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Digital sine wave generation

Discussion in 'Electronic Design' started by Tim Shoppa, Mar 30, 2006.

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  1. Tim Shoppa

    Tim Shoppa Guest

    While my obsession is still digital sine wave generation, I'd like to
    note that the resistor values in AOE 2nd Ed, 9.38, for the twisted-ring
    counter weighting resistors are not the best values to choose in the
    E96 series. They're not bad but not the best.

    If someone wants to check, I think Lancaster's _CMOS Cookbook_ has the
    same resistor values as AOE. Can't find my copy of Lancaster's book at
    this moment.

    What is in AOE is:

    Q1: 57.6K
    Q2: 30.9K
    Q3: 23.7K
    Q4: 22.1K
    Q5: 23.7K
    Q6: 30.9K
    Q7: 57.6K

    My search has shown that you get 80% less sum-of-squares error if you
    instead use these standard E96 values:

    Q1: 30.1K
    Q2: 16.2K
    Q3: 12.4K
    Q4: 11.5K
    Q5: 12.4K
    Q6: 16.2K
    Q7: 30.1K

    The matching with these values makes it worthwhile to use 0.1% or even
    better resistors (which are actually affordable and available these
    days!)

    It appears to me that a generic 74HC595 output has an output impedance
    of 40 or 50 ohms at room temperature and currents under 1mA and 5V Vcc,
    and at these impedances it is actually very slightly better to use the
    11.5K-12.4K-16.2K-30.1K range rather than the ten-times-higher
    115K-124K-162K-301K.

    Tim.
     
  2. Guest

    I think that you are about to find about Gibbs oscillations. If your
    values are sine x/x (sinc) you are going to find that you want to
    applying a Hamming raised cosine window (which it looks as if Win and
    Don have done) to eliminate the ringing in the frequency response that
    you get by truncating the shift register (Gibbs oscillations).

    http://www.daqarta.com/EEX06.HTM
     
  3. Tim Shoppa

    Tim Shoppa Guest

    I don't think that's what I'm writing about Bill. The synthesizer in
    question uses those 7 resistors (4 different values, really very
    clever) to synthesize a 16-step approximation to a sin wave. The values
    synthesized are "just" sin(pi/16),sin(3*pi/16), etc. The shift register
    runs at exactly 16 times the resulting sine wave frequency.

    sinc(x) comes in when you're windowing and the sampling rate is not a
    simple harmonic of the signal, right? Interesting stuff but not
    relevant to my much simpler application.

    Tim.
     
  4. Guest

    You are running a square wave through a shift register, and using the
    resistors to set up the coefficients for an FIR low pass filter. That's
    exactly what I was doing (in another context) and I spent a long time
    recalculating all the resistors and soldering in the new set.
    Beleive me (and Win Hill and Don Lancaster), it is relevant to your
    application.
     
  5. Tim Shoppa

    Tim Shoppa Guest

    OK, Bill, this is what I'm saying: to synthesize a sine wave with 16
    steps, choose the values so that what comes out is sin(phase) for 16
    equally spaced phases from 0 to 2pi. This is exactly the way the
    resistors were chosen in AOE. I'm only saying that I chose some E96
    resistors that implement this a factor of five better (in terms of
    mean-square error) than the set in AOE. The ratios are all the same to
    within a percent, it's just that my proposed set is better at achieving
    the ideal voltage ratios.

    Tell me how this is different than what's in AOE 9.38? Or the approach
    of any other DDS sine wave synthesizer?

    And then tell me how what you do is better. Are your first 14 harmonics
    zero? Or are you spreading out the badness so that some harmonics below
    the 14th are nonzero, but such that your synth plus low-pass filter
    combo is more effective at reducing THD in the end because your higher
    harmonics have been chosen to be lower?

    Tim.
     
  6. Guest

    Good question. Since I was filtering psuedo-random noise back in 1977,
    it is going to be difficult to answer this question. From what I can
    recall, the truncated sinc sequence gave a sharper low frequency
    cut-off (with some ringing) followed by a rather poor and oscillating
    cut-off at higher freqencies, while the raised-cosine windowed sequence
    did not cut off as sharply at low frequencies, but gave a pretty much
    monotonically increasing cut-off with increasing frequency.

    If you only want to synthesise a narrow range of sine wave frequencies,
    the amplitude of the high frequency harmonics isn't too important - you
    can always filter them out with a low pass filter. IIRR, my circuit did
    relay on a Sallen and Keys two-pole to get rid of the last of the high
    freqncy junk, with switchable capacitors to handle decade changes in
    frequency range, but I had to change the resistor values to get the
    customer happy.
     
  7. Tim Shoppa

    Tim Shoppa Guest

    Bill -

    I've come to the conclusion that you are solving a different problem
    (bandpass of a noisy signal) than the one I'm obsessed with
    (synthesizing a sine wave where the clock is an exact multiple of the
    output frequency).

    Nothing against what you did, I'm sure it was interesting, and it is
    related in an abstract way to what I'm doing (your parallel with FIR
    filters has some relevance in a sort-of upside down way, because I'm
    starting out with a digital clock that has no noise beyond jitter in
    the 10-20 ns range).

    But what I'm doing is hoplelessly more simplistic. JUST choosing
    optimal E96 resistor values to give good approximations for sin(phase)
    for equally spaced steps.

    Tim.
     
  8. Mochuelo

    Mochuelo Guest

    I understand, and sounds interesting. How do you connect the 7
    resistors to distinguish 16 different phase instants? What is AOE?

    Best,
     

  9. A book that is mentioned quite often in this newsgroup.


    --
    Service to my country? Been there, Done that, and I've got my DD214 to
    prove it.
    Member of DAV #85.

    Michael A. Terrell
    Central Florida
     
  10. The Phantom

    The Phantom Guest

    Could you post a schematic showing how the resistors are connected to the
    digital IC over on ABSE?
     
  11. Guest

    The Art of Electronics by Horowitz and (Win) Hill. ISBN: 0521370957. It
    is on special at Amazon at the momnent fpr $70 new, and from $45 used -
    there are five used copies on offer, and a bunch of sellers offering
    new copies from $62.

    Originally written as a tesxtbook for training physicists at Harvard in
    electronics, so aimed at the brigher student. Cambridge University in
    the U.K. used it as an undergraduate test for their electronic
    engineers.
     
  12. Guest

    Right, but what you are doing is producing a staircase apprixiation to
    a sine wave.

    If you take the staircase that you are going to synthesise, and do a
    Fourier analysis on it, you will find - as you intend - a large
    component at 1/16th of your clock frequency, very small components at
    the first few harmonics of that frequency - due to the tolerances on
    your resistors, and the limitations of working within the E96 grid,
    followed by quite large harmonics at the clock frequency and its
    harmonics.

    You cna understand this intuitively by subtracting your desired sine
    wave from your staircase approximation, which leaves you with a series
    of sawteeth, whose amplitude is at a maximum around 0 and 180 degrees,
    and whose polarity reverses from 0 to 180 degrees.

    At present you are tinker with the staircase to minimise the harmonics
    below the clock frequency. You can equally tinker to minimise the
    harmonics above the clock frequency.

    If your clock frequency is stable and never gets changed, then a fancy
    multipole filter is the better way of getting rid of the higner
    harmonics,

    Even better is to use an integrator to synthesise the sine wave as
    series of straight lines of variable slope - though the amplitude then
    depends on the value of your integrating capacitor, and you might need
    a precision rectifier and feedback loop to get an absolutely stable
    amplitude over temperature.
     
  13. Ken Smith

    Ken Smith Guest

    I'm not sure what he did but here's how I did it for less states:

    You make a "shift register" that does this:


    State Bits
    0 0 0 0 <----
    1 1 0 0 !
    2 0 1 0 !
    3 0 1 1 !
    4 1 1 1 Repeats
    5 0 1 1 !
    6 0 1 0 !
    7 1 0 0 !
    8 0 0 0 ---->

    Obviously, there is at least one bit hidden inside this part. In this
    circuit, you only need 3 resistors to get 8 points on the curve.

    If you want to use a switched capacitor filter to smoothen it out, a tap a
    few more up the chain will give the clock for it. The nice thing about
    that is that when you slew the clock speed, the internal time constants
    move with it and the node voltages are uneffected. The result is that it
    doesn't generate the low frequencies.
     
  14. http://www.tinaja.com/glib/msintro1.pdf

    --
    Many thanks,

    Don Lancaster voice phone: (928)428-4073
    Synergetics 3860 West First Street Box 809 Thatcher, AZ 85552
    rss: http://www.tinaja.com/whtnu.xml email:

    Please visit my GURU's LAIR web site at http://www.tinaja.com
     
  15. bogax

    bogax Guest

  16. bogax

    bogax Guest

  17. Mochuelo

    Mochuelo Guest

    Ok, I know which circuit you mean. When you talk about sum-of-squares
    error, you mean between the staircase and the sinusoid? If that's the
    case, that is not a good criterium. You should look at the spectrum of
    the staircase signal. If you have Matlab (or equivalent), you can
    compute the fft of your signal and see which resistor values give you
    the lowest THD, for instance. You can easily include the effect of the
    resistor tolerances to see if the circuit will be viable or not. The
    effect of the output impedance of the shift register stages is more
    difficult to be taken into account because the resistance is not
    constant. I would clock manually the circuit, measure the output
    voltage with a good DMM, solve a system of equations, and find the set
    of resistor values that more or less "calibrate" the circuit.

    But I don't like very much this structure. Using pairs of resistors
    that should have matching values (but in fact will never have) is not
    the most "compact" way of doing this.

    Best,
     
  18. The Phantom

    The Phantom Guest

    What voltage levels did you determine give the best approximation to the
    sine wave?
     
  19. The Phantom

    The Phantom Guest

    I analyzed the problem with some assumptions. I assumed the counter was
    some kind of logic like CMOS that can swing the outputs rail to ground with
    zero output impedance (I didn't consider the 40 or 50 ohms you mention
    below). To do the mathematical analysis I assumed the rail voltage was 2
    volts, so that the output sine wave's most negative voltage was zero, and
    the most positive voltage was 2 volts. In other words, a standard
    mathematical sine wave with a 1 volt offset. I assumed 16 steps. I
    computed the sum-of-squares error with only half of the sine wave
    (Sin[t]+1), namely, that portion from t=-pi/2 to t=pi/2. The
    sum-of-squares error was computed by integrating from the left edge of each
    step to the right edge and doing this for all 8 steps of the half sine
    wave, then adding the 8 results. The integrand was (Sin[t]+1-Vn)^2, where
    Vn was the output voltage of the circuit for the steps starting at t=-pi/2,
    n=1 to 8.
    For this set of resistors, I got a sum-of-squares error of .0211462.
    For this set of resistors, I got a sum-of-squares error of .0211649,
    slightly larger than the previous set.
    I found that the minimum sum-of-squares error that this circuit with the
    7 resistors can give is .0205938. If I use the following standard value
    resistors, the error is .0205970:

    Q1: 57.6K
    Q2: 36.5K
    Q3: 28.0K
    Q4: 26.1K
    Q5: 28.0K
    Q6: 36.5K
    Q7: 57.6K

    I also found that if one selects for the desired output voltages the
    value of (Sin[t]+1) at the time midpoints of the steps, the sum-of-squares
    error is .0206273.

    There is a subtlety that leads to a small error when comparing all these
    values. Each stepped approximation should be normalized by dividing by the
    magnitude of the fundamental component in the given stepped approximation
    before computing the sum-of-squares error, but I didn't do this (yet; I may
    later, but the error isn't large).

    With this much number crunching involved it's easy to make a mistake, but
    these are the results I got with a reasonable amount of care.

    Once you pass this stepped sine wave through almost any kind of low-pass
    filter I suspect the result from these various cases will be
    indistinguishable in terms of waveshape purity.
     
  20. The Phantom

    The Phantom Guest

    As far as I know, minimizing sum-of-squares gives the same result as
    minimizing THD.
     
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