quiz 2 - ECEN 487 QUIZ 2 NAME_KEY 1 a Define a sequence x n...

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ECEN 487 QUIZ 2 NAME:_________KEY_____________________ 1. a) Define a sequence ˆ x [ n ] with z-transform ˆ X ( z ) = ln{ X ( z )} , with an X ( z ) that has poles at z = 1.5 and z = 3 j , and a single zero at z = 1/ 4 . Is there an R.O.C. choice for which ˆ x [ n ] is two sided and for which the corresponding Fourier transform, ) ( ˆ ω j e X , exists (i.e. ) ( ˆ j e X is bounded for all ! )? If so, specify the R.O.C. or explain why not. (Hint: Recall that for any complex a , } arg{ } ln{ } ln{ a j a a + = ). ˆ X ( z ) z = 1/4 = ln X ( z ) { } z = 1/4 = ln{0 } = "# . So the zero at z = 1/4 becomes a pole. Thus a two sided sequence with a Fourier transform requires the R.O.C. to include the unit circle, so 1/ 4 ! z ! 3/ 2 . Therefore ˆ x [ n ] can be both right sided and have a Fourier transform,
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