z-Transform Example- The z-transform u(z of the unit step sequence u[n] can be obtained from X(z)= for az <1 1-az by setting a=1 for z<1 ROC is the annular region 1<=<oo Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 11 z-Transform • Example - The z-transform (z) of the unit step sequence [n] can be obtained from by setting = 1: • ROC is the annular region , for 1 1 1 ( ) 1 1 − = − − z z X z 1 z for 1 1 1 1 1 − = − − z z (z)
z-Transform Note: The unit step sequence u[n] is not absolutely summable, and hence its dtFt does not converge uniformly Example- Consider the anti-causal sequence yn]=-0"-n-1] 12 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 12 z-Transform • Note: The unit step sequence [n] is not absolutely summable, and hence its DTFT does not converge uniformly • Example - Consider the anti-causal sequence y[n] = − [−n −1] n
z-Transform Its z-transform is given by X(z)=∑-0 ∑ 1=-0 m=1 nn a Z Oz∑O m=0 for a 1-0z ROC is the annular region z<al 13 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 13 z-Transform • Its z-transform is given by • ROC is the annular region = − = − = − − =− − 1 1 ( ) m m m n n n Y z z z , for 1 1 1 1 1 − = − − z z z z z z m m m 1 1 0 1 1 − − = − − − = − = − z
z-Transform note: The z-transforms of the two sequences a"un and -a"[-n-1 are identical even though the two parent sequences are different Only way a unique sequence can be associated with a z-transform is by specifying its roc 14 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 14 z-Transform • Note: The z-transforms of the two sequences and are identical even though the two parent sequences are different • Only way a unique sequence can be associated with a z-transform is by specifying its ROC − [−n −1] n [n] n
z-Transform The dtFT G(e/o)of a sequence gn converges uniformly if and only if the roc of the z-transform G(z)of gln] includes the unit circle The existence of the dtft does not al ways imply the existence of the z-transform 15 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 15 z-Transform • The DTFT of a sequence g[n] converges uniformly if and only if the ROC of the z-transform G(z) of g[n] includes the unit circle • The existence of the DTFT does not always imply the existence of the z-transform ( ) j G e