1.2 Iir filter structures

IIR (Infinite Impulse Response) filter structures must be recursive (use feedback); an infinite number of coefficients could not otherwisebe realized with a finite number of computations per sample. $$H(z)=\frac{N(z)}{D(z)}=\frac{b_0+b_{1}z^{(-1)}+b_{2}z^{-2}++b_{M}z^{-M}}{1+a_{1}z^{(-1)}+a_{2}z^{-2}++a_{N}z^{-N}}$$ The corresponding time-domain difference equation is $$y(n)=-(a_{1}y(n-1))-a_{2}y(n-2)+-a_{N}y(n-N)+b_{0}x(0)+b_{1}x(n-1)++b_{M}x(n-M)$$

Direct-form i iir filter structure

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, $N(z)$ and $\frac{1}{D(z)}$ . If we reverse the order of the filters, the overall system is unchanged: The memory elements appear in the middleand store identical values, so they can be combined, to form the Direct-Form II IIR Filter Structure.

Direct-form ii iir filter structure

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

Flowgraph reversal gives the

Transpose-form iir filter structure

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

Obviously, since all these structures have identical frequency response, filter structures are not unique. Weconsider 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 somestructures behave much better than others with quantization.

Iir cascade form

The numerator and denominator polynomials can be factored $$H(z)=\frac{b_0+b_{1}z^{(-1)}++b_{M}z^{-m}}{1+a_{1}z^{(-1)}++a_{N}z^{-N}}=\frac{b_0\prod_{k=1}^M z-z_k}{z^{(M-N)}\prod_{i=1}^N z-p_k}$$ and implemented as a cascade of short IIR filters.

Parallel form

A rational transfer function can also be written as $$\frac{b_0+b_{1}z^{(-1)}++b_{M}z^{-m}}{1+a_{1}z^{(-1)}++a_{N}z^{-N}}=c_0^\prime +c_1^\prime z^{(-1)}++\frac{A_1}{z-p_1}+\frac{A_2}{z-p_2}++\frac{A_N}{z-p_N}$$ which by linearity can be implemented as

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

Other forms

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