On 2/7/12 12:55 PM, Tim Wescott wrote:> On Tue, 07 Feb 2012 17:27:21 +0000, Eric Jacobsen wrote: >...>> >> Tim and I aren't the only people who casually refer to images as >> aliases....>> >> But, yes, I think "image" is a better term for the artifacts in >> reconstruction, but "alias" is often used. > > I will try to be more precise in my terminology in the future -- but I'll > still think privately that "alias" is a perfectly valid term for the > phenomenon under discussion.if there is no undersampling, that is if there are no components of any image that falls under Fs/2, then there are no aliases. is that the phenomenon under discussion? it may be nit-picking, but i consider this terminology to be sorta like the definition of Nyquist (it's Fs/2 *not* necessarily the bandwidth of the signal about to be sampled). aliases are components of images that fold over around Nyquist. aliases are impostors. they masquerade as a component that *could* have been at that lower frequency from the beginning, which is why we hear them as if they *were* at that folded-over frequency. again, the LPF that comes after the D/A is *not* the "anti-aliasing filter", it is the "anti-imaging filter". the LPF that comes before the A/D is the anti-aliasing filter. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Spectrum compensation for zero-order hold DAC
Started by ●February 3, 2012
Reply by ●February 8, 20122012-02-08
Reply by ●February 8, 20122012-02-08
On 2012/02/07 09:53, Tim Wescott wrote:> On Tue, 07 Feb 2012 12:11:19 -0500, Jerry Avins wrote: > >> On 2/6/2012 8:36 PM, Tim Wescott wrote: >> >> ... >> >>> What the OP was overlooking is that _in the limit_ as f approaches Fs/2 >>> the signal in question has an amplitude _at f_ of 2/pi (I think that's >>> what it was, at least), as predicted. But the closer that f gets to >>> Fs/2 the longer you have to look to see the average, etc. >> >> That's true provided only that "the signal" is properly defined. Stay >> with the OP's simplification and sample a cosine. Stay with yours and >> keep an integer number of samples per cycle. (At f = fs/2, that means >> two samples per cycle.) The amplitude of the samples will be the same as >> the peak amplitude of a very low frequency. Properly defined, "the >> signal" doesn't mean the raw DAC output, but the output after the images >> are removed. The amplitude of _that_ is 2/pi. It is, after all, the >> magnitude of first term of the Fourier series of a square wave. > > Well, I guess that's what I'm trying to bring the OP to a realization > of. He appears to be looking at the DAC output on an oscilloscope and > taking the "signal amplitude" as the p-p amplitude of the DAC output, I'm > trying to show why that doesn't fit either with what he's asking or what > we're saying.Thanks, Tim. You succeeded. When you pointed out the aliases in my sampled "pure" sinusoid, I realized why the time domain amplitude of the composite signal was not equivalent to the amplitude of the desired pure sine. Since my application later does frequency analysis on these generated signals, the amplitude of the desired frequency is much more important than the amplitude of the distorted signal, although I do have to make sure that the op-amps don't hit the rails and further distort the signal. Even though the budget meant that the multiple reconstruction filters are missing, it's still important to be able to predict the resulting magnitude in the frequency domain. Thanks to everyone for the informative discussion. Brian Willoughby Sound Consulting
Reply by ●February 8, 20122012-02-08
On 2012/02/07 09:16, Jerry Avins wrote:> On 2/3/2012 6:05 PM, HardySpicer wrote: >> On Feb 4, 10:31 am, Jerry Avins<j...@ieee.org> wrote: >>> On 2/3/2012 7:44 AM, Brian Willoughby wrote: >>> >>>> I just came across a concept that doesn't completely make sense to me, >>>> and I am hoping that the group can help me to understand more. >>> >>> [confusion deleted (if only!)] >>> >>> The root of your confusion is a bad assumption early on. The >>> sample-and-hold does affect the response below Nyquist. A sample and >>> hold has no effect at DC, and puts out only DC when the input signal is >>> at the sample rate. In between it gently rolls off along a sin(x)/x >>> curve. (x = pi at the sample rate.) At the Nyquist rate, the output (of >>> the fundamental) sin(pi/2)/(pi/2), or about .637. that's roughly -4 dB. >>> >>> Jerry >>> -- >>> Engineering is the art of making what you want from things you can get. >>> ����������������������������������������������������������������������� >> >> You mean zero-order hold - not sample and hold... > > Bryan imagined that his ZOH is made by following an impulsive DAC with > an analog S&H. (Please pardon the alphabet soup.) I tried to stay with > his paradigm. I shouldn't have. > > JerryIt's much more than my imagination! There actually are multiple analog S&H circuits in my system. One reason for them is that they hide the discontinuity that occurs when this particular DAC changes from one code to the next. I was going to argue that zero-order hold is practically the same thing as sample and hold, apart from the usual nonidealities of analog circuits. When comparing digital SRC to D/A conversion, there might be some subtle distinctions, but they are largely the same. Brian Willoughby Sound Consulting
Reply by ●February 9, 20122012-02-09
On 2/8/2012 9:42 PM, Brian Willoughby wrote:> On 2012/02/07 09:16, Jerry Avins wrote: >> On 2/3/2012 6:05 PM, HardySpicer wrote: >>> On Feb 4, 10:31 am, Jerry Avins<j...@ieee.org> wrote: >>>> On 2/3/2012 7:44 AM, Brian Willoughby wrote: >>>> >>>>> I just came across a concept that doesn't completely make sense to me, >>>>> and I am hoping that the group can help me to understand more. >>>> >>>> [confusion deleted (if only!)] >>>> >>>> The root of your confusion is a bad assumption early on. The >>>> sample-and-hold does affect the response below Nyquist. A sample and >>>> hold has no effect at DC, and puts out only DC when the input signal is >>>> at the sample rate. In between it gently rolls off along a sin(x)/x >>>> curve. (x = pi at the sample rate.) At the Nyquist rate, the output (of >>>> the fundamental) sin(pi/2)/(pi/2), or about .637. that's roughly -4 dB. >>>> >>>> Jerry >>>> -- >>>> Engineering is the art of making what you want from things you can get. >>>> ����������������������������������������������������������������������� >>> >>> You mean zero-order hold - not sample and hold... >> >> Bryan imagined that his ZOH is made by following an impulsive DAC with >> an analog S&H. (Please pardon the alphabet soup.) I tried to stay with >> his paradigm. I shouldn't have. >> >> Jerry > > It's much more than my imagination! There actually are multiple analog > S&H circuits in my system. One reason for them is that they hide the > discontinuity that occurs when this particular DAC changes from one code > to the next. > > I was going to argue that zero-order hold is practically the same thing > as sample and hold, apart from the usual nonidealities of analog > circuits. When comparing digital SRC to D/A conversion, there might be > some subtle distinctions, but they are largely the same.Are those DACs home brew? the glitches produced by the guts of most commercial DACs are largely removed by the inherent filtering of analog output drivers. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 9, 20122012-02-09
On 2/8/12 11:21 PM, Jerry Avins wrote:> On 2/8/2012 9:42 PM, Brian Willoughby wrote: >> On 2012/02/07 09:16, Jerry Avins wrote: >>> On 2/3/2012 6:05 PM, HardySpicer wrote: >>>> On Feb 4, 10:31 am, Jerry Avins<j...@ieee.org> wrote: >>>>> On 2/3/2012 7:44 AM, Brian Willoughby wrote: >>>>> >>>>>> I just came across a concept that doesn't completely make sense to >>>>>> me, >>>>>> and I am hoping that the group can help me to understand more. >>>>> >>>>> [confusion deleted (if only!)] >>>>> >>>>> The root of your confusion is a bad assumption early on. The >>>>> sample-and-hold does affect the response below Nyquist. A sample and >>>>> hold has no effect at DC, and puts out only DC when the input >>>>> signal is >>>>> at the sample rate. In between it gently rolls off along a sin(x)/x >>>>> curve. (x = pi at the sample rate.) At the Nyquist rate, the output >>>>> (of the fundamental) sin(pi/2)/(pi/2), or about .637. that's roughly >>>>> -4 dB. >>>>> >>>> >>>> >>>> You mean zero-order hold - not sample and hold... >>> >>> Bryan imagined that his ZOH is made by following an impulsive DAC with >>> an analog S&H. (Please pardon the alphabet soup.) I tried to stay with >>> his paradigm. I shouldn't have. >>> >> >> It's much more than my imagination! There actually are multiple analog >> S&H circuits in my system. One reason for them is that they hide the >> discontinuity that occurs when this particular DAC changes from one code >> to the next.we used to call these S/H circuits after the DAC a "deglitcher". they may be needed for a conventional DAC (like an R-2R ladder), but are not necessary for the sigma-delta DACs, which are a horse of a different color. one problem is that when the S/H changes from "hold" back to "sample", the output will rapidly change to the new value and your op-amp circuits will go into slew rate limiting, which is a non-linear operation.>> I was going to argue that zero-order hold is practically the same thing >> as sample and hold, apart from the usual nonidealities of analog >> circuits. When comparing digital SRC to D/A conversion, there might be >> some subtle distinctions, but they are largely the same. > > Are those DACs home brew? the glitches produced by the guts of most > commercial DACs are largely removed by the inherent filtering of analog > output drivers. >well, those glitches aren't so much removed, but they're filtered. this is something that i had actually written about in 1988 in the AES journal. it was my first published paper. the title was "Effect of DAC deglitching on frequency response". it was obsolete pretty much right away because conventional DACs were soon to be replaced with "1-bit DACs" in audio playback. but the issue is a little more subtle than you might think. a problem was the S/H slewing at whatever was the slew rate limit (some number of volts/sec) no matter how big the step transition was. this is non-linear, like clipping followed by an integrator. to prevent that, you would put in an RC circuit so that the slope of the RC decay would never exceed the slew rate. then it was always linear. but, then the frequency response was a little bit messed up and not as simple as an RC LPF followed by the sinc(f) from the ZOH. just like modeling the ZOH, you have to come up with a hypothetical LTI system that turns the correctly scaled string of impulse functions (weighted by the sample values), into the output of the RC-limited, deglitched output. that was a little messy. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●February 9, 20122012-02-09
On Wed, 08 Feb 2012 23:21:20 -0500, Jerry Avins <jya@ieee.org> wrote:>On 2/8/2012 9:42 PM, Brian Willoughby wrote: >> On 2012/02/07 09:16, Jerry Avins wrote: >>> On 2/3/2012 6:05 PM, HardySpicer wrote: >>>> On Feb 4, 10:31 am, Jerry Avins<j...@ieee.org> wrote: >>>>> On 2/3/2012 7:44 AM, Brian Willoughby wrote: >>>>> >>>>>> I just came across a concept that doesn't completely make sense to me, >>>>>> and I am hoping that the group can help me to understand more. >>>>> >>>>> [confusion deleted (if only!)] >>>>> >>>>> The root of your confusion is a bad assumption early on. The >>>>> sample-and-hold does affect the response below Nyquist. A sample and >>>>> hold has no effect at DC, and puts out only DC when the input signal is >>>>> at the sample rate. In between it gently rolls off along a sin(x)/x >>>>> curve. (x = pi at the sample rate.) At the Nyquist rate, the output (of >>>>> the fundamental) sin(pi/2)/(pi/2), or about .637. that's roughly -4 dB. >>>>> >>>>> Jerry >>>>> -- >>>>> Engineering is the art of making what you want from things you can get. >>>>> ����������������������������������������������������������������������� >>>> >>>> You mean zero-order hold - not sample and hold... >>> >>> Bryan imagined that his ZOH is made by following an impulsive DAC with >>> an analog S&H. (Please pardon the alphabet soup.) I tried to stay with >>> his paradigm. I shouldn't have. >>> >>> Jerry >> >> It's much more than my imagination! There actually are multiple analog >> S&H circuits in my system. One reason for them is that they hide the >> discontinuity that occurs when this particular DAC changes from one code >> to the next. >> >> I was going to argue that zero-order hold is practically the same thing >> as sample and hold, apart from the usual nonidealities of analog >> circuits. When comparing digital SRC to D/A conversion, there might be >> some subtle distinctions, but they are largely the same. > >Are those DACs home brew? the glitches produced by the guts of most >commercial DACs are largely removed by the inherent filtering of analog >output drivers. > >JerryThe analog output "filtering" is often designed to be much, much broader bandwidth than the supported sample rate in order to facilitate things like using images of the output rather than the direct output energy <fs/2. As Robert mentioned, these days many DACs are pretty close to ideal ZOH circuits (up to practical limits), and we've measured pretty low EVM (essentially the opposite of SNR) on high-order modulation signals taken from images using only sinx/x correction predistortion, i.e., we didn't bother trying to correct any other effects. So the distortion or noise contribution to other effects was apparently pretty insignificant. This was with DACs that weren't expensive, either. So whatever they're doing inside a lot of DACs, it seems pretty reasonable for a lot of applications to model them as an ideal impulsive DAC followed by a ZOH (or S&H, whatever you want to call it). Eric Jacobsen Anchor Hill Communications www.anchorhill.com
Reply by ●February 9, 20122012-02-09
i'm curiouser and curiouser, Eric... On 2/9/12 5:20 PM, Eric Jacobsen wrote:> > The analog output "filtering" is often designed to be much, much > broader bandwidth than the supported sample rate in order to > facilitate things like using images of the output rather than the > direct output energy<fs/2.now why do we want to use the images of the output? why don't we just want to eliminate them?> As Robert mentioned, these days many DACs are pretty close to ideal > ZOH circuits (up to practical limits),i didn't realize i said that. really i meant to say that the ZOH effect is mostly obviated in DACs these days because the sample rate (3 MHz) of the 1-bit DACs is so high. oversampled DACs these days are pretty close to the ideal brick-wall filter with sinc() impulse response. if you send the numerical samples into the DAC: 0, 0, 0, 0, 0, 0, 0, 0, 0, A, 0, 0, 0, 0, 0, 0, 0, 0, 0 something that resembles a sinc() function proportional to A will come out (after the simple RC LPF). maybe if the LPF isn't good enough, it's a jaggy sinc() function that comes out. i'm thinking about audio, Eric. dunno about some other apps where conventional DACs, perhaps with S/H deglitchers, are used and ZOH modeling is indicated. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●February 9, 20122012-02-09
On Thu, 09 Feb 2012 22:21:19 -0500, robert bristow-johnson <rbj@audioimagination.com> wrote:> >i'm curiouser and curiouser, Eric... > >On 2/9/12 5:20 PM, Eric Jacobsen wrote: >> >> The analog output "filtering" is often designed to be much, much >> broader bandwidth than the supported sample rate in order to >> facilitate things like using images of the output rather than the >> direct output energy<fs/2. > >now why do we want to use the images of the output? why don't we just >want to eliminate them?For a modulator if you want the signal output at, say, 70 MHz, which used to be a very common IF frequency (so it made a lot of equipment interoperable), you can eliminate a mixing stage if you can digitize it directly at 70 MHz. Eliminating mixing is nice because it reduces cost, power consumption, and board area. Using a sampled IF also eliminates some distortion sources by getting rid of the analog mixer. To do so in a traditional sense one would need to run the DAC at double the 70 MHz plus margin for the signal bandwidth and the required rolloff room for the reconstruction filter. So you'd probably be looking in the region of a sample rate of close to 200 MHz. Making the sample rate high increases the cost of the DAC, the digital hardware that has to run that fast, and power consumption goes up as well. If you can take an image, instead, then, for example, you can cut the sample rate down to 100 MHz, and use an image at 70 MHz created by the spectral inverse of the desired signal at 30 MHz. This makes the DAC cheaper, the digital hardware cheaper, and the power consumption lower. The cost is a little trickier design work, especially in the digital compensation filter, etc.>> As Robert mentioned, these days many DACs are pretty close to ideal >> ZOH circuits (up to practical limits), > >i didn't realize i said that. really i meant to say that the ZOH effect >is mostly obviated in DACs these days because the sample rate (3 MHz) of >the 1-bit DACs is so high. oversampled DACs these days are pretty close >to the ideal brick-wall filter with sinc() impulse response. if you >send the numerical samples into the DAC: > > 0, 0, 0, 0, 0, 0, 0, 0, 0, A, 0, 0, 0, 0, 0, 0, 0, 0, 0 > >something that resembles a sinc() function proportional to A will come >out (after the simple RC LPF). maybe if the LPF isn't good enough, it's >a jaggy sinc() function that comes out. > >i'm thinking about audio, Eric. dunno about some other apps where >conventional DACs, perhaps with S/H deglitchers, are used and ZOH >modeling is indicated.For the comm case, like I described above, one cares a lot about correcting the sinx/x due to the ZOH (or S/H or whatever you want to call it) and any other significant distortion sources. Since the ZOH sinx/x is predictable it is correctable, and making a correction filter just based on the ZOH effect works quite well. It is possible (and normal practice) to measure the distortion in a signal as it comes out of the transmitter, and if you get that correction wrong or don't correct for significant distortion sources, you'll know about it in those tests. So I was just saying that those tests pass quite handily when the DAC is modelled as an impulsive DAC sampler and a ZOH, and the correction done accordingly. Eric Jacobsen Anchor Hill Communications www.anchorhill.com
Reply by ●February 10, 20122012-02-10
On 2/9/12 10:43 PM, Eric Jacobsen wrote:> On Thu, 09 Feb 2012 22:21:19 -0500, robert bristow-johnson > <rbj@audioimagination.com> wrote: > >> >> i'm curiouser and curiouser, Eric... >> >> On 2/9/12 5:20 PM, Eric Jacobsen wrote: >>> >>> The analog output "filtering" is often designed to be much, much >>> broader bandwidth than the supported sample rate in order to >>> facilitate things like using images of the output rather than the >>> direct output energy<fs/2. >> >> now why do we want to use the images of the output? why don't we just >> want to eliminate them? > > For a modulator if you want the signal output at, say, 70 MHz, which > used to be a very common IF frequency (so it made a lot of equipment > interoperable), you can eliminate a mixing stage if you can digitize > it directly at 70 MHz. Eliminating mixing is nice because it reduces > cost, power consumption, and board area. Using a sampled IF also > eliminates some distortion sources by getting rid of the analog mixer.so, what you're talking about is using a conventional DAC with images and picking the image you want to keep around 70 MHz, right? ...> For the comm case, like I described above, one cares a lot about > correcting the sinx/x due to the ZOH (or S/H or whatever you want to > call it) and any other significant distortion sources. Since the ZOH > sinx/x is predictable it is correctable, and making a correction > filter just based on the ZOH effect works quite well.how well does it work at frequencies of and close to integer multiples of Fs? these would be the images of DC (and frequencies very close to that). kinda hard to divide by zero after multiplying by it (by the sinc() function). multiplying by zero has a way of destroying information, never to be retrieved again. like a black hole. so, if you're keeping the 140th image up there at 70 MHz (say Fs is 500 kHz), besides being reduced by something like 1/(pi*140), the image up there will have a notch taken out at the midpoint of the image.> So I was just saying that those tests pass quite handily when the DAC > is modeled as an impulsive DAC sampler and a ZOH, and the correction > done accordingly.well, that's the only model that's an LTI and fits with the S/H output. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●February 10, 20122012-02-10
On Thu, 09 Feb 2012 23:21:00 -0500, robert bristow-johnson <rbj@audioimagination.com> wrote:>On 2/9/12 10:43 PM, Eric Jacobsen wrote: >> On Thu, 09 Feb 2012 22:21:19 -0500, robert bristow-johnson >> <rbj@audioimagination.com> wrote: >> >>> >>> i'm curiouser and curiouser, Eric... >>> >>> On 2/9/12 5:20 PM, Eric Jacobsen wrote: >>>> >>>> The analog output "filtering" is often designed to be much, much >>>> broader bandwidth than the supported sample rate in order to >>>> facilitate things like using images of the output rather than the >>>> direct output energy<fs/2. >>> >>> now why do we want to use the images of the output? why don't we just >>> want to eliminate them? >> >> For a modulator if you want the signal output at, say, 70 MHz, which >> used to be a very common IF frequency (so it made a lot of equipment >> interoperable), you can eliminate a mixing stage if you can digitize >> it directly at 70 MHz. Eliminating mixing is nice because it reduces >> cost, power consumption, and board area. Using a sampled IF also >> eliminates some distortion sources by getting rid of the analog mixer. > >so, what you're talking about is using a conventional DAC with images >and picking the image you want to keep around 70 MHz, right?Yes. Typically with a bandpass filter that rejects everything else.>... > >> For the comm case, like I described above, one cares a lot about >> correcting the sinx/x due to the ZOH (or S/H or whatever you want to >> call it) and any other significant distortion sources. Since the ZOH >> sinx/x is predictable it is correctable, and making a correction >> filter just based on the ZOH effect works quite well. > >how well does it work at frequencies of and close to integer multiples >of Fs? these would be the images of DC (and frequencies very close to >that). kinda hard to divide by zero after multiplying by it (by the >sinc() function). multiplying by zero has a way of destroying >information, never to be retrieved again. like a black hole.Yes, you tend to engineer it to avoid the nulls. And the further up in frequency you go the more attenuation you get from the decay of the sidelobes of the sinx/x response, so it's harder to get good performance if you take higher and higher images, and the DAC output bandwidth has to be wide enough to cover the range of interest.>so, if you're keeping the 140th image up there at 70 MHz (say Fs is 500 >kHz), besides being reduced by something like 1/(pi*140), the image up >there will have a notch taken out at the midpoint of the image.>> So I was just saying that those tests pass quite handily when the DAC >> is modeled as an impulsive DAC sampler and a ZOH, and the correction >> done accordingly. > >well, that's the only model that's an LTI and fits with the S/H output.And it means that any glitches that might be expected at the sample transitions are either already well-tamed by the DAC or don't contribute enough to be a problem. Eric Jacobsen Anchor Hill Communications www.anchorhill.com
