Direct form II of y(n)=−0.1y(n−1)+0.72y(n−2)…

System function

y(n) = −0.1y(n−1) + 0.72y(n−2) + 0.7x(n) − 0.252x(n−2)

Taking the z-transform: Y(z)[1 + 0.1z⁻¹ − 0.72z⁻²] = X(z)[0.7 − 0.252z⁻²]

H(z) = (0.7 − 0.252z⁻²)/(1 + 0.1z⁻¹ − 0.72z⁻²) = 0.7(1 − 0.36z⁻²)/((1 − 0.8z⁻¹)(1 + 0.9z⁻¹))

Poles: 0.8 and −0.9 (inside the unit circle, so the system is stable). Zeros: ±0.6.

Direct form II (canonical)

Split H(z) into an all-pole part followed by an all-zero part, sharing one delay line:

w(n) = x(n) − 0.1·w(n−1) + 0.72·w(n−2)
y(n) = 0.7·w(n) − 0.252·w(n−2)

(Here the feedback coefficients are −a₁ = −0.1 and −a₂ = +0.72.) Only two delays are needed (order N = 2).

x(n) ──►(+)───────────●──── ×0.7 ──────►(+)──► y(n)
         ▲            │ w(n)             ▲
         │          [z⁻¹]                │
         ├─ ×(−0.1) ◄─● w(n−1)           │
         │            │                  │
         │          [z⁻¹]                │
         └─ ×0.72 ◄───● w(n−2) ×(−0.252) ┘

Reading the diagram: the left adder forms w(n) = x(n) + (−0.1)w(n−1) + 0.72w(n−2); the right adder forms y(n) = 0.7w(n) + (−0.252)w(n−2). The feedback and feed-forward taps are both taken from the same two delays. There is no w(n−1) feed-forward tap, because the x(n−1) coefficient is 0.

Direct form II saves memory (2 delays instead of 4) and is called canonic.