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Unformatted text preview: UCLA Winter 20092010 Systems and Signals Lecture 13: Impulse trains, Periodic Signals, and Sampling May 12, 2010 EE102: Systems and Signals; Spr 0910, Lee 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 finite energy). We can generalize Fourier transform to such signals. Given a periodic signal f ( t ) with period T , f ( t ) has a Fourier Series. f ( t ) = X n = D n e jn t where D n = 1 T Z T f ( t ) e jn t dt and = 2 /T . EE102: Systems and Signals; Spr 0910, Lee 2 Fourier series resembles an inverse Fourier transform of f ( t ) , but it is a and not an R . We can make the connection much clearer using the Fourier transform for complex exponentials, and extended linearity: f ( t ) = X n = D n e jn t F ( j ) = X n = D n 2 (  n ) 2  2  2  2  D n Fourier Series Coefcients Fourier Transform F ( j ) The Fourier series coefficients and Fourier transform are the same! (with a scale factor of 2 ). EE102: Systems and Signals; Spr 0910, Lee 3 Example: Square Wave Consider the square wave f ( t ) = X n = rect( t 2 n ) This is the square pulse of width T = 1 defined on the interval of width = 2 and then replicated infinitely often. 1 2 3 1 2 3 t f ( t ) EE102: Systems and Signals; Spr 0910, Lee 4 The Fourier series from before (Lecture 7, page 38) is f ( t ) = X n = D n e j 2 nt/ = X n = D n e jnt with Fourier coefficients D n = T sinc n T = 1 2 sinc( n/ 2) so that f ( t ) = X n = 1 2 sinc( n/ 2) e jnt . The Fourier transform is then F ( j ) = X n = 1 2 sinc( n/ 2)(2 (  n )) EE102: Systems and Signals; Spr 0910, Lee 5 = X n = sinc( n/ 2) (  n ) Note that this can also be written: F ( j ) = X n = sinc( / 2 ) (  n ) . This is the Fourier transform of the rect, multiplied by an array of evenly spaced s. EE102: Systems and Signals; Spr 0910, Lee 6 1 / 2  8 8 4  4  12 12  8 8 4  4  12 12 1 2 sinc ( / 2 ) n = sinc ( n / 2 ) (  n ) EE102: Systems and Signals; Spr 0910, Lee 7 Impulse Trains Sampling Functions Define T ( t ) to be a sequence of unit functions spaced by T , T ( t ) = X n = ( t nT ) which looks like 2TT3T2T T 3T t T ( t ) 1 What do we get if we expand this function as a Fourier series over T/ 2 to T/ 2 ? EE102: Systems and Signals; Spr 0910, Lee 8 The Fourier coefficients are D n = 1 T Z T/ 2 T/ 2 f ( t ) e j 2 nt/T dt = 1 T Z T/ 2 T/ 2 ( t ) e j 2 nt/T dt = 1 T ....
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This note was uploaded on 10/21/2010 for the course EE ee102 taught by Professor Levan during the Spring '09 term at UCLA.
 Spring '09
 Levan

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