Final value theorem
Let x(n) be a signal causal, and let X(Z) be its Z-Transform. Then :
\(X(\infty)=\lim\limits_{Z \to 1}(1-Z^{-1}) = \lim\limits_{Z \to 1} \frac{Z-1}{Z} X(Z)= \lim\limits_{n \to \infty }x(n) \quad\) , and this limit is finite.
Example : Example :
\(x(n)=a^n\) with, |\(a|< 1\), we get , \(\lim\limits_{Z \to 1}(1-Z^{-1})X(Z)= \lim\limits_{Z \to 1} \frac{Z-1}{Z} \frac{Z}{Z-a}\)