Lecture 3

Lecture 3 - 1 Lecture 3 Discrete-Time Fourier Transform...

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Unformatted text preview: 1 Lecture 3 Discrete-Time Fourier Transform ECSE304 Signals and Systems II ECSE 304 Signals and Systems II Lecture 3: Discrete-Time 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 Discrete-Time Fourier Transform ECSE304 Signals and Systems II Development of the D-T Fourier Transform Examples of D-T Fourier Transforms Convergence Issues Properties of the D-T Fourier Transform Outline 3 Lecture 3 Discrete-Time 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 D-T sequence whose period tends to infinity Development of the D-T Fourier Transform [ ] x n [ ] x n [ ] x n [ ] x n = 1 2 [ ] N x n N [ ] x n N [ ] x n 4 Lecture 3 Discrete-Time Fourier Transform ECSE304 Signals and Systems II Starting with the D-T 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 D-T 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 Discrete-Time 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 D-T Fourier Transform N + ( ) j X e k a [ ] x n N 2 6 Lecture 3 Discrete-Time Fourier Transform ECSE304 Signals and Systems II Discrete-Time Fourier Transform Pair: is periodic with period X e x n e...
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This note was uploaded on 08/17/2011 for the course ECSE 304 taught by Professor Chenandbacsy during the Spring '11 term at McGill.

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Lecture 3 - 1 Lecture 3 Discrete-Time Fourier Transform...

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