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Unformatted text preview: UCLA Fall 2011 Systems and Signals Lecture 13: Introduction to Laplace Transform November 09, 2011 EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 1 Administration Review Session: Monday, Nov. 14. Second Midterm: Wednesday, Nov. 16, 10:00am12:00pm in class. No office hours on Tuesday, Nov. 15. TA extra office hour on Tuesday, Nov. 15, 4:00pm6:00pm. Location: TBA. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 2 Agenda Todays topics Laplace Transform EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 3 Limitations of the Fourier Transform So far weve considered Fourier Transforms (and Fourier series) of discrete and continuous signals. To be useful, Fourier transforms must exist, or be defined in a generalized sense. For many areas, this will be all you will need (communications, optics, image processing). For many signals and systems the Fourier transform is not enough: Signals that grow with time (your bank account, or the GDP of the US) Systems that are unstable (many mechanical or electrical systems). These are important problems. How can we analyze these? EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 4 Consider the signal f ( t ) = e 2 t u ( t ) This is an increasing exponential, so it doesnt have a Fourier transform. However, we can create a new function ( t ) = f ( t ) e t . If > 2 , then this is a decreasing exponential. It does have a Fourier transform. The Fourier transform represents ( t ) in terms of spectral components e jt . We can express f ( t ) as f ( t ) = ( t ) e t Each spectral component is multiplied by e t , so f ( t ) can be represented by spectral components e t e jt = e ( + j ) t . EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 5 t t ( t ) = f ( t ) e t f ( t ) 1 1 t t e j t e ( + j ) t EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 6 How big should we choose ? For f ( t ) = e 2 t , any > 2 will produce a decaying, Fourier transformable signal, and a different Fourier transform. If is the smallest value for which f ( t ) e t converges to zero, then any > will do j Region of Convergence This means the spectrum of f ( t ) is not unique. The part of the complex plane where the spectrum exists is the region of covergence . EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 7 Bilateral Laplace Transform The Fourier transform is: F ( j ) = Z  f ( t ) e jt dt f ( t ) = 1 2 Z  F ( j ) e jt d The Fourier transform of f ( t ) e t is F f ( t ) e t = Z  f ( t ) e t e jt dt = Z  f ( t ) e ( + j ) t dt = Z  f ( t ) e st dt = F ( s ) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 8 were s the complex frequency s = + j ....
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