Given in Fig. P2.8-1 are two digital-filter structures, or realizations, for second-order filters.

(a) Write the difference equation for the 3D structure of Fig. P2.8-1(a), expressing \(y(k)\) as a function of \(y(k-i)\) and \(e(k-i)\).

(b) Derive the filter transfer function \(Y(z) / E(z)\) for the 3D structure by taking the \(z\)-transform of the equation in part (a).

(c) Write the difference equation for the 1D structure of Fig. P2.8-1(b). Two equations are required, with one for \(f(k)\) and one for \(y(k)\).

(d) Derive the filter transfer function \(Y(z) / E(z)\) for the 1D structure by taking the \(z\)-transform of the equations in part (c) and eliminating \(F(z)\).

(e) From parts (b) and (d), relate the coefficients \(\alpha_{i}, \beta_{i}\) to \(a_{i}, b_{i}\) such that the two filters realize the same transfer function.

(f) Write a computer-program segment that realizes the 3D structure. This program should be of the form used in Example 2.10.

(g) Write a computer-program segment that realizes the 1D structure. This program should be of the form used in Example 2.10.

Fig. P2.8-1

e(k) T T B2 Bo (a) y(k) T 04 T y(k)