# [music-dsp] Filter design question

Luigi Castelli superbigio at yahoo.com
Fri Oct 12 14:47:00 EDT 2007

```> i hope pointers are acceptable in lieu of just doing this for you.

Absolutely. Pointer are more than acceptable but I'll be back with more
questions if I get stuck...  ;-)

Thanks Robert.

- Luigi

--- robert bristow-johnson <rbj at audioimagination.com> wrote:

>
> > ----- Original Message -----
> > From: "Luigi Castelli" <superbigio at yahoo.com>
> > To: music-dsp at music.columbia.edu
> > Subject: [music-dsp] Filter design question
> > Date: Fri, 12 Oct 2007 09:45:29 -0700 (PDT)
> >
> >
> > Hi,
> >
> > the transform function of a high order IIR lowpass filter is given
> by:
> >
> > H(z) = { [ (1 - alpha) / 2 ] * [ (1 + z^-1) / (1 - alpha*z^-1) ]
> }^K
> >
> > where K is the filter order, alpha is the filter parameter which
> can be
> > determined by the 3dB cutoff frequency wc, i.e. |H(wc)|^2 = 0.5
> >
> > How do I derive the close-form formula for alpha ?
>
> plug & chug!
>
> plug:  z^-1 = e^(-jw) = cos(w) - j*sin(w)
>
> collect real terms together and imag terms together.
>
> then get an expression for
>
>    |H(e^(jw))|^2
>
> and then set it to 1/2.  if you apply some common trig identities,
> you'll see that the sin(w) terms get squared and and added to
> cos^2(w) terms (go to 1) and you'll be left with only cos(w) terms
> that you can solve for.  fortunately the ^K power is applied to
> *both* numerator and denominator, otherwise you would have a bitch of
> a time solving a high order polynomial.  the magnitude of the stuff
> inside the {..}^K gets set to (1/2)^(1/K), so a closed form solution
> is still doable.
>
> > Thank you.
>
> i hope pointers are acceptable in lieu of just doing this for you.
>
> --
>
> r b-j                  rbj at audioimagination.com
>
> "Imagination is more important than knowledge."
>
> --
> dupswapdrop -- the music-dsp mailing list and website:
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> http://music.columbia.edu/cmc/music-dsp
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>

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