Prove the conjugation properties and Parseval?s relation for the z -transform given in Property Time Domain z-Domain

Prove the conjugation properties and Parseval?s relation for the z-transform given in

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Property Time Domain z-Domain ROC ROC: 7 < kl < ROC, ROC, At least the intersection of ROC and ROC, Notation X (2) X;(2) X2{z) x(n) x1(n) *2(n) ai X (2) +a, X;(2) Lincarity ajx1 (n) + azxz(n) Time shifting *X(2) x(n - k) That of X(2), except z = 0 if k > 0 and z = o if k < ở la'rz < Izl < Inlr Scaling in the z-domain X (a-'2) a"x(n) < lel < Time reversal X(2) x(-n) r* (n) ROC Conjugation Real part (,2).X IX(2) + X*(2)] |X(2) – X*(:")| Refx(n)} Includes ROC Imaginary part Imfx(n)) Includes ROC Differentiation in the 2-domain nx(n) (:)XP -z dz Convolution At least, the intersection of ROC, and ROC; (2)*X(2)'X r,1) = x;U) • x2(-1) R(2) = X,(z)X2(z-) Correlation At least, the intersection of ROC of Xi(z) and X;(2-) If x(n) causal Initial value theorem x (0) = lim X(z) -'dv At least rury < lzl < ru2 Multiplication A1(n)x2(n) Σεξ o) Parseval's relation