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notes_LaplaceTransforms - Notes on Laplace transform...

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Notes on Laplace transform, transfer function, and BIBO stability Dr. Prabir Barooah EML 4312 Spring 2009 January 23, 2009 1 Laplace transform The Laplace transform of a signal { y ( t ) } is defined by Y ( s ) = L ( y ( t )) Δ = integraldisplay 0 - y ( τ ) e - dτ, s C (1) The integral exists (has a well defined, finite value) only if the signal y ( t ) grows with t at a rate slower than the exponential e - st decays, or vice versa. The rate of decay depends on the complex number s . Therefore, for a given signal y ( t ), for the Laplace transform to exist, the value of s must be such that the integral above converges. The region of convergence of a Laplace transform Y ( s ) is the region in the complex plane C that s can take values in so that Y ( s ) is finite and well defined. Example 1. Let us evaluate the Laplace transform of the complex signal x ( t ) = e ( a + bj ) t Δ = braceleftBigg e a + bj t t 0 0 t < 0 By applying the definition of the Laplace transform, we get L ( y ( t )) = L ( x ( t )) = X ( s ) = integraldisplay 0 x ( t ) e - st dt = integraldisplay 0 e ( a + bj - s ) t dt = 1 a + bj s parenleftbigg lim τ + e ( a - Re ( s )+( b - Im ( s )) j ) τ 1 parenrightbigg = braceleftBigg 1 s - ( a + bj ) if Re ( s ) > a undefined otherwise Hence, the Laplace transform of the signal e ( a + bj ) t is 1 s - ( a + bj ) with a region of convergence Re ( s ) > a . In general, the Laplace transform of a signal e pt (where p is a complex number) is 1 s - p with a ROC given by Re ( s ) > Re ( p ). Note that the strict inequality is important. The Laplace transform Y ( s ) of a signal y ( t ) is a complex number, whose values depend on the complex argument s . Therefore Laplace transform can be thought of as a complex function, that maps one complex plane (the s -plane) to another complex plane (the Y ( s )- plane). For a complex number s 0 ROC, the Laplace transform Y ( s 0 ) is a complex number. Exercise: plot the values of Y ( s ) for a few complex numbers s . 1
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1.1 Useful properties of the Laplace transform The following properties of the Laplace transform will be very useful in analyzing LTI systems specified in terms of differential equations. To state one of the properties, we’ll need the unit step signal 1( t ), which is defined as: 1( t ) Δ = braceleftBigg 1 t 0 0 t < 0 (2) 1. Linearity : L ( ax ( t ) + by ( t )) = a L ( x ( t )) + b L ( y ( t )) , where { x ( t ) } , { y ( t ) } are signals are a, b are complex scalars. 2. Transform of a derivative signal: If Y ( s ) is the Laplace transform of a signal { y ( t ) } , then the transform of the signal ˙ y is L ( ˙ y ( t )) = sY ( s ) y (0) . This can be shown by integration by parts: L ( ˙ y ) = integraldisplay 0 e - st ˙ y ( t ) dt = bracketleftbigg e - st y ( t ) integraldisplay ( s ) e - st y ( t ) dt bracketrightbigg 0 = y (0) + sY ( s ) , where we have used the fact that, since e - st y ( t ) is integrable for all values of s in the ROC (otherwise the Laplace transform Y ( s ) would not exist 1 ), e - st y ( t ) 0 as t → ∞ . 3. Transform of a integral signal: If Y ( s ) is the Laplace transform of a signal { y ( t ) } , then L parenleftbiggintegraldisplay t 0 y ( η ) parenrightbigg = 1 s Y ( s ) It is left as an exercise for you to show that this true, using the previous result.
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