We want to implement a causal system H(z) with the pole–zero diagram shown in Figure P6....

We want to implement a causal system H(z) with the pole–zero diagram shown in Figure P6.23. For all parts of this problem, z₁, z₂, p₁, and p₂ are real, and a gain constant that is independent of frequency can be absorbed into a gain coefficient in the output branch of each flow graph.

(a) Draw the flow graph of the direct form II implementation. Determine an expression for each of the branch gains in terms of the variables z₁, z₂, p₁, and p₂.

(b) Draw the flow graph of an implementation as a cascade of 2^nd-order direct form II sections. Determine an expression for each of the branch gains in terms of the variables z₁, z₂, p₁, and p₂.

(c) Draw the flow graph of a parallel form implementation with 1^st-order direct form II sections. Specify a system of linear equations that can be solved to express the branch gains in terms of the variables z₁, z₂, p₁, and p₂.