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anti-aliasing

Discussion in 'Electronic Design' started by John Larkin, Nov 26, 2007.

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  1. John Larkin

    John Larkin Guest

    Suppose one (actually, me) were firing an adc at some fixed rate,
    ballpark 12 KHz in this case, and the input to the adc was a sine wave
    of fixed but unknown frequency. The adc will take a bunch of samples,
    ballpark 1000 maybe, and I want to compute the mean (ie, dc value) and
    the mean of the abs value of the samples (ie, the ac value). That
    works, but sometimes the input frequency aliases against the sample
    rate and messes up the data, like gives a big average dc value when
    there's really none there.

    (This is not a Nyquist issue; the sample rate may be above or below
    the sinewave frequency.)

    So maybe I can fuzz up the sample rate so that it can't alias against
    any constant sinewave frequency.

    I could add a pseudo-random delay after every adc sample shot; but the
    mean sample rate wouldn't change a lot. Or I could add successive
    delays, essentially sweeping the sample rate down.

    Any ideas?

    John
     
  2. Tim Wescott

    Tim Wescott Guest

    Randomly varying the sample points will randomly vary any aliasing going
    on, which means that instead of having a few really bad magic
    frequencies you'll have only slightly bad, but broad magic frequency
    ranges -- just how broad and how bad would depend on your sampling, and
    could either be found by analysis or simulation.

    If you couldn't live with the resulting error, sampling at two
    distinctly different rates, overlapped, then checking for one of the set
    of samples being aliased to DC may be more effective. The algorithm,
    however, would be way more complex.

    --

    Tim Wescott
    Wescott Design Services
    http://www.wescottdesign.com

    Do you need to implement control loops in software?
    "Applied Control Theory for Embedded Systems" gives you just what it says.
    See details at http://www.wescottdesign.com/actfes/actfes.html
     
  3. BobW

    BobW Guest

    If the granularity of your "pseudo-random delay" is small enough, and you
    take enough samples, then the answer will converge on the correct one (i.e.
    no effects of aliasing will be seen).

    How many samples to take as a function of delay granularity to give an
    acceptable amount of error? That's the hard part.

    You should probably repost on comp.dsp, or go to comp.arch.fpga and ask for
    Ray Andraka's help. He's smarter than everyone here -- combined!

    Bob
     
  4. Robert Baer

    Robert Baer Guest

    Issue #1: What time period is going to be used for the sampling? If
    the (presumedly) sine wave is of a sufficently low frequency WRT to full
    sample time, then any given sub-section of that sine would be samples,
    giving obvious errors.
    Possible solution: trigger start and stop of sample period from
    incoming signal, say at zero crossing + slope.
    If it is a complex waveform signal, that might not be a useable solution.
    Issue #2: if the input waveform is fast enough, the samples will
    (again) represent only a small time section of the presumedly repeating
    waveform, with the same obvious errors.
    In either case, "small" dithering of the sampling periodicity will
    not help; it might be useful when the input periodocity is near one of
    the harmonics of the sampling rate.
    For kicks, assume the nominal periods are the same; dithering would
    only allow one to "slip" left ot right of the assumed synch point,
    again achieving only part of the full waveform.
    Hell, ASS-u-ME the sampling rate is *exactly* twice the input
    waveform rate; i ask you the following nasty question: is it possible to
    recover the input waveform? The answer might make one think that
    Nyquist was a liar.
    So.
    If it is a given that the input frequency is unknown and possibly
    wideband, then use two samplers, where the rates are decidedly NOT
    harmonically related; maybe one sampling for a long time to recover low
    frequency waveforms, and the other some generic high speed sampler for
    comparison purposes.
    Ratio? Maybe 11.5 if free-running (both stable, temp comp), possibly
    to 31.5 (maybe more) if clocks generated from same crystal.
     
  5. John Larkin

    John Larkin Guest

    That's a separate issue. Obviously I need to sample for many cycles of
    any waveform, to avoid getting just a slice. Let's assume I'll sample
    for long enough to avoid that problem.
    Can't do that; my sample rate is generated in software and I have no
    trigger hardware.
    I don't understand that one. If the signal rate is well above the
    sample rate, I'll be sampling over many cycles, so the only problem is
    down-aliasing.
    I'm not trying to recover the waveform. All I want to do is measure
    its mean and the mean of its abs value.

    In the 2:1 case, I'd always get a mean of zero, and an AC value (mean
    of abs) that depends on the relative phases for that particular run.

    Sorry, I only have one adc. But if they differed, which one would you
    pick to believe?

    I'm thinking that if I add a pseudorandom delay after each sample, so
    that the sample period varies from the original to, say, twice as
    much, the spectrum of the sampling impulses will be spread over about
    a 2:1 range, so nothing can alias against it. It's a little easier to
    think about if the added delays had a gaussian distribution, but
    uniform might be OK too.

    John
     
  6. I think you might be going about it the wrong way.

    The DC terms are simply the integrals of the function.

    i.e., for a signal f(t)

    DC term = F(0) = int(f(t))
    abs DC term = G(0) = int(|f(t)|)

    If you cannot take the abs of a signal(by using rectification say, then you
    can also do it by clipping.

    F(0) + G(0) = int(f+(t))

    where f+(t) is the positive terms of f(t)(which can be simply gotten by
    rectification) and F(0) is simply determined by an integrator.

    Seems like you can do all the hard work using an op amp or two and your adc
    will simply digitize the result.

    This even goes along the line that you should just LP your signal because
    they do not effect the DC term. LP it enough so that your sample rate is 2x
    the highest frequency. (I suppose there is an issue of practicality here if
    your sample rate is extremely low... in that case just get a faster adc.)
     
  7. Guest

    If you check the number of times each ADC output showed up - creating
    a histogram - you could probably see when you were sampling at some
    multiple or sub-multiple of the sampling frequency.

    Enough completely random samples would give you an M-shaped histogram,
    with every voltage interval between the peak and the minimum of the
    sine wave eventually getting some hits. If your sampling is
    synchronous (sampling at an exact multiple of the sine-wave frequency
    or some integral fraction of the frequency), you won't get this
    pattern, but will rather see a number of isolated peaks in the
    histogram where the sampling hits at more or less the same phase time
    after time.

    If your sine wave amplitude could go to down to zero you wouldn't be
    able to distinguish the single peak you'd see on the histogram from
    the single peak that you'd get from perfectly synchronous sampling at
    the same phase point, but you could check that by changing the
    sampling frequency.

    You haven't given enough detail for me to be sure that is would be a
    practical approach, but it might give you what you seem to want.
     
  8. Fred Bloggs

    Fred Bloggs Guest

    Do you intend on using a sample zero mean, or closeness to it, as a
    numerical criterion for confidence in the estimate of the mean of
    abs(.)? Is there any analytical basis for this that you actually
    understand? What range of confidence are you looking for? 75%? 90%?
    99.9%? or what? Give us some numbers. There is a form of filtering
    derived from principles of artificial intelligence that is more or less
    applicable, it is called _____ _____ filtering and it's quite powerful.
    The very word "ideas" in this context actually means "guesses." That
    would be in keeping with your methodology. The success rate will be a
    function of something called the density of useful solutions...
     
  9. This will dither the error through the whole range. Although you will
    avoid the worst cases, the average error will be higher.
    I would lock the sampling to the incoming frequency. That can be done
    either by adjusting the sample rate directly or by resampling in the
    software.


    Vladimir Vassilevsky
    DSP and Mixed Signal Design Consultant
    http://www.abvolt.com
     
  10. Fred Bloggs

    Fred Bloggs Guest

    I'm pretty sure it is just a simplified version of sequential particle
    filtering, and very simplified if he knows the input is sine, there are
    no guesses about the form of the distribution whatsoever.
    http://en.wikipedia.org/wiki/Particle_filter
     
  11. John Larkin

    John Larkin Guest

    Right. It's sorta like a delta-sigma dac, where artifacts are
    pulverized and scattered to become a broadband noise floor.
    As I mentioned, the incoming frequency is unknown. I'm perfectly
    willing to take, say, 1000 samples to get, say, 2% measurement
    accuracy, if I can ensure that my sampling can't alias the sinewave
    input. One of the checks I'm making on sinewave quality, and on the
    signal chain producing it, is that there is actually a very small DC
    offset. I'm not willing to run a bunch of trial runs and pick the best
    one by some as-yet-unknown criterion; the BIST function I'm running
    already takes 7 seconds to execute.

    I was just wondering if the problem has come up before, and if anyone
    had ideas or algorithms for sample time splattering that guaranteed no
    aliasing.

    John
     
  12. John Larkin

    John Larkin Guest

    This is a BIST (self-test) function for a DDS synthesizer. Any
    significant mean value would be an indicator that something is wrong
    in the stuff that generates the sinewaves. That might be excess dc
    offset, clipping, or distortion.


    Is there any analytical basis for this that you actually
    Certainly.

    What range of confidence are you looking for? 75%? 90%?
    I'd like to verify that the sinewave amplitude is within, say, 2% of
    my target (that's the system spec) and that a 20 volt p-p sine has
    below, say, 200 mV of offset. If I don't have aliasing, I'm seeing
    offset measurements in the 50 mV range. The BIST subsystem is pretty
    simple: a relay diverts the signal from the customer connector into a
    12-bit ADC, and the rest is all firmware, including the ADC triggers.


    There is a form of filtering
    There's no filtering that can remove aliases once they're in the data.
    Where do you think new concepts come from? They come from generating a
    great number of ideas, and riffs on those ideas, and filtering them
    for quality. A filter with no input has no output.

    I wouldn't expect ideas from you, because you are hostile to the
    chaotic nature of the creative process. Some people are that way, and
    can do good work in other ways, but they won't allow themselves to
    design.

    John
     
  13. John Fields

    John Fields Guest

     
  14. Tim Wescott

    Tim Wescott Guest

    -- snip --
    This should work. The added delays wouldn't have to be Gaussian, you'd
    just want their spectrum to be fairly white.

    As I pointed out in my response to your original post, though, this will
    practically guarantee you some DC noise floor.

    --

    Tim Wescott
    Wescott Design Services
    http://www.wescottdesign.com

    Do you need to implement control loops in software?
    "Applied Control Theory for Embedded Systems" gives you just what it says.
    See details at http://www.wescottdesign.com/actfes/actfes.html
     
  15. Tim Wescott

    Tim Wescott Guest

    Ooh. Testy.

    Do you get to know what frequency the DDS is running at? If so, you
    could adjust your sampling rate to match, so you got a good picture of
    what it was doing. If the DDS always went fast enough, and the ADC
    sample time could be set precisely enough, you could even build up
    exactly one cycle of the output and analyze it for a heck of a lot more
    than just DC content.

    --

    Tim Wescott
    Wescott Design Services
    http://www.wescottdesign.com

    Do you need to implement control loops in software?
    "Applied Control Theory for Embedded Systems" gives you just what it says.
    See details at http://www.wescottdesign.com/actfes/actfes.html
     
  16. Phil Hobbs

    Phil Hobbs Guest

    Another approach would be to do it twice, with sampling rate F and
    F(1+1/N), where N is the number of samples. The lowest frequency that
    aliases both of those to DC is (N+1)*F. If you use three frequencies
    that are relatively prime, it'll be N**2 * F, which should be enough to
    cover most uses. This has the advantage of relying only on theory you
    can do in your sleep, which is important for a BIST routine.

    Cheers,

    Phil Hobbs
     
  17. Fred Bloggs

    Fred Bloggs Guest

    Well if your system is generating the sine wave then how does it come to
    pass that you don't know the frequency? If you're admitting the
    possibility of a harmonic or subharmonic output, what makes you think it
    will even be constant or a sine. You may need some analog to steer your
    processing in the large...like a filter bank or a voltage controlled
    variable center frequency band pass or something or IF swept mixer
    thing, dunno.

    -mystified
     
  18. John Larkin

    John Larkin Guest

    Well, Fred keeps insulting me because I have ideas and design things.
    He seems to think that the initial stages of a design, the explore
    possibilities phase, is impure and beneath his dignity.

    I've known lots of people who couldn't design because they feel
    insecure in the early stages, the deliberate uncertainty phase. These
    people, if thet are responsible for design (which they shouldn't be)
    tend to look for well-analyzed prior art, and if they can't find any,
    usually seize onto the first clumsy concept they can come up with, and
    then settle into brute-force analysis and implementation of a bad
    idea. Design is a creative, chaotic, psychological process that
    uptight people don't like and seldom respect.


    When I run autonomous BIST, I test a range of frequencies. I suppose I
    could fine-tune the sample rate, or the test frequencies, on a
    case-by-case basis, or do more number crunching and find one magical
    sample rate that won't alias *any* of my test frequencies. But there's
    another mode where the customer can program a waveform and ask for it
    to be measured (which we also use in testing) so it would be cool to
    have a general solution.

    John
     
  19. John Larkin

    John Larkin Guest

    Say, 2 KHz to 1 MHz. The flat-out sample rate is 12.4 KHz (software
    loop, bit-banging a 12-bit serial ADC) which I can slow down on a
    per-sample basis, randomly or swept or whatever. I usually take about
    1000 samples and average the mean and the software-rectified mean.

    John
     
  20. John Larkin

    John Larkin Guest

    For N=1000, (1+1/N) is only 1.001. That won't move an alias very far.
    The alias doesn't have to be to DC to mess me up, it just has to be to
    a low frequency relative to my total sampling run duration.

    If I did it twice, which measurement would I trust? Not the one with
    the lowest DC offset, because I'm trying to measure offset.

    Even three runs (which would take a lot of time) would be tricky...
    would I throw out the one that has the highest DC offset, and average
    the other two? Or pick the two that agree best?

    Too much work!

    John
     
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