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

Figure 1

Note that this is a cascade of two systems, N⁢z N z and 1D⁢z 1 D z . If we reverse the order of the filters, the overall system is unchanged: The memory elements appear in the middle and store identical values, so they can be combined, to form the Direct-Form II IIR Filter Structure.

Figure 2

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

Flowgraph reversal gives the

Figure 3

Usually we design IIR filters with N=M N M , but not always.

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

The numerator and denominator polynomials can be factored H⁢z= b 0 + b 1 ⁢z-1+…+ b M ⁢z−m1+ a 1 ⁢z-1+…+ a N ⁢z−N= b 0 ⁢∏ k =1Mz− z k zM−N⁢∏ i =1Nz− p k H z b 0 b 1 z … b M z m 1 a 1 z … a N z N b 0 k 1 M z z k z M N i 1 N z p k and implemented as a cascade of short IIR filters.

Figure 4

A rational transfer function can also be written as b 0 + b 1 ⁢z-1+…+ b M ⁢z−m1+ a 1 ⁢z-1+…+ a N ⁢z−N= c 0 ′+ c 1 ′⁢z-1+…+ A 1 z− p 1 + A 2 z− p 2 +…+ A N z− p N b 0 b 1 z … b M z m 1 a 1 z … a N z N c 0 c 1 z … A 1 z p 1 A 2 z p 2 … A N z p N which by linearity can be implemented as

Figure 5

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.