Map given poles and zeros, magnitude response

Reading: a real system has poles and zeros in complex-conjugate pairs. I read the poles as 0.45 ± 0.77i and −2 ± 0.3i, and the zeros as 1.2 ± 3i. No gain is given, so take K = 1:

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

Locations (polar form)

Pole-zero map

                  ^ Im
                  |         o  z1 = 1.2 + j3
                  |
           .------|------.
  x       /       |   x   \
----------+-------+-------+-----> Re
  x       \       |   x   /
           '------|------'
                  |
                  |         o  z1* = 1.2 − j3

(x = pole, o = zero; the circle is the unit circle |z| = 1, crossing the Re axis at −1 and +1; the vertical scale is compressed.) The poles p₁, p₁* = 0.45 ± 0.77i are just inside the unit circle. The poles p₂, p₂* = −2 ± 0.3i (far left) and both zeros are outside it.

Magnitude response

|H(e^jω)| = |e^jω − z₁||e^jω − z₁*| / (product of the distances to the four poles), evaluated at points on the unit circle. Computed (K = 1):

 7.04 |            *
 6.26 |            *
 5.48 |           ***
 4.69 |           ***                    ***
 3.91 |          *****                ******
 3.13 |         *******            *********
 2.35 |        ************   **************
 1.56 |   **********************************
 0.78 |*************************************
      +-------------------------------------
       0            pi/2          pi  (w)

• A sharp peak (about 7.0) near ω ≈ π/3 (60°), because the pole 0.45 + 0.77i (radius 0.89) is very close to the unit circle at that angle.
• A smaller rise towards ω = π (about 4.7), caused by the poles at −2 ± 0.3i (angle about 171°); they are outside the circle but are nearest to it at ω ≈ π.
• The lowest value (≈ 1.1) is at ω = 0. The zeros (radius 3.2) are far from the circle, so they produce no notch.

Stability: the poles with |p| = 2.02 > 1 lie outside the unit circle, so a causal realisation is unstable. The magnitude response above exists only for the (non-causal) ROC that contains the unit circle.