Difference equation need; pole-zero, magnitude response

Why we need a difference equation

• It is the time-domain description of a discrete system, giving a direct recipe for implementation: each output is a weighted combination of present/past inputs and past outputs, needing only adders, multipliers and unit delays.
• It represents recursive (IIR) systems, whose impulse response is infinite, with a finite number of coefficients.
• Taking its z-transform gives the system function H(z), from which poles, zeros, stability and the frequency response are obtained.
• It is easy to program on a DSP processor and is the basis of realisation structures (direct form, cascade, parallel, lattice).
• It is the discrete counterpart of the differential equation describing analog systems.

System function

Taking the z-transform of y[n] − 0.4y[n−1] + 0.2y[n−2] = x[n] + 0.1x[n−1] − 0.06x[n−2] with zero initial conditions (x[n−k] ↔ z⁻ᵏ X(z)):

H(z) = Y(z)/X(z) = (1 + 0.1z⁻¹ − 0.06z⁻²) / (1 − 0.4z⁻¹ + 0.2z⁻²)

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

• Poles (roots of D(z)): 0.200 + 0.400j and its conjugate (radius 0.447)
• Zeros (roots of N(z)): -0.300 (real); 0.200 (real)

All poles are inside the unit circle (largest radius 0.447), 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.

 1.51 |       ********
 1.34 |******************
 1.17 |********************
 1.00 |***********************
 0.84 |**************************
 0.67 |*******************************
 0.50 |*************************************
 0.33 |*************************************
 0.17 |*************************************
      +-------------------------------------
       0            pi/2          pi  (w)

Shape: low-pass with a small peak. |H(0)| = 1.3, a maximum of 1.51 near ω ≈ 0.30π (pole pair 0.2 ± 0.4j at angle 63.4°, radius 0.447), then a smooth fall to |H(π)| = 0.525. The zeros (−0.3, 0.2) are near the origin and hardly shape the response.