chapter2 - EE 303 S i g n a l A n a l y s i s a n d T r a n...

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1 Dr. Tommaso Melodia, EE 303 Signal Analysis and Transform Methods 1 2. Time-Domain Analysis of Continuous-Time Systems Dr. Tommaso Melodia Email: [email protected] . buffalo . edu Office: 215G Bonner Hall EE 303 Signal Analysis and Transform Methods
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2 Dr. Tommaso Melodia, EE 303 Signal Analysis and Transform Methods 2 Analysis of Linear Differential Systems We consider linear time invariant (LTI) systems Continuous-Time (LTIC) Time-Domain Analysis Frequency-Domain Analysis We consider Linear Differential Systems For practical systems: M N ) ( ... ) ( ... 1 1 1 1 1 1 1 1 t x b dt x d b dt x d b dt x d b t y a dt dy a dt y d a dt y d N N M M M N M M M N N N N N N N + + + + = = + + + + ! ! ! + ! ! ! ! !
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3 Dr. Tommaso Melodia, EE 303 Signal Analysis and Transform Methods 3 Linear Differential Systems Considering notation dt d D = ) ( ) ( ) ( ) ( t x D P t y D Q = ) ( ) ... ( ) ( ) ... ( 1 1 1 1 1 1 t x b D b D b D b t y a D a D a D N N M N M M N N N N M N + + + + = + + + + ! + ! ! ! ! ! ) ( ... ) ( ... 1 1 1 1 1 1 1 1 t x b dt x d b dt x d b dt x d b t y a dt dy a dt y d a dt y d N N M M M N M M M N N N N N N N + + + + = = + + + + ! ! ! + ! ! ! ! ! N N M N M M N N N N b D b D b D b D P a D a D a D D Q M N + + + + = + + + + = ! + ! ! ! ! ! 1 1 1 1 ... ) ( ... ) ( 1 1
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4 Dr. Tommaso Melodia, EE 303 Signal Analysis and Transform Methods 4 Zero-Input Response of LTIC System A linear combination of y o (t) and all its derivatives is 0 for all values of t Hence y o (t) and all its derivatives must have the same form TOTAL RESPONSE = ZERO-INPUT RESPONSE + ZERO-STATE RESPONSE 0 ) ( ) ( 0 ) ( ) ( ) ( ) ( ) ( = = = t y D Q t x when t x D P t y D Q 0 ) ( ) ... ( 0 1 1 1 = + + + + ! ! t y a D a D a D N N N N t ce t y ! = ) ( 0
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5 Dr. Tommaso Melodia, EE 303 Signal Analysis and Transform Methods 5 The solutions of Q( λ )=0 determine the solutions of Q(D)y 0 (t)=0 Solutions: Zero-input Response of LTIC System t N N t t t e c t y D e c t y D e c t Dy ce t y ! ! ! ! ! ! ! = = = = ) ( .... ) ( ) ( ) ( 0 2 0 2 0 0 0 ) ( ) ... ( 0 1 1 1 = + + + + ! ! t y a a a c N N N N " " " 0 ) ( = ! Q t t t N e e e ! ! ! ..., , , 2 1 ) )...( )( ( ) ( 2 1 N Q ! ! ! ! ! ! ! " " " =
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6 Dr. Tommaso Melodia, EE 303 Signal Analysis and Transform Methods 6 Zero-input Response of LTIC System Fact: If are solutions, then is the General Solution Constants c 1 , c 2 ,…, c N are determined by the auxiliary conditions (N additional constraints on the solution) t t t N e e e ! ! ! ..., , , 2 1 t N t t N e c e c e c t y ! ! ! + + + = ... ) ( 2 1 2 1 0
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7 Dr. Tommaso Melodia, EE 303 Signal Analysis and Transform Methods 7 Characteristic Polynomial of the System Characteristic polynomial of the system Characteristic equation of the System Characteristic values Eigenvalues Natural Frequencies Characteristic roots Characteristic modes Natural modes (modes) ) ( ! Q 0 ) ( = ! Q N ! ! ! ,..., , 2 1 t i e ! The zero-input response of the system is a linear combination of the characteristic modes The entire behavior of the system is dictated by its characteristic modes
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