The Z-Transform Fall 202, EE2 Digital Signal Processing Lecture 4 September 4, 202 Used for: Analysis of LTI systems Solving di erence equations Determining system stability Finding frequency response of stable systems Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing Eigen Functions of LTI System Eigen Functions of LTI System Consider an LTI system with an impulse response h[n]: Consider an LTI system with an impulse response h[n]: We already showed that x[n] =e j!n are eigen-functions. What if x[n] =z n = re j!n We already showed that x[n] =e j!n are eigen-functions. What if x[n] =z n = re j!n Calculate using Convolution: y[n] = = X h[k]z n k= X k= k! h[k]z k z n = H(z)z n Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing
Eigen Functions of LTI System The Z Transform x[n] =z n are also eigen-functions of LTI systems. Transfer Function X H(z) = h[n]z n n= Bilateral Z-Transform X x[n]z n n= Since z = re j! X X (z) z=e j! = x[n]e j!n = DT FT {x[n]} n= H(z) exists for larger classs of h[n] thanh(e j! ) Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing Example : Right-sided sequence x[n] = a The ROC is the set of values of z for which the sum X X a n z n = (az ) n n=0 n=0 X n= x[n]z n converges. Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing
Example : Right-sided sequence x[n] = a X X a n z n = (az ) n Example 2: x[n] = 2 + n=0 n=0 Recall: So, +x + x 2 + = x, if x < az, ROC = {z : z > a } Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing Example 2: x[n] = 2 + 2 z + + z Example 2: x[n] = 2 + 2 z + + z ROC = {z : z > 2 } \ {z : z > } = {z : z > 2 } Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing
Example : Left-sided sequence x[n] = a n u[ n ] Example : Left-sided sequence x[n] = a n u[ n ] X n= a n z n = X m= a m z m = X (a z) m m=0 Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing Example : Left-sided sequence x[n] = a n u[ n ] X n= a n z n = X m= If a z <, i.e., z < a then, a m z m = a z a z = a z = az X (a z) m m=0 Expression X [z] = az the same as Example ROC = {z : z < a } is di erent The Z-transform without ROC does not uniquely define a sequence! Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing
Example 4: x[n] = 2 n u[ n ] + Example 4: x[n] = 2 n u[ n ] + 2 z + + z same as example 2 Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing Example 4: x[n] = 2 n u[ n ] + 2 z + + z same as example 2 ROC = {z : z < 2 } \ {z : z > } = {z : < z < 2 } Example 5: x[n] = 2 n u[ n ] ROC = {z : z > 2 } \ {z : z < } = 0 Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing
Example 5: x[n] = 2 n u[ n ] Example 7: Finite sequence x[n] =a u[ n + M ] ROC = {z : z > 2 } \ {z : z < } = 0 Example 6: x[n] =a n, two sided a 6= 0 ROC = {z : z > a} \ {z : z < a} = 0 Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing Example 7: Finite sequence x[n] =a u[ n + M ] X [z] = MX n=0 a n z n Finite, always converges = am z M az zero cancels the pole MY = ( ae j 2 k M z ) k= Example 7: Finite sequence x[n] =a u[ n + M ] X [z] = MX n=0 a n z n Finite, always converges = am z M az zero cancels the pole MY = ( ae j 2 k M z ) k= ROC = {z : z > 0} Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing
Properties of the ROC Section.2 A ring or a disk in the Z-plane, centered at the origin 2 DTFT converges i ROC includes the unit circle ROC can t contain poles Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing Properties of the ROC Properties of the ROC 4 For finite duration sequences, ROC is the entire Z-plane, except possibly z =0orz = 4 For finite duration sequences, ROC is the entire Z-plane, except possibly z =0orz = +z + z 2 ROC excludes z =0 +z + z 2 ROC excludes z =0 +z + z 2 ROC excludes z = Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing
Properties of the ROC Properties of the ROC 5 For right-sided sequences (x[n] =0 8 n < N for some N ) ROC extends outward from the outer-most pole to infinity. Examples,2 5 For right-sided sequences (x[n] =0 8 n < N for some N ) ROC extends outward from the outer-most pole to infinity. Examples,2 6 For left-sided sequences (x[n] =0 8 n > N 2 for some N 2 ) ROC extends inwards from the inner-most pole to zero. Example Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing Properties of the ROC Several Properties of the Z Transform 5 For right-sided sequences (x[n] =0 8 n < N for some N ) ROC extends outward from the outer-most pole to infinity. Examples,2 6 For left-sided sequences (x[n] =0 8 n > N 2 for some N 2 ) ROC extends inwards from the inner-most pole to zero. Example 7 For two-sided sequences, ROC is a ring Inner-bound: Pole with largest magnitude that contributes for n > 0 Outer-bound: Pole with smallest magnitude that contributes for n < 0 Examples 4,5,6 x[n n d ] $ z n d X (z) z n 0 x[n] $ X ( z z 0 ) dx (z) nx[n] $ z dz x[ n] $ X (z ) x[n] y[n] $ X (z)y (z) Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing
Several Properties of the Z Transform x[n n d ] $ z n d X (z) z n 0 x[n] $ X ( z z 0 ) dx (z) nx[n] $ z dz x[ n] $ X (z ) In general, we can invert by contour integration within the ROC: x[n] = I X (z)z n 2 j Ways to avoid it: Inspection (known transforms) Properties of the Z-transform Power series expansion Partial fraction expansion Residue theorem C x[n] y[n] $ X (z)y (z) ROC at least ROC x \ ROC y Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing In general, we can invert by contour integration within the ROC: x[n] = I X (z)z n 2 j C Example: Long division 2+z 2 z, Ways to avoid it: Inspection (known transforms) Properties of the Z-transform Power series expansion Partial fraction expansion Residue theorem Most useful is the inverse of rational polynomials Why? B(z) A(z) Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing
Example: Long division 2+z 2 z, ROC = {z : z > 2 } x[n] right/left sequencs? Arrange num/denum in ascending powers of z What about ROC = {z : z < 2 }? Example: Long division 2+z 2 z, ROC = {z : z > 2 } x[n] right/left sequencs? Arrange num/denum in ascending powers of z What about ROC = {z : z < 2 }? 2 z ) 2 + 2z + z 2 + 2 z + 2 + z 2 z 2z 2z z 2 z 2 z 2 2 z 2 z Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing Example: Partial Fraction Expansion 2+2z + z 2 + 2 z +... X = x[n]z n n= = = 4 z + 8 z 2 4 z 2 z A 4 z + A 2 2 z x[n] = 2 [n]+2 [n ] + [n 2] + 2 [n ] +... Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing
Example: Partial Fraction Expansion = = 4 z + 8 z 2 4 z 2 z A 4 z + A 2 2 z Partial fraction expansion: From the tables: 4 z + 2 2 z Find A and A 2 A = ( A 2 = ( 4 z )X (z) z= 4 2 z )X (z) z= 2 = =2 Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing Partial fraction expansion: 4 z + 2 2 z From the tables: apple x[n] = ( 4 )n + 2( 2 ) because right sided Partial fraction expansion in general: b 0 + b z + + b M z M a 0 + a z + + a N z N Suppose real and unrepeated poles d,, d N If M < N, NX k= A k d k z Like example Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing
If M N, ) x[n] = XN B r z r + M r=0 M NX k= XN B r [n r]+ r=0 A k d k z NX A k dk k= If M N, ) x[n] = XN B r z r + M r=0 M NX k= XN B r [n r]+ r=0 A k d k z NX A k dk k= If d k is a repeated pole of order S, replace SX m= C m ( d k z ) m A k d k z with Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing Example Example +2z + z 2 2 z + 2 z M = N =2 2 A = B 0 + 2 z + A 2 z Matching coe cients: A = 9, A 2 = 8, B 0 =2 +2z + z 2 2 z + 2 z M = N =2 2 A = B 0 + 2 z + A 2 z Matching coe cients: A = 9, A 2 = 8, B 0 =2 ROC = {z : z > } ) x[n] = 2 [n] 9( 2 )+8u[n] Fall 202, EE2 Digital Signal Processing Fall 202, EE2 Digital Signal Processing