Real-time Bilinear Transform Library

[OygoSound]

Hey devs-

I was wondering if anyone of you are familiar with a C/C++ library or header that does a bilinear transformation, ideally converting up to a 4th-order transfer function? Any help would be greatly appreciated

THANKS!

[neotec]

Well ... do it 'per-hand':

Code: Select all

1   z - 1
- * -----
k   z + 1

as² + bs + c


 a    z² - 2z + 1    b    z - 1
--- * ----------- + --- * ----- + c
 k²   z² + 2z + 1    k    z + 1

 a    z² - 2z + 1   bk     z² - 1      ck²   z² + 2z + 1
--- * ----------- + -- * ----------- + --- * -----------
 k²   z² + 2z + 1   k²   z² + 2z + 1    k²   z² + 2z + 1
 
az² - 2az + a + bkz² - bk + ck²z² + 2ck²z + ck²

z²(a0 + a1*k + a2*k²) + z(2*a2*k² - 2*a0) + (a0 - a1*k + a2*k²)
---------------------------------------------------------------
z²(b0 + b1*k + b2*k²) + z(2*b2*k² - 2*b0) + (b0 - b1*k + b2*k²)


         1
LP: -----------
    s² + 2s + 1

         s²
HP: -----------
    s² + 2s + 1

         s
BP: -----------
    s² + 2s + 1

       s² + 1
BR: -----------
    s² + 2s + 1


k = tan(PI * FC/FS)

Common:
    B0 = 1 + 2 / Q * k + k²
    B1 = 2 * k² - 2
    B2 = 1 - 2 / Q * k + k²

LP:
	A0 = k²
	A1 = 2 * k²
	A2 = k²

HP:
	A0 = 1
	A1 = -2
	A2 = 1
	
BP:
	A0 = k
	A1 = 0
	A2 = -k
	
BR:
	A0 = 1 + k²
	A1 = 2*k² - 2
	A2 = 1 + k²

[OygoSound]

so what you meant by 'per-hand' was copy/paste from the internet

Anyway, what I needed was a solution for a 4th order polynomial, and I needed it to run real-time, so I basically followed this method:
1) break up the 4th ordered analog prototype into four 1st ordered sections using Ferrari's method http://en.wikipedia.org/wiki/Quartic_fu ... s_solution
2) do a bilinear transformation using the s=2*FS(1-z)/(1+z) substitution
3) combine the results into two cascaded 2nd ordered sections

[neotec]

Copy'n'paste from the internet? Huh? Well ... if you think so ... nevermind.

[Music Engineer]

Copy'n'paste from the internet? Huh? Well ... if you think so ... nevermind.

as a new member, he probably did not expect someone being so kind to elaborately derive all these equations. for fourth order, i'd certainly prefer to let the algebra be done by some computer algebra system such as maxima. i think, it's going to be messy and error-prone to do by hand.

[neotec]

Copy'n'paste from the internet? Huh? Well ... if you think so ... nevermind.

as a new member, he probably did not expect someone being so kind to elaborately derive all these equations. for fourth order, i'd certainly prefer to let the algebra be done by some computer algebra system such as maxima. i think, it's going to be messy and error-prone to do by hand.

Yeah, you're possibly right.

I did this by hand to get known to it. In my opinion it is good to know how things work. And if you do this z-transform using variables/placeholders, then you only have to do it once and can simply insert different values and get directly usable filters.

[OygoSound]

thank you, neotec and robin, for your responses. and i agree with you, neotec, about doing things by hand, its certainly the more erudite approach to engineering.

[neotec]

Sorry, if my post was a bit ... don't know what it was, though sry.

As of my experience, searching for a library, which is free and easy to use will take you (mostly) longer, than understanding it and doing it on your own. But if anybody knows such a lib (which is free and easy to use) please post it here.

[mystran]

Copy'n'paste from the internet? Huh? Well ... if you think so ... nevermind.

as a new member, he probably did not expect someone being so kind to elaborately derive all these equations. for fourth order, i'd certainly prefer to let the algebra be done by some computer algebra system such as maxima. i think, it's going to be messy and error-prone to do by hand.

Having recently derived (and then debugged after it behaved a bit weird) some fancy filters, I'd say the that sort of stuff tends to be messy and error prone even when done in a CAS. Doing it without a CAS tends to be twice as messy and error prone, so definitely +1 on your suggestion.

[asomers]

This implementation might be too complete for your needs but the BLT is in there http://code.google.com/p/dspfilterscpp/