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# 102_1_lecture11_student - UCLA Fall 2011 Systems and...

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UCLA Fall 2011 Systems and Signals Lecture 11: Fourier Transform Theorems November 2, 2011 EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 1

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Administration MATLAB Project Opportunity to do hands-on signal processing Will be handed out on Monday (Nov. 8) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 2
Introduction Review of Fourier Transform basics Fourier Transform Properties ? The Convolution Theorem ? EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 3

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Review LTI systems allow us to compute output for arbitrary input via convolution - but it is often tedious If an input can be expressed as a sum of scaled, complex exponentials (with complex frequency s k = σ k + k ) and the transfer function H ( s ) of the LTI system is known, then the input-output relationship can be succinctly expressed as: x ( t ) = X k = -∞ a k e js k t y ( t ) = X k = -∞ H ( s k ) a k e k t Output is the sum of the complex exponentials present in the input, scaled by H ( s ) . EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 4
Many periodic signals can be expressed in terms of Fourier series coeﬃcients: x ( t ) = X k = -∞ a k e jkω 0 t The number of Fourier series coeﬃcients a k could be inﬁnite Frequencies present in x ( t ) are harmonically related - multiples of ω 0 . Frequency spectrum is discrete. Many aperiodic signals can be expressed using Fourier Transform: x ( t ) = 1 2 π Z -∞ X ( ) e jωt dt The frequency spectrum is now continuous - inﬁnite number of frequencies is needed. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 5

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We can represent signals in terms of their time-domain behavior ( x ( t ) ) or frequency content ( a k or X ( ) ). We can alternate between the frequency and time domain using analysis and synthesis equations. Primarily interested in Fourier Transform - it allows frequency representation of broader class of signals (not just periodic) Many Fourier Series properties have analogous Fourier Transform properties Frequency representation using Fourier Transform makes LTI system analysis simple we will see this today! EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 6
Review example . Compute Fourier transform of x ( t ) = e - at u ( t ) . 2 1 0 1 2 3 4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 t f(t) e - at u ( t ) t x ( t ) Hint: Use Fourier Transform analysis equation: X ( ) = Z -∞ x ( t ) e - jωt dt EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 7

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Linearity Linear combination of two signals x 1 ( t ) and x 2 ( t ) is a signal of the form ax 1 ( t ) + bx 2 ( t ) . Linearity Theorem: The Fourier transform is linear; that is, given two signals x 1 ( t ) and x 2 ( t ) and two complex numbers a and b , then ax 1 ( t ) + bx 2 ( t ) aX 1 ( ) + bX 2 ( ) . This follows from linearity of integrals: Z -∞ ( ax 1 ( t ) + bx 2 ( t )) e - jωt dt = a Z -∞ x 1 ( t ) e - dt + b Z -∞ x 2 ( t ) e - dt = aX 1 ( ) + bX 2 ( ) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 8
This easily extends to ﬁnite combinations. Given signals x k ( t ) with Fourier transforms X k ( ) and complex constants q k , k = 1 , 2 ,...K , then K X k =1 q k x k ( t ) K X k =1 q k X k ( ) .

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102_1_lecture11_student - UCLA Fall 2011 Systems and...

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