The difference equation above is implemented directly as written by the Direct-Form I IIR Filter Structure.

Note that this is a cascade of two systems,

Summary: The Direct Forms I and II, the Transpose Form, Parallel Form, and Cascade Form are the most common basic structures for implementing IIR filters.

IIR (Infinite Impulse Response) filter structures must be recursive
(use feedback); an infinite number of coefficients could not otherwise
be realized with a finite number of computations per sample.

The difference equation above is implemented directly as written by the Direct-Form I IIR Filter Structure.

Note that this is a cascade of two systems,

This structure is canonic: (i.e., it requires the minimum number of memory elements).

Flowgraph reversal gives the

Usually we design IIR filters with

Obviously, since all these structures have identical frequency response, filter structures are not unique. We consider many different structures because

- Depending on the technology or application, one might be more convenient than another
- The response in a practical realization, in which the data and coefficients must be quantized, may differ substantially, and some structures behave much better than others with quantization.

The numerator and denominator polynomials can be factored

A rational transfer function can also be written as

The cascade and parallel forms are of interest because they are much less sensitive to coefficient quantization than higher-order structures, as analyzed in later modules in this course.

There are many other structures for IIR filters,
such as wave digital filter
structures, lattice-ladder, all-pass-based forms, and so forth.
These are the result of extensive research to find structures
which are computationally efficient *and*
insensitive to quantization error. They all represent various
tradeoffs; the best choice in a given context is not yet fully
understood, and may never be.