This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1 Lecture 3 DiscreteTime Fourier Transform ECSE304 Signals and Systems II ECSE 304 Signals and Systems II Lecture 3: DiscreteTime Fourier Transform Reading: O and W, Secs. 5.1 through 5.7 Boulet, Chapter 12 Richard Rose McGill University Dept. of Electrical and Computer Engineering 2 Lecture 3 DiscreteTime Fourier Transform ECSE304 Signals and Systems II Development of the DT Fourier Transform Examples of DT Fourier Transforms Convergence Issues Properties of the DT Fourier Transform Outline 3 Lecture 3 DiscreteTime Fourier Transform ECSE304 Signals and Systems II Given a finite duration a periodic sequence, , such that outside of the range Construct a periodic signal, , of period that is equal to for one period We develop the Fourier Transform based on the Fourier Series of a DT sequence whose period tends to infinity Development of the DT Fourier Transform [ ] x n [ ] x n [ ] x n [ ] x n = 1 2 [ ] N x n N [ ] x n N [ ] x n 4 Lecture 3 DiscreteTime Fourier Transform ECSE304 Signals and Systems II Starting with the DT Fourier Series Pair We can replace with by defining the interval of summation to be Defining the function: The coefficients are scaled samples of Where is the spacing of samples in frequency N n N N 1 2 a N x n e N x n e k jk N n n N jk N n n = = = = + 1 1 2 2 [ ] [ ] X e x n e j j n n ( ) : [ ] = = + a N X e N X e k j N k j k = = 1 1 2 ( ) ( ) ~ [ ] x n a e k jk N n k N = = 2 a N x n e k jk N n n N = = 1 2 ~ [ ] Development of the DT Fourier Transform [ ] x n [ ] x n 2 / N = k a ( ) j X e Period of constructed periodic signal How is related to when N gets really large? ( ) j X e k a (continuous) 5 Lecture 3 DiscreteTime Fourier Transform ECSE304 Signals and Systems II Observe limiting behavior as by substituting for in DTFS expansion for : As Summation over intervals of width tends to an integral over a total interval of integration of width The above equation becomes 1 1 [ ] ( ) ( ) 2 jk n jk jk n jk jk n k k N k N k N x n a e X e e X e e N = = = = = = N + d k 2 = N d ~ [ ] [ ] x n x n x n X e e d j j n [ ] ( ) = z 1 2 2 Development of the DT Fourier Transform N + ( ) j X e k a [ ] x n N 2 6 Lecture 3 DiscreteTime Fourier Transform ECSE304 Signals and Systems II DiscreteTime Fourier Transform Pair: is periodic with period X e x n e...
View
Full
Document
This note was uploaded on 08/17/2011 for the course ECSE 304 taught by Professor Chenandbacsy during the Spring '11 term at McGill.
 Spring '11
 ChenandBacsy

Click to edit the document details