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Is this legit?

Discussion in 'Electronic Design' started by Don Lancaster, May 27, 2013.

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  1. http://en.wikipedia.org/wiki/File:Comparison_Butterworth_Legendre_Chebyshev.svg

    It should not take fifty years for something this revolutionary to happen.

    Ultimate high frequency slopes of fiters without stopband zeros should
    be similar regardless of passband frequencies and damping.

    When all is said and done a fourth order active lowpass consisting of
    two cascaded second order seconds has four control variables, two
    frequency setters and two damping setters. It knows nothing about any
    exotic math used to adjust a theory behind those four variables.

    There's a rumor that the Acrive Filter Cookbook missed this.

    Is there any point in plotting a thousand possible fourth order lowpass
    filters in hope of finding something significantly better than
    Butterworth yet remains monotonic?

    --
    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
     
  2. Bill Sloman

    Bill Sloman Guest

    But since most of us design filters from cook-book recipes, it could easily take that long to filter through to general knowledge.

    That what what the Legendre filter was invented for (in 1958, if this reference is to be believed).

    http://www.crbond.com/papers/lopt.pdf
    There are an infinite number of Chebyshev low pass filters. Williams and Taylor
    (ISBN-0-07-07-0434-1) give worked examples with up to 1dB of pass-band ripple. Tolerating more ripple could presumably offer a sharper cut-off.

    Williams and Taylor doesn't make explicit reference to the Legendre polynomials, but my mathematical skills aren't up to working out if they are covered anyway.

    Presumably, the Legendre filter screws up the phase response even worse than the Butterworth does.
     
  3. Martin Brown

    Martin Brown Guest

    Qualitatively it is right. Butterworth is flattest in the passband,
    Chebeshevs is rough in passband with steepest attenuation outside.
    Legendre is one of many other compromises that sit in between.

    They are practical engineering versions of limiting forms of elliptic
    filters the mathematics of which were largely worked out by Cauer and
    others. The Wiki for this includes mathematics of other filters in the
    family and graphs which show their responses warts and all.

    http://en.wikipedia.org/wiki/Wilhelm_Cauer
    http://en.wikipedia.org/wiki/Elliptic_filter

    AFAIK Elliptic is the sharpest cutoff you can get but you pay for it in
    ripple in both passband and stopband. There is no free lunch!

    You can trade flat passband (and/or stopband attenuation) for a steeper
    transitional zone. It has been known for more than 50 years at least to
    mathematicians and those who design filters and programs that design
    filters. The practical problem is that to realise them in analogue you
    need insanely close tolerance components for the higher orders.

    That restriction is now lifted for DSP although then you have to worry
    about numerical stability of the filter implementation.
    Actually the whole point about it is that it *requires* exotic
    mathematics to understand exactly why such tradeoffs are possible.

    Requiring students to work out the behaviour of

    -L-+-L-
    C
    ---+---

    followed by two and three consecutive stages quickly shows that it isn't
    at all as simple as you think. The later stages affect the first stage
    impedance and there is interesting optimisation to be done.
    Not really. ISTR the Butterworth is the steepest of the monotonic
    filters. You have to trade some ripple somewhere to go steeper.

    The Legendre filter is closest to what you asked for. You might find
    this paper helpful if you have IEEE subs.

    http://ieeexplore.ieee.org/xpl/logi...ieee.org/stamp/stamp.jsp?tp=&arnumber=5268868

    They seem to have parameterised the problem in a form more amenable to
    electronic engineers way of thinking about the problem.
     
  4. o pere o

    o pere o Guest

    The graph doesn't seem right to me. Had to do it myself for n=3:
    https://www.dropbox.com/s/wtnttry49d7vtkg/legendre.png

    Legendre lies between But and Cheb, as expected.

    Pere
     

  5. .... which I called the "slight dips" filter.

    I suspect there is no discernible difference between Legendre and SD.
    Except one takes hairy and unstable math, the other simply averages four
    analog values.

    That "just plain wrong" plot got me off on this tangent.

    There is also the question as to whether true fourth order solutions
    have any advantage over cascaded second order ones.

    --
    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
     
  6. Yeah, an even worse step response than a butterworth.

    George H.
     
  7. Bill Sloman

    Bill Sloman Guest

    Williams and Taylor is better, if harder to read.

    Obviously not.
    Legendre is supposed to be monotonic.
    There's nothing hairy or unstable about the math. Some implementations may be both, but a pair of Sallen and Keys second order sections won't be either.
    There's no evidence that the plot was actually wrong.
    Cascaded - non-identical but non-interacting - second order sections are the most practical way of realising fourth order solutions, though not usually the cheapest. I like Sallen and Keys sections with a little bit of well-controlled gain - E96 resistors are the cheapest way of getting idiosyncratic component ratios.
     
  8. josephkk

    josephkk Guest

    I think you need to consider phase response as well in the passband. In
    some measurement situations and use situations it is very important.

    ?-)
     
  9. So the Wiki picture is wrong... the Butterworth should be rolling off
    faster at the higher frequencies. I wonder how much different it
    really is.
    Do your books give any values? Can you have a two pole Legendre
    filter?
    (From my limited understanding a two pole filter can be described by a
    corner frequency and a Q.... OK for filters with no zero's... no stop
    band ripple.)

    George H.
     
  10. amdx

    amdx Guest

  11. Bill Sloman

    Bill Sloman Guest

    The Wiki picture probably isn't wrong - it's merely confined to the region near the pass-band where the differences matter.
    I don't think so.
    There aren't enough adjustable parameters in two-pole filters to allow the finer differences between filters - you are confined to Bessel, Butterworthand Chebyshev. Williams and Taylor's table of parameters for the more interesting filters start at the third and fourth order filters.
     
  12. Grin... Yup, dat's all I know :^)
    For a physicist the simple harmonic oscillator is enough.
    Beyond that engineers take over..
    ....ducking of cover...


    Williams and Taylor's table of parameters for the more interesting
    filters start at the third and fourth order filters.

    That makes sense,
    I guess I like simple things, two poles means fewer parameters to
    fit... Easier to make.
    I've been testing all these two pole SV filters, made with opamps,
    switched R's and two matched caps. (f_max <100 kHz.) They've been
    great, I don't know how they'll age... all npo ceramic caps... Q from
    0.7 to 10.

    George H.
     
  13.  
  14. Fred Abse

    Fred Abse Guest

    GROAN!
     
  15. miso

    miso Guest

    I think you missed the point of the original post. Leave out elliptic
    filters since we don't want transmission zeroes.

    The point was at infinity, N poles is N poles, so dammit, the falloff
    rate should be the same for all classes of filters. I think at infinity,
    that is true. But for practical purposes, you aren't at infinity. The
    poles are not all at the same frequency, so as you look at the response
    with increasing frequency, you are getting further away from DIFFERENT
    poles.
     
  16. miso

    miso Guest

    You need to be specific as to what a "direct high order solution" is. If
    we are talking ladder filters, you have less component sensitivity than
    cascaded biquads. But in continuous time active filters, ladders always
    have more components than a cascade of biquads simply because you need
    the inverted phase signal.

    If I may quote Dan Senderovich, "It is hard to move a ladder."
     
  17. PeteD

    PeteD Guest

    The shape of the Chebychev's ripple says it's a 4th order. Given that
    the Butterworth has 60 dB loss at a normalized frequency of 10, it's
    only a 3rd order.
     
  18. PeteD

    PeteD Guest

     
  19. Jon Lark

    Jon Lark Guest

    ..
    Regarding your last question "Is there any point..."
    ..
    Apparently so. A good example is the Linkwitz-Riley crossover network. It's not "optimal" in any classic sense. Mr Riley, while fooling around with filters, discovered that by cascading two 2nd order Butterworth filters, heobtained a 4th order filter, later dubbed the "Linkwitz Riley" filter. He then discovered that a L-R lowpass combined with a L-R Highpass results in a loudspeaker crossover network with near ideal characteristics: The two filters combine to produce an exact flat voltage response throughout the entire spectrum. The Highpass and Lowpass outputs remain exactly 360 degrees out of phase (ie, in phase) throughout the entire frequency spectrum. Thisphase response eliminates the "Lobing" problem in loudspeaker design. A fine example of tinkering that produced useful results.
    ..
    Regarding the Legendre filter. It has been proven to be the monotonic filter that is optimum in the sense that it has maximum attenuation in the stopband. Its response is steeper than the Butterworth (and all other filtersthat are monotonic in the passband and stopband) at frequencies close to the characteristic frequency, but it eventuall attains a slope of 6n per octave, where n is the order of the filter.
     
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