Lec 13

Lec 13 - UCLA Winter 2009-2010 Systems and Signals Lecture...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: UCLA Winter 2009-2010 Systems and Signals Lecture 13: Impulse trains, Periodic Signals, and Sampling May 12, 2010 EE102: Systems and Signals; Spr 09-10, 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 09-10, 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 09-10, 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 09-10, 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 09-10, 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 09-10, 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 09-10, 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 2T-T-3T-2T 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 09-10, 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 ....
View Full Document

This note was uploaded on 10/21/2010 for the course EE ee102 taught by Professor Levan during the Spring '09 term at UCLA.

Page1 / 34

Lec 13 - UCLA Winter 2009-2010 Systems and Signals Lecture...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online