Lecture 5

Lecture 5 - Lecture 5 The z-Transform ECSE304 Signals and...

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1 Lecture 5 – The z-Transform ECSE304 Signals and Systems II ECSE 304 Signals and Systems II Lecture 5: The z-Transform Reading: O and W, Secs. 10.1, 10.2, and 10.5 Boulet, Chapter 13 Richard Rose McGill University Dept. of Electrical and Computer Engineering 2 Lecture 5 – The z-Transform ECSE304 Signals and Systems II • Development of the z-Transform • Region of Convergence (ROC) • Properties of the z-Transform Outline
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3 Lecture 5 – The z-Transform ECSE304 Signals and Systems II • In Lecture 2, we found that the response if a LTI system to a complex exponential is a scaled version of the same complex exponential: is referred to as the z-Transform of assuming that the sum converges The Bi-Lateral z-Transform nn j n z re ω = zH z z () yn hkxn k hkz zh k z Hzz k nk k k n [] [][ ] =− = = = =−∞ +∞ +∞ +∞ ∑∑ hn Hz 4 Lecture 5 – The z-Transform ECSE304 Signals and Systems II •T h e z-Transform is also defined for a D-T signal • The z-Transform reduces to the D-T Fourier Transform when evaluated on the unit circle : The Bi-Lateral z-Transform xn Xz xnz n n : = +∞ j z e = Xe xne j ze jn n j ( ) [ ] ωω == = +∞ 1 Unit Circle
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5 Lecture 5 – The z-Transform ECSE304 Signals and Systems II • O and W Example 10.1: Consider the signal • For convergence of : The region of convergence (ROC) is the range of values of for which , or The Bi-Lateral z-Transform - Example xn aun n [] = Xz az a z nn n n n () ( ) == = +∞ = +∞ ∑∑ 0 1 0 X z az n n = +∞ <∞ 1 0 az < 1 1 za > a z az z n n ( ) , = > = +∞ 1 0 1 1 1 z 6 Lecture 5 – The z-Transform ECSE304 Signals and Systems II • For example, suppose • The z-Transform is then The Bi-Lateral z-Transform – Example (Continued) [] 0 . 5 [] n un = . n = 1 1 , 0 . 5 10 . 5 0 . 5 z z zz > −− Region of Convergence 0.5 z >
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7 Lecture 5 – The z-Transform ECSE304 Signals and Systems II • O and W Example 10.2: Consider the signal • The region of convergence (ROC) is the range of values of for which , or The Bi-Lateral z-Transform – Example (Continued) [] [ 1 ] n xn au n =− − − z 1 1 10 () [ 1 ] 1( ) nn n n n Xz z az +∞ =−∞ ∞∞ −− == ∑∑ = − 1 1 < z a < 11 () 1 , z X zz a a z z a = = < 8 Lecture 5 – The z-Transform ECSE304 Signals and Systems II • For example, suppose • The z-Transform is then The Bi-Lateral z-Transform – Example (Continued) Region of Convergence 0.5 z < ( 0 . 5)[ 1 ] n u n 1 , 0 . 5 12 . 5 0 . 5 z z z = = < ( 0 . 1 ] n
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9 Lecture 5 – The z-Transform ECSE304 Signals and Systems II •E x am p l e : Consider the signal: The Bi-Lateral z-Transform – Example (Continued) xn un u n nn [] [ ] = F H G I K J + F H G I K J −− 1 3 2 1 2 1 N N N 1 0 00 1 11 22 1 1 3 3 () [] 2 [ 1 ] 32 2 1 2( 2 ) 1 3 14 4 12 1 33 1 ) 6 1 ( 3 n n n zz z z Xz z z z +∞ =−∞ +∞ == +∞ +∞ << > > ⎡⎤ ⎛⎞ =+ ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦ ∑∑ = + = ±²³ ²´ , 12) z z • Compute z-Transform: 1 1 0 1( ) n az a z −∞ =
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Lecture 5 - Lecture 5 The z-Transform ECSE304 Signals and...

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