1.2 Inverse z-Transform by Partial- 1.3 Partial-Fraction Expansion 1.3 Partial-Fraction Expansion Fraction Expansion Using MATLAB Using MATLAB Let H(z)= 5.5+2.121 .[r.p.c]=residuez(num,den)develops the .[num,den]=residuez(r.p.c)converts a z- partial-fraction expansion of a rational z- 1+0.82+0.22 transform expressed in a partial-fraction transform with numerator and denominator expansion form to its rational form 2.75+0.25i 2.75-0.257i coefficients given by vectors num and den 1-(-0.4+02021-(-0.4-0.202 Vector r contains the residues ↑ Vector p contains the poles (2.75+0.250-0.4+0.20m Vector c contains the constants (2.75-0.250-0.4-0.2i)(m 19 20 21 1.4 Inverse z-Transform via 1.4 Inverse z-Transform via Long Division Long Division 1.5 Inverse z-Transform Using MATLAB The z-transform G(z)of a causal sequence Example (g(n))can be expanded in a power series in ●Consider X(z)= 1+221 .The function impz can be used to find the 1+0.42-0.122 inverse of a rational z-transform G(z) In the series expansion,the coefficient The function computes the coefficients of the Long division of the numerator by the multiplying the term z-"is then the n-th denominator yields power series expansion of G(z) sample g(n) X(z)=1+1.6z-0.52z2+0.4z3-0.224z4+… The number of coefficients can either be user .For a rational z-transform expressed as a specified or determined automatically ●Hence ratio of polynomials in z-,the power series (m)}={1,1.6-0.52,0.4,-0.2224,.}n≥0 expansion can be obtained by long division. 中 -0 23 2419 1.2 Inverse z-Transform by Partial￾Fraction Expansion 1 1 2 1 1 5.5 2.1 ( ) 1 0.8 0.2 2.75 0.25 2.75 0.25 1 ( 0.4 0.2 ) 1 ( 0.4 0.2 ) z H z z z i i iz iz                  Let (2.75 0.25 )( 0.4 0.2 ) ( ) n   i i un (2.75 0.25 )( 0.4 0.2 ) ( ) n   i i un 20 1.3 Partial-Fraction Expansion Using MATLAB [r,p,c]= residuez(num,den) develops the partial-fraction expansion of a rational z￾transform with numerator and denominator coefficients given by vectors num and den Vector r contains the residues Vector p contains the poles Vector c contains the constants l 21 1.3 Partial-Fraction Expansion Using MATLAB [num,den]=residuez(r,p,c) converts a z￾transform expressed in a partial-fraction expansion form to its rational form 22 1.4 Inverse z-Transform via Long Division The z-transform G(z) of a causal sequence {g(n)} can be expanded in a power series in zˉ1 In the series expansion, the coefficient multiplying the term zˉn is then the n-th sample g(n) For a rational z-transform expressed as a ratio of polynomials in zˉ1, the power series expansion can be obtained by long division. 23 1.4 Inverse z-Transform via Long Division Example Example Consider Long division of the numerator by the denominator yields Hence {x(n)}={ 1, 1.6, ˉ0.52, 0.4, ˉ0.2224,…} nı0 1 1 2 1 2 ( ) 1 0.4 0.12 z X z z z        1 23 4 Xz z z z z ( ) 1 1.6 0.52 0.4 0.224         n=0 24 1.5 Inverse z-Transform Using MATLAB The function impz can be used to find the inverse of a rational z-transform G(z) The function computes the coefficients of the power series expansion of G(z) The number of coefficients can either be user specified or determined automatically