Lattice coefficients of fourth order FIR

Method

The FIR lattice of order M uses the stage equations

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

The reflection coefficient of the last stage is Km = am(m) (the last coefficient of the polynomial Am(z)). The lower-order polynomials come from the step-down recursion

a(m−1)(i) = [ am(i) − Km · am(m−i) ] / (1 − Km²),   i = 1 … m−1

Computation

A4(z) = 1 + 3.1z⁻¹ + 5.5z⁻² + 4.2z⁻³ + 2.3z⁻⁴

• A4: coefficients {1, 3.1, 5.5, 4.2, 2.3}, so K4 = a4(4) = 2.3
• A3: coefficients {1, 1.5291, 1.6667, 0.683}, so K3 = a3(3) = 0.683
• A2: coefficients {1, 0.7325, 1.1664}, so K2 = a2(2) = 1.1664
• A1: coefficients {1, 0.3381}, so K1 = a1(1) = 0.3381

Lattice coefficients: K1 = 0.3381, K2 = 1.1664, K3 = 0.683, K4 = 2.3

Structure

One lattice stage (repeated for m = 1 … 4 with K1 … K4):

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

Cascade (f and g pass together from stage to stage):

        +------+     +------+     +------+     +------+
x[n]-+->|      |-f1->|      |-f2->|      |-f3->|      |-f4-> y[n] = f4[n]
     |  |  K1  |     |  K2  |     |  K3  |     |  K4  |
     +->|      |-g1->|      |-g2->|      |-g3->|      |
        +------+     +------+     +------+     +------+

(The upper path carries f and the lower path carries g; f0 = g0 = x[n].)

Check: simulating the 4-stage lattice with a unit impulse gives {1, 3.1, 5.5, 4.2, 2.3}. ✓ (Since |K2| and |K4| exceed 1, some zeros lie outside the unit circle.)