3 - 2/18/10 ECE 483 Homework #3 Solution Problem 1. Given a...

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Unformatted text preview: 2/18/10 ECE 483 Homework #3 Solution Problem 1. Given a continuous-time signal e ( t ) = e- 3 t , t 0. (a) Find the signal e * ( t ) obtained by passing e ( t ) through an ideal sampler with sampling period T = 0 . 5 s, and its Laplace transform E * ( s ). (b) Suppose e * ( t ) is subsequently passed through a zero-order data hold to produce the signal e ( t ). Find e ( t ) and its Laplace transform E ( s ). (c) Repeat (a)(b) for the signal f ( t ) = e ( t 1) = e- 3( t- 1) u ( t 1), t 0. (d) Repeat (a)(b) for the signal g ( t ) = e ( t . 1) = e- 3( t- . 1) u ( t . 1), t 0. Solution: (a) e * ( t ) = k =0 e ( kT ) ( t kT ) = k =0 e- 1 . 5 k ( t . 5 k ), which is a sequence of impulses; E * ( s ) = k =0 e- 1 . 5 k e- . 5 ks = 1 1- e- 1 . 5- . 5 s . (b) E ( s ) = E * ( s ) 1- e- sT s = 1- e- . 5 s s (1- e- 1 . 5- . 5 s ) , and e ( t ) = e ( kT ) = e- 1 . 5 k , for kT t < ( k + 1) T, k = 0 , 1 , . . ., which is a stair-like signal. (c) For the given f ( t ), we have f * ( t ) = summationdisplay k =0 f ( kT ) ( t kT ) = summationdisplay k =2 e- 3(0 . 5 k- 1) ( t kT ) = summationdisplay k =2 e- 1 . 5 k +3 ( t . 5 k ) F * ( s ) = summationdisplay k =2 e- 1 . 5 k +3 e- . 5 ks = e- s 1 e- 1 . 5- . 5 s f ( t ) = braceleftBigg if 0 t < 2 T e- 3(0 . 5 k- 1) if kT t < ( k + 1) T, k = 2 , 3 , . . . F ( s ) = F * ( s ) 1 e- T s s = e- s (1 e- . 5 s ) s (1 e- 1 . 5- . 5 s ) . Note that since the time delay 1 = 2 T is an integer multiple of the sampling period T = 0 . 5, the time delay property of star transform applies, i.e., F * ( s ) = e- 2 sT E * ( s ). 1 (d) For the given g ( t ), we have g * ( t ) = summationdisplay k =0 g ( kT ) ( t kT ) = summationdisplay k =1 e- 3(0 . 5 k- . 1) ( t kT ) = summationdisplay k =1 e- 1 . 5 k +0 . 3 ( t . 5 k ) G * ( s ) = summationdisplay...
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This note was uploaded on 04/15/2011 for the course ECE 483 taught by Professor Evens during the Spring '08 term at Purdue University-West Lafayette.

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3 - 2/18/10 ECE 483 Homework #3 Solution Problem 1. Given a...

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