Difference equation example; poles/zeros response

Difference equation

A difference equation relates the present output to present and past inputs and past outputs. A linear constant-coefficient difference equation of order N is

Σ(k=0 to N) aₖ y[n − k] = Σ(k=0 to M) bₖ x[n − k]

Example: y[n] = 0.5y[n − 1] + x[n]. For the input x[n] = δ[n], y[0] = 1, y[1] = 0.5, y[2] = 0.25, …, so h[n] = (0.5)ⁿ u[n] and H(z) = 1/(1 − 0.5z⁻¹). It is the standard way to describe and implement a digital filter.

Magnitude response from the given poles and zeros

Reading: real filters have conjugate pairs, so I take poles at 0.45 ± 0.77i and 2 ± 0.7i, and zeros at 1.2 ± 0.43i. No gain is given, so K = 1:

H(z) = (z − z₁)(z − z₁*) / [(z − p₁)(z − p₁*)(z − p₂)(z − p₂*)]

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                  ...........
               ....    |   x....          x
              ..       |       .. o
              .        |        ..
-------------.---------|---------.-------------
              .        |        ..
              ..       |       .. o
               ....    |   x....          x
                  ...........
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(x = pole, o = zero, dotted circle = unit circle; Re axis horizontal, Im axis vertical; range ±2.4)

|H(e^(jω))| = (distances to the two zeros) / (distances to the four poles), computed with K = 1:

 2.09 |            *
 1.86 |            *
 1.63 |           ***
 1.40 |           ***
 1.16 |          *****
 0.93 |          ******
 0.70 |         *********
 0.47 |       ***************
 0.23 |*************************************
      +-------------------------------------
       0            pi/2          pi  (w)

Observations:

• A sharp resonance peak (about 2.09) at ω ≈ π/3 (60°) because the pole pair 0.45 ± 0.77i (radius 0.89) lies close to the unit circle at that angle.
• A small dip (about 0.16) near ω ≈ 0.08π, and the response then settles to a flat, low value (about 0.2) at high frequencies.
• The zeros at angle 19.7° almost coincide with the poles at angle 19.3° (outside the circle), so their effects nearly cancel and no extra peak appears near ω ≈ 0.1π.
• The poles at radius 2.12 are outside the unit circle, so a causal realisation is unstable; the response above is for the ROC containing the unit circle (non-causal system). The pole pair at radius 0.89 alone would be stable.