Techniques for synthesizing
two pole filters are well known. A number
of techniques introduce unpleasant sounding
transients in the
filter
response when the frequency or damping of the filter is rapidly
changed. We will demonstrate a
difference equation for a
digital
filter in which both the frequency and the damping can be changed
without producing discontinuities in the filter output. The technique
is based on the well known property of the product of
complex numbers.
In polar form, the magnitude of the product of two numbers is the
product of their magnitudes and the angle of the product is the sum of
their angles. Successive multiplies can produce a rotating vector
whose real or imaginary parts are samples of constant amplitude
sine
waves or of exponentially damped sine waves. The frequency and
damping of the resulting waves can be changed without changing the
amplitude of the waves. These properties can be used to make a
digital filter whose input, frequency, and damping can all be
functions of time in a useful way. Two alternative structures are
additionally proposed, having better numerical properties for low-cost
fixed-point implementations. A program to demonstrate some musical
applications of these filters will be shown.
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