Series programming of a third-order transfer function

In series programming, $X(z)$ (the given pulse transfer function) is split into cascaded low-order sections. Each section is realized by standard programming.

Step 1: Sections in powers of $z^{-1}$

$$X(z) = \underbrace{\frac{1 + z^{-1}}{1 - 2z^{-1}}}_{G_1(z)}\cdot\underbrace{\frac{1 + 3z^{-1} + 7z^{-2}}{1 + z^{-1} + 9z^{-2}}}_{G_2(z)}$$

Both quadratics have complex roots, so each is kept as one second-order factor.

Step 2: Difference equations

Let $u(k)$ be the input and $y(k)$ the output.

Section 1:

• $w_1(k) = u(k) + 2w_1(k-1)$
• $y_1(k) = w_1(k) + w_1(k-1)$

Section 2:

• $w_2(k) = y_1(k) - w_2(k-1) - 9w_2(k-2)$
• $y(k) = w_2(k) + 3w_2(k-1) + 7w_2(k-2)$

Step 3: Block diagram

Section 1:

u(k)-->(S)------------o--[1]-------->(S)--> y1(k)
        ^             |               ^
        |          [z^-1]             |
        |             | w1(k-1)       |
        +<[2]---------o--[1]----------+

Section 2:

y1(k)-->(S)------------o--[1]-------->(S)--> y(k)
         ^             |               ^
         |          [z^-1]             |
         |             | w2(k-1)       |
        (S)<[-1]-------o--[3]-------->(S)
         ^             |               ^
         |          [z^-1]             |
         |             | w2(k-2)       |
         +<[-9]--------o--[7]----------+

u(k) --> [ G1(z) ] --y1(k)--> [ G2(z) ] --> y(k)

Three delay elements are used. Each section can be scaled and checked separately.

Note: The poles $z = 2$ and $z = -0.5 \pm j2.96$ lie outside the unit circle, so the system is unstable. The series structure is still a valid realization.