102_1_lecture10

102_1_lecture10 - UCLA Fall 2010 Systems and Signals...

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Unformatted text preview: UCLA Fall 2010 Systems and Signals Lecture 10: The Continuous Time Fourier Transform October 27, 2010 EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 1 Introduction to Fourier Transforms Fourier transform as a limit of Fourier series Inverse Fourier transform: The Fourier integral theorem Examples: the rect function, one-sided exponential Symmetry properties EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 2 Administration Homework 4 due. Submit now or after class to Engineering IV 66-144 (TA Office Hour room dropbox) Midterm Nov. 1 10:00 am - 12:00pm 4 problems One sheet of notes allowed EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 3 Fourier Series (Review) We can write a periodic, continuous time signal x ( t ) as a sum of harmonically related complex exponentials: x ( t ) = X k =- a k e jk t where = 2 T is the fundamental frequency. a k are called Fourier series coefficients. In general, a k are complex. Some signals may require an infinite number of complex exponentials. Allows us to talk about frequency content (spectral representation) of a signal EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 4 What to do if a signal is not periodic? Aperiodic signals are extremely common - we need to know their frequency content! Recall a periodic square wave example from few lectures ago: t T 1 T x(t)-T 1-T This signal has Fourier series coefficients given by: a k = 1 k sin( k T 1 ) Notice that if T increases, decreases and its harmonics become more closely spaced. EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 5 Figures below illustrate this property. Fourier series representation of the signal is shown with T 1 = 2 and T = 16 , 32 , and 64 .-8-6-4-2 2 4 6 8-0.02 0.02 0.04 0.06 0.08-8-6-4-2 2 4 6 8-0.05 0.05 0.1 0.15-8-6-4-2 2 4 6 8-0.1-0.05 0.05 0.1 0.15 0.2 0.25-8-6-4-2 2 4 6 8-0.02 0.02 0.04 0.06 0.08-8-6-4-2 2 4 6 8-0.05 0.05 0.1 0.15 EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 6-8-6-4-2 2 4 6 8-0.02 0.02 0.04 0.06 0.08 As the period increases, more and more spectral coefficients are needed EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 7 Fourier Transforms Given a continuous time signal x ( t ) , define its Fourier transform as the function of a real : X ( j ) = Z - x ( t ) e- jt dt if the integral makes sense. This is similar to the expression for the Fourier series coefficients. We can interpret this as the result of expanding x ( t ) as a Fourier series in an interval [- T/ 2 ,T/ 2) , and then letting T . The Fourier series for x ( t ) in interval [- T/ 2 ,T/ 2) : x ( t ) = X k =- a k e jk t EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 8 where a k = 1 T Z T/ 2- T/ 2 x ( t ) e- jk t dt....
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This note was uploaded on 04/20/2011 for the course EE 102 taught by Professor Levan during the Fall '08 term at UCLA.

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102_1_lecture10 - UCLA Fall 2010 Systems and Signals...

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