The parallel realization of IIR filters 

The transfer function H(z) can be written as a sum of transfer functions H₁(z), H₂(z), ..., H_k(z) obtained by the partial fraction expansion:
 

Thus
 

and is shown in the block diagram shown below. The outputs y₁(n), y₂(n), ..., y_k(n) are not dependent on each other; they are not coupled as in case of the cascade structure.

Based on whether H(z)/z or H(z) is the beginning point for the partial fractions we have parallel forms I and II. Both of these methods are shown below.

 


Parallel Form I
This corresponds to expanding H(z)/z inspite of H(z) into partial fractions.

This structure is obtained when the H (z) are all selected to be of the form
 

here the quadratic terms are used for every pair of complex conjugate poles such that the coefficients of corresponding H_i(z) will be real. Every H_i(z) is then realized as a DF I or

DF II. The special cases of H_i(z) are as follows

 

Parallel Form II
It corresponds to expanding H(z) directly into the partial fractions.
This structure is obtained when the H_i(z) are all selected to be of the form

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