15.4 Iir coefficient quantization analysis

Coefficient quantization is an important concern with IIR filters, since straigthforward quantization often yields poor results, and becausequantization can produce unstable filters.

Sensitivity analysis

The performance and stability of an IIR filter depends on the pole locations, so it is important to know howquantization of the filter coefficients $a_k$ affects the pole locations $p_j$ . The denominator polynomial is $$D(z)=1+\sum_{k=1}^N a_{k}z^{-k}=\prod_{i=1}^N 1-p_{i}z^{(-1)}$$ We wish to know $\frac{\partial^1{p}_i()}{\partial a_k}$ , which, for small deviations, will tell us that a $$ change in $a_k$ yields an $=\frac{\partial^1{p}_i()}{\partial a_k}$ change in the pole location. $\frac{\partial^1{p}_i}{\partial a_k}$ is the sensitivity of the pole location to quantization of $a_k$ . We can find $\frac{\partial^1{p}_i}{\partial a_k}$ using the chain rule. $$(z, , \frac{\partial^{1}A(z)}{\partial a_k})=(z, , \frac{\partial^{1}A(z)}{\partial z}\frac{\partial^{1}z}{\partial a_k})$$ $$$$ $$\frac{\partial^1{p}_i}{\partial a_k}=\frac{(z, , \frac{\partial^{1}A(z_i)}{\partial a_k})}{(z, , \frac{\partial^{1}A(z_i)}{\partial z})}$$ which is

How can we reduce this high sensitivity to IIR filter coefficient quantization?

Solution

Cascade or parallel form implementations! The numerator and denominator polynomials can be factored off-line at very high precision and grouped intosecond-order sections, which are then quantized section by section. The sensitivity of the quantization is thus thatof second-order, rather than $N$ -th order, polynomials. This yields major improvements in the frequency response of theoverall filter, and is almost always done in practice.

Note that the numerator polynomial faces the same sensitivity issues; the cascade form also improves the sensitivity of the zeros, because they arealso factored into second-order terms. However, in the parallel form, the zeros are globally distributed across the sections, so they suffer fromquantization of all the blocks. Thus the cascade form preserves zero locations much better than the parallel form, which typically meansthat the stopband behavior is better in the cascade form, so it is most often used in practice.

Quantized pole locations

In a direct-form or transpose-form implementation of a second-order section, the filter coefficients are quantized versions of the polynomial coefficients. $$D(z)=z^2+a_{1}z+a_2=(z-p)(z-\overline{p})$$ $$p=\frac{(-a_1, \sqrt{a_1^2-4a_2})}{2}$$ $$p=re^i$$ $$D(z)=z^2-2r\cos +r^2$$ So $$a_1=-(2r\cos )$$ $$a_2=r^2$$ Thus the quantization of $a_1$ and $a_2$ to $B$ bits restricts the radius $r$ to $r=\sqrt{k{}_B}$ , and $a_1=-(2\Re (p))=k{}_B$ The following figure shows all stable pole locations after four-bit two's-complement quantization.

In the "normal-form" structures, a state-variable based realization, the poles are uniformly spaced.

The normal form has a number of other advantages; both eigenvalues are equal, so it minimizes the norm of $Ax$ , which makes overflow less likely, and it minimizes the output variance due to quantization of the statevalues. It is sometimes used when minimization of finite-precision effects is critical.