Filter Realization
After a filter is designed, it must be realized by developing a signal flow diagram that describes the filter in terms of operations on sample sequences.
A given transfer function may be realized in many ways. Consider how a simple expression such as could be evaluated – one could also compute the equivalent . In the same way, all realizations may be seen as "factorizations" of the same transfer function, but different realizations will have different numerical properties. Specifically, some realizations are more efficient in terms of the number of operations or storage elements required for their implementation, and others provide advantages such as improved numerical stability and reduced round-off error. Some structures are better for fixed-point arithmetic and others may be better for floating-point arithmetic.
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—Aldous Huxley (1894–1963)