The Z-Plane as Design Space
The Z-transform converts a filter's coefficient sequence into a polynomial (or rational function) in the complex variable z. The transfer function H(z) has:
- Zeros: values z_k where H(z_k) = 0
- Poles: values p_k where H(z) → ∞ (denominator roots for recursive filters)
Evaluating H(z) on the unit circle z = e^{i2πf} gives the frequency response H(f). The unit circle is the boundary where time-domain stability & frequency-domain analysis intersect.
The Distance-Product Formula
|H(f)| = ∏_k |e^{i2πf} − z_k| / ∏_k |e^{i2πf} − p_k|
Reading response from the plot:
- Zero ON the circle: distance = 0 → complete null
- Zero INSIDE the circle: distance > 0 → partial attenuation near that angle
- Pole NEAR the circle: small denominator → large gain (peak)
- Pole OUTSIDE the circle: filter unstable (IIR only)
Designing Zeros for Nulls
To null frequency f_0 completely: place a zero at z_0 = e^{i2πf_0}.
To null both f_0 & its conjugate frequency (for a real-coefficient filter): place zeros at e^{±i2πf_0}. Complex zeros must come in conjugate pairs for real coefficients.
Each zero adds one factor to the numerator: (z − z_0). A filter that nulls N frequencies has N zeros.
Poles Boost Response
A pole at z = p contributes a factor 1/(z − p) to H(z). Near the unit circle point closest to p, |e^{i2πf} − p| is small, making |H(f)| large. The closer the pole to the unit circle, the sharper the peak.
Stability Boundary
For a recursive (IIR) filter, the system is stable if & only if all poles lie strictly inside the unit circle (|p| < 1). A pole at |p| = 1 produces sustained oscillation (marginally stable). A pole at |p| > 1 produces growing oscillation (unstable).
The unit circle serves as the stability boundary in the Z-plane, just as the imaginary axis serves as the stability boundary in the Laplace s-plane for continuous-time systems.
Hamming's Feedback Shower Story
Hamming illustrated stability with a shower that required finding the right temperature. The pipe delay meant his corrections arrived late — he kept overshooting. The feedback loop became unstable. IIR filters face the same risk: too much feedback (poles too close to or outside the unit circle) and the output diverges.
Stability from Pole Locations
A second-order IIR filter has the transfer function:
H(z) = 1 / (1 − a₁z⁻¹ − a₂z⁻²) = z² / (z² − a₁z − a₂)
The poles are the roots of z² − a₁z − a₂ = 0.
Stability: |p₁| < 1 and |p₂| < 1 for both roots.
The Graphical Design Method
Experienced filter designers sketch pole-zero plots before computing anything. The geometry reveals the response shape instantly.
Design Rules of Thumb
1. Nulls at unwanted frequencies: place zeros on the unit circle at those angles.
2. Passband with gain: place poles near (but inside) the unit circle at the desired passband angle.
3. Real coefficients: ensure all complex zeros & poles appear in conjugate pairs.
4. Stability check: verify all poles satisfy |p| < 1 before computing coefficients.
5. Transition width: poles closer to the unit circle → sharper transition but less stability margin.
The graphical method converts the engineering specification (pass these frequencies, stop those, with this ripple) into a geometric constraint (place poles & zeros here), then reads off the polynomial coefficients.