Property

Time Domain

z-Domain

ROC

Notation x(n)

x1(n)

x2(n)

X(z)

X1(z)

X2(z)

r2 < |z| < r1
Linearity a1x1(n) + a2x2(n) a1X1(z) + a2X2(z) The intersection of ROC1 and ROC2
Time Shifting x(n-k) z-k X(z) except z=0, if k>0
Scaling in z-Domain anx(n) X(a-1z) |a|r2 < |z| < |a|r1
Time reversal x(-n) X(z-1) 1/r1 < |z| < 1/r2
Conjugation x*(n) X*(z*) ROC
Real Part Re{x(n)} 0.5 [ X(z) + X*(z*) ] Includes ROC
Imaginary part Im{x(n)} 0.5 j [ X(z) – X*(z*) ] Includes ROC
Differentiation nx(n) -z dX(z) / dz r2 < |z| < r1
Convolution x1(n) * x2(n) X1(z)X2(z) Intersection of ROC1 and ROC2