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102_1_lecture10_student - UCLA Fall 2011 Systems and...

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UCLA Fall 2011 Systems and Signals Lecture 10: The Continuous Time Fourier Transform October 31, 2011 EE102: Systems and Signals; Fall 2011, 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 2011, Jin Hyung Lee 2
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Fourier Series (Review) Many continuous time, periodic signals can be represented as sums of complex exponentials. Formally, if the signal is periodic in T ( x ( t ) = x ( t + T ) ), and has a fundamental frequency ω 0 = 2 π T , we can write: x ( t ) = X k = -∞ a k e jkω 0 t This is the synthesis equation. Complex exponentials are summed to produce the time-domain signal. Complex exponentials are scaled by { a k } - Fourier series coefficients . Notice that some signals require an infinite number of Fourier coefficients! EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 3
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Notice also that the frequency of complex exponentials included in the equation is an integer multiple of fundamental frequency. That is, ω = 0 . If we included other frequencies, the signal would not be periodic in T! Fourier series coefficients (how much each complex exponentials contributes to the signal) are determined using the analysis equation: a k = 1 T Z T x ( t ) e - jkω 0 t dt The integral is over one period of x ( t ) . Can integrate from 0 to T , - ± to T - ± , - T/ 2 to 2 , etc – all are equivalent. In practice choose simpler limits. Memorize these equations. Be very clear on the sign and variables in the exponent. Remember 1 T factor in the analysis equation! EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 4
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Examples. 1. Given signal has a 0 6 = 0 . Can this signal be odd? 2. Find the Fourier coefficients a k of x ( t ) = 1 + cos(2 πt ) , periodic in T = 1 . 3. Find the time-domain representation of x ( t ) , if its single nonzero Fourier coefficient is a 2 = 1 . EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 5
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Solution: EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 6
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We usually use shorthand notation for computing Fourier series: x ( t ) FS --→ a k Last class, introduced many properties of Fourier series: here, x ( t ) a k , y ( t ) b k , z ( t ) c k . Linearity z ( t ) = Ax ( t ) + By ( t ) c k = Aa k + Bb k Time Shifting y ( t ) = x ( t - t 0 ) b k = a k e - jkω 0 t 0 Time Reversal y ( t ) = x ( - t ) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 7
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b k = a - k Conjugation y ( t ) = ± x ( t ) ² * b k = a * - k Parseval’s relation 1 T Z T | x ( t ) | 2 dt = X k = -∞ | a k | 2 Ability to apply these properties will save you significant time. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 8
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Example. Suppose a signal is periodic and even: x ( t ) = x ( t + T ) and x ( t ) = x ( - t ) . Suppose further that x ( t ) is real. Could it be that a k 6 = a - k ? Could a k have an imaginary component? Using time reversal property: x ( t ) = x ( - t ) 7→ a k = a - k .
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This note was uploaded on 03/30/2012 for the course ELEC ENGR 102 taught by Professor Lee during the Fall '11 term at UCLA.

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102_1_lecture10_student - UCLA Fall 2011 Systems and...

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