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EE3TP4_13b_FourierSeriesProperties_v2_Lecture18

# EE3TP4_13b_FourierSeriesProperties_v2_Lecture18 - Fourier...

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Fourier Series Expansion 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) T = 2π/ω 0 ω 0 = fundamental frequency (rad/sec)

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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 Complex Exponential Form
Fourier Series Trigonometric Form f t = a 0 n = 1 [ a n cos n 0 t  b n sin n 0 t ] a n = 2 T T f t cos n 0 t dt n 0 b n = 2 T T f t sin n 0 t dt n 0 a 0 = 1 T T f t dt We can derive these results in the same way as the complex exponential case using orthogonal functions!

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f(t) = 1 for t = [-T/2, T/2]

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Fourier Series Properties

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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
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