! z-transform. " Tie up loose ends. " Regions of convergence properties. ! Inverse z-transform. " Inspection. " Partial fraction

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1 Lecture Outline ESE 53: Digital Signal Processing Lec 6: January 3, 207 Inverse z-transform! z-transform " Tie up loose ends " gions of convergence properties! Inverse z-transform " Inspection " Partial fraction " Power series expansion! z-transform of difference equations 2 z-transform z-transform! Define the forward z-transform of x[n] as! The core basis functions of the z-transform are the complex exponentials z n with arbitrary z C; these are the eigenfunctions of LTI systems for infinite-length signals! otation abuse alert: We use X(#) to represent both the DTFT X(ω) and the z-transform X(z); they are, in fact, intimately related 3 4 Transfer Function of LTI System! We can use the z-transform to characterize an LTI system gion of Convergence (ROC)! and relate the z-transforms of the input and output 5 6

For finite duration sequences, ROC is the entire z- plane, except possibly z=0, z= " Example 6! What is the z-transform of x 6 [n]? ROC? x 6 [n] = a n u[n]u[ n + ] finite length sequence 9 0 visit: ROC Example 6 visit: ROC Example 6!

2 gion of Convergence (ROC) Properties of ROC! For right-sided sequences: ROC extends outward from the outermost pole to infinity " Examples,2! For left-sided: inwards from inner most pole to zero " Example 3! For two-sided, ROC is a ring - or do not exist " Examples 4,5 7 8 Properties of ROC visit: ROC Example 6! For finite duration sequences, ROC is the entire z- plane, except possibly z=0, z= " Example 6! What is the z-transform of x 6 [n]? ROC? x 6 [n] = a n u[n]u[ n + ] finite length sequence 9 0 visit: ROC Example 6 visit: ROC Example 6! What is the z-transform of x 6 [n]? ROC?! What is the z-transform of x 6 [n]? ROC? x 6 [n] = a n u[n]u[ n + ] x 6 [n] = a n u[n]u[ n + ] X 6 (z) = a z az = ( ae j 2π z ) = Zero cancels pole Zero cancels pole X 6 (z) = a z az = ( ae j 2π z ) = =00 2 2

3 Formal Properties of the ROC Formal Properties of the ROC! PROPERTY : " The ROC will either be of the form 0 < r R < z, or z < r L <, or, in general the annulus, i.e., 0 < r R < z < r L <.! PROPERTY 2: " The Fourier transform of x[n] converges absolutely if and only if the ROC of the z-transform of x[n] includes the unit circle.! PROPERTY 3: " The ROC cannot contain any poles.! PROPERTY 4: " If x[n] is a finite-duration sequence, i.e., a sequence that is zero except in a finite interval - < < n < 2 <, then the ROC is the entire z- plane, except possibly z = 0 or z =.! PROPERTY 5: " If x[n] is a right-sided sequence, the ROC extends outward from the outermost finite pole in X(z) to (and possibly including) z =.! PROPERTY 6: " If x[n] is a left-sided sequence, the ROC extends inward from the innermost nonzero pole in X(z) to (and possibly including) z=0.! PROPERTY 7: " A two-sided sequence is an infinite-duration sequence that is neither right sided nor left sided. If x[n] is a two-sided sequence, the ROC will consist of a ring in the, bounded on the interior and exterior by a pole and, consistent with Property 3, not containing any poles.! PROPERTY 8: " The ROC must be a connected region. 3 4 Example: ROC from Pole-Zero Plot Example: ROC from Pole-Zero Plot! How many possible ROCs? ROC : right-sided a b c a b c 5 6 Example: ROC from Pole-Zero Plot Example: ROC from Pole-Zero Plot ROC 2: left-sided ROC 3: two-sided a b c a b c 7 8 3

4 Example: ROC from Pole-Zero Plot Example: Pole-Zero Plot ROC 4: two-sided! H(z) for an LTI System " How many possible ROCs? a b c -2 / Example: Pole-Zero Plot BIBO Stability visited! H(z) for an LTI System " How many possible ROCs? " What if system is stable? H(e jω ) = = h[] e jω -2 / Example: Pole-Zero Plot! H(z) for an LTI System " How many possible ROCs? " What if system is causal? Inverse z-transform -2 /2 23 4

5 Inverse z-transform Inverse z-transform! Ways to avoid it: " Inspection (nown transforms) " Properties of the z-transform " Partial fraction expansion " Power series expansion Inspection Properties of z-transform! Linearity: ax [n]+ bx 2 [n] ax (z) + bx 2 (z)! Time shifting: x[n] X (z) x[n n d ] z n d X (z)! ultiplication by exponential sequence x[n] X (z) z n 0 x[n] X z z Properties of z-transform Partial Fraction Expansion! Time versal: x[n] X (z) x[ n] X (z )! Differentiation of transform: x[n] X (z)! Convolution in Time: y[n] = x[n] h[n] Y (z) = X (z)h(z) dx (z) nx[n] z dz ROC Y at least ROC x ROC H! Let b z z! zeros and poles at nonzero locations = z z b z z

6 Partial Fraction Expansion Partial Fraction Expansion! If < and the poles are st order b z! Factored numerator/denominator z = z z b 0 = = b z z ( c z ) ( d z )! where b 0 = = ( c z ) A = ( d z ) = d z A = ( d z )X (z) z=d 3 32 Example: 2 nd- Order z-transform Example: 2 nd- Order z-transform! 2 nd -order = two poles! 2 nd -order = two poles 4 z, ROC = z : 2 < z 2 z 4 z, ROC = z : 2 < z 2 z A + 4 z 2 z Example: 2 nd- Order z-transform Example: 2 nd- Order z-transform! 2 nd -order = two poles A = ( d z )X (z) z=d! 2 nd -order = two poles Right sided A + 4 z 2 z z, ROC = z : 2 < z 2 z A = ( 4 z )X (z) = ( 2 z )X (z) z=/4 z=/2 ( = 4 z ) ( 4 z )( 2 z ) ( = 2 z ) ( 4 z )( 2 z ) z=/4 z=/2 = = 2 x[n] = n 4 u[n]+ 2 n 2 u[n]

7 Partial Fraction Expansion Example: Partial Fractions! If and the poles are st order A B r z r + r=0 = d z! Where B is found by long division! ==2 and poles are first order + 2z + z z + 2 z 2, (+ z ) 2 = ( 2 z )( z ) ROC = { z : < z } A = ( d z )X (z) z=d Example: Partial Fractions Example: Partial Fractions! ==2 and poles are first order + 2z + z z + 2 z 2, (+ z ) 2 = ( 2 z )( z ) ROC = { z : < z }! ==2 and poles are first order A B z, 2 z ROC = { z : < z } 2 z z + z 2 + 2z + z 2 3z + 2 5z A B z 2 z Example: Partial Fractions Example: Partial Fractions! ==2 and poles are first order! ==2 and poles are first order A B z, 2 z +5z 2 + ( 2 z )( z ) ROC = { z : < z } 2 z z + z 2 + 2z + z 2 3z + 2 5z z, 2 z x[n] = 2δ[n] 9 n 2 u[n]+8u[n] ROC = { z : < z }

8 Power Series Expansion Example: Finite-Length Sequence! Expansion of the z-transform definition! Poles and zeros? n= x[n]z n =!+ x[ 2]z 2 + x[ ]z + x[0]+ x[]z + x[2]z 2 +! z 2 2 z (+ z )( z ) = z 2 2 z + 2 z Example: Finite-Length Sequence Example: Finite-Length Sequence! Poles and zeros?! Poles and zeros? n= z 2 2 z (+ z )( z ) = z 2 2 z + 2 z x[n]z n =!+ x[ 2]z 2 + x[ ]z + x[0]+ x[]z + x[2]z 2 +! n= z 2 2 z (+ z )( z ) = z 2 2 z + 2 z, n = 2 2, n = x[n] =, n = 0 x[n]z n 2, n = 0, else =!+ x[ 2]z 2 + x[ ]z + x[0]+ x[]z + x[2]z 2 +! Example: Finite-Length Sequence Example: Finite-Length Sequence! Poles and zeros?! Poles and zeros? z 2 2 z (+ z )( z ) z 2 2 z (+ z )( z ) = z 2 2 z + 2 z = z 2 2 z + 2 z, n = 2 2, n = x[n] =, n = 0 2, n = 0, else = δ[n + 2] 2 δ[n +] δ[n]+ δ[n ] 2, n = 2 2, n = x[n] =, n = 0 2, n = 0, else = δ[n + 2] 2 δ[n +] δ[n]+ δ[n ]

9 minder: Difference Equations Difference Equation to z-transform! Accumulator example y[n] = n = y[n] = x[n]+ x[] n = x[] y[n] = x[n]+ y[n ] y[n] y[n ] = x[n] y[n ] = b m x[n m] y[n ] = b m x[n m] a y[n] = y[n ] + = b x[n m]! Difference equations of this form behave as causal LTI systems " when the input is zero prior to n=0 " Initial rest equations are imposed prior to the time when input becomes nonzero " i.e y[-]=y[-+]= =y[-]= Difference Equation to z-transform Difference Equation to z-transform a y[n] = y[n ] + a = 0 b 0 x[n m] a y[n] = y[n ] + a = 0 b 0 x[n m] a Y (z) = z Y (z) + = 0 b z X (z) a Y (z) = z Y (z) + = 0 b z X (z) a b z Y (z) = 0 z X (z) Y (z) = 0 ( b ) z ( ) z X (z) 5 52 Difference Equation to z-transform a y[n] = y[n ] + a = 0 b 0 x[n m] Example: st -Order System y[n] = ay[n ]+ x[n] H(z) = ( b ) z ( ) z a Y (z) = z Y (z) + = 0 a b z Y (z) = 0 z X (z) Y (z) = 0 H(z) = b ( b ) z ( ) z z X (z) ( b ) z ( ) z X (z) 53 b 0 H(z) = az a 54 9

10 Example: st -Order System y[n] = ay[n ]+ x[n] H(z) = h[n] = a n u[n] az H(z) = ( b ) z ( ) z Why right sided? Big Ideas! z-transform " Draw pole-zero plots " ust specify region of convergence (ROC) " ROC properties! z-transform properties " Similar to DTFT! Inverse z-transform " Avoid it! " Inspection, properties, partial fractions, power series! Difference equations easy to transform Admin! HW 2 due Friday at midnight! Shlesh Office hours Location Change " T 6-8pm Th -3pm at Education Commons Rm 235 " Updated on course website and piazza 57 0

ESE 531: Digital Signal Processing

ESE 531: Digital Signal Processing Lec 6: January 31, 2017 Inverse z-transform Lecture Outline! z-transform " Tie up loose ends " Regions of convergence properties! Inverse z-transform " Inspection " Partial