Filter slopes

Tom May tom at
Tue Nov 2 00:26:39 CET 1999

Brian.Dekok at 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:

6dB/oct filter response is  -----

12dB/oct filter response is -------

The sum is -------  which for large s becomes 1/s which is a 6dB rolloff.

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.


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