Nyquist Sampling

Started by RF January 12, 2008
I'd like to ask a couple very basic questions.  Being hands-on rather than 
mathematically-focused I'm trying to visualize sampling in a project I'm 
planning to start.

1. By sampling say a 3.1 kHz band-limited voice channel at the Nyquist rate 
am I guaranteed to capture *all* the information in the channel?  Is nothing 
lost (ignoring ADC resolution)?  More to the point: To know what the 
original analog did between sample points can I simply interpolate 
post-sampling and be able to see *everything* that I could see by looking at 
the original analog, limited only by ADC resolution?

2. Is the following a valid way to envision how Nyquist sampling captures 
all information in an analog waveform: a) we know that the waveform is 
completely represented by its Fourier expansion, b) we know the highest term 
in the expansion since the channel is band-limited, and c) every component 
is sinusoidal and therefore predictable if defined by two sample points per 
period.

Thanks for help with this simple question.


"RF" <nospam@nospam.com> writes:

> I'd like to ask a couple very basic questions. Being hands-on rather than > mathematically-focused I'm trying to visualize sampling in a project I'm > planning to start. > > 1. By sampling say a 3.1 kHz band-limited voice channel at the Nyquist rate > am I guaranteed to capture *all* the information in the channel? Is nothing > lost (ignoring ADC resolution)? More to the point: To know what the > original analog did between sample points can I simply interpolate > post-sampling and be able to see *everything* that I could see by looking at > the original analog, limited only by ADC resolution?
Yes to all. -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://www.digitalsignallabs.com
On Jan 12, 3:44&#2013266080;pm, "RF" <nos...@nospam.com> wrote:
> I'd like to ask a couple very basic questions. &#2013266080;Being hands-on rather
than
> mathematically-focused I'm trying to visualize sampling in a project I'm > planning to start. > > 1. By sampling say a 3.1 kHz band-limited voice channel at the Nyquist rate > am I guaranteed to capture *all* the information in the channel? &#2013266080;Is
nothing
> lost (ignoring ADC resolution)? &#2013266080;More to the point: To know what the > original analog did between sample points can I simply interpolate > post-sampling and be able to see *everything* that I could see by looking at > the original analog, limited only by ADC resolution? > > 2. Is the following a valid way to envision how Nyquist sampling captures > all information in an analog waveform: a) we know that the waveform is > completely represented by its Fourier expansion, b) we know the highest term > in the expansion since the channel is band-limited, and c) every component > is sinusoidal and therefore predictable if defined by two sample points per > period. > > Thanks for help with this simple question.
RF - Pretty close. 1] 3.1 Khz voice is not to be truly bandlimited, so a little is lost. Can be made not important. 2] Precise 2 point per period will not define sinus or cosinus. Takes just a little more. Regards, Kamar Ruptan Dsp Guru
On Sat, 12 Jan 2008 12:44:47 -0800, RF wrote:

> I'd like to ask a couple very basic questions. Being hands-on rather > than mathematically-focused I'm trying to visualize sampling in a > project I'm planning to start. > > 1. By sampling say a 3.1 kHz band-limited voice channel at the Nyquist > rate am I guaranteed to capture *all* the information in the channel? > Is nothing lost (ignoring ADC resolution)?
Your question is under-specified. If your voice channel is perfectly bandlimited, then yes you can capture all the information in the channel. Alas, if your voice channel is perfectly bandlimited then you'll have to wait an infinite amount of time before your first sample is valid. Since most of us aren't that patient, most voice channels aren't perfectly bandlimited.
> More to the point: To know > what the original analog did between sample points can I simply > interpolate post-sampling and be able to see *everything* that I could > see by looking at the original analog, limited only by ADC resolution?
Even to the extent that your information is good, you can't use simple interpolation. You can use reconstruction filters that use more than a few surrounding samples to get back the original analog signal, at least to the extent that it has been corrupted by the aliasing of the less than perfectly bandlimited signal.
> 2. Is the following a valid way to envision how Nyquist sampling > captures all information in an analog waveform: a) we know that the > waveform is completely represented by its Fourier expansion, b) we know > the highest term in the expansion since the channel is band-limited, and > c) every component is sinusoidal and therefore predictable if defined by > two sample points per period. > > Thanks for help with this simple question.
Yes, except for (c) every component is sinusoidal and therefore predictable if defined by _more than_ two sample points per period. _How much_ more than two sample points per period depends strongly on how close to a perfect bandlimiting filter your reconstruction filter is, which gets back to how patient you are. See my article at http://www.wescottdesign.com/articles/Sampling/ sampling.html for more detailed (and possibly less flippant) discussion on this topic. -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
Look at this reference, it explains how information is maintained during
analog to digital conversion. 
 
http://www.dspguide.com/ch3/2.htm