EE3TP4_14_FourierTransform_v3_Lecture 21

EE3TP4_14_FourierTransform_v3_Lecture 21 - 4.3 Fourier...

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4.3 Fourier Transform Recall : Fourier Series represents a periodic signal as a sum of sinusoids Note : Because the FS uses “harmonically related” frequencies k ϖ 0 , it can only create periodic signals x t = k =−∞ c k e k t or complex sinusoids e jk ω 0 t With arbitrary discrete frequencies… NOT harmonically related x t = k =−∞ c k e k t The problem with is that it cannot include all possible frequencies! Q: Can we modify the FS idea to handle non -periodic signals? A: Yes! What about ? That will give some non-periodic signals but not some that are important!! These overheads were originally developed by Mark Fowler at Binghamton University, State University of New York.
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How about: x t = 1 −∞ X ω e jωt Called the “Fourier Integral ” also, more commonly, called the Inverse Fourier Transform Plays the role of c k Plays the role of e jk ω 0 t Integral replaces sum because it can “add up over the continuum of frequencies”! Given x ( t ) how do we get X (ω)? X ω = −∞ x t e jωt dt Note: X (ω) is complex-valued function of ω (- , ) | X (ω)| X ω Yes … this will work for any practical non -periodic signal!! Called the Fourier Transform of x ( t ) Need to use two plots to show it
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Comparison of FT and FS Fourier Series : Used for periodic signals Fourier Transform : Used for non-periodic signals (although we will see later that it can also be used for periodic signals) FS coefficients c k are a complex-valued function of integer k FT X (ω) is a complex-valued function of the variable ω (- , ) Synthesis Analysis Fourier Series Fourier Series Fourier Coefficients x t = n =−∞ c k e jk ω 0 t c k = 1 T t 0 t 0 T x t e jk ω 0 t dt x t = 1 −∞ X ω e jωt X ω = −∞ x t e jωt dt Fourier Transform
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Synthesis Viewpoints : We need two plots to show these x t = n =−∞ c k e jk ω 0 t | X (ω)| shows how much there is in the signal at frequency ω. X (ω) shows how much phase shift is needed at frequency ω . x t = 1 −∞ X ω e jωt We need two plots to show these FS: | c k | shows how much there is of the signal at frequency k ϖ 0 c k shows how much phase shift is needed at frequency k 0 FT:
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Some FT Notation: x t ↔ X ω 1. If X ( ϖ ) is the Fourier transform of x ( t )… then we can write this in several ways: X ω = F { x t } 2. F { } is an “operator” that operates on x ( t ) to give X ( ) F -1 { } is an “operator” that operates on X ( ) to give x ( t ) x t = F 1 { X ω } 3.
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We can think of the Fourier transform as a limiting case of a Fourier series expansion as the fundamental period, T, goes to infinity. Lets see how that works .
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This note was uploaded on 02/01/2011 for the course ECE 3TP4 taught by Professor Drterrencetodd during the Fall '10 term at McMaster University.

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EE3TP4_14_FourierTransform_v3_Lecture 21 - 4.3 Fourier...

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