Initial and Final value Theorems (Z-Transforms)

Initial  value Theorem (Z-Transforms):-

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

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

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

$\lim_{Z\rightarrow&space;\infty&space;}X_{+}(Z)&space;=&space;\lim_{Z\rightarrow&space;\infty&space;}\left&space;[&space;x(0)&space;+x(1)\&space;Z^{-1}+&space;x(2)\&space;Z^{-2}+x(3)\&space;Z^{-3}+......&space;\right&space;]$

$\lim_{Z\rightarrow&space;\infty&space;}X_{+}(Z)&space;=&space;x(0)$ .

Final  value Theorem (Z-Transforms):-

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

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

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

$\lim_{z\rightarrow&space;1}\left&space;[&space;(Z-1)\&space;X_{+}(Z)&space;-x(0)&space;\right&space;]=\lim_{z\rightarrow&space;1}&space;\lim_{k\rightarrow&space;\infty&space;}\sum_{n=0}^{k}&space;\left&space;[&space;\left&space;\{&space;x(1)-x(0)&space;\right&space;\}+\left&space;\{&space;x(2)-x(1)&space;\right&space;\}Z^{-1}+........+\left&space;\{&space;x(k+1)-x(k)&space;\right&space;\}Z^{-k}&space;\right&space;]$

$\lim_{z\rightarrow&space;1}\left&space;[&space;(Z-1)\&space;X_{+}(Z)&space;-x(0)&space;\right&space;]=&space;\lim_{z\rightarrow&space;1}\left&space;\{&space;x(\infty&space;)-x(0)&space;\right&space;\}$.

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

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Author: vikramarka

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