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Stick Diagram Design

When designing a filter with a prescribed magnitude response by the Bode ``stick diagram'' method, we think in terms of poles and zeros ``breaking'' at certain frequencies. For example, in the term

$\displaystyle H_n(j\omega) = \frac{-p_n}{j\omega-p_n}

which is scaled to have unity gain at $ \omega=0$ , the pole is said to ``break'' at frequency $ \omega=\vert p_n\vert$ . This is easily seen to be the $ -3$ dB point of the term, since

$\displaystyle H_n(jp_n) = \frac{-p_n}{j (\pm p_n)-p_n} = \frac{1}{1\pm j}

which has magnitude $ 1/\sqrt{2} \approx -3$ dB. Thus, the gain of the term is approximately constant out to $ \omega=\vert p_n\vert$ , where it reaches its $ -3$ dB, or ``half power'' frequency, followed by its asymptotic roll-off of $ -6$ dB per octave. A zero term $ H_m(j\omega) =
(j\omega-z_m)/(-z_m) = 1-j\omega/z_m$ similarly starts out flat, reaches magnitude-gain $ +3$ dB at its break frequency $ \omega=\vert z_m\vert$ , and asymptotically approaches $ +6$ dB per octave for $ \omega\gg
\vert z_m\vert$ .

The Bode design procedure is then to start at dc ($ \omega=0$ ) and map out break-frequencies for poles and zeros so as to follow the desired response as closely as desired. Since this tool is commonly applied in control-system design, there is also usually consideration for the phase plot as well, which has similarly simple behavior.17

The basic Bode ``stick diagram'' consists only of straight line segments, each having slope given by some integer number of nepers per neper (or integer multiple of $ \pm 6$ dB/octave, etc.), with the knowledge that the actual response will be a smoothed version of the stick diagram, traversing the $ \pm 3$ dB points at line-segment intersections corresponding to isolated breaking zeros and poles, respectively.

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``Spectral Tilt Filters'', by Julius O. Smith IIIand Harrison F. Smith, adapted from publication arXiv:1606.06154 [cs.CE].
Copyright © 2021-06-18 by Julius O. Smith IIIand Harrison F. Smith
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University