Cascade realization of the IIR filters 

Many different realizations are their depending on how we choose to write and rearrange the given transfer function. 2 very important ways of decomposing transfer function are parallel and cascade decompositions.

  In cascade realization H(z) is broken up into the product of transfer functions H₁, H₂,

..., H_k, each a rational expression in z^-1 which is given as follows:
 
such that Y(z) can be written as follows
 
 

Although H(z) could be broken up in several different ways, the most common cascade realization is to require each of the k product H's to be a biquadratic section. In a number of cases the design procedure yields the product of biquadratic expressions so no further work is essential to put H(z) in the needed form. The product terms Hi(z) could take various forms, depending upon the actual problem. Some of the possible forms are

 

Each of the H_i(z) can be realized using the direct form I or II.

  The different structures are obtained by changing the ordering of sections and by changing the zero pole pairings. In practice because of the finite word-length effects, each such type of cascade realization behaves differently from others.

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