Standard and series realization of a filter

Divide by $10z^3$:

$$G(z) = \frac{6z^2 + 7z}{10z^3 + 6z^2 + 7z + 5} = \frac{0.6z^{-1} + 0.7z^{-2}}{1 + 0.6z^{-1} + 0.7z^{-2} + 0.5z^{-3}}$$

(a) Standard programming

Introduce an intermediate variable $W(z) = X(z)/(\text{denominator})$:

$$\begin{aligned} w(k) &= x(k) - 0.6w(k-1) - 0.7w(k-2) - 0.5w(k-3) \\ y(k) &= 0.6w(k-1) + 0.7w(k-2) \end{aligned}$$

x(k)-->(S)------------o
        ^             |
        |          [z^-1]
        |             | w(k-1)
       (S)<[-0.6]-----o--[0.6]------>(S)--> y(k)
        ^             |               ^
        |          [z^-1]             |
        |             | w(k-2)        |
       (S)<[-0.7]-----o--[0.7]--------+
        ^             |
        |          [z^-1]
        |             | w(k-3)
        +<[-0.5]------o

Only $n = 3$ delays are used, the minimum number.

(b) Series programming

Factor the denominator. Its real root, found numerically, is $z = -0.6697$:

$$10z^3 + 6z^2 + 7z + 5 = 10(z + 0.6697)(z^2 - 0.0697z + 0.7467)$$

The quadratic has complex roots $0.0348 \pm j0.8634$, so it is kept as one second-order block. The numerator is $6z^2 + 7z = 6z(z + 1.1667)$. So

$$G(z) = 0.6\cdot\underbrace{\frac{1}{1 + 0.6697z^{-1}}}_{G_1}\cdot\underbrace{\frac{z^{-1} + 1.1667z^{-2}}{1 - 0.0697z^{-1} + 0.7467z^{-2}}}_{G_2}$$

Check: $0.6(z^{-1} + 1.1667z^{-2}) = 0.6z^{-1} + 0.7z^{-2}$, and $(1 + 0.6697z^{-1})(1 - 0.0697z^{-1} + 0.7467z^{-2}) = 1 + 0.6z^{-1} + 0.7z^{-2} + 0.5z^{-3}$. ✔

Difference equations:

• $w_1(k) = x(k) - 0.6697w_1(k-1)$, and $y_1(k) = 0.6w_1(k)$
• $w_2(k) = y_1(k) + 0.0697w_2(k-1) - 0.7467w_2(k-2)$
• $y(k) = w_2(k-1) + 1.1667w_2(k-2)$

Block $G_1$ (with gain 0.6):

x(k)-->(S)------------o--[1]-------->(S)->[0.6]-> y1(k)
        ^             |
        |          [z^-1]
        |             | w1(k-1)
        +<[-0.6697]---o

Block $G_2$:

y1(k)-->(S)------------o
         ^             |
         |          [z^-1]
         |             | w2(k-1)
        (S)<[0.0697]---o--[1]-------->(S)--> y(k)
         ^             |               ^
         |          [z^-1]             |
         |             | w2(k-2)       |
         +<[-0.7467]---o--[1.1667]-----+

The series form also uses 3 delays. Each block is first- or second-order, so it is less sensitive to coefficient rounding.