hw2_1 - t ∈-5 5(c y t = x t-3 on t ∈-5 5(d y t = x(2...

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ASE 330M Linear System Analysis Unique Number: 12495, Spring 2006 Homework #2 Math Review Date given: February 7, 2006 Date Due: February 16, 2006 1. The systems given below have input x ( t ) and output y ( t ) respectively. Determine whether each of them is: (i) causal, (ii) linear, (iii) time invariant, and (iv) memoryless. Assume zero initial energy for the system. Make sure to provide elaborate justiﬁcations to your answers. (a) y ( t ) = cos( x ( t )) (b) y ( t ) = - x ( t ) + e - t - 1 (c) ˙ y ( t ) = ty ( t ) x ( t ) (d) y ( t ) = x (2 - t ) (e) y ( t ) = ( t + 3) x ( t - 2) (f) y ( t ) = R t -∞ ( t - τ ) x ( τ ) (g) y ( t ) = e - t ˙ x ( t ) (h) y ( t ) = x ( at ) for both cases a < 1 and a > 1. 2. A system initially at rest is speciﬁed by its input-output relationship as y ( t ) = x 2 ( t ) ˙ x ( t ) Discuss whether it is a linear system or not. 3. Consider a signal x ( t ) deﬁned as follows: x ( t ) = 0 , if | t |≥ 1 1 + t, if - 1 < t < 0 1 - t, if 0 t < 1 Use Matlab to plot the function x ( t ) with respect to t on the interval [ - 5 , 5]. Using the same Matlab tool, generate plots of the following functions: (a) y ( t ) = sin( x ( t )) on t [ - 2 π, 2 π ]; (b) y ( t ) = x (2 t ) on

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Unformatted text preview: t ∈ [-5 , 5] (c) y ( t ) = x ( t-3) on t ∈ [-5 , 5] (d) y ( t ) = x (2 t-3) on t ∈ [-5 , 5] (e) y ( t ) = x (-t ) on t ∈ [-5 , 5] 1 In all your plots, use a step-size of h = 0 . 01 between adjacent values in time. Submit all your Matlab scripts. 4. Consider a complex valued function H ( ω ) = 1 3 + jω Perform the following tasks: (a) Express H ( ω ) in polar coordinate form such that you express the magnitude | H ( ω ) | and angle arg( H ( ω )) in terms of the variable ω . (b) Use Matlab loglog tool to plot the magnitude | H ( ω ) | as a function of ω on the range ω ∈ [10-3 , 10 5 ]. Use suitably small steps in ω to ensure a smooth plot. (c) Use Matlab semilogx tool to represent the angle arg( H ( ω )) computed in degrees as a function of ω on the range ω ∈ [10-3 , 10 5 ]. Use suitably small steps in ω to ensure a smooth plot. 2...
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hw2_1 - t ∈-5 5(c y t = x t-3 on t ∈-5 5(d y t = x(2...

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