z-Transform Example The finite energy sequence Sinon o n LPIn SInc 0o <n<oo has a dtft given by 0≤0≤0 Lp(evo 0,0,<(≤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 i.e 1=-0 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 equence Transform ROC All values of I-(rcos wo)z (r cos on)u[nI z>r -(2r cos Wo)z-I+rez (r sin co)z ("sino0)l1_c0m2-1+r2z-2 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 oflti discrete-time systems we are concerned with in this course. all pertinent z-transforms are rational functions of z That is, they are ratios of two polynomials In z P(=)P0+p12+…+pM=1 (M-1) G(二) + pM D()do+d1=-1+…+dk21(-1)+ 小令个 19 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(z)is N An alternate representation of a rational transform is as a ratio of two polynomials in M-1 (N-M)poz +p12 +.+PM-12+pM N N d1z-+…+dN-12+dN 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 .... .... ( ) ( )