Partial fraction expansion method is possible only when Z transform definition which is rational in nature, That means they are expressed of two polynomials.

X(Z) = N(Z) / D(Z)

^{= 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}

^{ b}

_{0}

^{ , b}

_{1}

^{ , }

^{b}

_{2}

^{ }

^{ , b}

_{M}

^{ = }Coefficients of numerator

^{ a}

_{0}

^{ , a}

_{1}

^{ ,}

^{a}

_{2}

^{ }.....

^{a}

_{N}

^{ = }

^{ }Coefficients of Denominator

M = Degree of numerator

N = Degree of denominator

N(Z) = Numerator polynomial

D(Z) = Denominator polynomial

**Step to follow the partial fraction expansion method :**

**Step 1:**Check whether the given function is the proper form or not. The function is said to be in proper form when the following conditions are satisfied.

- The coefficient
^{a}_{0}in the above equation should be equal to 1. If^{a}_{0 }not equal to 1 then the polynomial is adjusted accordingly. - In equation
^{a}_{0 }not equal to 1 and the degree of numerator should be less than the degree of the denominator(M<N). If this condition is satisfied then the long division is carried out to make M<N.

**Step 2:**Multiply the numerator and denominator by Z

^{N}. That means to convert the function in term of the positive power of Z.

**Step 3 :**Obtain the equation X(Z) / Z

**Step 4:**Factorize the denominator and obtain the roots. Then the denominator will be in the form

_{1}) (Z - P

_{2}) (Z - P

_{3})............(Z -P

_{N})

Here P

_{1}P_{2 }P_{3 }........P_{N }is called as poles.**Step 5 :**Write down the equation in partial expansion form as follows :

X(Z) / Z = A

_{1}/ Z - P_{1}+ A_{2}/ Z - P_{2}+ .............+ A_{N}/ Z - P_{N}
A

_{1, }A_{2, }A_{3 .....}A_{N}are coefficient. the coefficient A_{K }is calculated as
A

_{K}= (Z - P_{K}). X(Z) / Z | where Z=P_{K}**Step 6 :**By calculating transfer of a

_{1, }a

_{2, }a

_{3 .....}a

_{n }z to R.H.S of equation. Now we standard from pair to obtain inverse Z transform

x(n) = IZT { Z / Z - P

that means causal sequence

_{K}} = (P_{K})^{n}u(n) if ROC : |Z| > | P_{K}|that means causal sequence

**AND**

x(n) = IZT { Z / Z - P

that means anticausal sequence

Now let us check it out the example of partial fraction method to learn more details about this article.

_{K}} = - (P_{K})^{n}u(-n-1) if ROC : |Z| < | P_{K}|that means anticausal sequence

Now let us check it out the example of partial fraction method to learn more details about this article.