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)

^{= b}₀ ^{+  b}₁  Z^-1  + ^b₂  Z^{-2   + …….+ b}_M Z^{-M  /  a}₀ ^{+  a}₁  Z^-1  + ^a₂  Z^{-2   + …….+ a}_N Z^-N

  ^b₀ ^{,  b}₁  ^, b₂   ^{, b}_M ⁼  Coefficients of numerator 

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

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

Here P₁  P₂  P₃  ........P_N is called as poles.

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

X(Z) / Z =  A₁ /  Z - P₁  + A₂ / Z - P₂  + .............+   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_K  } = (P_K)ⁿ u(n) if ROC : |Z| > | P_K |
that means causal sequence 

AND 

x(n) = IZT { Z /  Z - P_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.