Lattice coefficients of all-pole IIR; stability

Method

For the all-pole system H(z) = 1/A₃(z), with A₃(z) = 1 − 0.01z⁻¹ − 0.23z⁻² + 0.5z⁻³, the reflection coefficients are those of the polynomial A₃(z), found by the step-down recursion

K_m = a_m(m), a_(m−1)(i) = [a_m(i) − K_m a_m(m−i)] / (1 − K_m²)

Computation

• A₃ = {1, −0.01, −0.23, 0.5}: K₃ = a₃(3) = 0.5
• 1 − K₃² = 0.75. a₂(1) = (−0.01 − 0.5(−0.23))/0.75 = 0.105/0.75 = 0.14; a₂(2) = (−0.23 − 0.5(−0.01))/0.75 = −0.225/0.75 = −0.3. A₂ = {1, 0.14, −0.3}: K₂ = −0.3
• 1 − K₂² = 0.91. a₁(1) = (0.14 − (−0.3)(0.14))/0.91 = 0.182/0.91 = 0.2. A₁ = {1, 0.2}: K₁ = 0.2

Lattice coefficients: K₁ = 0.2, K₂ = −0.3, K₃ = 0.5.

Lattice structure (all-pole)

Stage equations, from m = 3 down to 1 (input enters at f₃):

f3(n) = x(n)
f(m−1)(n) = fm(n) − Km · g(m−1)(n−1)
gm(n)     = Km · f(m−1)(n) + g(m−1)(n−1)
y(n) = f0(n) = g0(n)

f(m)[n] ─────(+)────────●──► f(m-1)[n]
               ▲────────┼─────────┐
                        │         │ -Km
                        │ Km      │
             ┌──────────┘         │
             ▼                    │
g(m)[n] ◄────(+)◄─────────────────●◄─[z⁻¹]◄─ g(m-1)[n]

Three stages in cascade:

          +------+     +------+     +------+
x[n]=f3-->| K3   |---->| K2   |---->| K1   |--> y[n]=f0=g0
          | =0.5 |     |=-0.3 |     | =0.2 |
          +------+     +------+     +------+

(the f path moves left to right, the g path returns right to left; g0 = f0 at the end).

Stability

An all-pole lattice is stable if and only if all |K_m| < 1. Here |0.2|, |−0.3|, |0.5| are all less than 1, so the system is stable.

Check with the poles: the roots of z³ − 0.01z² − 0.23z + 0.5 are −0.886 and 0.448 ± 0.603j (radius 0.751). All have magnitude < 1. ✓