Initial and Final value Theorems (Z-Transforms)

Initial  value Theorem (Z-Transforms):-

If uni-lateral Z-Transform of x[n] is X_{+}(Z)  then  x(0) = \lim_{z\rightarrow \infty } X_{+}(Z) .

X_{+}(Z) = \sum_{n=0}^{\infty } x[n] \ Z^{-n} .

X_{+}(Z) = x(0) +x(1)\ Z^{-1}+ x(2)\ Z^{-2}+x(3)\ Z^{-3}+......

\lim_{Z\rightarrow \infty }X_{+}(Z) = \lim_{Z\rightarrow \infty }\left

\lim_{Z\rightarrow \infty }X_{+}(Z) = x(0) .

Final  value Theorem (Z-Transforms):-

If uni-lateral Z-Transform of x[n] is X_{+}(Z)  then  x(\infty ) = \lim_{z\rightarrow 1}(Z-1) \ X_{+}(Z) .

Z\left \{ x(n+1) \right \}-Z\left \{ x(n) \right \} = \lim_{k\rightarrow \infty }\sum_{n=0}^{k} (x[n+1]-x[n]) \ Z^{-n} .

Z\ X_{+}(Z) -x(0)-X_{+}(Z)= \lim_{k\rightarrow \infty }\sum_{n=0}^{k} (x[n+1]-x[n]) \ Z^{-n}.

\lim_{z\rightarrow 1}\left =\lim_{z\rightarrow 1} \lim_{k\rightarrow \infty }\sum_{n=0}^{k} \left

\lim_{z\rightarrow 1}\left = \lim_{z\rightarrow 1}\left \{ x(\infty )-x(0) \right \}.

\lim_{z\rightarrow 1}(Z-1)\ X_{+}(Z) = x(\infty ).

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Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.