Is this legit?

Looks funny. The Butterworth is dropping about 15 dB/octave but the Legendre is dropping about 32.

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).

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.

The graph doesn't seem right to me. Had to do it myself for n=3:

Legendre lies between But and Cheb, as expected.

Pere

Would you happen to have n=4 handy?

I have three copies of the Active Filter Cookbook: one at work, one at home, and one up here in the cabin in Truckee. Good stuff for a quick filter.

Working on another edition?

I don't see any big advantage to the Legendre filter. It just gives up passband flatness for a few dBs of stopband rejection. A 0.1 dB Chebychev does that already.

... 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.

Speaking of active filters...

Quite often, you can have another pole or two from decoupling caps or HF correction caps or additional RC. Those real poles come for free, and they could be used to improve response.

I don't know of any analytical design method that would allow to calculate filter with a number of extra real poles; so I do that by optimization.

Vladimir Vassilevsky DSP and Mixed Signal Designs

Direct high order solutions could be nightmare as far as component values and tolerances. The advantages are less of parts and perhaps higher dynamic range.

Vladimir Vassilevsky DSP and Mixed Signal Designs

I am thinking that perhaps the "improvement" this filter is due to GBW limitation of the op amp?

What's worse is to have a free active real pole or two and try to merge that into an LC filter design!

..

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ly take that long to filter through to general knowledge.

erence is to be believed).

Taylor

pple. Tolerating more ripple could presumably offer a sharper cut-off.

omials, but my mathematical skills aren't up to working out if they are cov ered anyway.

han the Butterworth does.

Yeah, an even worse step response than a butterworth.

George H.

I have to mention that whenever low waveform (not harmonic) distortion is needed, one requires linear phase response, which leads one to a Bessel filter. They are widely used in radar systems.

Joe Gwinn

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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 eithe r.

There's no evidence that the plot was actually wrong.

Cascaded - non-identical but non-interacting - second order sections are th e most practical way of realising fourth order solutions, though not usuall y the cheapest. I like Sallen and Keys sections with a little bit of well-c ontrolled gain - E96 resistors are the cheapest way of getting idiosyncrati c component ratios.

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happen.

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

?-)

I checked my books -- the Butterworth is maximally flat in the passband, which is different from the Legendre filter.

Yes, once you get a good ways away from the corner frequency, they're all going to have a slope of (20dB) * (pole count). The difference is in how soon that slope is reached, and how much lower (or higher) the line is from a Butterworth.

It'd be interesting to know how bad the phase response is, how sensitive the filter is to changes in the pole positions, and, of course, how sensitive practical realizations are to component value variations.

ow

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.

ttdesign.com- Hide quoted text -

FWIW,

Don't forget this Chebychev optimized for second harmonic reduction.

Mikek

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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, Butterworth and Chebyshev. Williams and Taylor's table of parameters for the more inte resting filters start at the third and fourth order filters.

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e finer differences between filters - you are confined to Bessel, Butterwor th and Chebyshev.

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