Next 
Prev 
Up 
Top

Index 
JOS Index 
JOS Pubs 
JOS Home 
Search
TwoPole
Partial Fraction Expansion
Note that every real twopole resonator can be broken up into a sum of
two complex onepole resonators:

(B.7) 
where
and
are constants (generally complex). In this
``parallel onepole'' form, it can be seen that the peak gain is no
longer equal to the resonance gain, since each onepole frequency
response is ``tilted'' near resonance by being summed with the
``skirt'' of the other onepole resonator, as illustrated in
Fig.B.9. This interaction between the positive and negativefrequency
poles is minimized by making the resonance sharper (
),
and by separating the pole frequencies
. The
greatest separation occurs when the resonance frequency is at
onefourth the sampling rate (
). However,
lowfrequency resonances, which are by far the most common in audio
work, suffer from significant overlapping of the positive and
negativefrequency poles.
To show Eq.(B.7) is always true, let's solve in general for
and
given
and
. Recombining the righthand side
over a common denominator and equating numerators gives
which implies
The solution is easily found to be
where we have assumed
im
, as necessary to have a
resonator in the first place.
Breaking up the twopole real resonator into a parallel sum of two
complex onepole resonators is a simple example of a partial
fraction expansion (PFE) (discussed more fully in §6.8).
Note that the inverse z transform of a sum of onepole transfer
functions can be easily written down by inspection. In particular,
the impulse response of the PFE of the twopole resonator (see
Eq.(B.7)) is clearly
Since
is real, we must have
, as we found above
without assuming it. If
, then
is a real sinusoid
created by the sum of two complex sinusoids spinning in opposite
directions on the unit circle.
Next 
Prev 
Up 
Top

Index 
JOS Index 
JOS Pubs 
JOS Home 
Search
[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]