Question

2.2-1. The rectangular rules for numerical integration are illustrated in Fig. P2.2-1. The left-side rule is depicted in Fig. P2.2-1(a), and the right-side rule is depicted in Fig. P2.2-1(b). The integral of \( x(t) \) is approximated by the sum of the rectangular areas shown for each rule. Let \( y(k T) \) be the numerical integral of \( x(t), 0 \leq t \leq k T \). (a) Write the difference equation relating \( y(k+1), y(k) \), and \( x(k) \) for the left-side rule. (b) Find the transfer function \( Y(z) / X(z) \) for part (a). (c) Write the difference equation relating \( y(k+1), y(k) \), and \( x(k+1) \) for the right-side rule. (d) Find the transfer function \( Y(z) / X(z) \) for part (c). (e) Express \( y(k) \) as a summation on \( x(k) \) for the left-side rule. (f) Express \( y(k) \) as a summation on \( x(k) \) for the right-side rule. (a) (b) FIGURE P2.2-1 Rectangular rules for integration: (a) left side; (b) right side.

          2.2-1. The rectangular rules for numerical integration are illustrated in Fig. P2.2-1. The left-side rule is depicted in Fig. P2.2-1(a), and the right-side rule is depicted in Fig. P2.2-1(b). The integral of \( x(t) \) is approximated by the sum of the rectangular areas shown for each rule. Let \( y(k T) \) be the numerical integral of \( x(t), 0 \leq t \leq k T \).
(a) Write the difference equation relating \( y(k+1), y(k) \), and \( x(k) \) for the left-side rule.
(b) Find the transfer function \( Y(z) / X(z) \) for part (a).
(c) Write the difference equation relating \( y(k+1), y(k) \), and \( x(k+1) \) for the right-side rule.
(d) Find the transfer function \( Y(z) / X(z) \) for part (c).
(e) Express \( y(k) \) as a summation on \( x(k) \) for the left-side rule.
(f) Express \( y(k) \) as a summation on \( x(k) \) for the right-side rule.
(a)
(b)

FIGURE P2.2-1 Rectangular rules for integration: (a) left side; (b) right side.

2.2-1. The rectangular rules for numerical integration are illustrated in Fig. P2.2-1. The left-side rule is depicted in Fig. P2.2-1(a), and the right-side rule is depicted in Fig. P2.2-1(b). The integral of x(t) is approximated by the sum of the rectangular areas shown for each rule. Let y(k T) be the numerical integral of x(t), 0 ≤ t ≤ k T.
(a) Write the difference equation relating y(k+1), y(k), and x(k) for the left-side rule.
(b) Find the transfer function Y(z) / X(z) for part (a).
(c) Write the difference equation relating y(k+1), y(k), and x(k+1) for the right-side rule.
(d) Find the transfer function Y(z) / X(z) for part (c).
(e) Express y(k) as a summation on x(k) for the left-side rule.
(f) Express y(k) as a summation on x(k) for the right-side rule.
(a)
(b)

FIGURE P2.2-1 Rectangular rules for integration: (a) left side; (b) right side.

Added by Joshua B.

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