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IF THE POLES OF THE LOCATED AT. b) H(근) のUSE-MATLAS TO PLOT IHI»)! oAT KIN OF A FILTER IS IT? 171 53 lilustrative Examples 5.3 ILLUSTRATIVE EXAMPLES Example 5.3-1: The following difference equation is known as the first- order trapezoidal integration rule in numerical analysis: n) - ) n-O (531) Find: The transfer function. The impulse response. The output if the input is the unit step sequence xtn) - 1, n z 0. Solution: (a) (b) (c) (a) The ZT of (5.3-1) yields which results in the transfer function 1) Y(z) H(2) -05 (53-2) (b) To obtain h(n), we need to evaluate the I2ZT of H(2) in (5.3-2). We do so by means of the residue method, and hence form (5.3-3) 0.52+ 1 CASE 1. n 0 results in two poles at z 0 and z 1, since 366 Thus 173 5.3 Inustrative Examples . Thus y(n)-R-1 where R-i is the residue of G(z) at z 1. It is evaluated using (3.5-15c) to obtain the desired result dz (5.3-6) - 0.5(2n 1) nz 0 From (5.3-6) it is apparent that y(O) 0.5. y(1)-15, y(2)-2.5, y(3)= 3.5, etc. which demonstrates that the trapezoidal integration rule (5.3-1) does indeed integrate the input unit-step sequence. We also observe that this process represents a marginally stable system since the corresponding H(z) in (5.3-2) has a pole on the unit circle; that is, z 1 Example 5.3-2: Consider the second-order IIR system y(n) = Kiyin-1) + Korn-2) + x(n), 0 (53-7) n where K, and K, are constants. (a) Evaluate H(z). (b) Find hin), n 2 0 for the following cases: (1) K3/4 + K2 0 (3) K1/4 + K20 Discuss stability conditions related to these cases. Solution: (a) Assuming zero initial conditions, the ZT of (5.3-7) yields from which we obtain (5.3-8)

Answer 1

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c) Matlab Code: