Initial value & Final value Theorems in 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: Lakshmi Prasanna Ponnala

Completed M.Tech in Digital Electronics and Communication Systems. View all posts by Lakshmi Prasanna Ponnala

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