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Sine Lookup Table with Linear Interpolation

Discussion in 'Electronic Design' started by rickman, Mar 16, 2013.

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  1. rickman

    rickman Guest

    I've been studying an approach to implementing a lookup table (LUT) to
    implement a sine function. The two msbs of the phase define the
    quadrant. I have decided that an 8 bit address for a single quadrant is
    sufficient with an 18 bit output. Another 11 bits of phase will give me
    sufficient resolution to interpolate the sin() to 18 bits.

    If you assume a straight line between the two endpoints the midpoint of
    each interpolated segment will have an error of

    Without considering rounding, this reaches a maximum at the last segment
    before the 90 degree point. I calculate about 4-5 ppm which is about
    the same as the quantization of an 18 bit number.

    There are two issues I have not come across regarding these LUTs. One
    is adding a bias to the index before converting to the sin() value for
    the table. Some would say the index 0 represents the phase 0 and the
    index 2^n represents 90 degrees. But this is 2^n+1 points which makes a
    LUT inefficient, especially in hardware. If a bias of half the lsb is
    added to the index before converting to a sin() value the value 0 to
    2^n-1 becomes symmetrical with 2^n to 2^(n+1)-1 fitting a binary sized
    table properly. I assume this is commonly done and I just can't find a
    mention of it.

    The other issue is how to calculate the values for the table to give the
    best advantage to the linear interpolation. Rather than using the exact
    match to the end points stored in the table, an adjustment could be done
    to minimize the deviation over each interpolated segment. Without this,
    the errors are always in the same direction. With an adjustment the
    errors become bipolar and so will reduce the magnitude by half (approx).
    Is this commonly done? It will require a bit of computation to get
    these values, but even a rough approximation should improve the max
    error by a factor of two to around 2-3 ppm.

    Now if I can squeeze another 16 dB of SINAD out of my CODEC to take
    advantage of this resolution! lol

    One thing I learned while doing this is that near 0 degrees the sin()
    function is linear (we all knew that, right?) but near 90 degrees, the
    sin() function is essentially quadratic. Who would have thunk it?
  2. this *should* be a relatively simple issue, but i am confused

    10 bits or 1024 points. since you're doing linear interpolation, add
    one more, copy the zeroth point x[0] to the last x[1024] so you don't
    have to do any modulo (by ANDing with (1023-1) on the address of the
    second point. (probably not necessary for hardware implementation.)

    x[n] = sin( (pi/512)*n ) for 0 <= n <= 1024
    so a 21 bit total index. your frequency resolution would be 2^(-21) in
    cycles per sampling period or 2^(-21) * Fs. those 2 million values
    would be the only frequencies you can meaningful
    do you mean "+" instead of the first "-"? to be explicit:

    ( sin((pi/512)*(n+1)) + sin((pi/512)*n) )/2 - sin((pi/512)*(n+0.5))

    that's the error in the middle. dunno if it's the max error, but it's
    might be.
    at both the 90 and 270 degree points. (or just before and after those
    do you mean biasing by 1/2 of a point? then your max error will be *at*
    the 90 and 270 degree points and it will be slightly more than what you
    had before.
    if you assume an approximate quadratic behavior over that short segment,
    you can compute the straight line where the error in the middle is equal
    in magnitude (and opposite in sign) to the error at the end points.
    that's a closed form solution, i think.

    dunno if that is what you actually want for a sinusoidal waveform
    generator. i might think you want to minimize the mean square error.
    Newton? Leibnitz? Gauss?

    sin(t + pi/2) = cos(t)
  3. Guest

    why not skip the lut and just do a full sine approximation

    something like this, page 53-54 1/Using the ADSP-2100 Family Volume 1.pdf

  4. In the early days of MOS ROMs, there were commercial such ROMs,
    I believe from National. (And when ROMs were much smaller than today.)

    The data sheet has (I can't find it right now, but it is around
    somewere) the combination of ROMs and TTL adders to do the
    I presume the ROM designers had all this figured out.
    I don't remember anymore. But since you have the additional bit to
    do the interpolation, it should be easy.
    I am not sure how you are thinking about doing it. I believe that
    some of the bits that go into the MSB ROM also go into the lower ROM
    to select the interpolation slope, and then additional bits to select
    the actual value. Say you have a 1024x10 ROM for the first one,
    then want to interpolate that. How many bits of linear interpolation
    can be done? (That is, before linear isn't close enough any more.)
    Then the appropriate number of low bits and high bits, but not the
    in between bits, go into the interpolation ROM, which is then added
    to the other ROMs output.
    Well, cos() is known to be quadratic near zero.

    I once knew someone with a homework problem something like:

    Take a string all the way around the earth at the equator, and add
    (if I remember right) 3m. (Assuming the earth is a perfect sphere.)
    If the string is at a uniform height around the earth, how far from
    the surface is it?

    Now, pull it up at one point. How far is that point above the surface?
    More specifically, as I originally heard it, is it higher than the
    height of a specific nine story library?

    As I remember it, the usual small angle approximations to trig.
    functions aren't enough to do this. The next term is needed.
    The 3m added might not be right, as the answer is close enough
    to the certain building height to need the additional term.

    -- glen
  5. Robert Baer

    Robert Baer Guest

    Sounds like you are making excellent improvements on the standard
    ho-hum algorithms; the net result will be superior to anything done out
    there (commercially).
    With the proper offsets, one needs only 22.5 degrees of lookup
    ("bounce" off each multiple of 45 degrees).
  6. See:

    The description starts on page 273.

    -- glen
  7. Nobody

    Nobody Guest

    An n-bit table has 2^n+1 entries for 2^n ranges. Range i has endpoints of
    table and table[i+1]. The final range has i=(1<<n)-1, so the last
    entry in the table is table[1<<n], not table[(1<<n)-1].
    sin((pi/2)+x) = sin((pi/2)-x) = cos(x), and the Maclaurin series for
    cos(x) is:

    cos(x) = 1 - (x^2)/2! + (x^4)/4! - ...
  8. rickman

    rickman Guest

    So you are suggesting a table with 2^n+1 entries? Not such a great idea
    in some apps, like hardware. What is the advantage? Also, why 10 bit
    address for a 1024 element table? My calculations indicate a linear
    interpolation can be done with 4 ppm accuracy with a 256 element LUT.
    I'm not completely finished my simulation, but I'm pretty confident this
    much is corrrect.

    No, that is the phase sent to the LUT. The total phase accumulator can
    be larger as the need requires.

    Yes, thanks for the correction. The max error? I'm not so worried
    about that exactly. The error is a curve with the max magnitude near
    the middle if nothing further is done to minimize it.

    I'm talking about the LUT. The LUT only considers the first quadrant.

    No, not quite right. There is a LUT with points spaced at 90/255
    degrees apart starting at just above 0 degrees. The values between
    points in the table are interpolated with a maximum deviation near the
    center of the interpolation. Next to 90 degrees the interpolation is
    using the maximum interpolation factor which will result in a value as
    close as you can get to the correct value if the end points are used to
    construct the interpolation line. 90 degrees itself won't actually be
    represented, but rather points on either side, 90±delta where delta is
    360° / 2^(n+1) with n being the number of bits in the input to the sin

    Yes, it is a little tricky because at this point we are working with
    integer math (or technically fixed point I suppose). Rounding errors is
    what this is all about. I've done some spreadsheet simulations and I
    have some pretty good results. I updated it a bit to generalize it to
    the LUT size and I keep getting the same max error counts (adjusted to
    work with integers rather than fractions) ±3 no matter what the size of
    the interpolation factor. I don't expect this and I think I have
    something wrong in the calculations. I'll need to resolve this.

    We are talking about the lsbs of a 20+ bit word. Do you think there
    will be much of a difference in result? I need to actually be able to
    do the calculations and get this done rather than continue to work on
    the process. Also, each end point affects two lines, so there are
    tradeoffs, make one better and the other worse? It seems to get
    complicated very quickly.

    How does that imply a quadratic curve at 90 degrees? At least I think
    like the greats!
  9. rickman

    rickman Guest

    I'm not sure this would be easier. The LUT an interpolation require a
    reasonable amount of logic. This requires raising X to the 5th power
    and five constant multiplies. My FPGA doesn't have multipliers and this
    may be too much for the poor little chip. I suppose I could do some of
    this with a LUT and linear interpolation... lol
  10. rickman

    rickman Guest

    I can't say I follow this. How do I get a 22.5 degree sine table to
    expand to 90 degrees?

    As to my improvements, I can see where most implementations don't bother
    pushing the limits of the hardware, but I can't imagine no one has done
    this before. I'm sure there are all sorts of apps, especially in older
    hardware where resources were constrained, where they pushed for every
    drop of precision they could get. What about space apps? My
    understanding is they *always* push their designs to be the best they
    can be.
  11. Seems what they instead do is implement

    sin(M)cos(L)+cos(M)sin(L) where M and L are the more and less
    significant bits of theta. Also, cos(L) tends to be almost 1,
    so they just say it is 1.

    Well, you can fix it in various ways, but you definitely want 2^n.
    Seems like that is one of the suggestions, but not done in the ROMs
    they were selling. Then the interpolation has to add or subtract,
    which is slightly (in TTL) harder.

    The interpolated sine was done in 1970 with 128x8 ROMs. With larger
    ROMs, like usual today, you shouldn't need it unless you want really
    high resolution.
    -- glen
  12. rickman

    rickman Guest

    Yeah, a friend of mine knows these polynomials inside and out. I never
    learned that stuff so well. At the time it didn't seem to have a lot of
    use and later there were always better ways of getting the answer. From
    looking at the above I see the X^2 term will dominate over the higher
    order terms near 0, but of course the lowest order term, 1, will be the
    truly dominate term... lol In other words, the function of cos(x) near
    0 is just the constant 1 to the first order approximation.
  13. rickman

    rickman Guest

    Interesting. This tickles a few grey cells. Two values based solely on
    the MS portion of x and a value based solely on the LS portion. Two
    tables, three lookups a multiply and an add. That could work. The
    value for sin(M) would need to be full precision, but I assume the
    values of sin(L) could have less range because sin(L) will always be a
    small value...

    I think one of the other posters was saying to add an entry for 90
    degrees. I don't like that. I could be done but it complicates the
    process of using a table for 0-90 degrees.

    I'm not using an actual ROM chip. This is block ram in a rather small
    FPGA with only six blocks. I need two channels and may want to use some
    of the blocks for other functions.

    Thanks for the advice to everyone.
  14. Guest

    I see, I assumed you had an MCU (or fpga) with multipliers

    considered CORDIC?

  15. rickman

    rickman Guest

    I should learn that. I know it has been used to good effect in FPGAs.
    Right now I am happy with the LUT, but I would like to learn more about
    CORDIC. Any useful referrences?
  16. rickman

    rickman Guest

    It could also be a difficult problem to solve without some sort of
    exhaustive search. Each point you fudge affects two curve segments and
    each curve segment is affected by two points. So there is some degree
    of interaction.

    Anyone know if there is a method to solve such a problem easily? I am
    sure that my current approach gets each segment end point to within ±1
    lsb and I suppose once you measure THD in some way it would not be an
    excessive amount of work to tune the 256 end points in a few passes
    through. This sounds like a tree search with pruning.

    I'm assuming there would be no practical closed form solution. But
    couldn't the THD be calculated in a simulation rather than having to be
    measured on the bench?
  17. Jon Kirwan

    Jon Kirwan Guest

    You already know why. Just not wearing your hat right now.
    Two terms of taylors at 0 of cos(x) is x + x^2/2. Two at
    sin(x) is 1 + x. One is quadratic, one is linear.

  18. rickman

    rickman Guest

    I don't think you are clear on this. The table and linear interp are as
    good as they can get with the exception of one or possibly two lsbs to
    minimize the error in the linear interp. These is no way to correct for
    this by adding a second harmonic, etc. Remember, this is not a pure
    table lookup, it is a two step approach.

    Ok, if you are compensating for the rest of the hardware, that's a
    different matter...
  19. Tim Williams

    Tim Williams Guest

    Oops, should be:
    cos(x) ~= 1 - x^2/2
    sin(x) ~= x (- x^3/6)
    Mixed up your odd/even powers :)

    Of course, the cubic is less important than the quadratic, so sin(x) ~= x is
    sufficient for most purposes.

    I suppose you could potentially synthesize the functions using low power
    approximations (like these) and interpolating between them over modulo pi/4
    segments. Probably wouldn't save many clock cycles relative to a CORDIC,
    higher order polynomial, or other, for a given accuracy, so the standard
    space-time tradeoff (LUT vs. computation) isn't changed.

  20. Jon Kirwan

    Jon Kirwan Guest

    hehe. yes. But it gets the point across, either way. Detail I
    didn't care a lot about at the moment.
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