BIQUAD

WHAT YOU ARE LOOKING AT

The notebook this comes from is not really about biquads. It is about not trusting your own algebra. It writes a signal flow graph down as three equations, hands them to sympy, and asks what transfer function falls out:

u1 = -w0*vo + vi*k0/w0
u2 = -w0/Q*vo + w0*u1/s + k1*vi + k2*s*vi
vo = u2/s

Do that by hand and you get

H(s) = (k2·s² + k1·s + k0) / (s² + (w0/Q)·s + w0²)

which is the standard biquad. There is no computer algebra in this page — that would mean shipping a CAS to a browser — so the derivation was done once by hand and then checked numerically against the three equations above at two hundred random points. They agree to fifteen digits.

What is worth having interactively is the other half: what that expression does. The denominator alone fixes the two poles, and therefore the ringing, the peaking and the settling. The numerator only places zeros, and that is the whole difference between a low-pass, a band-pass, a high-pass, a notch and an all-pass. Press the buttons and watch the poles stay exactly where they are.

THINGS WORTH TRYING

• Cycle LP → BP → HP → Notch → Allpass and keep your eye on the s-plane. The red crosses never move. Five completely different frequency responses, one set of poles. This is why filter design is usually taught as "place the poles, then pick the numerator".
• Raise Q. The poles swing towards the jω axis, the magnitude peak grows, the step response rings for longer, and the group delay spikes at f0. At Q = 0.5 the poles are real and coincident and there is no ringing at all; below that they split apart on the real axis.
• Take Q negative — drag it below the marked line. The poles cross into the right half plane and the step response runs away. The Bode plot still looks perfectly reasonable, which is the point: a magnitude plot cannot tell you a system is unstable.
• Select Allpass and look at the magnitude: flat at 0 dB everywhere. All that changes is phase, 0 to −360°, and the group delay bump at f0. Allpass sections are how you fix phase without touching amplitude — and why "it has no effect on the magnitude" is not the same as "it has no effect".
• In Free mode, put a zero exactly on a pole by matching the numerator to the denominator (k0/w0² = 1, k1/(w0/Q) = 1, k2 = 1). The response goes flat: the zero cancels the pole. In a real circuit that cancellation is never exact, which is the whole reason pole-zero doublets are something to worry about.

THE NOTEBOOK

Source: jupyter/biquad.ipynb, which uses sympy and sympy.physics.control to expand the flow graph. Its first cell does the same for a simpler single-integrator graph. For the discrete-time equivalent of all this, see the IIR example, where the same ideas live on the z-plane and the unit circle replaces the jω axis.