MIT MAS 160 - Relation to Discrete-Time Fourier Transform (9 pages)

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Relation to Discrete-Time Fourier Transform



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Relation to Discrete-Time Fourier Transform

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Lecture Notes


Pages:
9
School:
Massachusetts Institute of Technology
Course:
Mas 160 - Signals, Systems and Information for Media Technology

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MIT OpenCourseWare http ocw mit edu MAS 160 MAS 510 MAS 511 Signals Systems and Information for Media Technology Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms Z transforms Part I MIT MAS 160 510 Additional Notes Spring 2003 R W Picard 1 Relation to Discrete Time Fourier Transform Consider the following discrete system written three di erent ways y n b 1 y n 1 b1 y n 1 a 1 x n 1 a0 x n a2 x n 2 Y z b 1 zY z b1 z 1 Y z a 1 zX z a0 X z a2 z 2 X z Y z a 1 z a0 a2 z 2 H z X z b 1 z 1 b1 z 1 1 Simple substitution nds the Z transform for a discrete system represented by a linear constant coe cient di erence equation LCCDE Simply replace y n with Y z x n with X z and shifts of n0 with multiplication by z n0 That s almost all there is to it Set z rej and let r 1 for the moment Then a shift in time by n0 becomes a multiplication in the Z domain by ej n0 This should look familiar given what you know about Fourier analysis Now here s the formula for the Z transform shown next to the discrete time Fourier transform of x n Z transform DTFT X z X ej n x n z n x n e j n n where we have used the notation X ej instead of the equivalent X to emphasize similarity with the Z transform Substituting z rej in the Z transform X z x n r n e j n n reveals that the Z transform is just the DTFT of x n r n If you know what a Laplace transform is X s then you will recognize a similarity between it and the Z transform in that the Laplace transform is the Fourier transform of x t e t Hence the Z transform generalizes the DTFT 1 in the same way that the Laplace transform generalizes the Fourier transform Whereas the Laplace transform is used widely for continuous systems the Z transform is used widely in design and analysis of discrete systems You may have noticed that in class we ve already sneaked in use of the Z plane in talking about the unit circle The Z plane contains all values of z whereas the unit circle contains only z ej The Z



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