lecture13n - Stanford University Winter 2009-2010 Signal...

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Stanford University Winter 2009-2010 Signal Processing and Linear Systems I Lecture 13: Impulse trains, Periodic Signals, and Sampling February 19, 2010 EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 1 Fourier Transforms of Periodic Signals So far we have used Fourier series to handle periodic signals, they do not have a Fourier transform in the usual sense (not ±nite energy). We can generalize Fourier transform to such signals. Given a periodic signal f ( t ) with period T 0 , f ( t ) has a Fourier Series. f ( t )= ° n = −∞ D n e jn ω 0 t where D n = 1 T ± T 0 f ( t ) e ω 0 t dt and ω 0 =2 π /T . EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 2
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Fourier series resembles an inverse Fourier transform of f ( t ) ,butitisa ° and not an ± . We can make the connection much clearer using the Fourier transform for complex exponentials, and extended linearity: f ( t )= ² n = −∞ D n e jn ω 0 t F ( j ω ² n = −∞ D n 2 πδ ( ω n ω 0 ) ! ! ! 0 2 ! 0 2 ! 0 ! 0 0 ! 0 2 ! 0 2 ! 0 ! 0 0 D n Fourier Series Coef±cients Fourier Transform F ( j ! ) The Fourier series coe cients and Fourier transform are the same! (with a scale factor of 2 π ). EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 3 Example: Square Wave Consider the square wave f ( t ² n = −∞ rect( t 2 n ) This is the square pulse of width T =1 de±ned on the interval of width τ =2 and then replicated in±nitely often. 0 1 2 3 1 2 3 t f ( t ) EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 4
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The Fourier series from before (Lecture 7, page 38) is f ( t )= ° n = −∞ D n e j 2 π nt/ τ = ° n = −∞ D n e j π nt with Fourier coe cients D n = T τ sinc ± n T τ ² = 1 2 sinc( n/ 2) so that f ( t ° n = −∞ 1 2 sinc( n/ 2) e j π nt . The Fourier transform is then F ( j ω ° n = −∞ 1 2 sinc( n/ 2)(2 πδ ( ω n π )) EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 5 = π ° n = −∞ sinc( n/ 2) δ ( ω n π ) Note that this can also be written: F ( j ω π ° n = −∞ sinc( ω / 2 π ) δ ( ω n π ) . This is the Fourier transform of the rect, multiplied by an array of evenly spaced δ ’s. EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 6
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1 / 2 ! ! ! 8 ! 8 ! 4 ! 4 ! 12 ! 12 ! 0 8 ! 8 ! 4 ! 4 ! 12 ! 12 ! 0 1 2 sinc ( ! / 2 " ) ! " # n = " sinc ( n / 2 ) $ ( % n ! ) EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 7 Impulse Trains – Sampling Functions Defne δ T ( t ) to be a sequence oF unit δ Functions spaced by T , δ T ( t )= ° n = −∞ δ ( t nT ) which looks like 2T -T -3T -2T T 0 3T t ! T ( t ) 1 What do we get iF we expand this Function as a ±ourier series over T/ 2 to 2 ? EE102A:Signal Processing and Linear Systems I; Win 09-10, Pauly 8
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The Fourier coe cients are D n = 1 T ° T/ 2 2 f ( t ) e j 2 π nt/T dt = 1 T ° 2 2 δ ( t ) e j 2 π nt/T dt = 1 T . All of the coe cients are the same!
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lecture13n - Stanford University Winter 2009-2010 Signal...

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