The output of our digital filter is . We have not so far
provided a mechanism to input a series of samples of a time function
into the filter. An input is necessary if we wish to use the filter
in a traditional way--that is, to ``filter'' samples of a sound wave.
A number of possibilities exist for adding input samples to our
difference equation. We have chosen to add input samples to .
With this modification, Equations (2-3) become

where are samples of the input to the filter. The transfer function of the filter is then

(14) | |||

(15) |

This is a so-called ``all pole'' filter, since its two zeros are at and therefore do not affect the magnitude response. This choice of input can be interpreted as driving the poles in

We have compared both the sound and the spectrum of our filter output with the outputs of other filter difference equations realizations and we have detected no differences.

We may alternatively add the input samples
on the right-hand side of Eq.(13)
to obtain the transfer function

(16) | |||

(17) |

This choice of input introduces a finite zero at , which is the real part of the location of both poles. This can be interpreted as driving the poles in

(18) | |||

(19) | |||

(20) |

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

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