Magnitude response, y−0.3y+0.225y

System function

Taking the z-transform of y[n] − 0.3y[n−1] + 0.225y[n−2] = x[n] + 0.5x[n−1] with zero initial conditions (x[n−k] ↔ z⁻ᵏ X(z)):

H(z) = Y(z)/X(z) = (1 + 0.5z⁻¹) / (1 − 0.3z⁻¹ + 0.225z⁻²)

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

• Poles (roots of D(z)): 0.150 + 0.450j and its conjugate (radius 0.474)
• Zeros (roots of N(z)): -0.500 (real); z = 0 (order 1)

All poles are inside the unit circle (largest radius 0.474), 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.

 1.80 |      *********
 1.60 |*****************
 1.40 |*******************
 1.20 |*********************
 1.00 |***********************
 0.80 |*************************
 0.60 |*****************************
 0.40 |*************************************
 0.20 |*************************************
      +-------------------------------------
       0            pi/2          pi  (w)

Shape: low-pass with a mild peak. |H(0)| = 1.62, a maximum of 1.80 near ω ≈ 0.30π (the poles 0.15 ± 0.45j have angle 71.6°, radius 0.474), then a steady fall to |H(π)| = 0.328 (the zero at −0.5 attenuates high frequencies).