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). The signal input is marked Vin; the LP, HP and BP outputs give the lowpass, highpass, and bandpass filtered signals respectively. For simplicity, we set: :R_{f1} = R_{f2} :C_1 = C_2 :R_1=R_2 Then: :F_0 = \frac{1}{2\pi R_{f1}C_1} :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: :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 Rf1 = Rf2 = 10k, C1 = C2 = 15nF and R1 = R2 = 10k. ==Applications==