### Z transform partial fraction expansion

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)

= b0 +  b1  Z-1  + b2  Z-2   + …….+ bM Z-M  /  a0 +  a1  Z-1  + a2  Z-2   + …….+ aN Z-N

b0 ,  b1  b2   , bM =  Coefficients of numerator

a0 , a1  ,  a2  ..... aN  =  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 a0 in the above equation should be equal to 1. If  a not equal to 1 then the polynomial is adjusted accordingly.
• In equation a 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 ZN. 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

(Z - P1 ) (Z - P2 ) (Z - P3 )............(Z -PN )

Here P1  P P ........Pis called as poles.

Step 5 :  Write down the equation in partial expansion form as follows :

X(Z) / Z =  A1 /  Z - P1  + A2 / Z - P2  + .............+   AN / Z - PN

A1, A2,  A3 .....AN  are coefficient. the coefficient Ais  calculated as

AK =  (Z - PK ). X(Z) / Z  | where Z=PK

Step 6 : By calculating  transfer  of a1, a2,  a3 .....an   z to R.H.S of equation. Now we standard from pair to obtain inverse Z transform

x(n) = IZT { Z /  Z - PK  } = (PK)n u(n) if ROC : |Z| > | PK |
that means causal sequence

AND
x(n) = IZT { Z /  Z - PK  } = - (PK)n u(-n-1) if ROC : |Z| < | PK |
that means anticausal sequence