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.

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.