Equations (1-3) can be modified to compute the response of a finite filter simply by changing to some value different from unity. From Eq.(1), will be raised to the th power, and will be

If is less than one, Eq.(4) shows the homogeneous solution to Eq.(1) is a sinusoid times a decaying exponential. This solution is the response of a two pole filter.

Equations (2-3) are unchanged except that the
coefficients and become

If is the input to a digital-to-analog converter operating at a sampling rate of samples per second, then and can be written

where

Equations (7-10) are valid for less than . For between and , the generated frequency will be reflected about .

Negative values of will generate exponentially increasing sinusoids which can be musically useful.

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