Homework 4 - Christopher Belcastro I.D. - 904424264 ECE...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Christopher Belcastro I.D. - 904424264 ECE 2704 Homework #4 2 October 2006 1) a. Find the transfer function: s 2Y ( s) - y (0) - sy (0) + 4 sY ( s ) - 4 y (0) + 3Y ( s) = X ( s ) Y ( s ) [ s 2 + 4 s + 3] = X ( s ) + y (0) - sy (0) - 4 y (0) Y ( s ) 1 + y (0) - sy (0) - 4 y (0) = X (s) X ( s) ( s 2 + 4 s + 3) b. Poles at -1 & -3 . . . Assume : y (0) = 0, y (0) = 0 . c. . A B 1 + y (0) - sy (0) - 4 y (0) L -1 + = 2 X ( s ) ( s + 4s + 3) s + 3 s + 1 L -1 { H ( s)} = A( s + 1) + B ( s + 3) s = -3 s = -1 - 2A = 1 2B = 1 so A = - so B = 1 2 1 2 1 -1 1 1 L -1 2 + 2 = - e -3t u s (t ) + e -t u s (t ) 2 s + 3 s + 1 2 -4 t d. x (t ) = e u s (t ) X (s) = 1 s+4 1 s+4 s 2Y ( s) - 1 + 4sY ( s ) + 3Y ( s ) - 2 s - 8 = Y (s) s 2 + 4s + 3 = Y (s) s 2 + 4s + 3 = [ [ [ ] ] 1 + 2s + 9 s+4 ( s + 4)(2 s + 9) 2 s( s + 4)(2 s + 9) 9( s + 4)(2 s + 9) + + ( s + 4)(2s + 9) ( s + 4)(2 s + 9) ( s + 4) 2 (2 s + 9) (2 s + 9) + 2 s( s + 4)(2 s + 9 ) + 9( s + 4)(2 s + 9) ( s + 4)(2 s + 9) Y (s) s 2 + 4s + 3 = 1 + 2 s 2 + 8s + 9 s + 36 2 s 2 + 17 s + 37 Y (s) = = ( s + 3)( s + 4)( s + 1) ( s + 3)( s + 4)( s + 1) 2 2 s + 17 s + 37 A B C Y (s) = = + + ( s + 3)( s + 4)( s + 1) s + 3 s + 1 s + 4 2 s 2 + 17 s + 37 = A( s + 1)( s + 4) + B ( s + 3)( s + 4) + C ( s + 3)( s + 1) at s = - 3 18 - 51 + 37 = -2 A 2 - 17 + 37 = 6 B 32 - 68 + 37 = 3C Y (s) = 11 1 -2 + 3 + 3 s + 3 s +1 s + 4 A = -2 22 11 B= = 6 3 1 C= 3 at s = -1 at s = -4 L-1 {Y ( s )} = -2e -3t + 11 -t 1 -4t e + e 3 3 11 1 y (t ) = - 2e -3t + e -t + e - 4t u s (t ) 3 3 e. Convolution Integral: 1 1 h (t ) = - e -3t u s (t ) + e -t u s (t ) 2 2 x (t ) = e - 4 t u s (t ) 1 1 y (t ) = - e -3( t - ) + e -( t - ) u s ( t - ) e -4 u s ( )d 2 2 - 12.1.2 ) y (t ) = - h ( t - ) x ( )d e at sin kt = k (s - a) 2 + k 2 -6 ( t - ) y (t ) = - [e sin( (t - )) u s ( t - ) x ( ) d u s (t ) H (s ) = ( s + 6) 2 + 2 The impulse response functions is: h (t ) = e -6t sin ( t ) u s ( t ) x ...
View Full Document

This note was uploaded on 04/01/2008 for the course ECE 2704 taught by Professor Djstilwell during the Fall '08 term at Virginia Tech.

Ask a homework question - tutors are online