Pole-zero and magnitude response, y−0.4y+0.1y

System function

Taking the z-transform of y[n] − 0.4y[n−1] + 0.1y[n−2] = x[n] + 0.6x[n−1] with zero initial conditions (x[n−k] ↔ z⁻ᵏ X(z)):

H(z) = Y(z)/X(z) = (1 + 0.6z⁻¹) / (1 − 0.4z⁻¹ + 0.1z⁻²)

Multiplying numerator and denominator by z^2: H(z) = N(z)/D(z) with

• Poles (roots of D(z)): 0.200 + 0.245j and its conjugate (radius 0.316)
• Zeros (roots of N(z)): -0.600 (real); z = 0 (order 1)

All poles are inside the unit circle (largest radius 0.316), so the causal system is stable.

Pole-zero plot

                       |
                       |
                       |
                    .......
               ......  |   ......
            ...        |        ...
          ...          |          ...
         ..            |            ..
        ..             |  x          ..
        .              |              .
--------.-----o--------o--------------.--------
        .              |              .
        ..             |  x          ..
         ..            |            ..
          ...          |          ...
            ...        |        ...
               ......  |   ......
                    .......
                       |
                       |
                       |

(x = pole, o = zero, dotted circle = unit circle; Re axis horizontal, Im axis vertical; range ±1.5)

Magnitude response

Put z = e^(jω): |H(e^(jω))| = (product of distances from e^(jω) to the zeros) / (product of distances to the poles). The response is large where e^(jω) passes close to a pole and small near a zero.

 2.29 |*******
 2.03 |***********
 1.78 |**************
 1.52 |****************
 1.27 |*******************
 1.02 |**********************
 0.76 |**************************
 0.51 |*******************************
 0.25 |*************************************
      +-------------------------------------
       0            pi/2          pi  (w)

Shape: low-pass. The maximum is |H(0)| = 2.286 and the response falls smoothly (no resonance, because the poles at radius 0.316 are far from the unit circle) to |H(π)| = 0.267. The zero at −0.6 lies near ω = π and keeps the high-frequency gain small.