Help understanding audio sampling

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.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

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



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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.
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