A *Bode Plot* of a filter frequency response
separately graphs the log-magnitude and phase versus
log-frequency.^{16}We are only concerned here with log-magnitude plots, and will omit
consideration of the Bode phase plot, which happen to behave as
expected naturally. The usual choice of log-magnitude units is
*decibels* (dB)
(relative to an arbitrary reference, such as
), and the
log-frequency axis is typically either in *octaves*
(
) or *decades*
(
). Thus, a single pole is said to give a
roll-off of
dB per octave or
dB per decade. Octaves are
typical in audio signal processing while decades are typical in the
field of automatic control.

Figure 1 illustrates the Bode plot and its associated ``stick diagram'' (comprised of asymptotic gains) for a single pole at . We see that the response is flat for low frequencies, drops to dB at the break frequency , and approaches the dB per decade asymptote, reaching the asymptote quite well by one decade above the break frequency at .

For a general filter transfer function having poles and zeros

the Bode plot can be expressed as

where , , , and is typically or . For mathematical simplicity, however, we'll choose instead and , giving

where . In this choice of units, integrators give a magnitude roll-off of ``nepers per neper'', while differentiators give a slope of in the Bode magnitude plot

Our problem is to find poles and zeros of to minimize some norm of the error

where denotes the derivative of with respect to , is the desired slope of the log-magnitude frequency-response versus log frequency , and denotes a real, nonnegative weighting function.

As a specific example, for the Chebyshev norm and a uniform weighting between frequencies and , we have

That is, we wish to minimize the worst-case deviation between the desired slope and the achieved slope over a specific (audio) band .

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