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.

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.