State variable filter

The example given below can produce simultaneous lowpass, highpass and bandpass outputs from a single input. This is a second-order (biquad) filter. Its derivation comes from rearranging a high-pass filter's transfer function, which is the ratio of two quadratic functions. The rearrangement reveals that one signal is the sum of integrated copies of another. That is, the rearrangement reveals a state variable filter structure. By using different states as outputs, different kinds of filters can be produced. In more general state variable filter examples, additional filter orders are possible with more integrators (i.e., more states).

Schematic for a state variable filter

The signal input is marked V_in; the LP, HP and BP outputs give the lowpass, highpass and bandpass filtered signals respectively.

For simplicity, we set:

${\displaystyle R_{f1}=R_{f2}}$

${\displaystyle C_{1}=C_{2}}$

${\displaystyle R_{1}=R_{2}}$

Then:

${\displaystyle F_{0}={\frac {1}{2\pi R_{f1}C_{1}}}}$

${\displaystyle Q=\left(1+{\frac {R_{4}}{R_{q}}}\right)\left({\frac {1}{2+{\frac {R_{1}}{R_{g}}}}}\right)}$

The pass-band gain for the LP and HP outputs is given by:

${\displaystyle A_{HP}=A_{LP}={\frac {R_{1}}{R_{G}}}}$

It can be seen that the frequency of operation and the Q factor can be varied independently. This and the ability to switch between different filter responses make the state-variable filter widely used in analogue synthesizers.

Values for a resonance frequency of 1 kHz are R_f1 = R_f2 = 10k, C₁ = C₂ = 15nF and R₁ = R₂ = 10k

State variable filters are frequently used for modifying frequency response in audio signal processing. At low Q settings they are often used in parametric equaliser circuits, and at high or variable Q settings to create resonant filter modules in analog synthesizers. For manual control of frequency, R1 and R2 in the section above may be replaced by a dual potentiometer; and for voltage control, the devices U2 and U3 may be replaced by voltage controlled amplifiers or operational transconductance amplifiers.

• Linear filter
• Biquad filter

• Texas Instruments' UAF42 Universal Active Filter datasheet
• Analog Devices Interactive Design Tools