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Shannon limit question

Started by Richard Owlett February 5, 2005
I read that band limiting the input to an A/D eliminates aliasing. I 
also read that using a very high sampling rate makes that filter very 
simple.

Suppose I:
1. filter with a single stage low pass RC filter with 3db pint at .5 MHz
2. sample at 44.1 MHz
3. do simple sample rate conversion by saving every thousandth sample

Would I have a nice artifact free 44.1 kHz digitized signal?

The question came to mind when reading "Digital Dharma of Audio A/D 
Converters" ( http://www.rane.com/note137.html ). Particularly section 
quoted below.

"Shannon studied Nyquist's work closely and came up with a deceptively 
simple addition. He observed (and proved) that if you restrict the input 
signal's bandwidth to less than one-half the sampling frequency then no 
errors due to aliasing are possible. So bandlimiting your input to no 
more than one-half the sampling frequency guarantees no aliasing. Cool 
... only it's not possible.

In order to satisfy the Shannon limit (as it is called -- Harry gets a 
"criteria" and Claude gets a "limit") you must have the proverbial 
brick-wall, i.e., infinite-slope filter. Well, this isn't going to 
happen, not in this universe. You cannot guarantee that there is 
absolutely no signal (or noise) greater than the Nyquist frequency. 
Fortunately there is a way around this problem. In fact, you go all the 
way around the problem and look at it from another direction.

If you cannot restrict the input bandwidth so aliasing does not occur, 
then solve the problem another way: Increase the sampling frequency 
until the aliasing products that do occur, do so at ultrasonic 
frequencies, and are effectively dealt with by a simple single-pole 
filter. This is where the term "oversampling" comes in. For full 
spectrum audio the minimum sampling frequency must be 40 kHz, giving you 
a useable theoretical bandwidth of 20 kHz -- the limit of normal human 
hearing. Sampling at anything significantly higher than 40 kHz is termed 
oversampling. In just a few years time, we have seen the audio industry 
go from the CD system standard of 44.1 kHz, and the pro audio 
quasi-standard of 48 kHz, to 8-times and 16-times oversampling 
frequencies of around 350 kHz and 700 kHz respectively. With sampling 
frequencies this high, aliasing is no longer an issue."
"Richard Owlett" <rowlett@atlascomm.net> wrote in message 
news:1109luhj7342pe7@corp.supernews.com...
>I read that band limiting the input to an A/D eliminates aliasing. I also >read that using a very high sampling rate makes that filter very simple. > > Suppose I: > 1. filter with a single stage low pass RC filter with 3db pint at .5 MHz > 2. sample at 44.1 MHz > 3. do simple sample rate conversion by saving every thousandth sample > > Would I have a nice artifact free 44.1 kHz digitized signal? > > The question came to mind when reading "Digital Dharma of Audio A/D > Converters" ( http://www.rane.com/note137.html ). Particularly section > quoted below. > > "Shannon studied Nyquist's work closely and came up with a deceptively > simple addition. He observed (and proved) that if you restrict the input > signal's bandwidth to less than one-half the sampling frequency then no > errors due to aliasing are possible. So bandlimiting your input to no more > than one-half the sampling frequency guarantees no aliasing. Cool ... only > it's not possible. > > In order to satisfy the Shannon limit (as it is called -- Harry gets a > "criteria" and Claude gets a "limit") you must have the proverbial > brick-wall, i.e., infinite-slope filter. Well, this isn't going to happen, > not in this universe. You cannot guarantee that there is absolutely no > signal (or noise) greater than the Nyquist frequency. Fortunately there is > a way around this problem. In fact, you go all the way around the problem > and look at it from another direction. > > If you cannot restrict the input bandwidth so aliasing does not occur, > then solve the problem another way: Increase the sampling frequency until > the aliasing products that do occur, do so at ultrasonic frequencies, and > are effectively dealt with by a simple single-pole filter. This is where > the term "oversampling" comes in. For full spectrum audio the minimum > sampling frequency must be 40 kHz, giving you a useable theoretical > bandwidth of 20 kHz -- the limit of normal human hearing. Sampling at > anything significantly higher than 40 kHz is termed oversampling. In just > a few years time, we have seen the audio industry go from the CD system > standard of 44.1 kHz, and the pro audio quasi-standard of 48 kHz, to > 8-times and 16-times oversampling frequencies of around 350 kHz and 700 > kHz respectively. With sampling frequencies this high, aliasing is no > longer an issue."
Hi Richard : leaving aside all the imperfections in a real ADC; sampling at 44.1 MHz and then just using every thousandth sample is just the same as sampling at 44.1kHz so I think you may need to do a slightly more sophisticated decimation. Best of Luck - Mike
Richard,

Even if your RC filter was a brick wall at 0.5 MHz you would have to keep 
your sampling rate at a minimum of 1 MHz to prevent any aliasing.  That is, 
after your 44.1 MHz ADC the best you could do is take 1 of every 44 samples 
in order to have an artifact free signal (and that's with ideal brick wall 
RC and ideal A/D).

Typically the way people do this sort of thing is that they oversample with 
the A/D in order to relax the restriction on the RC filter.  Since you won't 
have a brickwall at 0.5 MHz you sample faster than 1 MHz to make sure you 
don't have any aliasing.  You then DIGITALLY filter since you can get pretty 
close to a brick wall filter with an IIR filter (or maybe a couple 
cascaded).  At that point you can then downsample to your desired sampling 
rate.

Brad

"Richard Owlett" <rowlett@atlascomm.net> wrote in message 
news:1109luhj7342pe7@corp.supernews.com...
>I read that band limiting the input to an A/D eliminates aliasing. I also >read that using a very high sampling rate makes that filter very simple. > > Suppose I: > 1. filter with a single stage low pass RC filter with 3db pint at .5 MHz > 2. sample at 44.1 MHz > 3. do simple sample rate conversion by saving every thousandth sample > > Would I have a nice artifact free 44.1 kHz digitized signal? > > The question came to mind when reading "Digital Dharma of Audio A/D > Converters" ( http://www.rane.com/note137.html ). Particularly section > quoted below. > > "Shannon studied Nyquist's work closely and came up with a deceptively > simple addition. He observed (and proved) that if you restrict the input > signal's bandwidth to less than one-half the sampling frequency then no > errors due to aliasing are possible. So bandlimiting your input to no more > than one-half the sampling frequency guarantees no aliasing. Cool ... only > it's not possible. > > In order to satisfy the Shannon limit (as it is called -- Harry gets a > "criteria" and Claude gets a "limit") you must have the proverbial > brick-wall, i.e., infinite-slope filter. Well, this isn't going to happen, > not in this universe. You cannot guarantee that there is absolutely no > signal (or noise) greater than the Nyquist frequency. Fortunately there is > a way around this problem. In fact, you go all the way around the problem > and look at it from another direction. > > If you cannot restrict the input bandwidth so aliasing does not occur, > then solve the problem another way: Increase the sampling frequency until > the aliasing products that do occur, do so at ultrasonic frequencies, and > are effectively dealt with by a simple single-pole filter. This is where > the term "oversampling" comes in. For full spectrum audio the minimum > sampling frequency must be 40 kHz, giving you a useable theoretical > bandwidth of 20 kHz -- the limit of normal human hearing. Sampling at > anything significantly higher than 40 kHz is termed oversampling. In just > a few years time, we have seen the audio industry go from the CD system > standard of 44.1 kHz, and the pro audio quasi-standard of 48 kHz, to > 8-times and 16-times oversampling frequencies of around 350 kHz and 700 > kHz respectively. With sampling frequencies this high, aliasing is no > longer an issue."