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102_1_lecture10

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

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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

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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

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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 jkω 0 t where ω 0 = 2 π 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 sin( 0 T 1 ) Notice that if T increases, ω 0 decreases and its harmonics become more closely spaced. EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 5

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Figures below illustrate this property. Fourier series representation of the signal is shown with T 1 = 2 and T = 16 , 32 , and 64 . 8 0.08 0.15 -8 -6 -4 -2 0 2 4 6 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 8 0.08 -8 -6 -4 -2 0 2 4 6 -0.05 0 0.05 0.1 0.15 EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 6
-8 -6 -4 -2 0 2 4 6 8 -0.02 0 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

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Fourier Transforms Given a continuous time signal x ( t ) , define its Fourier transform as the function of a real ω : X ( ) = Z -∞ x ( t ) e - jωt 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 jkω 0 t EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 8
where a k = 1 T Z T/ 2 - T/ 2 x ( t ) e - jkω 0 t dt. Define the truncated Fourier transform: X T ( ) = Z T 2 - T 2 x ( t ) e - jωt dt so that a k = 1 T X T ( jkω 0 ) . The Fourier series is then x T ( t ) = X k = -∞ 1 T X T ( jkω 0 ) e jkω 0 t As T → ∞ , then 2 π T = ω 0 0 . Suppose that k increases with T so that 0 ω where ω is fixed (this is only approximate except as limit).

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

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