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Chap1_properties_sol

Chap1_properties_sol - 1 a dy 6y t = 4 x t dt This is an...

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1. a) dy dt y t x t + = 6 4 ( ) ( ) This is an ordinary differential equation with constant coefficients, therefore, it is linear and time- invariant. It contains memory and it is causal. b) dy dt ty t x t + = 4 2 ( ) ( ) This is an ordinary differential equation. The coefficients of 4t and 2 do not depend on y or x, so the system is linear. However, the coefficient 4t is not constant, so it is time-varying. The system is also causal and has memory c) This is a difference equation with constant coefficients; therefore, it is linear and time-invariant. It is noncausal since the output depends on future values of x. Specifically, let x[n] = u[n], then y[-1] = 1. d) y(t) = sin(x(t)) check linearity: y t x t 1 1 ( ) sin( ( )) = y t x t 2 2 ( ) sin( ( )) = Solution to an input of a x t a x t 1 1 2 2 ( ) ( ) + is sin( ( ) ( )) a x t a x t 1 1 2 2 + . This is not equal to a y t a y t 1 1 2 2 ( ) ( ) + . As a counter example, consider x t 1 ( ) = π and x t 2 2 ( ) / = π , a a 1 2 1 = = the system is causal since the output does not depend on future values of time, and it is memoryless the system is time-invariant e) dy dt y t x t + = 2
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