Long Division (or) Power Series Expansion Method:- ece4uplp

Power Series ExpansionMethod:-

$X(Z) = \sum_{n=-\infty }^{\infty }x[n]\ Z^{-n}$

$X(Z)$ can be expressed either in positive powers of Z or negative powers of Z.

if the sequence is causal $X(Z)$ has negative powers of Z similarly the Non-causal sequence $X(Z)$ negative powers of Z.

Let $X(Z)=\frac{N(Z)}{D(Z)} =\frac{b_{0}+b_{1}Z^{-1}+b_{2}Z^{-2}+...........+b_{M}Z^{-M}}{a_{0}+a_{1}Z^{-1}+a_{2}Z^{-2}+...........+a_{N}Z^{-N}}$ .

$X(Z)$ is causal it has ROC $\left | Z \right |> \left | r \right |$ then $X(Z)$ can be expressed as

$X(Z) = x(0)+x(1)Z^{-1}+x(2)Z^{-2}+x(3)Z^{-3}+........$

$X(Z)$ is non-causal it has ROC $\left | Z \right |<\left | r \right |$ then $X(Z)$ can be expressed as

$X(Z)=x(0)+x(-1)Z^{1}+x(-2)Z^{2}+x(-3)Z^{3}+...........$

Partial fraction Method:-

Let $X(Z)=\frac{N(Z)}{D(Z)} =\frac{b_{0}+b_{1}Z^{-1}+b_{2}Z^{-2}+...........+b_{M}Z^{-M}}{a_{0}+a_{1}Z^{-1}+a_{2}Z^{-2}+...........+a_{N}Z^{-N}}$ and $a_{o}=1$

if $M<N$ , $X(Z)$ is a proper function .

if $M\geq N$ , $X(Z)$ is improper function so convert the improper function to proper function as

$X(Z)=c_{o}+c_{1}Z^{-1}+c_{2}Z^{-2}+.........+c_{M-N}Z^{-(M-N)}+ \frac{N_{1}(Z)}{D(Z)}$ .

$X(Z) = polynomial + rational \ proper \ function$ .

express $X(Z )$ into powers of Z as follows

$X(Z ) = \frac{b_{o}Z^{N}+b_{1}Z^{N-1}+b_{2}Z^{N-2}+.......+b_{M}Z^{N-M}}{Z^{N}+a_{1}Z^{N-1}+a_{2}Z^{N-2}+.......+a_{N}}$

then divide $X(Z)$ by Z

$\frac{X(Z)}{Z} = \frac{b_{o}Z^{N-1}+b_{1}Z^{N-2}+b_{2}Z^{N-3}+.......+b_{M}Z^{N-M-1}}{Z^{N}+a_{1}Z^{N-1}+a_{2}Z^{N-2}+.......+a_{N}}$ .

Now express $\frac{X(Z)}{Z}$ into partial fractions using different cases and find out the inverse Z-transform for the function

$X(Z ) = Z.(partial fraction \ expansion )$ .

Convolution Method:-

express $X(Z)$ as a product of two functions $X_{1}(Z)$ and $X_{2}(Z)$ as follows $X(Z) =X_{1}(Z) . X_{2}(Z)$

then find the inverse Z- transforms of individual functions

$x_{1}[n]\leftrightarrow X_{1}(Z)$

$x_{2}[n]\leftrightarrow X_{2}(Z)$

by using convolution method find convolution of $x_{1}[n]$ and $x_{2}[n]$

i.e, $x[n] = x_{1}[n] * x_{2}[n]$

now $x[n]$ is the inverse Z-transform of $X(Z)$ .

Cauchy Residue Theorem:-

$f(Z)$ a function in Z if the derivative $\frac{df(Z)}{dZ}$ exists on and inside contour C and $f(Z)$ has no poles at $Z=Z_{o}$ then.

$\frac{1}{2\pi j}\oint_{c} \frac{f(Z)dZ}{Z-Z_{o}}=\left\{\begin{matrix} f(Z_{o}) \ if \ Z_{o} \ is\ inside \ C\\ 0 \ if \ Z_{o}\ is\ outside \ C \end{matrix}\right.$ .

if $(k+1)^{th}$ the derivative of $f(Z)$ exists on and has no poles at $Z=Z_{o}$ then.

$\frac{1}{2\pi j}\oint_{c} \frac{f(Z)}{(Z-Z_{o})^k}dZ=\left\{\begin{matrix}\frac{1}{(k-1)!} \frac{d^{k-1}f(Z)} {dZ^{k-1}}\ if \ Z_{o} \ is\ inside \ C\\ 0 \ if \ Z_{o}\ is\ outside \ C \end{matrix}\right.$ .

the values on the right hand side are called Residue’s of the pole $Z=Z_{o}$ .

if there are n no of poles inside C $\frac{1}{2\pi j}\oint_{c} \frac{f(Z)dZ}{(Z-Z_{1})(Z-Z_{2})(Z-Z_{3})..(Z-Z_{n})}=\sum_{i=1}^{n}\lim_{Z\rightarrow Z_{i}} \left$ .

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Author: Lakshmi Prasanna Ponnala

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

Long Division (or) Power Series Expansion Method:-

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