r ∈ Z {\displaystyle r\in \mathbb {Z} } X ( z k ) {\displaystyle X(z^{k})} X k ( z ) = ∑ n = − ∞ ∞ x k ( n ) z − n = ∑ r = − ∞ ∞ x ( r ) z − r k = ∑ r = − ∞ ∞ x ( r ) ( z k ) − r = X ( z k ) {\displaystyle {\begin{aligned}X_{k}(z) =\sum _{n=-\infty }^{\infty }x_{k}(n)z^{-n}\\ =\sum _{r=-\infty }^{\infty }x(r)z^{-rk}\\ =\sum _{r=-\infty }^{\infty }x(r)(z^{k})^{-r}\\ =X(z^{k})\end{aligned}}} R 1 k {\displaystyle R^{\frac {1}{k}}} സമയവ്യതിയാനം x [ n − k ] {\displaystyle x[n-k]} z − k X ( z ) {\displaystyle z^{-k}X(z)} Z { x [ n − k ] } = ∑ n = 0 ∞ x [ n − k ] z − n = ∑ j = − k ∞ x [ j ] z − ( j + k ) j = n − k = ∑ j = − k ∞ x [ j ] z − j z − k = z − k ∑ j = − k ∞ x [ j ] z − j = z − k ∑ j = 0 ∞ x [ j ] z − j x [ β ] = 0 , β 0 = z − k X ( z ) {\displaystyle {\begin{aligned}Z\{x[n-k]\} =\sum _{n=0}^{\infty }x[n-k]z^{-n}\\ =\sum _{j=-k}^{\infty }x[j]z^{-(j+k)} j=n-k\\ =\sum _{j=-k}^{\infty }x[j]z^{-j}z^{-k}\\ =z^{-k}\sum _{j=-k}^{\infty }x[j]z^{-j}\\ =z^{-k}\sum _{j=0}^{\infty }x[j]z^{-j} x[\beta ]=0,\beta 0\\ =z^{-k}X(z)\end{aligned}}} ROC, except z = 0 if k z = ∞ if k 0 സീ മണ്‌ഡലവികാസം a n x [ n ] {\displaystyle a^{n}x[n]} X ( a − 1 z ) {\displaystyle X(a^{-1}z)} Z { a n x [ n ] } = ∑ n = − ∞ ∞ a n x ( n ) z − n = ∑ n = − ∞ ∞ x ( n ) ( a − 1 z ) − n = X ( a − 1 z ) {\displaystyle {\begin{aligned}{\mathcal {Z}}\left\{a^{n}x[n]\right\} =\sum _{n=-\infty }^{\infty }a^{n}x(n)z^{-n}\\ =\sum _{n=-\infty }^{\infty }x(n)(a^{-1}z)^{-n}\\ =X(a^{-1}z)\end{aligned}}} | a | r 2 | z | | a | r 1 {\displaystyle |a|r_{2} |z| |a|r_{1}} സമയവിപരീതനം x [ − n ] {\displaystyle x[-n]} X ( z − 1 ) {\displaystyle X(z^{-1})} Z { x ( − n ) } = ∑ n = − ∞ ∞ x ( − n ) z − n = ∑ m = − ∞ ∞ x ( m ) z m = ∑ m = − ∞ ∞ x ( m ) ( z − 1 ) − m = X ( z − 1 ) {\displaystyle {\begin{aligned}{\mathcal {Z}}\{x(-n)\} =\sum _{n=-\infty }^{\infty }x(-n)z^{-n}\\ =\sum _{m=-\infty }^{\infty }x(m)z^{m}\\ =\sum _{m=-\infty }^{\infty }x(m){(z^{-1})}^{-m}\\ =X(z^{-1})\\\end{aligned}}} 1 r 1 | z | 1 r 2 {\displaystyle {\tfrac {1}{r_{1}}} |z| {\tfrac {1}{r_{2}}}} Complex conjugation x ∗ [ n ] {\displaystyle x^{*}[n]} X ∗ ( z ∗ ) {\displaystyle X^{*}(z^{*})} Z { x ∗ ( n ) } = ∑ n = − ∞ ∞ x ∗ ( n ) z − n = ∑ n = − ∞ ∞ [ x ( n ) ( z ∗ ) − n ] ∗ = [ ∑ n = − ∞ ∞ x ( n ) ( z ∗ ) − n ] ∗ = X ∗ ( z ∗ ) {\displaystyle {\begin{aligned}{\mathcal {Z}}\{x^{*}(n)\} =\sum _{n=-\infty }^{\infty }x^{*}(n)z^{-n}\\ =\sum _{n=-\infty }^{\infty }\left[x(n)(z^{*})^{-n}\right]^{*}\\ =\left[\sum _{n=-\infty }^{\infty }x(n)(z^{*})^{-n}\right]^{*}\\ =X^{*}(z^{*})\end{aligned}}} വാസ്തവികസംഖ്യ ഭാഗം Re ⁡ { x [ n ] } {\displaystyle \operatorname {Re} \{x[n]\}} 1 2 [ X ( z ) + X ∗ ( z ∗ ) ] {\displaystyle {\tfrac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]} സാങ്കൽപിക സംഖ്യാ ഭാഗം Im ⁡ { x [ n ] } {\displaystyle \operatorname {Im} \{x[n]\}} 1 2 j [ X ( z ) − X ∗ ( z ∗ ) ] {\displaystyle {\tfrac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]} അവകലനം n x [ n ] {\displaystyle nx[n]} − z d X ( z ) d z {\displaystyle -z{\frac {dX(z)}{dz}}} Z { n x ( n ) } = ∑ n = − ∞ ∞ n x ( n ) z − n = z ∑ n = − ∞ ∞ n x ( n ) z − n − 1 = − z ∑ n = − ∞ ∞ x ( n ) ( − n z − n − 1 ) = − z ∑ n = − ∞ ∞ x ( n ) d d z ( z − n ) = − z d X ( z ) d z {\displaystyle {\begin{aligned}{\mathcal {Z}}\{nx(n)\} =\sum _{n=-\infty }^{\infty }nx(n)z^{-n}\\ =z\sum _{n=-\infty }^{\infty }nx(n)z^{-n-1}\\ =-z\sum _{n=-\infty }^{\infty }x(n)(-nz^{-n-1})\\ =-z\sum _{n=-\infty }^{\infty }x(n){\frac {d}{dz}}(z^{-n})\\ =-z{\frac {dX(z)}{dz}}\end{aligned}}} Convolution x 1 [ n ] ∗ x 2 [ n ] {\displaystyle x_{1}[n]*x_{2}[n]} X 1 ( z ) X 2 ( z ) {\displaystyle X_{1}(z)X_{2}(z)} Z { x 1 ( n ) ∗ x 2 ( n ) } = Z { ∑ l = − ∞ ∞ x 1 ( l ) x 2 ( n − l ) } = ∑ n = − ∞ ∞ [ ∑ l = − ∞ ∞ x 1 ( l ) x 2 ( n − l ) ] z − n = ∑ l = − ∞ ∞ x 1 ( l ) [ ∑ n = − ∞ ∞ x 2 ( n − l ) z − n ] = [ ∑ l = − ∞ ∞ x 1 ( l ) z − l ] [ ∑ n = − ∞ ∞ x 2 ( n ) z − n ] = X 1 ( z ) X 2 ( z ) {\displaystyle {\begin{aligned}{\mathcal {Z}}\{x_{1}(n)*x_{2}(n)\} ={\mathcal {Z}}\left\{\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)\right\}\\ =\sum _{n=-\infty }^{\infty }\left[\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)\right]z^{-n}\\ =\sum _{l=-\infty }^{\infty }x_{1}(l)\left[\sum _{n=-\infty }^{\infty }x_{2}(n-l)z^{-n}\right]\\ =\left[\sum _{l=-\infty }^{\infty }x_{1}(l)z^{-l}\right]\!