State variable filter

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "State variable filter" – news · newspapers · books · scholar · JSTOR (December 2019) (Learn how and when to remove this message)

A state variable filter is a versatile type of active filter used in electronic circuits, prized for its ability to provide precise control over filter parameters, especially the quality factor (Q factor). The design typically consists of one or more integrator stages connected in a feedback configuration, enabling simultaneous output of multiple filter responses such as low-pass, high-pass, and band-pass from a single input signal.^[1]

The classical implementation often employs operational amplifiers configured as integrators. By manipulating voltages corresponding to the input signal and its successive integrals, these filters achieve highly accurate frequency response tuning. Another common approach employs a multiplying digital-to-analog converter (MDAC) for the filter core, enabling digitally controlled adjustments.^[2]

The well-known KHN biquad state variable filter exemplifies the core principle. It is a second-order filter circuit capable of producing low-pass, high-pass, and band-pass outputs simultaneously from a single input, providing flexibility for various signal processing applications.^[3]

This is accomplished by utilizing two integrator stages arranged so that the input signal and its first and second integrals are combined in feedback, yielding outputs at different frequency responses corresponding to each integral. The advantage lies in the ability to independently control the center frequency and Q factor, allowing for precise tuning.

In the schematic, the input voltage is denoted as V_in. The outputs labeled LP, HP, and BP correspond to the low-pass, high-pass, and band-pass filtered signals respectively.^[4][5]

Mathematically, assuming selected component values for convenience:

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

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

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

The operating frequency (\(f_0\)) and quality factor (Q) are described by:

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

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

The passband gain for low-pass and high-pass outputs is:

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

This independence of frequency and Q adjustment and the simultaneous provision of multiple response outputs make the state variable filter highly adaptable. For instance, a commonly used set of component values for around 1 kHz center frequency includes:

• ${\displaystyle R_{f1}=R_{f2}=10\,k\Omega }$
• ${\displaystyle C_{1}=C_{2}=15\,nF}$
• ${\displaystyle R_{1}=R_{2}=10\,k\Omega }$

State variable filters are extensively used in audio processing and analogue signal conditioning because of their stability and flexibility. They're popular in parametric equaliser circuits, where varying the Q allows boosting or attenuating precise frequency bands.^[6]

In synthesizers, state variable filters enable dynamic control over resonance and cutoff frequency, often via voltage control, giving musicians the ability to sculpt evolving timbres.^[7]

Modern digital adaptations of state variable filters offer digitally adjustable parameters, enabling precise tuning and integration into software-defined radio, digital audio workstations, and programmable signal processors.^[8]

• Active filter
• Biquad filter
• Butterworth filter

• Introduction to the State Variable Filter - All About Circuits
• Understanding Audio Filters with the State Variable Filter