Revision as of 10:20, 11 November 2011

Practice Problem on time-invariance of a CT system

Question

Y(t) = x(t − 1) − x(1 − t)

It is Time Invariant? Justify.

Answer

No.

S₁ = Y(t) = x(t − 1) − x(1 − t)

S₂ = Y(t) = x(t − t_o)

x(t) − > S1 − > S2 − > x(t − t_o − 1) − x(1 − t + t_o)

x(t) − > S2 − > S1 − > x(t − t_o − 1) − x(1 − t − t_o)

x(t − t_o − 1) − x(1 − t + t_o) = / = x(t − t_o − 1) − x(1 − t − t_o)

Comments

Please comment on this answer. Are there any mistakes? Is it clear? Could it be improved?

Comment 1

Actually this is where I'm unsure on HW3 Q1b.  It would definitely be improved and more clear if the student showed intermediate steps.  

x(t) −> [S1] −> y(t) = x(t - 1) - x(1 - t) --> [S2] −> z₁(t) = y(t - t_o) = x((t - t_o) -1) - x(1 - (t - t_o))

                                                                --> z₁(t) = x(t − t_o − 1) − x(1 − t + t_o)
x(t) −> [S2] −> y(t) = x(t - t_o) --> [S1] −> z₂(t) = y(t - 1) - y(1 - t) = x((t - 1) - t_o) - x((1 - t) - t_o)

                                                               --> z₂(t) = x(t − t_o − 1) − x(1 − t − t_o)

Comment 2

Write it here

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