Lecture 10.ppt - Signals and Systems Lecture 10 1 Introduction Laplace transform is another method to transform a signal from time domain to frequency

Lecture 10.ppt - Signals and Systems Lecture 10 1...

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1 Signals and Systems Lecture 10
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2 Introduction Laplace transform is another method to transform a signal from time domain to frequency domain (s-domain) The basic idea of Laplace transform comes from the Fourier transform As we have seen in the previous lecture, not many functions have their Fourier transform such as t , t 2 , e t etc.
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3 The Fourier transform formula: The Laplace transform formula is the modification of the above formula, that is, the term j is replaced by s s is equal to + j , where is a large positive real number The Laplace transform formula: However, the Laplace transform only support the function f ( t ) which domain t ≥ 0 dt e t f t f F t j ) ( )} ( { ) ( F 0 ) ( )} ( { ) ( dt e t f t f s F st L
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4 Using definition, find the Laplace transform of (a) (b) (c) (d) Example 1 ) ( ) ( t u t f ) ( ) ( 5 t u e t g t ) ( ) ( t u t t i ) ( cos ) ( t u t t v
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5 Solution (a) s s s e s e dt e dt e t u s F st st st 1 1 0 1 ) ( ) ( 0 0 0 (b) 5 1 ) 5 ( 1 0 ) 5 ( 1 ) 5 ( ) ( ) 5 ( 0 ) 5 ( 0 ) 5 ( 0 5 s s s e s e dt e dt e e s G s t s t s st t
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6 (c) 2 2 0 2 0 0 0 1 1 ) 0 0 ( ) ( s s e s e dt s e s te dt te s I st st st st
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7 (d) ) ( cos ) 1 0 ( ) )( cos ( ) cos ( sin ) 0 0 ( ) ( sin sin cos ) ( 2 0 2 0 0 0 0 0 0 s V s s tdt e s s dt se t s t e s tdt e s dt se t t e tdt e s V st st st st st st st 1 ) ( 2 s s s V
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8 Properties of L -transform Linearity L { af ( t ) ± bg ( t )} = a L { f ( t )} ± b L { g ( t )} a
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