Real DSP.
Real Hardware.
Real Results.

Before we can describe what a digital filter does, we need two transforms. The Fourier transform decomposes a signal into frequency components. The Z transform generalizes this to discrete-time systems, mapping filter coefficients directly to frequency-domain behavior, connecting the numbers you write in C to the response you hear on the RadioSonic board.

This lesson focuses on building intuition, not re-deriving the mathematics. By the end you should be able to look at a pole-zero plot and make a reasonable prediction about a filter's frequency response before running a single line of code.

Demo code: FFT of a two-tone test signal

import numpy as np
import matplotlib.pyplot as plt

Fs = 44100        # RadioSonic sample rate (Hz)
N  = 4096         # FFT length
t  = np.arange(N) / Fs

x = np.sin(2*np.pi*500*t) + 0.5*np.sin(2*np.pi*2000*t)
X   = np.fft.rfft(x, N)
f   = np.fft.rfftfreq(N, 1/Fs)
mag = 20*np.log10(np.abs(X)/N + 1e-12)

plt.plot(f, mag)
plt.xlabel("Frequency (Hz)"); plt.ylabel("Magnitude (dBFS)")
plt.show()

The Z transform and what poles & zeros mean

For a discrete-time filter, the Z transform produces a rational function H(z). Numerator roots are zeros, frequencies the filter nulls. Denominator roots are poles, frequencies it resonates at. For a stable IIR filter, all poles must lie strictly inside the unit circle. This is the geometric picture you will use in Section 4 when we implement biquad sections on the board.