Direct form I and II of 3y[n]+y[n−1]+2y[n−4]

Divide the equation by 3 so that the coefficient of y[n] is 1:

y[n] = −(1/3)y[n−1] − (2/3)y[n−4] + (2/3)x[n] + (1/3)x[n−3]

So H(z) = (2/3 + (1/3)z⁻³) / (1 + (1/3)z⁻¹ + (2/3)z⁻⁴).

Feed-forward coefficients: b₀ = 2/3, b₃ = 1/3 (b₁ = b₂ = 0). Feedback coefficients: a₁ = 1/3, a₄ = 2/3 (a₂ = a₃ = 0), entering with a minus sign.

Direct Form I

Uses separate delay lines for input (3 delays) and output (4 delays): 7 delays in all.

 x[n] ──────►[×2/3]──────────┐
   │z⁻¹                      │
 x[n-1]                      │
   │z⁻¹                      ▼
 x[n-2]                    ( Σ )────┬──► y[n]
   │z⁻¹                      ▲       │
 x[n-3] ────►[×1/3]──────────┤       │ (y[n] enters the
                             │       │  output delay chain)
 y[n-1] ────►[×−1/3]─────────┤       ▼
   │z⁻¹   (y[n-1] = y[n] delayed)  z⁻¹ chain
 y[n-2]                      │
   │z⁻¹                      │
 y[n-3]                      │
   │z⁻¹                      │
 y[n-4] ────►[×−2/3]─────────┘

• Input side: x[n] → ×2/3 into the sum. x[n] → z⁻¹ → z⁻¹ → z⁻¹ → ×1/3 into the sum.
• Output side: y[n] → z⁻¹ → ×(−1/3) into the sum; y[n] → z⁻¹ → z⁻¹ → z⁻¹ → z⁻¹ → ×(−2/3) into the sum.
• y[n] = sum of the four branches.

Direct Form II (canonic)

Introduce w[n] between the poles and zeros:

• w[n] = x[n] − (1/3)w[n−1] − (2/3)w[n−4]
• y[n] = (2/3)w[n] + (1/3)w[n−3]

The two delay lines share the same delays, so only max(3, 4) = 4 delays are needed.

 x[n]──►(+)──────────────┬─►[×2/3]──►(+)──► y[n]
         ▲ w[n]          │             ▲
         │               ▼ z⁻¹ (w[n-1]) │
         ├──[×−1/3]──────┤             │
         │               ▼ z⁻¹ (w[n-2]) │
         │               ▼ z⁻¹ (w[n-3])─┴─[×1/3]
         │               ▼ z⁻¹ (w[n-4])
         └──[×−2/3]──────┘

Taps: w[n−1] feeds back through −1/3; w[n−4] feeds back through −2/3; w[n] and w[n−3] feed forward through 2/3 and 1/3.

Answer: DF-I needs 7 delays; DF-II needs 4 delays. Coefficients: b₀ = 2/3, b₃ = 1/3; feedback −1/3 (z⁻¹) and −2/3 (z⁻⁴).