EE113 midterm solutions

# EE113 midterm solutions - Problem 1 For parts(a(b and(c...

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Unformatted text preview: Problem 1. For parts (a), (b) and (c) determine whether or not the system is i. linear ii. time-invariant iii. BIBO stable, i.e., bounded input-bounded output stable a. y ( n ) = x ( n ) + x ( n − 1) + 1 . b. y ( n ) = cos [ x ( n )] . c. y ( n ) = [ x ( n )] n . Note: You do not need to show a lot of work on this problem if you can quickly recognize the answer. Solution. a. y ( n ) = x ( n ) + x ( n − 1) + 1 . Clearly Non-linear and Time-invariant. Also BIBO stable: | x ( n ) | < M ∀ n ⇒ | y ( n ) | < 2 M + 1 ∀ n . b. y ( n ) = cos [ x ( n )] . Clearly Non-linear and Time-invariant. BIBO stable because | y ( n ) | = | cos[ x ( n )] | ≤ 1 ∀ n . c. y ( n ) = [ x ( n )] n . Clearly Non-linear. Not Time-invariant because if y ( n ) = T [ x ( n )] = [ x ( n )] n , then T [ x ( n − k )] = [ x ( n − k )] n 6 = y ( n − k ) = [ x ( n − k )] n − k Not BIBO stable: if x ( n ) = 2 ∀ n (bounded), then y ( n ) = 2 n is un- bounded as n → ∞ . Problem 2. Consider the system described by the following difference equa- tion: y ( n ) − 11 10 y ( n − 1) + 1 10 y ( n − 2) = x ( n ) , where, x ( n ) = (1 / 3) n u ( n ) , y ( − 1) = 1 , y ( − 2) = 0 . Find a closed form expression for y ( n ). 1 Solution. Method I: Time-domain y ( n ) = y h ( n ) + y p ( n ) The characteristic equation is λ 2 − 11 10 λ − 1 10 = 0. The roots are λ = 1 10 , 1. Therefore y h ( n ) = C 1 1 10 n + C 2 (1) n x ( n ) = (1 / 3) n u ( n ) ⇒ y p ( n ) = K (1 / 3) n u ( n ) For n ≥ 2, y p ( n ) − 11 10 y p ( n − 1) + 1 10 y p ( n − 2) = x ( n ) K 1 3 n − 11 10 K 1 3 n − 1 + 1 10 K 1 3 n − 2 = 1 3 n K − 11 10 K (3) + 1 10 K (9) = 1 ⇒ K = − 5 7 = 0 . 7143 y ( n ) = C 1 1 10 n + C 2 − 5 7 1 3 n To get C 1 and C 2 , we first find y (0) and y (1) by propagating the difference equation. y (0) = 11 10 y ( − 1) − 1 10 y ( − 2) + x (0) = 11 10 − 0 + 1 = 21 10 y (1) = 11 10 y (0) − 1 10 y ( − 1) + x (1) = 11 10 · 21 10 − 1 10 + 1 3 = 763 300 Using the expression for y ( n ) we get the equations:...
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EE113 midterm solutions - Problem 1 For parts(a(b and(c...

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