z-Transform Example- The finite energy sequence lpIn]sinOn ocsinc/acn 00<1<00 has a dtft given by 0≤o≤0 LP(evo 0.,0n<0≤兀 which converges in the mean-square sense 16 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 16 z-Transform • Example - The finite energy sequence has a DTFT given by which converges in the mean-square sense sin [ ] sinc , c c c LP n n h n n n = = − = c j c LP H e 0, 1, 0 ( )
z-Transform However, hp[n] does not have a z-transform as it is not absolutely summable for any value ofr 1.e LpIn Some commonly used z-transform pairs are listed on the next slide 17 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 17 z-Transform • However, does not have a z-transform as it is not absolutely summable for any value of r, i.e. • Some commonly used z-transform pairs are listed on the next slide h [n] LP [ ] n LP n h n r − =− = r
Table 3. 8: Commonly Used z- Transform Pairs Sequence "-Transform ROC All values of (r cos wo)z (r" cos on)u[n] (2r cos 0)2+ (r sin co)z (r"sin won)u[n - (2r cos wo)2+r-z 18 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 18 Table 3.8: Commonly Used zTransform Pairs
Rational z-Transforms In the case of lti discrete-time systems we are concerned with in this course all pertinent z-transforms are rational functions That is, they are ratios of two polynomials G(二) P(z)P0+p12+…+pM-12 (M-1)+M D=)d0+421+…+dN=12(M+)+dMzN Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 19 Rational z-Transforms • In the case of LTI discrete-time systems we are concerned with in this course, all pertinent z-transforms are rational functions of • That is, they are ratios of two polynomials in : −1 z −1 z N N N N M M M M d d z d z d z p p z p z p z D z P z G z − − − − − − − − − − + + + + + + + + = = ( ) ( ) .... .... ( ) ( ) ( ) 1 1 1 0 1 1 1 1 0 1
Rational z- Transforms The degree of the numerator polynomial p(z) is M and the degree of the denominator polynomial d()is N an alternate representation of a rational z- transform is as a ratio of two polynomials in G()=2(N-M)02+P2=+…+PM+PM N d02+d1+…+dN-12+d 20 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 20 Rational z-Transforms • The degree of the numerator polynomial P(z) is M and the degree of the denominator polynomial D(z) is N • An alternate representation of a rational ztransform is as a ratio of two polynomials in z: N N N N M M M M N M d z d z d z d p z p z p z p G z z + + + + + + + + = − − − − − 1 1 0 1 1 1 0 1 .... .... ( ) ( )