The Laplace Transform Montek Singh Thurs., Feb. 19, 2002 3:30-4:45 pm, SN115 1 What we will learn The notion of a complex frequency Representing a signal in the frequency domain Manipulating signals in the frequency domain 2 Complex Exponential Functions Complex exponential = est, where s = + j Examples:
<0, =0 >0, =0 =0, =0 <0 >0 Re(est) =0 3 Some Useful Equalities e st e ( j )t et e jt et (cos t j sin t ) 1 j t cos t (e e jt )
2 1 j t j t sin t (e e ) 2j 4 The Laplace Transform: Overview Key Idea: Represent signals as sum of complex exponentials since all exponentials have the form Aest, it suffices to know the value of A for each s, to completely represent the original signal i.e., representation transformed from t to s domain Benefits: Complex operations in the time domain get transformed into simpler operations in the s-domain e.g., convolution, differentiation and integration in time
algebraic operations in the s-domain! Even fairly complex differential equations can be transformed into algebraic equations 5 The Laplace Transform F ( s ) L[ f (t )] F(s) = Laplace Transform of f(t): st F ( s ) f (t )e dt j 1 st f (t )
F ( s ) e ds 2j j 1-to-1 correspondence between a signal and its Laplace Transform Frequently, only need to consider time t > 0: F ( s) f (t )e st dt 0 6 Example 1: The Unit Impulse Function f (t ) (t )
F ( s ) (t )e st dt (t )e s 0 dt 1 F(s) = 1 everywhere! 7 Example 2: The Unit Step Function f (t ) 1 for t 0 st F ( s ) e dt 0
st e s 0 1 , for 0 s 8 Some Useful Transform Pairs L (t ) 1 1 L u (t ) , 0 s n!
n L t n 1 , 0, n 0 s 1 at Le , a sa s L cos t 2 , 0 2 s L sin t 2 , 0 2 s
9 Properties of the Laplace Transform (1) L f1 (t ) f 2 (t ) L f1 (t ) L f 2 (t ) F1 ( s ) F2 ( s ) L af (t ) aL f (t ) aF ( s) L e at f (t ) F ( s a ) as L f (t a )u (t a) e F ( s ) 1 s L f (at ) F a a
10 Properties of the Laplace Transform (2) d L f (t ) sF ( s) f (0) dt d2 df 2 L 2 f (t ) s F ( s ) sf (0) (0) dt dt t F (s) L f ( )d s 0
t F ( s) f 1 (0) L f ( )d s s t lim 1 where f (0) f ( )d t 0 11
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