Lecture 6

# Lecture 6 - Lecture 6 The Inverse z-Transform ECSE304...

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1 Lecture 6 – The Inverse z-Transform ECSE304 Signals and Systems II ECSE 304 Signals and Systems II Lecture 6: The Inverse z-Transform Reading: O and W, Section 10.3 Richard Rose McGill University Dept. of Electrical and Computer Engineering 2 Lecture 6 – The Inverse z-Transform ECSE304 Signals and Systems II • Development of the Inverse z-Transform – Contour Integral • Practical Methods for Obtaining a Sequence from its z-Transform – Partial Fraction Expansion – Power Series Expansion Outline

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3 Lecture 6 – The Inverse z-Transform ECSE304 Signals and Systems II • Obtain the expression for a sequence, , in terms of its z- Transform, , by interpreting the z-Transform as the Fourier Transform of an exponentially weighted sequence: • Start with , where is inside the unit circle Take DTFT of the exponentially weighted sequence • Take the inverse Fourier Transform: Development of the Inverse z-Transform xn X z [] () Z z re j = ω {} ( ) j nj n n n X re xnr e xnr ωω −− =−∞ == = F { } 1 ( ) = F n n Xz 4 Lecture 6 – The Inverse z-Transform ECSE304 Signals and Systems II • Solve for using the inverse DTFT expression: • … Corresponds to evaluating along contour in the ROC with fixed and varying over a interval Development of the Inverse z-Transform 1 2 2 ( ) 2 1 2 n jj n n r r Xr e e d e r e d π = = = F x n z re j = r 2 Definition of the inverse DTFT
5 Lecture 6 – The Inverse z-Transform ECSE304 Signals and Systems II • Change the variable of integration from to z: • … and plug into the expression for the inverse DTFT Development of the Inverse z-Transform xn X re

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Lecture 6 - Lecture 6 The Inverse z-Transform ECSE304...

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