Whatever poles are chosen for the least-damped part, and however they
are computed (provided they are stable), the damped part can be
computed from the full impulse response and parametric part using
*inverse filtering*, as illustrated in the computed examples
above. The inverse filter is formed from zeros equal to the estimated
resonant poles. When the inverse filter is applied to the full
resonator impulse, a ``residual'' signal is formed which is defined as
the impulse response of the leftover, more damped modes. The residual
is in exactly the nonparametric form needed for commuting with the
string and convolving with the string excitation signal, such as a
``pluck'' signal. Feeding the residual signal to the parametric
resonator gives the original resonator impulse response to an
extremely high degree of accuracy. The error is due only to numerical
round-off error during the inverse and forward filtering computations.
In particular, the least-damped resonances need not be accurately
estimated for this to hold. When there is parametric estimation error,
the least-damped components will fail to be completely removed from
the residual signal; however, the residual signal through the
parametric resonator will always give an exact reconstruction of the
original body impulse response, to within roundoff error. This is
similar to the well known feature of linear predictive coding that
feeding the prediction error signal to the LP model always gives back
the original signal
[300].

The parametric resonator need not be restricted to all-pole filters,
however, although all-pole filters (plus perhaps zeros set manually to the
same angles but contracted radii) turn out to be very convenient and simple
to work with. Many filter design techniques exist which can produce a
parametric part having any prescribed number of poles and zeros, and
weighting functions can be used to ``steer'' the methods toward the
least-damped components of the impulse response. The equation-error method
illustrated in Fig. 8.13 is an example of a method
which can also compute zeros in the parametric part as well as poles.
However, for inverse filtering to be an option, the zeros must be
constrained to be *minimum phase* so that their inverses will be stable
poles.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University