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Normalized Second-Order Continuous-Time Lowpass Filter

The transfer function of a normalized second-order lowpass can be written as

$\displaystyle H_l(s) \;=\;\frac{1}{\tilde{s}^2+\frac{1}{Q}\tilde{s}+ 1}

where the normalization maps the desired -3dB frequency $ \omega_c $ to 1, i.e.,

$\displaystyle \tilde{s}\;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\frac{s}{\omega_c },

and the ``quality factor'' $ Q$ is defined as

$\displaystyle Q \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\frac{\omega_c }{2\alpha}

where $ \alpha$ is a convenient definition for bandwidth in radians per second, given by minus the real part of the complex-conjugate pole locations $ p$ and $ \pc$ in the $ s$ plane:1

$\displaystyle p,\pc\;=\;-\alpha \pm j\sqrt{\omega_c ^2-\alpha^2}

Here we assume $ 0<\alpha<\omega_c $ , so that the poles have nonzero imaginary parts.

A second-order Butterworth lowpass filter is obtained for $ Q=1/\sqrt{2}$ . Larger $ Q$ values give the ``corner resonance'' effect often used in music synthesizers.

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``Digital State-Variable Filters'', by Julius O. Smith III.
Copyright © 2021-02-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University