102_1_lecture13_student

# 102_1_lecture13_student - UCLA Fall 2011 Systems and...

This preview shows pages 1–10. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Administration Review Session: Monday, Nov. 14. Second Midterm: Wednesday, Nov. 16, 10:00am-12:00pm in class. No office hours on Tuesday, Nov. 15. TA extra office hour on Tuesday, Nov. 15, 4:00pm-6:00pm. 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 + ) t . EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 5

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 0 is the smallest value for which f ( t ) e - σ 0 t converges to zero, then any σ > σ 0 will do σ 0 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Bilateral Laplace Transform The Fourier transform is: F ( ) = Z -∞ f ( t ) e - jωt dt f ( t ) = 1 2 π Z -∞ F ( ) e jωt 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 + ) 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 + . The complex frequency s includes: the oscillation component that we’re used to, plus a decay/growth component σ .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern