hwsln-421-s06-01

Hwsln-421-s06-01 - ECE 421 Spring 2006 Solutions to HW Assignment#1 Problem 1 For the given time-domain expression x1(t the Laplace transform is 2

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ECE 421, Spring 2006 Solutions to HW Assignment #1 Problem 1 For the given time-domain expression x 1 ( t ) , the Laplace transform is X 1 ( s )= 2 s 3 4 s +3 + 3 s 2 = 2( s +3)( s 2) 4 s 3 ( s 2) + 3 s 3 ( s +3) s 3 ( s s 2) (1) = 2 s 2 +2 s 12 4 s 4 +8 s 3 s 4 +9 s 3 s 3 ( s s 2) (2) = ¡ s 4 17 s 3 2 s 2 2 s +12 ¢ s 3 ( s s 2) = ( s 0 . 8225) ( s +17 . 1212) ( s +0 . 4719 ± j 0 . 7934) s 3 ( s s 2) (3) Problem 2 X 2 a ( s 1 s (4 s +1) = 0 . 25 s ( s . 25) = A 1 a s + A 2 a ( s . 25) (4) A 1 a =[ sX 2 a ( s )] | s =0 = 0 . 25 ( s . 25) ¯ ¯ ¯ ¯ s =0 =1 (5) A 2 a ( s . 25) X 2 a ( s )] | s = 0 . 25 = 0 . 25 s ¯ ¯ ¯ ¯ s = 0 . 25 = 1 (6) x 2 a ( t )=1 e 0 . 25 t e t/ 4 , for t 0 (7) X 2 b ( s 3 s 2 ( s . 25) = A 1 b s 2 + A 2 b s + A 3 b ( s . 25) (8) A 3 b ( s . 25) X 2 b ( s )] | s = 0 . 25 = 3 s 2 ¯ ¯ ¯ ¯ s = 0 . 25 =48 (9) A 1 b = £ s 2 X 2 b ( s ) ¤¯ ¯ s =0 = 3 ( s . 25) ¯ ¯ ¯ ¯ s =0 =12 (10) A 2 b = d ds £ s 2 X 2 b ( s ) ¤ ¯ ¯ ¯ ¯ s =0 = d ds 3 ( s . 25) ¸¯ ¯ ¯ ¯ s =0 = " ( s . 25) · 0 3 · 1 ( s . 25) 2 ¯ ¯ ¯ ¯ s =0 = 48 (11) x 2 b ( t )=12 t 48 + 48 e 0 . 25 t for t 0 (12) X 2 c ( s 13 s ( s 2 +4 s + 13) = A 1 c s + A 2 c s + A 3 c ( s 2 s +13) (13) A 1 c sX 2 c ( s )] | s =0 = 13 ( s 2 s + 13) ¯ ¯ ¯ ¯ s =0 (14) 13 = A 1 c ¡ s 2 s +13 ¢ + A 2 c s 2 + A 3 c s (15) 1
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s 2 :0 = A 1 c + A 2 c A 2 c = A 1 c = 1 s 1 = 4 A 1 c + A 3 c A 3 c = 4 A 1 c = 4 s 0 :1 3 = 1 3 A 1 c 1=1 X 2 c ( s )= 1 s s +4 ( s 2 s + 13) ¸ = 1 s " s ( s +2) 2 +3 2 # (16) = 1 s " s +2 ( s 2 2 + 2 ( s 2 2 # = 1 s " s ( s 2 2 +0 . 6667 · 3 ( s 2 2 # (17) x 2 c ( t )=1 e 2 t [cos (3 t )+0 . 6667 sin (3 t )] (18) Problem 3 This problems makes use of the very important fact that when a sinusoidal input signal is applied to a linear, time-invariant (LTI) system, the output signal is a sinusoid of the same frequency. The magnitude
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This note was uploaded on 10/25/2010 for the course ECE 421 taught by Professor Cook,g during the Spring '08 term at George Mason.

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Hwsln-421-s06-01 - ECE 421 Spring 2006 Solutions to HW Assignment#1 Problem 1 For the given time-domain expression x1(t the Laplace transform is 2

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