It is easy to realize a filter transfer function in state-space form
by means of the so-called *controller-canonical form*, in which
transfer-function coefficients appear directly in the matrices of the
state-space form. In our case, the state-space model becomes

where
,
,
, and
for a normalized
Butterworth lowpass filter. *I.e.*,

This system is diagrammed in Fig.3. Note that due to the chain
of integrators in controller-canonical form, we also have available
the bandpass and highpass outputs as shown in the figure. Each
integrator is typically implemented (in analog circuits) by means of
an *operational amplifier* (``op amp'') having a capacitor in its
feedback loop.^{1}

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