Forums

Sampling, Again -- Updates

Started by Tim Wescott December 20, 2010
On 12/21/2010 04:59 PM, John Larkin wrote:
> On Wed, 22 Dec 2010 11:10:41 +1100, John Monro > <johnmonro@optusnet.com.au> wrote: > >> On 21/12/2010 8:15 AM, Tim Wescott wrote: >>> I'm starting a new thread (from "Sampling: What Nyquist >>> Didn't Say, and What to Do About It"), because the old one >>> rapidly filled up with all sorts of interesting stuff that I >>> didn't want to detract from. >>> >>> I've posted a new version. It uses Bitstream fonts for Roman >>> -- used because it was mentioned, and because it was there. >>> It's still a serif font which isn't optimum for monitor >>> viewing, but I want the thing to look good when it's printed >>> (and I'm lazy about figuring out how to tell Lyx/LaTeX how >>> to use sans!). >>> >>> It's 12-point, so you won't have to squint to see it, or >>> have as much trouble scanning across the line. It certainly >>> looks better in Evince, and I'm about to find out how it >>> looks in Adobe, on my wife's computer upstairs. >>> >>> And Randy, I've changed the discussion of subsampling to >>> make it more clear -- I hope that if it doesn't fully answer >>> your difficulties (I think you thought I was claiming to >>> sample at an effectively infinite rate) it does explain what >>> I'm thinking more fully. >>> >>> THANK YOU ALL who responded to the previous thread, and >>> please don't feel shy if you see something that I still >>> haven't caught! I need to add an acknowledgements section >>> for all the kind folks on USENET who critique my work. >>> >> >> Tim, >> Thanks for making available a clear and informative paper. I >> would like to make a comment at this late stage about the >> section on signal reconstruction, mainly about terminology. >> >> You explain that reconstruction is done by interpolating the >> output signal but unfortunately I feel that you have not >> clearly identified where the interpolaton occurs. >> >> You say that the first step in interpolation is the >> generation of a stepped waveform by the zero-hold D/A and >> you mention the stepped waveform of Figure 5 as being a >> picture of an interpolated signal. In my view this should be >> changed to say that generating the stepped waveform is the >> first step in the reconstruction process, interpolaton being >> the second step. > > The ideal reconstruction is an impulse generator (not the ZOH) > followed by an ideal lowpass filter. That avoids the sinc frequency > error thingie you get from the ZOH.
Mathematically ideal, perhaps, but hard to find in practice. Taking the sinc frequency characteristic* into account when one designs one's analog filters is absolutely necessary, of course. * Which is only an error if you don't want it, or can't easily work around it. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
On Tue, 21 Dec 2010 18:02:01 -0800, Tim Wescott <tim@seemywebsite.com>
wrote:

>On 12/21/2010 04:59 PM, John Larkin wrote: >> On Wed, 22 Dec 2010 11:10:41 +1100, John Monro >> <johnmonro@optusnet.com.au> wrote: >> >>> On 21/12/2010 8:15 AM, Tim Wescott wrote: >>>> I'm starting a new thread (from "Sampling: What Nyquist >>>> Didn't Say, and What to Do About It"), because the old one >>>> rapidly filled up with all sorts of interesting stuff that I >>>> didn't want to detract from. >>>> >>>> I've posted a new version. It uses Bitstream fonts for Roman >>>> -- used because it was mentioned, and because it was there. >>>> It's still a serif font which isn't optimum for monitor >>>> viewing, but I want the thing to look good when it's printed >>>> (and I'm lazy about figuring out how to tell Lyx/LaTeX how >>>> to use sans!). >>>> >>>> It's 12-point, so you won't have to squint to see it, or >>>> have as much trouble scanning across the line. It certainly >>>> looks better in Evince, and I'm about to find out how it >>>> looks in Adobe, on my wife's computer upstairs. >>>> >>>> And Randy, I've changed the discussion of subsampling to >>>> make it more clear -- I hope that if it doesn't fully answer >>>> your difficulties (I think you thought I was claiming to >>>> sample at an effectively infinite rate) it does explain what >>>> I'm thinking more fully. >>>> >>>> THANK YOU ALL who responded to the previous thread, and >>>> please don't feel shy if you see something that I still >>>> haven't caught! I need to add an acknowledgements section >>>> for all the kind folks on USENET who critique my work. >>>> >>> >>> Tim, >>> Thanks for making available a clear and informative paper. I >>> would like to make a comment at this late stage about the >>> section on signal reconstruction, mainly about terminology. >>> >>> You explain that reconstruction is done by interpolating the >>> output signal but unfortunately I feel that you have not >>> clearly identified where the interpolaton occurs. >>> >>> You say that the first step in interpolation is the >>> generation of a stepped waveform by the zero-hold D/A and >>> you mention the stepped waveform of Figure 5 as being a >>> picture of an interpolated signal. In my view this should be >>> changed to say that generating the stepped waveform is the >>> first step in the reconstruction process, interpolaton being >>> the second step. >> >> The ideal reconstruction is an impulse generator (not the ZOH) >> followed by an ideal lowpass filter. That avoids the sinc frequency >> error thingie you get from the ZOH. > >Mathematically ideal, perhaps, but hard to find in practice. Taking the >sinc frequency characteristic* into account when one designs one's >analog filters is absolutely necessary, of course. > >* Which is only an error if you don't want it, or can't easily work >around it. >
The ZOH sinc response can be corrected either digitally prior to the DAC or in the reconstruction filter, whichever provides the best implementation tradeoff. I don't know of any practical way to implement an ideal impulse generator. Eric Jacobsen Minister of Algorithms Abineau Communications http://www.abineau.com
On Wed, 22 Dec 2010 20:24:41 +1100, John Monro wrote:

> On 22/12/2010 1:00 PM, Tim Wescott wrote: >> On 12/21/2010 04:10 PM, John Monro wrote: >>> On 21/12/2010 8:15 AM, Tim Wescott wrote: >>>> I'm starting a new thread (from "Sampling: What Nyquist Didn't Say, >>>> and What to Do About It"), because the old one rapidly filled up with >>>> all sorts of interesting stuff that I didn't want to detract from. >>>> >>>> I've posted a new version. It uses Bitstream fonts for Roman -- used >>>> because it was mentioned, and because it was there. It's still a >>>> serif font which isn't optimum for monitor viewing, but I want the >>>> thing to look good when it's printed (and I'm lazy about figuring out >>>> how to tell Lyx/LaTeX how to use sans!). >>>> >>>> It's 12-point, so you won't have to squint to see it, or have as much >>>> trouble scanning across the line. It certainly looks better in >>>> Evince, and I'm about to find out how it looks in Adobe, on my wife's >>>> computer upstairs. >>>> >>>> And Randy, I've changed the discussion of subsampling to make it more >>>> clear -- I hope that if it doesn't fully answer your difficulties (I >>>> think you thought I was claiming to sample at an effectively infinite >>>> rate) it does explain what I'm thinking more fully. >>>> >>>> THANK YOU ALL who responded to the previous thread, and please don't >>>> feel shy if you see something that I still haven't caught! I need to >>>> add an acknowledgements section for all the kind folks on USENET who >>>> critique my work. >>>> >>>> >>> Tim, >>> Thanks for making available a clear and informative paper. I would >>> like >>> to make a comment at this late stage about the section on signal >>> reconstruction, mainly about terminology. >>> >>> You explain that reconstruction is done by interpolating the output >>> signal but unfortunately I feel that you have not clearly identified >>> where the interpolaton occurs. >>> >>> You say that the first step in interpolation is the generation of a >>> stepped waveform by the zero-hold D/A and you mention the stepped >>> waveform of Figure 5 as being a picture of an interpolated signal. In >>> my >>> view this should be changed to say that generating the stepped >>> waveform >>> is the first step in the reconstruction process, interpolaton being >>> the >>> second step. >>> >>> Interpolation, the generation of intermediate values between known >>> data >>> points, occurs when the stepped analog signal reaches the analog >>> output >>> reconstruction filter. It is the analog components that generate the >>> intermediate data values, on a continuous basis. You have mentioned >>> the >>> output filter, and I think it would be useful to identify this filter >>> as >>> the spot where interpolation occurs. >> >> I thought about that before I used the term, and I like my way. The >> reason I used it is because, in the context of that paper (and as I >> generally think about sampling), the sampled time signal _does not >> exist_ in between the sampling instants. That's a rather strict way to >> consider things, but I find it keeps me from miss-applying >> continuous-time wisdom to sampled time systems. Since the signal >> _doesn't_ exist, the action of a ZOH is, indeed, to interpolate. >> >> The usage is different from what some people would use, but I (a) don't >> think it's all that far off, (b) think it's more accurate, and (c) >> think that if someone can't handle Author A using different terminology >> from Author B, then they can't handle engineering. >> > Tim, > Using a single sample to decide the values existing at later instants > (up to 1/fs later in this case) is extrapolation. It is interpolation > only when you use two or more samples to decide the values existing > between samples. Regards, > John
Ahhh. Right. Dangit. -- http://www.wescottdesign.com
On 22/12/2010 1:00 PM, Tim Wescott wrote:
> On 12/21/2010 04:10 PM, John Monro wrote: >> On 21/12/2010 8:15 AM, Tim Wescott wrote: >>> I'm starting a new thread (from "Sampling: What Nyquist >>> Didn't Say, and What to Do About It"), because the old one >>> rapidly filled up with all sorts of interesting stuff that I >>> didn't want to detract from. >>> >>> I've posted a new version. It uses Bitstream fonts for Roman >>> -- used because it was mentioned, and because it was there. >>> It's still a serif font which isn't optimum for monitor >>> viewing, but I want the thing to look good when it's printed >>> (and I'm lazy about figuring out how to tell Lyx/LaTeX how >>> to use sans!). >>> >>> It's 12-point, so you won't have to squint to see it, or >>> have as much trouble scanning across the line. It certainly >>> looks better in Evince, and I'm about to find out how it >>> looks in Adobe, on my wife's computer upstairs. >>> >>> And Randy, I've changed the discussion of subsampling to >>> make it more clear -- I hope that if it doesn't fully answer >>> your difficulties (I think you thought I was claiming to >>> sample at an effectively infinite rate) it does explain what >>> I'm thinking more fully. >>> >>> THANK YOU ALL who responded to the previous thread, and >>> please don't feel shy if you see something that I still >>> haven't caught! I need to add an acknowledgements section >>> for all the kind folks on USENET who critique my work. >>> >> >> Tim, >> Thanks for making available a clear and informative paper. >> I would like >> to make a comment at this late stage about the section on >> signal >> reconstruction, mainly about terminology. >> >> You explain that reconstruction is done by interpolating >> the output >> signal but unfortunately I feel that you have not clearly >> identified >> where the interpolaton occurs. >> >> You say that the first step in interpolation is the >> generation of a >> stepped waveform by the zero-hold D/A and you mention the >> stepped >> waveform of Figure 5 as being a picture of an interpolated >> signal. In my >> view this should be changed to say that generating the >> stepped waveform >> is the first step in the reconstruction process, >> interpolaton being the >> second step. >> >> Interpolation, the generation of intermediate values >> between known data >> points, occurs when the stepped analog signal reaches the >> analog output >> reconstruction filter. It is the analog components that >> generate the >> intermediate data values, on a continuous basis. You have >> mentioned the >> output filter, and I think it would be useful to identify >> this filter as >> the spot where interpolation occurs. > > I thought about that before I used the term, and I like my > way. The reason I used it is because, in the context of that > paper (and as I generally think about sampling), the sampled > time signal _does not exist_ in between the sampling > instants. That's a rather strict way to consider things, but > I find it keeps me from miss-applying continuous-time wisdom > to sampled time systems. Since the signal _doesn't_ exist, > the action of a ZOH is, indeed, to interpolate. > > The usage is different from what some people would use, but > I (a) don't think it's all that far off, (b) think it's more > accurate, and (c) think that if someone can't handle Author > A using different terminology from Author B, then they can't > handle engineering. >
Tim, Using a single sample to decide the values existing at later instants (up to 1/fs later in this case) is extrapolation. It is interpolation only when you use two or more samples to decide the values existing between samples. Regards, John
On 22/12/2010 1:14 PM, Eric Jacobsen wrote:
> On Tue, 21 Dec 2010 18:02:01 -0800, Tim Wescott<tim@seemywebsite.com> > wrote: > >> On 12/21/2010 04:59 PM, John Larkin wrote: >>> On Wed, 22 Dec 2010 11:10:41 +1100, John Monro >>> <johnmonro@optusnet.com.au> wrote: >>> >>>> On 21/12/2010 8:15 AM, Tim Wescott wrote: >>>>> I'm starting a new thread (from "Sampling: What Nyquist >>>>> Didn't Say, and What to Do About It"), because the old one >>>>> rapidly filled up with all sorts of interesting stuff that I >>>>> didn't want to detract from. >>>>> >>>>> I've posted a new version. It uses Bitstream fonts for Roman >>>>> -- used because it was mentioned, and because it was there. >>>>> It's still a serif font which isn't optimum for monitor >>>>> viewing, but I want the thing to look good when it's printed >>>>> (and I'm lazy about figuring out how to tell Lyx/LaTeX how >>>>> to use sans!). >>>>> >>>>> It's 12-point, so you won't have to squint to see it, or >>>>> have as much trouble scanning across the line. It certainly >>>>> looks better in Evince, and I'm about to find out how it >>>>> looks in Adobe, on my wife's computer upstairs. >>>>> >>>>> And Randy, I've changed the discussion of subsampling to >>>>> make it more clear -- I hope that if it doesn't fully answer >>>>> your difficulties (I think you thought I was claiming to >>>>> sample at an effectively infinite rate) it does explain what >>>>> I'm thinking more fully. >>>>> >>>>> THANK YOU ALL who responded to the previous thread, and >>>>> please don't feel shy if you see something that I still >>>>> haven't caught! I need to add an acknowledgements section >>>>> for all the kind folks on USENET who critique my work. >>>>> >>>> >>>> Tim, >>>> Thanks for making available a clear and informative paper. I >>>> would like to make a comment at this late stage about the >>>> section on signal reconstruction, mainly about terminology. >>>> >>>> You explain that reconstruction is done by interpolating the >>>> output signal but unfortunately I feel that you have not >>>> clearly identified where the interpolaton occurs. >>>> >>>> You say that the first step in interpolation is the >>>> generation of a stepped waveform by the zero-hold D/A and >>>> you mention the stepped waveform of Figure 5 as being a >>>> picture of an interpolated signal. In my view this should be >>>> changed to say that generating the stepped waveform is the >>>> first step in the reconstruction process, interpolaton being >>>> the second step. >>> >>> The ideal reconstruction is an impulse generator (not the ZOH) >>> followed by an ideal lowpass filter. That avoids the sinc frequency >>> error thingie you get from the ZOH. >> >> Mathematically ideal, perhaps, but hard to find in practice. Taking the >> sinc frequency characteristic* into account when one designs one's >> analog filters is absolutely necessary, of course. >> >> * Which is only an error if you don't want it, or can't easily work >> around it. >> > > The ZOH sinc response can be corrected either digitally prior to the > DAC or in the reconstruction filter, whichever provides the best > implementation tradeoff. I don't know of any practical way to > implement an ideal impulse generator. > > > Eric Jacobsen > Minister of Algorithms > Abineau Communications > http://www.abineau.com
Eric, A crude way to minimise the sinc response is to drive the D/A output low for say half or three-quarters of each sample period. The catch is that the analog output drops by half or three quarters, which may not be convenient. To do the process properly you insert zeros betwen signal samples and then pass the signal through a LP filter before going to the D/A. This process is of course sample-rate up-conversion. Regards, John
Tim Wescott wrote:
> On 12/20/2010 02:34 PM, Randy Yates wrote: >> On 12/20/2010 04:15 PM, Tim Wescott wrote: >>> [...] >>> I've posted a new version. It uses Bitstream fonts for Roman -- used >>> because it was mentioned, and because it was there. It's still >>> a serif font which isn't optimum for monitor viewing, but I want the >>> thing to look good when it's printed (and I'm lazy about >>> figuring out how to tell Lyx/LaTeX how to use sans!). >>> >>> It's 12-point, so you won't have to squint to see it, or have as much >>> trouble scanning across the line. It certainly looks better in >>> Evince, and I'm about to find out how it looks in Adobe, on my wife's >>> computer upstairs. >>> >>> And Randy, I've changed the discussion of subsampling to make it more >>> clear -- I hope that if it doesn't fully answer your >>> difficulties (I think you thought I was claiming to sample at an >>> effectively infinite rate) it does explain what I'm thinking more >>> fully. >>> >>> THANK YOU ALL who responded to the previous thread, and please don't >>> feel shy if you see something that I still haven't caught! I >>> need to add an acknowledgements section for all the kind folks on >>> USENET who critique my work. >> >> Tim, >> >> Appearance-wise this looks EXCELLENT to my eyes! The fonts are now >> vector, at least all the ones I checked. >> >> FYI, a nice sans font I like to use is palatino. Simply place >> \usepackage{palatino} >> near the top, e.g.: >> >> \documentclass[english,11pt]{article} >> \usepackage[T1]{fontenc} >> \usepackage[utf8]{inputenc} >> \usepackage{babel} >> \usepackage{palatino} >> >> I'm reading the content now... > > I think that for the moment I'm going to register any comments, and > unless someone comes up with something really tremendously bad, I'm > going to move on to all the _other_ grotty papers I have that need to be > improved. They're all old OpenOffice documents that have been printed > out as HTML, with all sorts of problems. > > So -- I'd rather have a bunch of pretty good documents than one really > excellent one. > > I'm trying to decide if I want to do a paper next, or if I want to start > in on all the homework problems I did for the first 1/2 of "Applied > Control Theory for Embedded Systems". >
Tim, I have your book, and have recommended it to others. One thing I don't think you cover is Phelan's pseudo-derivative feedback (PDF) loops. have the nice feature of allowing essentially perfect rejection of load disturbances, at least if the system is really polynomial (as opposed to one containing time delays or thermal conduction, say). Phelan's approach is basically to take whatever the controlled variable is and turn it into a one-pole rolloff using a proportional or proportional-derivative loop, and then wrap an integrating loop around that. That way the inner loop controls the load disturbance rejection and the outer one keeps the setpoint stable. Phelan had an exceptionally well-developed ego, which isn't always helpful in getting one's ideas adopted widely, but it does seem as though his approach should be better known. Cheers Phil Hobbs -- Dr Philip C D Hobbs Principal ElectroOptical Innovations 55 Orchard Rd Briarcliff Manor NY 10510 845-480-2058 email: hobbs (atsign) electrooptical (period) net http://electrooptical.net
On Dec 22, 10:57&#2013266080;am, Phil Hobbs
<pcdhSpamMeSensel...@electrooptical.net> wrote:
> Tim Wescott wrote: > > On 12/20/2010 02:34 PM, Randy Yates wrote: > >> On 12/20/2010 04:15 PM, Tim Wescott wrote: > >>> [...] > >>> I've posted a new version. It uses Bitstream fonts for Roman -- used > >>> because it was mentioned, and because it was there. It's still > >>> a serif font which isn't optimum for monitor viewing, but I want the > >>> thing to look good when it's printed (and I'm lazy about > >>> figuring out how to tell Lyx/LaTeX how to use sans!). > > >>> It's 12-point, so you won't have to squint to see it, or have as much > >>> trouble scanning across the line. It certainly looks better in > >>> Evince, and I'm about to find out how it looks in Adobe, on my wife's > >>> computer upstairs. > > >>> And Randy, I've changed the discussion of subsampling to make it more > >>> clear -- I hope that if it doesn't fully answer your > >>> difficulties (I think you thought I was claiming to sample at an > >>> effectively infinite rate) it does explain what I'm thinking more > >>> fully. > > >>> THANK YOU ALL who responded to the previous thread, and please don't > >>> feel shy if you see something that I still haven't caught! I > >>> need to add an acknowledgements section for all the kind folks on > >>> USENET who critique my work. > > >> Tim, > > >> Appearance-wise this looks EXCELLENT to my eyes! The fonts are now > >> vector, at least all the ones I checked. > > >> FYI, a nice sans font I like to use is palatino. Simply place > >> \usepackage{palatino} > >> near the top, e.g.: > > >> \documentclass[english,11pt]{article} > >> \usepackage[T1]{fontenc} > >> \usepackage[utf8]{inputenc} > >> \usepackage{babel} > >> \usepackage{palatino} > > >> I'm reading the content now... > > > I think that for the moment I'm going to register any comments, and > > unless someone comes up with something really tremendously bad, I'm > > going to move on to all the _other_ grotty papers I have that need to be > > improved. They're all old OpenOffice documents that have been printed > > out as HTML, with all sorts of problems. > > > So -- I'd rather have a bunch of pretty good documents than one really > > excellent one. > > > I'm trying to decide if I want to do a paper next, or if I want to start > > in on all the homework problems I did for the first 1/2 of "Applied > > Control Theory for Embedded Systems". > > Tim, > > I have your book, and have recommended it to others. &#2013266080;One thing I don't > think you cover is Phelan's pseudo-derivative feedback (PDF) loops. > have the nice feature of allowing essentially perfect rejection of load > disturbances, at least if the system is really polynomial (as opposed to > one containing time delays or thermal conduction, say). > > Phelan's approach is basically to take whatever the controlled variable > is and turn it into a one-pole rolloff using a proportional or > proportional-derivative loop, and then wrap an integrating loop around > that. &#2013266080;That way the inner loop controls the load disturbance rejection > and the outer one keeps the setpoint stable. > > Phelan had an exceptionally well-developed ego, which isn't always > helpful in getting one's ideas adopted widely, but it does seem as > though his approach should be better known. > > Cheers > > Phil Hobbs > > -- > Dr Philip C D Hobbs > Principal > ElectroOptical Innovations > 55 Orchard Rd > Briarcliff Manor NY 10510 > 845-480-2058 > > email: hobbs (atsign) electrooptical (period) nethttp://electrooptical.net- Hide quoted text - > > - Show quoted text -
"> I have your book, and have recommended it to others." Excellent, I was looking for an excuse to buy Tim's book. George H.
On Mon, 20 Dec 2010 23:01:34 +0100, Jeroen <jeroen@nospam.please>
wrote:

>On 12/20/2010 10:30 PM, glen herrmannsfeldt wrote: >> In comp.dsp Tim Wescott <tim@seemywebsite.com> wrote: >>> I'm starting a new thread (from "Sampling: What Nyquist Didn't Say, and >>> What to Do About It"), because the old one rapidly filled up with all >>> sorts of interesting stuff that I didn't want to detract from. >> >> (snip) >> >> I do wonder how many people who discuss Nyquist have ever actually >> read his paper. I used to have a copy of it. It isn't hard to >> find in most university libraries, (at least ones old enough to >> have been around). > >Anyway, he only sort-of states his criterion, without proof. It was >Claude Shannon who actually gave a -remarkably short- demonstration >in his 1949 paper "Communication in the presence of noise". > >Neither were first, by any means. For example, V. Kotelnikov stated >basically the same thing, much more verbosely, in the context of >interpolation theory. And before that, in 1915, E.T Whittaker wrote >something similar, and before that there was this Japanese fellow... >You get the idea.
Hi Jeroen, From the paper "The Origins of the Sampling Theorem" by Hans Dieter L&#2013266172;ke: "Lastly, it should be mentioned that the sampling theorem is also treated in 1949 in the Japanese book Hakei Denso (Signal Transmission) by I. Someya. Hence, the term &#2013266067;Someya&#2013266066;s Theorem&#2013266068; may be found in some Japanese literature." See Ya', [-Rick-]
Tim Wescott wrote:
> On Wed, 22 Dec 2010 20:24:41 +1100, John Monro wrote: > >> On 22/12/2010 1:00 PM, Tim Wescott wrote: >>> On 12/21/2010 04:10 PM, John Monro wrote: >>>> On 21/12/2010 8:15 AM, Tim Wescott wrote: >>>>> I'm starting a new thread (from "Sampling: What Nyquist Didn't Say, >>>>> and What to Do About It"), because the old one rapidly filled up with >>>>> all sorts of interesting stuff that I didn't want to detract from. >>>>> >>>>> I've posted a new version. It uses Bitstream fonts for Roman -- used >>>>> because it was mentioned, and because it was there. It's still a >>>>> serif font which isn't optimum for monitor viewing, but I want the >>>>> thing to look good when it's printed (and I'm lazy about figuring out >>>>> how to tell Lyx/LaTeX how to use sans!). >>>>> >>>>> It's 12-point, so you won't have to squint to see it, or have as much >>>>> trouble scanning across the line. It certainly looks better in >>>>> Evince, and I'm about to find out how it looks in Adobe, on my wife's >>>>> computer upstairs. >>>>> >>>>> And Randy, I've changed the discussion of subsampling to make it more >>>>> clear -- I hope that if it doesn't fully answer your difficulties (I >>>>> think you thought I was claiming to sample at an effectively infinite >>>>> rate) it does explain what I'm thinking more fully. >>>>> >>>>> THANK YOU ALL who responded to the previous thread, and please don't >>>>> feel shy if you see something that I still haven't caught! I need to >>>>> add an acknowledgements section for all the kind folks on USENET who >>>>> critique my work. >>>>> >>>>> >>>> Tim, >>>> Thanks for making available a clear and informative paper. I would >>>> like >>>> to make a comment at this late stage about the section on signal >>>> reconstruction, mainly about terminology. >>>> >>>> You explain that reconstruction is done by interpolating the output >>>> signal but unfortunately I feel that you have not clearly identified >>>> where the interpolaton occurs. >>>> >>>> You say that the first step in interpolation is the generation of a >>>> stepped waveform by the zero-hold D/A and you mention the stepped >>>> waveform of Figure 5 as being a picture of an interpolated signal. In >>>> my >>>> view this should be changed to say that generating the stepped >>>> waveform >>>> is the first step in the reconstruction process, interpolaton being >>>> the >>>> second step. >>>> >>>> Interpolation, the generation of intermediate values between known >>>> data >>>> points, occurs when the stepped analog signal reaches the analog >>>> output >>>> reconstruction filter. It is the analog components that generate the >>>> intermediate data values, on a continuous basis. You have mentioned >>>> the >>>> output filter, and I think it would be useful to identify this filter >>>> as >>>> the spot where interpolation occurs. >>> >>> I thought about that before I used the term, and I like my way. The >>> reason I used it is because, in the context of that paper (and as I >>> generally think about sampling), the sampled time signal _does not >>> exist_ in between the sampling instants. That's a rather strict way to >>> consider things, but I find it keeps me from miss-applying >>> continuous-time wisdom to sampled time systems. Since the signal >>> _doesn't_ exist, the action of a ZOH is, indeed, to interpolate. >>> >>> The usage is different from what some people would use, but I (a) don't >>> think it's all that far off, (b) think it's more accurate, and (c) >>> think that if someone can't handle Author A using different terminology >>> from Author B, then they can't handle engineering. >>> >> Tim, >> Using a single sample to decide the values existing at later instants >> (up to 1/fs later in this case) is extrapolation. It is interpolation >> only when you use two or more samples to decide the values existing >> between samples. Regards, >> John > > Ahhh. Right. Dangit. > > >
Depends on where you put you time origin. If it's in the middle of the sample, a ZOH is an interpolator. Cheers Phil Hobbs -- Dr Philip C D Hobbs Principal ElectroOptical Innovations 55 Orchard Rd Briarcliff Manor NY 10510 845-480-2058 email: hobbs (atsign) electrooptical (period) net http://electrooptical.net
On 23/12/2010 8:07 AM, Phil Hobbs wrote:
> Tim Wescott wrote: >> On Wed, 22 Dec 2010 20:24:41 +1100, John Monro wrote: >> >>> On 22/12/2010 1:00 PM, Tim Wescott wrote: >>>> On 12/21/2010 04:10 PM, John Monro wrote: >>>>> On 21/12/2010 8:15 AM, Tim Wescott wrote: >>>>>> I'm starting a new thread (from "Sampling: What >>>>>> Nyquist Didn't Say, >>>>>> and What to Do About It"), because the old one rapidly >>>>>> filled up with >>>>>> all sorts of interesting stuff that I didn't want to >>>>>> detract from. >>>>>> >>>>>> I've posted a new version. It uses Bitstream fonts for >>>>>> Roman -- used >>>>>> because it was mentioned, and because it was there. >>>>>> It's still a >>>>>> serif font which isn't optimum for monitor viewing, >>>>>> but I want the >>>>>> thing to look good when it's printed (and I'm lazy >>>>>> about figuring out >>>>>> how to tell Lyx/LaTeX how to use sans!). >>>>>> >>>>>> It's 12-point, so you won't have to squint to see it, >>>>>> or have as much >>>>>> trouble scanning across the line. It certainly looks >>>>>> better in >>>>>> Evince, and I'm about to find out how it looks in >>>>>> Adobe, on my wife's >>>>>> computer upstairs. >>>>>> >>>>>> And Randy, I've changed the discussion of subsampling >>>>>> to make it more >>>>>> clear -- I hope that if it doesn't fully answer your >>>>>> difficulties (I >>>>>> think you thought I was claiming to sample at an >>>>>> effectively infinite >>>>>> rate) it does explain what I'm thinking more fully. >>>>>> >>>>>> THANK YOU ALL who responded to the previous thread, >>>>>> and please don't >>>>>> feel shy if you see something that I still haven't >>>>>> caught! I need to >>>>>> add an acknowledgements section for all the kind folks >>>>>> on USENET who >>>>>> critique my work. >>>>>> >>>>>> >>>>> Tim, >>>>> Thanks for making available a clear and informative >>>>> paper. I would >>>>> like >>>>> to make a comment at this late stage about the section >>>>> on signal >>>>> reconstruction, mainly about terminology. >>>>> >>>>> You explain that reconstruction is done by >>>>> interpolating the output >>>>> signal but unfortunately I feel that you have not >>>>> clearly identified >>>>> where the interpolaton occurs. >>>>> >>>>> You say that the first step in interpolation is the >>>>> generation of a >>>>> stepped waveform by the zero-hold D/A and you mention >>>>> the stepped >>>>> waveform of Figure 5 as being a picture of an >>>>> interpolated signal. In >>>>> my >>>>> view this should be changed to say that generating the >>>>> stepped >>>>> waveform >>>>> is the first step in the reconstruction process, >>>>> interpolaton being >>>>> the >>>>> second step. >>>>> >>>>> Interpolation, the generation of intermediate values >>>>> between known >>>>> data >>>>> points, occurs when the stepped analog signal reaches >>>>> the analog >>>>> output >>>>> reconstruction filter. It is the analog components that >>>>> generate the >>>>> intermediate data values, on a continuous basis. You >>>>> have mentioned >>>>> the >>>>> output filter, and I think it would be useful to >>>>> identify this filter >>>>> as >>>>> the spot where interpolation occurs. >>>> >>>> I thought about that before I used the term, and I like >>>> my way. The >>>> reason I used it is because, in the context of that >>>> paper (and as I >>>> generally think about sampling), the sampled time signal >>>> _does not >>>> exist_ in between the sampling instants. That's a rather >>>> strict way to >>>> consider things, but I find it keeps me from miss-applying >>>> continuous-time wisdom to sampled time systems. Since >>>> the signal >>>> _doesn't_ exist, the action of a ZOH is, indeed, to >>>> interpolate. >>>> >>>> The usage is different from what some people would use, >>>> but I (a) don't >>>> think it's all that far off, (b) think it's more >>>> accurate, and (c) >>>> think that if someone can't handle Author A using >>>> different terminology >>>> from Author B, then they can't handle engineering. >>>> >>> Tim, >>> Using a single sample to decide the values existing at >>> later instants >>> (up to 1/fs later in this case) is extrapolation. It is >>> interpolation >>> only when you use two or more samples to decide the >>> values existing >>> between samples. Regards, >>> John >> >> Ahhh. Right. Dangit. >> >> >> > Depends on where you put you time origin. If it's in the > middle of the sample, a ZOH is an interpolator. > > Cheers > > Phil Hobbs
Phil, Not he way I look at it, which is as follows: If a process uses known samples to determine signal values at times OUTSIDE the interval covered by the known samples then the process is extrapolation. The prefix 'extra,' meaning 'outside,' indicates this. If the process uses known samples to determine the signal values INSIDE the interval covered by the known values then the process is interpolation. Again, note the prefix. The Zero Hold circuit uses the last known sample to determine that the signal will have this sample value at any later time, potentially for all of eternity, but in practice up until the instant that the next sample arrives. As far as I can see, none of this depends on my choice of a t=0 point. If we choose t=0 to be half-way between samples as you suggest then extrapolation starts at t = -T/2 and continues up to t = +T/2. (T being the sampling interval.) For all of this time the Zero Hold circuit is producing values which are outside the interval covered by the known values, and so extrapolation is taking pleace. Regards, John