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Author Topic: worth a read: Sampling theory  (Read 9123 times)

Andy Peters

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Jon Hodgson

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Re: worth a read: Sampling theory
« Reply #1 on: September 29, 2006, 08:18:37 PM »

Quote:

The Nyquist theorem states that if you have a signal that is perfectly band limited to a bandwidth of f0 then you can collect all the information there is in that signal by sampling, as long as your sample rate is 2f0 or more


WRONG.

It states that you can collect all the information as long as your sample rate is MORE than 2f0

And where does he get the idea that you have to sample at 660Hz to capture a 300Hz signal? 601Hz would do (taking into consideration his assumption that the signal has no noise and therefore no components over 300 Hz which could alias).

The rest of it appears ok, though much of it is only useful for control and monitoring applications (such as the bit on sub nyquist sampling) and not audio.
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danlavry

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Re: worth a read: Sampling theory
« Reply #2 on: October 04, 2006, 08:03:29 PM »

Jon Hodgson wrote on Sat, 30 September 2006 01:18

Quote:

The Nyquist theorem states that if you have a signal that is perfectly band limited to a bandwidth of f0 then you can collect all the information there is in that signal by sampling, as long as your sample rate is 2f0 or more


WRONG.

It states that you can collect all the information as long as your sample rate is MORE than 2f0

And where does he get the idea that you have to sample at 660Hz to capture a 300Hz signal? 601Hz would do (taking into consideration his assumption that the signal has no noise and therefore no components over 300 Hz which could alias).

The rest of it appears ok, though much of it is only useful for control and monitoring applications (such as the bit on sub nyquist sampling) and not audio.


I agree. I scanned the paper fast and it is basically OK.

I too have an issue only with the section "what Nyquist did not say". Nyquist said that you need to sample FASTER THEN twice the highest frequency content, and that is a very important to understand. If anyone can not guarantee that condition, and there is energy at frequencies over 1/2 the sample rate, they are responsible for poor results. In practice, we simply can not have a brick wall filter. In practice, we may need to have some margins. In practice, the removal of high frequencies may cause unwanted side effects (such as phase issues), and that too will require some margins. Nyquist statements are regarding conversion of a signal AFTER it has been processed to assure that there is no energy (or low enough energy to be acceptable) above half the sampling.

In practice, I have little issue with leaving some serious margin, say even as much as 50-100% over what the theory states (so 96KHz sampling should be more then enough for the ear).  

Anyone that tries to sample up to nyquist, with zero margin, has a problem separating a theoretical limit from real world design. While there are no brick wall filters, the subject of real world filters is well developed.

Regards
Dan Lavry
http://www.lavryengineering.com
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mpd

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Re: worth a read: Sampling theory
« Reply #3 on: October 04, 2006, 09:01:26 PM »

danlavry wrote on Wed, 04 October 2006 20:03


I too have an issue only with the section "what Nyquist did not say". Nyquist said that you need to sample FASTER THEN twice the highest frequency content, and that is a very important to understand.


I hate to be pedantic, but Nyquist said the following:

"Let x_c(t) be a bandlimited signal with

X_c(jΩ) = 0, for all |Ω| > Ω_N

then x_c is uniquely determined by its samples x[n] = x_c(nT), n=0, +-1,+-2,..., if

Ω_s = 2pi/T > 2Ω_N"

The above is verbatim from Oppenheim and Schafer's book, which is the standard text on DSP theory.

X_c(jΩ) = 0 means that there is nothing outside the frequency band of interest.

Ω_s is the sampling frequency.

Ω_N is the bandwidth of the signal.

the n=0,+-1,+-2... is periodic sampling.

Assuming digital audio, Ω_s is the highest frequency component.  There is a lot of work and information on the net (mainly related to digital communications) that makes use of bandlimited sampling.  This is sometimes called subsampling or undersampling.

If the "X_c(jΩ) = 0" part is violated, then you will have aliasing.  All real systems will have some degree of aliasing, even if it is minute.

Practically, clock jitter means that the "n=0,+-1,+-2..." part is violated.  All real systems will have clock jitter, even if is is minute.

Nyquist also assumes infinite precision.  Real systems use quantization.

What sets converters apart is how they handle the aliasing, jitter, and quantization issues.


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Jon Hodgson

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Re: worth a read: Sampling theory
« Reply #4 on: October 05, 2006, 05:55:18 AM »

mpd wrote on Thu, 05 October 2006 02:01

danlavry wrote on Wed, 04 October 2006 20:03


I too have an issue only with the section "what Nyquist did not say". Nyquist said that you need to sample FASTER THEN twice the highest frequency content, and that is a very important to understand.


I hate to be pedantic, but Nyquist said the following:

"Let x_c(t) be a bandlimited signal with

X_c(jΩ) = 0, for all |Ω| > Ω_N

then x_c is uniquely determined by its samples x[n] = x_c(nT), n=0, +-1,+-2,..., if

Ω_s = 2pi/T > 2Ω_N"

The above is verbatim from Oppenheim and Schafer's book, which is the standard text on DSP theory.

X_c(jΩ) = 0 means that there is nothing outside the frequency band of interest.

Ω_s is the sampling frequency.

Ω_N is the bandwidth of the signal.

the n=0,+-1,+-2... is periodic sampling.

Assuming digital audio, Ω_s is the highest frequency component.  There is a lot of work and information on the net (mainly related to digital communications) that makes use of bandlimited sampling.  This is sometimes called subsampling or undersampling.

If the "X_c(jΩ) = 0" part is violated, then you will have aliasing.  All real systems will have some degree of aliasing, even if it is minute.

Practically, clock jitter means that the "n=0,+-1,+-2..." part is violated.  All real systems will have clock jitter, even if is is minute.

Nyquist also assumes infinite precision.  Real systems use quantization.

What sets converters apart is how they handle the aliasing, jitter, and quantization issues.





I hate to be pedantic, but you just confirmed exactly what Dan said about Nyquist's theorem.
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mpd

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Re: worth a read: Sampling theory
« Reply #5 on: October 05, 2006, 01:34:43 PM »

Jon Hodgson wrote on Thu, 05 October 2006 05:55


I hate to be pedantic, but you just confirmed exactly what Dan said about Nyquist's theorem.



Partially, but I don't think I was making myself clear.

Nyquist didn't just say that you need to sample at more than twice the bandwidth of the signal of interest.  There are conditions.

One of these conditions is that there can be nothing outside the  band of interest.  The other condition is that the sampling has to be precisely periodic.  Both of these are unavoidable in any real world system.

The issues of aliasing and clock jitter are direct results of the inability to fully meet the criteria that Nyquist defined.
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kraster

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Re: worth a read: Sampling theory
« Reply #6 on: October 06, 2006, 10:29:01 PM »

mpd



The issues of aliasing and clock jitter are direct results of the inability to fully meet the criteria that Nyquist defined.



Dan Lavry



Anyone that tries to sample up to nyquist, with zero margin, has a problem separating a theoretical limit from real world design




I have to agree with Jon here. Respectfully, I think both you and Dan are stating the same thing.

I would imagine thatt he inability of real world systems to meet the exact criteria of the Theorem are exactly the issues that are involved in converter design. The Theorem is a sound base(excuse the pun)on which to build the electronics. Remember that the Theorem will also show you what will happen when you violate its conditions.




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