HW3sol - ECEN 314: Signals and Systems Solutions to HW 3...

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ECEN 314: Signals and Systems Solutions to HW 3 Problem 1.27 (a) y ( t ) = x ( t - 2) + x (2 - t ) Let us check for linearity. x 1 ( t ) y 1 ( t ) = x 1 ( t - 2) + x 1 (2 - t ) x 2 ( t ) y 2 ( t ) = x 2 ( t - 2) + x 2 (2 - t ) ax 1 ( t ) + bx 2 ( t ) = x 3 ( t ) y 3 ( t ) = x 3 ( t - 2) + x 3 (2 - t ) = ax 1 ( t - 2) + bx 2 ( t - 2) + ax 1 (2 - t ) + bx 2 (2 - t ) = a ( x 1 ( t - 2) + x 1 (2 - t )) + b ( x 2 ( t - 2) + x 2 (2 - t )) = ay 1 ( t ) + by 2 ( t ) Hence linear. Let us check for time-invariance. x 1 ( t ) y 1 ( t ) = x 1 ( t - 2) + x 1 (2 - t ) x 1 ( t - t o ) = x 2 ( t ) y 2 ( t ) = x 2 ( t - 2) + x 2 (2 - t ) = x 1 ( t - t o - 2) + x 2 (2 - t - t o ) 6 = y 1 ( t - t o ) Note that y 1 ( t - t o ) = x 1 ( t - t o - 2) + x 1 (2 - t + t o ). Hence time-variant. Suppose | x ( t ) | < B . Then y ( t ) < B + B = 2 B (because | x ( t - 2) | < B and | x (2 - t ) | < B ). Hence stable. Not memoryless as the present output at time t depends on t - 2. Non-Causal because y(-1)=x(-3)+x(3). So depends on future inputs. (b) y ( t ) = [cos(3 t )] x ( t ) 1
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Let us check for linearity. x 1 ( t ) y 1 ( t ) = [cos(3 t )] x 1 ( t ) x 2 ( t ) y 2 ( t ) = [cos(3 t )] x 2 ( t ) ax 1 ( t ) + bx 2 ( t ) = x 3 ( t ) y 3 ( t ) = [cos(3 t )] x 3 ( t ) = [cos(3 t )] ( ax 1 ( t ) + bx 2 ( t ) ) = ay 1 ( t ) + by 2 ( t ) Hence linear. Let us check for time-invariance. x 1 ( t ) y 1 ( t ) = [cos(3 t )] x 1 ( t ) x 1 ( t - t o ) = x 2 ( t ) y 2 ( t ) = [cos(3 t )] x 2 ( t ) = [cos(3 t )] x 1 ( t - t o ) 6 = y 1 ( t - t o ) Note that y 1 ( t - t o ) = [cos(3( t - t o ))] x 1 ( t - t o ). Hence time-variant. Stable as | y ( t ) | < , when | x ( t ) | < B . Memoryless as the output at time
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This note was uploaded on 01/01/2012 for the course ECEN 314 taught by Professor Halverson during the Spring '08 term at Texas A&M.

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HW3sol - ECEN 314: Signals and Systems Solutions to HW 3...

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