ece_351_hwset1_more_sol_fall_20

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California State University, Northridge *

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351

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Electrical Engineering

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Oct 30, 2023

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1 ECE 351 Homework Set 1: More Solutions (60 65)
2 Problem 60 a. h[n] = (1/4) n u[n] S | h[n] | = 1 + (1/4) + (1/16) + ... = 1/(1-1/4) = 4/3 <   BIBO Stable b. y[n + 2] 5.5 y[n + 1] 3 y[n] = 2 x[n] char. eq: E 2 5.5E 3 = 0  g = 6, -.5 not asymptotically stable not BIBO stable (see p. 314)
3 Problem 61 a. y[n] = x[n] * h[n] = x[n] * { d [n] + d [n 1] + d [n 2] + d [n 3] }/4 = { x[n] + x[n 1] + x[n 2] + x[n - 3] }/4 b. y[n] = x[n] * h[n] = (-.9) n u[n] * { d [n] + d [n 1] + d [n 2] + d [n 3] }/4 = (1/4) {(-.9) n u[n] + (-.9) n-1 u[n-1] + (-.9) n-2 u[n-2] + (-.9) n-3 u[n-3] } = (1/4) (-.9) n { u[n] + (-.9) -1 u[n-1] + (-.9) -2 u[n-2] + (-.9) -3 u[n-3] } = (1/4) (-.9) n { u[n] 1.1111 u[n-1] + 1.2346 u[n-2] - 1.3717 u[n-3] }
4 Problem 61, continued c. y[n] = x[n] * h[n] = {5, 5, 5, 5, 5, 5, 5} * { d [n] + d [n 1] + d [n 2] + d [n 3] }/4 =(1/4) [ {5 5 5 5 5 5 5} 1 st copy {5 5 5 5 5 5 5} 2 nd copy {5 5 5 5 5 5 5} {5 5 5 5 5 5 5}] = (1/4) { 5 10 15 20 20 20 20 15 10 5} sum = { 5/4, 10/4, 15/4, 5, 5, 5, 5, 15/4, 10/4, 5/4}
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