[music-dsp] Complex coefficients on a filter

[Martin Eisenberg]

> I finally get it. But could you describe formally what happens in
> the z-domain when I drop the complex part?

Hmm, I don't know how to get any more formal than I've already been...
In short, the base filter is averaged with a replica that has all
poles and zeros conjugated. The justification is all there. Note that
I made a slight mistake and lost the minus sign in the exponent of z
when finding the Z transform of a conjugated time series; fortunately,
correcting that does not change the result.

> Filter with zero at Pi/2: H(z)=(z-j)
>
> When I drop the complex part the filter becomes H(z)=(z-j)+X=1
> where X is the mirrored filter that's added in the process.

That would be H(z) = (z-j + X)/2 = z, so X = z+j = conj(conj(z)-j),
the conjugate of z-j evaluated at the conjugate of z.


Martin