Filter slopes

[Tom May]

Brian.Dekok at ca.jdsunph.com writes:

> Can filters slopes be obtained in other multiples than 6dB/oct? For
> example, I feed a signal to a first-order and a second-order filter
> in parallel, with the same fc. If the outputs are mixed together,
> what is the final slope of the filtered signal?

You'll probably get thousands of answers, but here goes.  Intuitively,
it will be only 6dB/oct because at high frequencies, the 12dB/oct
filter is letting negligible signal through and the 6dB/oct is only
filtering at 6dB/oct.

Here's the frequency domain math:

                              1
6dB/oct filter response is  -----
                            (s+1)

                               1
12dB/oct filter response is -------
                            (s+1)^2

             s+2
The sum is -------  which for large s becomes 1/s which is a 6dB rolloff.
           (s+1)^2

It intially rolls of a bit more steeply than a 6dB/oct filter because
of the double pole at s = -1, but the zero at s = -2 counteracts that.

fTom.