FIRST ORDER IIR

WHAT YOU ARE LOOKING AT

A second order infinite impulse response filter is one line of code: y[n] = b*x[n-1] + 2a*y[n-1] - (a²+b²)*y[n-2]. Each output feeds back the two previous outputs, and it is that feedback that makes the impulse response infinite and puts a complex conjugate pole pair in the transfer function

H(z) = b·z / (z² − 2a·z + a²+b²)

with a zero at the origin and poles at z = a ± jb — the same complex frequency the lecture's z-plane discussion uses. On the z-plane panel the circle is |z| = 1; the filter is stable as long as |a + jb| < 1. The bottom middle panel is the measured output spectrum, and the yellow trace is |H(f)| evaluated straight from b and a — they should lie on top of each other, which is a good way to convince yourself that the loop and the algebra agree.

THINGS WORTH TRYING

• Push a or b until |a + jb| reaches 1. The poles land on the unit circle and the filter turns into an oscillator at the pole angle; past it the output grows without bounds. Watch the |z| readout.
• Hold |z| roughly constant while trading a against b. The pole angle — and with it the resonant peak — sweeps across the band while the decay rate stays put. At b = 0 the poles collapse onto the real axis and the resonance disappears.
• Set f_1 low, around bin 100. The tone now sits in the passband and comes through with the DC gain instead of being attenuated. Compare the input and output spectra: the noise above the corner has dropped, the tone has not.
• Set the sampling period P to 1 to remove the pulse train. The images vanish and you see the filter on its own, which makes the 40 dB/decade roll-off easy to read off the right-hand panel.

THE PYTHON

#- Second-order IIR filter, poles at z = a +/- jb
b = 0.25
a = 0.85
z = a + 1j*b
z_abs = np.abs(z)
y = np.zeros(N)
for i in range(2,N):
    y[i] = b*x_sn[i-1] + 2*a*y[i-1] - (a*a + b*b)*y[i-2]

Source: ex/iir.py. This page makes the a coefficient a slider so you can watch the pole move. There are faster ways to do this in Python; see scipy.signal.iirfilter.