampling works as there have been several holes in my head as to
how it works for many years. I have never actually had sampling
explained to me, so what I am going to do is write out an
understanding which I have come up with in my head and will ask anyone
to correct me or further say anything they wish (including telling me
to go away or where else I could go for information). :

The two key values I find with regards to converting the analog signal
to it's digital representation are the sampling rate and bit depth. If
a sampling rate of 10kHz is used, if one zoomed in on the X axis the
tick marks would be 1/10,000th of a second apart. Take one sample of
that, and the Y axis would be divided into single bits, set by the
number in the bit depth (8, 16, 24 .. ). So for each 1/10,000th of a
second you have n bits representing the waveform. That makes me think
of a question... wouldn't a signal of greater amplitude yield a lesser
percentage of representation of the overall signal with the number of
bits versus a lower amplitude with the same number of bits distributed
over the same Y axis?

What happens to the information between the single bit samples on the
waveform? What happens to the information between the time on the X
axis? Someone told me that since the human ear can only hear
frequencies up to 20kHz that sampling rates above this are
unnecessary, but I don't buy it even though I can't put my finger on
it. I can't get it out of my head that digital representation of an
analog signal is going to throw away some data. A finite
representation of an infinite source. My audio soft/hardware offers
sampling setups up to 96kHz and 32 bits per sample. It just seems to
me that there is no limit to which one could push these numbers and
still not reach the original.

Thanks
 - Ritual
<Ritual> wrote in message news:e6t223d9b34qavo5pqr5d6silvkg1q9l75@4ax.com...
> > Please forgive me if this is the wrong place to ask such a question as > this, but I really don't know offhand where else to inquire.
Try rec.audio.pro (read the FAQ first).
Hi,

These are all very good questions, and this is the appropriate place to
ask them. 

For the bulk of them, I refer you to Chris Bores' excellent tutorials:

  http://www.bores.com/courses/intro/basics/index.htm

I will handle one question here:

Ritual <Ritual> writes:
> [...] > Someone told me that since the human ear can only hear > frequencies up to 20kHz that sampling rates above this are > unnecessary, but I don't buy it even though I can't put my finger on > it. I can't get it out of my head that digital representation of an > analog signal is going to throw away some data. A finite > representation of an infinite source. My audio soft/hardware offers > sampling setups up to 96kHz and 32 bits per sample. It just seems to > me that there is no limit to which one could push these numbers and > still not reach the original.
You need to clearly distinguish between sampling in time and amplitude quantization. Time sampling is established by the sample rate. Amplitude quantization is established by the number of bits. In theory you can do either one without the other. However, in all real digital systems that we can build, both must done. In the general (theory) sense, you can sample in time where each sample has infinite resolution. In other words, you can sample in time without performing amplitude quantization. It is an absolute fact, with no subjective discussion possible, that the theoretical sampling in time of a signal that is bandlimited to B Hz at a sample rate greater than 2*B samples/second without amplitude quantization (i.e., with infinite resolution in amplitude) preserves perfectly all information in the original signal, i.e., the discrete-time signal can be converted perfectly back into the original continuous-time analog signal. However, once we perform amplitude quantization, what you say is true: We are representing an infinite-resolution source by a finite resolution, and therefore we are necessarily throwing some information away. However, this "argument" against properly-designed digital systems is specious since any real-world analog system also throws information away. That is because any real-world analog system will always add some noise to the input signal, and the noise reduces the information available from the original signal. By a properly-designed digital system, I mean one in which dither is properly applied. It has been known for about 20-30 years that, when such digital systems are constructed, the effect of amplitude quantization is to add benign wideband noise. The only possible arguments against digital is that the technology isn't sufficiently advanced. For example, when CDs first came out around 1982, it was hard to find a true 16 bit D/A converter. There was also a lot of controversy on the subject of clock jitter and its impact on the digital signal. Several other areas of the technology, i.e., the *implementation* of digital, have been and are under scrutiny. However, in theory, a time- and amplitude-quantized digital signal is identical to the original except for wideband noise whose amplitude depends on the number of bits used in the quantizer. -- % Randy Yates % "Midnight, on the water... %% Fuquay-Varina, NC % I saw... the ocean's daughter." %%% 919-577-9882 % 'Can't Get It Out Of My Head' %%%% <yates@ieee.org> % *El Dorado*, Electric Light Orchestra http://home.earthlink.net/~yatescr
Ritual wrote:
> Please forgive me if this is the wrong place to ask such a question as > this, but I really don't know offhand where else to inquire. I am a > musician who often works with WAV files and am trying to understand > how sampling works as there have been several holes in my head as to > how it works for many years.
This is one of the good places to ask. Others have given you sound explanations and good on-line references. I want to add a bit. The bits that represent the samples are digits of a number that describes the sample's size. "Bit" stands for "Binary digIT"; the collection of bits in a sample together form a number. More bits is like more decimal digits in a number, allowing it to be closer to the actual value. The signal "between the bits" isn't there. It's like doing your taxes to the nearest dollar instead of to the nearest penny. "Close enough for government work." The signal between the samples _is_ there if the samples are frequent enough. You can't reproduce it by just "connecting the dots" made by the samples, though. To reproduce it, you have to pass the samples through a digital-to-analog converter, and then remove from the audio all the frequencies that would make the samples _not_ frequent enough. "Frequent enough" is easy to know. the frequency or sampling must be higher than twice the highest frequency in the signal being sampled. To keep the system simple, 2.5 times is better. Some terms might be helpful. The sampling frequency is often abbreviated Fs or F_s (Eff sub ess). The highest frequency allowed in a signal samples at Fs is Fs/2, the "Nyquist rate". Good luck. Write again to clarify what you read. Jerry -- Engineering is the art of making what you want from things you can get. &macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;
Ritual wrote:
> Please forgive me if this is the wrong place to ask such a question as > this, but I really don't know offhand where else to inquire. I am a > musician who often works with WAV files and am trying to understand > how sampling works as there have been several holes in my head as to > how it works for many years. I have never actually had sampling > explained to me, so what I am going to do is write out an > understanding which I have come up with in my head and will ask anyone > to correct me or further say anything they wish (including telling me > to go away or where else I could go for information). : > > The two key values I find with regards to converting the analog signal > to it's digital representation are the sampling rate and bit depth. If > a sampling rate of 10kHz is used, if one zoomed in on the X axis the > tick marks would be 1/10,000th of a second apart. Take one sample of > that, and the Y axis would be divided into single bits, set by the > number in the bit depth (8, 16, 24 .. ). So for each 1/10,000th of a > second you have n bits representing the waveform.
That is correct.
> That makes me think > of a question... wouldn't a signal of greater amplitude yield a lesser > percentage of representation of the overall signal with the number of > bits versus a lower amplitude with the same number of bits distributed > over the same Y axis?
If I understand your statement right, you are seeing things in reverse. When you design equipment to do the analog to digital conversion you have to choose an upper limit to the signal amplitude that can be converted; anything greater than that upper limit will get clipped. The bit depth determines how finely you can determine what was actually recorded -- so for a 16-bit ADC you _must_ separate the input into 65536 steps. Small signals are proportionally 'rougher' than large signals for a given piece of equipment, but large signals are more likely to clip.
> > What happens to the information between the single bit samples on the > waveform?
Information about voltages between the possible ADC output values is lost.
> What happens to the information between the time on the X > axis?
That is lost also.
> Someone told me that since the human ear can only hear > frequencies up to 20kHz that sampling rates above this are > unnecessary, but I don't buy it even though I can't put my finger on > it.
Do some web searches on the Nyquist Sampling Theorem. I have an article on this at http://www.wescottdesign.com/articles/Sampling/sampling.html; I recommend it with reservations because it's intended more for people who _think_ they understand sampling but misuse the Nyquist/Shannon theorem rather than people who are entirely new to the subject. It may help you, however.
> I can't get it out of my head that digital representation of an > analog signal is going to throw away some data.
That is correct.
> A finite > representation of an infinite source. My audio soft/hardware offers > sampling setups up to 96kHz and 32 bits per sample. It just seems to > me that there is no limit to which one could push these numbers and > still not reach the original.
That is also correct. However, you don't have to match the original exactly, you only need to match it well enough. With audio what is "well enough" is a subject for infinite debate, but I can guarantee you that there isn't an analog recording media, or even just a transmitting media, out there that will do a perfect job of reproducing an audio signal either. Nor are there perfect microphones or perfect speakers. As long as the digital part of your system is outperforming the microphones you're doing plenty good. Stepping down from that, if you're outperforming the combination of microphone, speaker, amplifiers and ears involved, then your digital part is doing just fine. Deciding just what you want to set as a quality goal is part of the endless debate on audio quality that I mentioned above (and if you have someone involved in your debate who dips their speaker cables in liquid nitrogen -- just shoot them, it'll help you arrive at consensus much faster). -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Posting from Google? See http://cfaj.freeshell.org/google/ Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
Ritual wrote:
> Please forgive me if this is the wrong place to ask such a question as > this, but I really don't know offhand where else to inquire. I am a > musician who often works with WAV files and am trying to understand > how sampling works as there have been several holes in my head as to > how it works for many years. I have never actually had sampling > explained to me, so what I am going to do is write out an > understanding which I have come up with in my head and will ask anyone > to correct me or further say anything they wish (including telling me > to go away or where else I could go for information). : > > The two key values I find with regards to converting the analog signal > to it's digital representation are the sampling rate and bit depth. If > a sampling rate of 10kHz is used, if one zoomed in on the X axis the > tick marks would be 1/10,000th of a second apart. Take one sample of > that, and the Y axis would be divided into single bits, set by the > number in the bit depth (8, 16, 24 .. ). So for each 1/10,000th of a > second you have n bits representing the waveform.
That is correct.
> That makes me think > of a question... wouldn't a signal of greater amplitude yield a lesser > percentage of representation of the overall signal with the number of > bits versus a lower amplitude with the same number of bits distributed > over the same Y axis?
If I understand your statement right, you are seeing things in reverse. When you design equipment to do the analog to digital conversion you have to choose an upper limit to the signal amplitude that can be converted; anything greater than that upper limit will get clipped. The bit depth determines how finely you can determine what was actually recorded -- so for a 16-bit ADC you _must_ separate the input into 65536 steps. Small signals are proportionally 'rougher' than large signals for a given piece of equipment, but large signals are more likely to clip.
> > What happens to the information between the single bit samples on the > waveform?
Information about voltages between the possible ADC output values is lost.
> What happens to the information between the time on the X > axis?
That is lost also.
> Someone told me that since the human ear can only hear > frequencies up to 20kHz that sampling rates above this are > unnecessary, but I don't buy it even though I can't put my finger on > it.
Do some web searches on the Nyquist Sampling Theorem. I have an article on this at http://www.wescottdesign.com/articles/Sampling/sampling.html; I recommend it with reservations because it's intended more for people who _think_ they understand sampling but misuse the Nyquist/Shannon theorem rather than people who are entirely new to the subject. It may help you, however.
> I can't get it out of my head that digital representation of an > analog signal is going to throw away some data.
That is correct.
> A finite > representation of an infinite source. My audio soft/hardware offers > sampling setups up to 96kHz and 32 bits per sample. It just seems to > me that there is no limit to which one could push these numbers and > still not reach the original.
That is also correct. However, you don't have to match the original exactly, you only need to match it well enough. With audio what is "well enough" is a subject for infinite debate, but I can guarantee you that there isn't an analog recording media, or even just a transmitting media, out there that will do a perfect job of reproducing an audio signal either. Nor are there perfect microphones or perfect speakers. As long as the digital part of your system is outperforming the microphones you're doing plenty good. Stepping down from that, if you're outperforming the combination of microphone, speaker, amplifiers and ears involved, then your digital part is doing just fine. Deciding just what you want to set as a quality goal is part of the endless debate on audio quality that I mentioned above (and if you have someone involved in your debate who dips their speaker cables in liquid nitrogen -- just shoot them, it'll help you arrive at consensus much faster). -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Posting from Google? See http://cfaj.freeshell.org/google/ Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
> Ritual <Ritual> writes:
> > My audio soft/hardware offers > > sampling setups up to 96kHz and 32 bits per sample.
Those are called "marketing numbers". The higher they can make them, the more you will pay and the more hype you will end up with (and actually might believe, given the right snake-oil salesman). If you care to pass on this observation, go back to records and tapes.
> > It just seems to > > me that there is no limit to which one could push these numbers and > > still not reach the original.
The original is relatively accurately represented by 16 bits at 44.1Khz. Upping the sample rate to 48 khz increases the top end representation slightly - in some cases, enough to actually comprehend. The average frequencies inclusive to human hearing rest well within the limitations of the 44.1khz sampling rate. -- David Morgan (MAMS) http://www.m-a-m-s DOT com Morgan Audio Media Service Dallas, Texas (214) 662-9901 _______________________________________ http://www.artisan-recordingstudio.com

David Morgan (MAMS) wrote:

>>>My audio soft/hardware offers >>>sampling setups up to 96kHz and 32 bits per sample. > > > Those are called "marketing numbers". The higher they can make > them, the more you will pay and the more hype you will end up with > (and actually might believe, given the right snake-oil salesman).
Well said.
> The original is relatively accurately represented by 16 bits at 44.1Khz. > Upping the sample rate to 48 khz increases the top end representation > slightly - in some cases, enough to actually comprehend.
18kHz vs 20kHz.
> The average > frequencies inclusive to human hearing rest well within the limitations > of the 44.1khz sampling rate.
The 16 bit accuracy is enough if the signal is normalized properly. This is generally not the case when doing the recording. The dynamic range of 120dB or so is required, which implies the 24 bit data. Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
<Ritual> wrote in message news:e6t223d9b34qavo5pqr5d6silvkg1q9l75@4ax.com...
> > Please forgive me if this is the wrong place to ask such a question as > this, but I really don't know offhand where else to inquire. I am a > musician who often works with WAV files and am trying to understand > how sampling works as there have been several holes in my head as to > how it works for many years. I have never actually had sampling > explained to me, so what I am going to do is write out an > understanding which I have come up with in my head and will ask anyone > to correct me or further say anything they wish (including telling me > to go away or where else I could go for information). : > > The two key values I find with regards to converting the analog signal > to it's digital representation are the sampling rate and bit depth. If > a sampling rate of 10kHz is used, if one zoomed in on the X axis the > tick marks would be 1/10,000th of a second apart. Take one sample of > that, and the Y axis would be divided into single bits, set by the > number in the bit depth (8, 16, 24 .. ). So for each 1/10,000th of a > second you have n bits representing the waveform. That makes me think > of a question... wouldn't a signal of greater amplitude yield a lesser > percentage of representation of the overall signal with the number of > bits versus a lower amplitude with the same number of bits distributed > over the same Y axis? > > What happens to the information between the single bit samples on the > waveform? What happens to the information between the time on the X > axis? Someone told me that since the human ear can only hear > frequencies up to 20kHz that sampling rates above this are > unnecessary, but I don't buy it even though I can't put my finger on > it. I can't get it out of my head that digital representation of an > analog signal is going to throw away some data. A finite > representation of an infinite source. My audio soft/hardware offers > sampling setups up to 96kHz and 32 bits per sample. It just seems to > me that there is no limit to which one could push these numbers and > still not reach the original. > > Thanks > - Ritual
Regarding the bit depth and its effect on the result: First, I'm going to assume that you have set up your system so that the maximum levels go through the system as you desire - just like a level needle / meter on an analog recording device. - Some really high peaks will be scrunched (limited) by the equipment. This causes distortion. So, we generally set the equipment so that the degree of limiting is, well, "limited" to what's acceptable. If *no* distortion of this type is acceptable then we probably set the gain so that all the peaks we see are at about 50% or so of the top of the range. This just means that statistically the occurrence of a distorted peak is much lower. Otherwise, we set the gain so that the more or less common peaks are near the top - the needle flips up into the red range now and then. I'm sure audio folks better than I have their favorite rules of thumb in this regard. **In a digital representation, almost the same thing happens - except the limiting is abrupt and absolute - whereas in an analog system the limiting may be a bit softer and may result in a bit less harmonic energy being generated when the limit is exceeded. The number of values that can be represented by 16, 24 and 32 bits are: 16 - 65,536 24 - 16,777,216 32 - 4,294,967,296 and these values are spread over both positive and negative values of the waveform so that absolute levels are represented by half these numbers. Now, going back to the limiting situation above, a well-adjusted system will have actual peaks that approach (and sometimes exceed) the values that can be represented. Here's an example: A 16-bit system has 32,768 absolute values that can be represented. We will reference the peak to 1 volt rms or 1.414 volts peak. 1.414 volts peak divided by 32,768 is 43.2 microvolts. This is the quantization level or distance in amplitude between sample values. Now, let's assume that there's a value in the continuous waveform of 0.690534... (which is 16,000 times 43.2 microvolts). When sampled it will be represented by 16,000 levels exactly. No information is lost. However, that perfect situation doesn't happen very often. In fact the statistically expected value / location for the input level is half way between the digital levels. This means the average "error" will be 43.2/2 or 21.6 microvolts in our example system. How do we deal with this "error" so we can analyze it? Generally we say that there is the "perfect" part of the sample (16,000 from our example above) and then what I called the "error". Because the "error" is random from sample to sample, it's readily described and handled as random noise. It's as if the samples were perfect but a certain level of random noise was added to the result. Notice how this is very much equivalent to passing a signal through an amplifier. I'm sure you've done this: Short out the input of an amplifier and then turn up the gain until you hear the random noise output (or, if it's not a very "quiet" system, you'll hear hum). Without hum, you'll be hearing the "noise floor" of the system - the random noise that's being introduced by the electronics. Tape hiss is another example - although a bit different in spectral content. Since your system is made up of analog and digital components, if the noise introduced by sampling is somewhat less than the noise introduced otherwise then the sampling noise will be unnoticeable. The bottom line is that you can choose the bit depth in a particular system such that quantization noise is unnoticeable. Conversely, you can choose the bit depth so that the quantization noise is clearly noticeable! From: http://en.wikipedia.org/wiki/Signal-to-noise_ratio The formula is then: 16-bit audio has a dynamic range, thus SNR of 96 dB. Each extra quantization bit increases the dynamic range by roughly 6 dB. If your system has an SNR of X, then you might choose the bit depth to yield by itself (X-3) dB and since you're stuck with adjusting by whole bits then you'll be adjusting in 6dB increments.e In reality you probably can only choose between 8, 16, 24, 32, right? So 48dB SNR difference between each if you choose to trust this approximation. I hope this helps. Fred begin 666 37ae33390c8f37488888d261ee252f79.png MB5!.1PT*&@H````-24A$4@```8@````7" `````T;X?8````"7!(67,```!X M````> "=]5I@````"79P06<```&(````%P!$Q:2>```%-4E$051HWNU96Y4D M(0R-!2R4!2Q@H2Q@`0M8J._]*PM8P (6L, "":^N]VQ/G]ESAH_IQU"Y26Y> MT!!^UX]8\.N"_XF(93WYGW^G/I]#NJ& Y]P*4!_!3D08H#6;T'T42]FTZO)N M%N)UNY>7.MHY;K9:YT>SK%4`5UN@AI2 OH"T#Y]DW5S-U.Q](;V'?\%^F!$. 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Tim Wescott wrote:
> Ritual wrote:
...
>> What happens to the information between the single bit samples on the >> waveform? > > Information about voltages between the possible ADC output values is lost.
That is the case with any measurement. Counting can be exact, but any measurement is of limited accuracy. \
>> What happens to the information between the time on the X >> axis? > > That is lost also.
I have to disagree. Only information above half the sample rate is lost. (There better not be any anformation above half the sample rate. It would corrupt the entire signal with aliases.) Information about the entire continuous and properly sampled signal is retained in the samples. That's fundamental to digital processing.
>> Someone told me that since the human ear can only hear >> frequencies up to 20kHz that sampling rates above this are >> unnecessary, but I don't buy it even though I can't put my finger on >> it.
Sampling above 40 KHz is needed to reproduce 20 KHz. Some of the system needs to be analog (anti-alias prefilter and reconstruction postfilter). The higher the sample rate, the easier these are to build. ...
>> I can't get it out of my head that digital representation of an >> analog signal is going to throw away some data.
Sure, but so does any other system of representation and recording. It's not possible to copy something perfectly by any means. There are always losses, distortions, and artifacts.
>> A finite >> representation of an infinite source. My audio soft/hardware offers >> sampling setups up to 96kHz and 32 bits per sample. It just seems to >> me that there is no limit to which one could push these numbers and >> still not reach the original.
You can reach the limits of audible frequencies at sampling rates far below 96 KHz. You can exceed the accuracy of tape or vinyl with numbers smaller that 32 bits. There are four important points of comparison for audio: frequency response, dynamic range, signal-to-noise ratio, and distortion. Digital systems can match or outdo analog in all of them. Jerry -- Engineering is the art of making what you want from things you can get. &macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;&macr;