EE3TP4_13_FourierSeriesDetails_v2_Lecture 17

EE3TP4_13_FourierSeriesDetails_v2_Lecture 17 - Fourier...

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Fourier Analysis
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3.2 & 3.3 Fourier Series Expansion In the last set of notes we looked at building signals using: x t = k =− N N c k e jk ω 0 t N = finite integer In general we need an infinite number of terms: x t = A 0 k = 1 N A k cos 0 t θ k x t = k =−∞ c k e jk ω 0 t Fourier Series (Complex Exponential Form) x t = A 0 k = 1 A k cos 0 t θ k Fourier Series (Trigonometric Form) These are 3 different forms for the same expression. x t = a 0 k = 1 [ a k cos 0 t  b k sin 0 t ] Fourier Series (Trigonometric Form)
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“ . .. the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour. “ Jean Baptiste Joseph Fourier About the Fourier Series Expansion Laplace Lagrange
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Q: Does this now let us form any periodic signal? A: No! Although Fourier thought so! Dirichlet showed that there are some that can’t be written in terms of a FS! Sufficient conditions for the FS expansion to exist . .. The function must 1. have a finite number of extrema in any given interval, 2. have a finite number of discontinuities in any given interval, 3. be absolutely integrable over a period, 4. be bounded. But … those will never show up in practice! So we can write any practical periodic signal as a FS with infinite # of terms!
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We can now break virtually any periodic signal into a sum of simple things… and we already understand how these simple things travel through an LTI system! So, instead of: h t x t y t = x t ∗ h t We break x ( t ) into its FS components and find how each component goes through. (See Chapter 5)
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To do this kind of convolution-evading analysis we need to be able to solve the following: Given time-domain signal model x ( t ) Find the FS coefficients { c k } “Time-domain” model “Frequency-domain model” Converting “time-domain” signal model into a “frequency-domain” signal model
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Fourier Series x t = k =−∞ c k e jk ω 0 t c k = 1 T t 0 t 0 T x t e jk ω 0 t dt T = 2π/ω 0 ω 0 = fundamental frequency (rad/sec) This is true for a very wide class of periodic signals!
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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_13_FourierSeriesDetails_v2_Lecture 17 - Fourier...

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