This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 2/18/10 ECE 483 Homework #3 Solution Problem 1. Given a continuoustime 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 zeroorder 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 stairlike 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...
View
Full
Document
This note was uploaded on 04/15/2011 for the course ECE 483 taught by Professor Evens during the Spring '08 term at Purdue UniversityWest Lafayette.
 Spring '08
 evens

Click to edit the document details