Complementary IIR Filters

[Chris Townsend]

At 07:56 AM 6/24/98 -0700, you wrote:
>Hello:
>
>>I'm trying to design complementary IIR filters (i.e. magnitude response
>>adds to one) so I can make a two band crossover network.  Of course I could
>>just use standard low pass and high pass filters with the same cutoff
>>frequencies, but in general there will be phase discrepancies between the
>>two filters such that if the bands get mixed back together a notch will
>>occur at the cutoff.  Any of you know of a good way to do this?  
>
>You have the signal with magnitude response of one in the system input (or
>delayed version thereof), so it may do just as well to employ one filter
>and subtract that filter output from the input for the complementary filter
>signal.  This probably takes fewer computrons, too.

Others have suggested this, and even though it does work, it doesn't work
all that well, because if one subtracts the input from the recursively
filtered response there is generally a non-linear phase difference between
the two responses and this requires an all pass filter to compensate.  I
find without this compensation that you do get a high pass filter by
subtracting the input from a low pass filter, but it will be far from
ideal.  Specifically,  the complementary version (the high pass in this
case) seems to have roughly a 6dB per octave slope regardless of the order
of the original (low pass) filter, and there will be a bump around the
cutoff frequency (about 3db for second order, 6dB for 4th order and 18db
for 8th order filters).  So if your specification are extremely loose this
will work, but for me I figure maximally flat 12-18dB per octave filters
would be nice.  Please let me know if any of you have gotten better results
than this, but I think the phase discrepancies will always cause a
significant deviation from the ideal response.



Chris




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Chris Townsend
DSP Engineer
Arboretum Systems, Inc.
http://www.arboretum.com
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"Hyperprism is a wonderful invention" - Brian Eno
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