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Unformatted text preview: Page 1 BME 343: Biomedical Engineering Signal and Systems Analysis Homework 2 Do the following problems from the textbook. The point value of each problem is given below and 1 bonus points will be given for following the rules. Problem 2.2 7 2.2 8 2.3 4 2.4 3 2.4 5 2.4 8 2.4 14 2.4 24 2.4 32 2.4 35 2.4 39 2.6 6 2.7 2 Point Value 6 6 6 7 8 8 8 10 10 7 11 6 6 2.27 The characteristic polynomial is 2 ( 1)( 5 6). + + + The characteristic equation is 2 ( 1)( 5 6) + + + = or ( 1)( 2)( 3) 0. + + + = The characteristic roots are 1, 2 and 3. The characteristic modes are t e , 2 t e and 3 t e . Therefore, 2 3 1 2 3 ( ) t t t y t c e c e c e = + + and ' 2 3 1 2 3 ( ) 2 3 t t t y t c e c e c e =  ' 2 3 1 2 3 ( ) 4 9 t t t y t c e c e c e = + + Setting t=0, and substituting initial conditions yields 1 2 3 1 2 3 1 2 3 2 2 3 1 4 9 5 c c c c c c c c c + + =  =  + + = 1 2 3 6, 7, 3 c c c = =  = Therefore, 2 3 ( ) 6 7 3 t t t y t e e e = + Page 2 BME 343: Biomedical Engineering Signal and Systems Analysis 2.28 The zeroinput response for a LTIC system is given as ( ) 2 3. t y t e = + Since two modes are visible, the system must have, at least, the characteristic roots 1 = and 2 1 =  . (a) No, it is not possible for the systems characteristic equation to be 1 + = since the required mode at = is missing. (b) Yes, it is possible for the systems characteristic equation to be 2 3( ) + = since this equation has the two required roots 1 = and 2 1 =  . (c) Yes, it is possible for the systems characteristic equation to be 2 ( 1) + = . This equation supports a general zeroinput response of 1 2 3 ( ) t t y t c c e c te = + + . By letting 1 3 c = , 2 2 c = , and 3 c = , the observed zeroinput response is possible. 2.34 The characteristic equation is 2 2 6 9 ( 3) 0. + + = + = Therefore, 3 1 2 ( ) ( ) t n y t c c t e = + and ' 3 1 2 2 ( ) [ 3( ) ] t n y t c c t c e =  + + Setting t = 0, and substituting ' (0) 0, (0) 1 n n y y = = , we obtain 1 1 2 3 1 c c c = + = 1 2 0, 1 c c = = 3 ( ) t n y t te = Hence, ' 3 ( ) ( ) [ ( ) ( )] ( ) 0 ( ) [(2 9) ( )] ( ) [2 ( ) 9 ( )] ( ) (2 3 ) ( ) n n n n t h t b t P D y t u t t D y t u t y t y t u t t e u t...
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This note was uploaded on 09/28/2009 for the course BME 343 taught by Professor Emelianov during the Fall '09 term at University of Texas at Austin.
 Fall '09
 Emelianov

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