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ECE633F09_HW3solutions - ECE633 Signals and Systems I Fall...

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ECE633 Signals and Systems I, Fall 2009 – Homework 3 Solutions 1 From: Linear Systems and Signals, 2 nd ed . B. P. Lathi, Oxford University Press, 2005. 1.6-1 Write the input-output relationship for an ideal integrator. Determine the zero-input and zero-state components of the response. ( ) ( ) ( ) ( ) ( ) N ( ) 0 0 0 zero-input zero-state 0 t t t y t x d x d x d y x d τ τ τ τ τ τ τ τ −∞ −∞ = = + = + ±²³²´ 1.7-1 For the systems described by the following equations, with the input ( ) x t and the output ( ) y t , determine which of the systems are linear and which are nonlinear. (a) ( ) ( ) 2 2 dy y t x t dt + = (e) ( ) ( ) 2 2 dy y t x t dt + = (b) ( ) ( ) 2 3 dy ty t t x t dt + = (f) ( ) ( ) ( ) sin 2 dy dx t y t x t dt dt + = + (c) ( ) ( ) 3 2 y t x t + = (g) ( ) ( ) 2 dy dx y t x t dt dt + = (d) ( ) ( ) 2 dy y t x t dt + = (h) ( ) ( ) t y t x d τ τ −∞ = (a) ( ) ( ) ( ) ( ) 2 2 1 2 1 2 1 2 2 2 dy dy y t y t x t x t dt dt + + + = + [ ] ( ) ( ) ( ) ( ) 2 1 2 1 2 1 2 2 d y y y t y t x t x t dt + + + + non-linear (b) ( ) ( ) ( ) ( ) 2 2 1 2 1 2 1 2 3 3 dy dy ty t ty t t x t t x t dt dt + + + = + [ ] ( ) ( ) ( ) ( ) 1 2 2 1 2 1 2 3 d y y t y t y t t x t x t dt + + + = + linear (c) ( ) ( ) ( ) ( ) 1 2 1 2 3 3 2 2 y t y t x t x t + + + = + ( ) ( ) ( ) ( ) 1 2 1 2 3 2 y t y t x t x t + + +
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