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Lecture 8

Lecture 8 - Lecture 8 The Unilateral z-Transform ECSE304...

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1 Lecture 8 – The Unilateral z-Transform ECSE304 Signals and Systems II ECSE 304 Signals and Systems II Lecture 8: Unilateral z-Transform and Time - Frequency Analysis of D-T Systems Reading: O and W, Sections 10.4 and 10.9 Richard Rose McGill University Dept. of Electrical and Computer Engineering 2 Lecture 8 – The Unilateral z-Transform ECSE304 Signals and Systems II • Definition and Properties of the Unilateral z-Transform • Solving Linear Constant-Coefficient Difference Equations with Non-Zero Initial Conditions • Time and Frequency Analysis of Discrete Time Systems Outline

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3 Lecture 8 – The Unilateral z-Transform ECSE304 Signals and Systems II • The unilateral z-Transform of a sequence is defined as • Adopt shorthand notation for and its unilateral z-Transform • The unilateral z-Transform differs from the bilateral z- Transform in that the summation is carried out only over positive values of n, even if is nonzero for – Side effect: ROC of unilateral z-Transform is always the exterior of a circle in the z plane Unilateral z-Transform 0 () : [] n n zx n z +∞ = = X x n x n {} xn z ↔= UZ XU Z 0 n < 4 Lecture 8 – The Unilateral z-Transform ECSE304 Signals and Systems II • The unilateral and bilateral z-Transform of sequence are not necessarily the same, even for right-sided sequences: • Example: Consider the sequence – Bilateral z-Transform: Apply the time shifting property to the z- Transform of – Unilateral z-Transform: – The unilateral and bilateral transforms are not equal Unilateral z-Transform a un n [ ] =+ + 1 1 Xz z az z za , = = > 1 1 2 11 00 1 ( ) , 1 nn n z a a z a az + ∞+ +− == => ∑∑ X n aun
5 Lecture 8 – The Unilateral z-Transform ECSE304 Signals and Systems II • Inverse unilateral z-Transform : – Partial Fraction Expansion: Select all ROCs of individual terms to be regions of the z-plane that are exterior to disks Example: The inverse z-Transform of can only be considered for – Power Series Expansion: The series must be in negative powers of z Example: In performing long division on: … the expansion can only include negative powers of z • This follows from the definition of the unilateral z-Transform stating that the summation only includes negative powers of z Unilateral z-Transform 1 1 () 1 z az = X za > 12 2 1 1 1 ... 1 az a z az −− = ++ + 6 Lecture 8 – The Unilateral z-Transform ECSE304 Signals and Systems II • For a unilateral z-Transform pair many of the properties are identical to bilateral z-transform properties: – Linearity, scaling in the z-domain, time-expansion, conjugation, and differentiation • Several Properties differ from the bilateral case:

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Lecture 8 - Lecture 8 The Unilateral z-Transform ECSE304...

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