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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 Today’s topics • Laplace Transform EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 3 Limitations of the Fourier Transform So far we’ve 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 doesn’t have a Fourier transform. However, we can create a new function φ ( t ) = f ( t ) e σt . If σ > 2 , then this is a decreasing exponential. It does have a Fourier transform. The Fourier transform represents φ ( t ) in terms of spectral components e jωt . We can express f ( t ) as f ( t ) = φ ( t ) e σt Each spectral component is multiplied by e σt , so f ( t ) can be represented by spectral components e σt e jωt = e ( σ + jω ) t . EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 5 t t φ ( t ) = f ( t ) e σ t f ( t ) 1 1 t t e j ω t e ( σ + j ω ) t EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 6 How big should we choose σ ? For f ( t ) = e 2 t , any σ > 2 will produce a decaying, Fourier transformable signal, and a different Fourier transform. If σ is the smallest value for which f ( t ) e σ t converges to zero, then any σ > σ will do σ σ j ω 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 ( jω ) = Z ∞∞ f ( t ) e jωt dt f ( t ) = 1 2 π Z ∞∞ F ( jω ) e jωt dω The Fourier transform of f ( t ) e σt is F f ( t ) e σt = Z ∞∞ f ( t ) e σt e jωt dt = Z ∞∞ f ( t ) e ( σ + jω ) 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 = σ + jω ....
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 Jin Hyung Lee

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