# 13.3 Data quantization in iir filters

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Quantization errors in an IIR filter structure are filtered by portions of the structure before reaching the output. The total quantization noise variance at the output is the sum of the variances of the individual filtered quantizer noises as seen at the filter output.

Finite-precision effects are much more of a concern with IIR filters than with FIR filters, since the effects are more difficult to analyze andminimize, coefficient quantization errors can cause the filters to become unstable, and disastrous things like large-scale limit cycles can occur.

## Roundoff noise analysis in iir filters

Suppose there are several quantization points in an IIR filter structure. By our simplifying assumptions about quantization errorand Parseval's theorem, the quantization noise variance ${}_{\mathrm{y,i}}^{2}$ at the output of the filter from the $i$ th quantizer is

${}_{\mathrm{y,i}}^{2}=\frac{1}{2\pi }\int_{-\pi }^{\pi } \left|{H}_{i}(w)\right|^{2}S{S}_{{n}_{i}}(w)\,d w=\frac{{}_{{n}_{i}}^{2}}{2\pi }\int_{-\pi }^{\pi } \left|{H}_{i}(w)\right|^{2}\,d w={}_{{n}_{i}}^{2}\sum$ h i n 2
where ${}_{{n}_{i}}^{2}$ is the variance of the quantization error at the $i$ th quantizer, $S{S}_{{n}_{i}}(w)$ is the power spectral density of that quantization error, and $H{H}_{i}(w)$ is the transfer function from the $i$ th quantizer to the output point.Thus for $P$ independent quantizers in the structure, the total quantization noise variance is ${}_{y}^{2}=\frac{1}{2\pi }\sum_{i=1}^{P} {}_{{n}_{i}}^{2}\int_{-\pi }^{\pi } \left|{H}_{i}(w)\right|^{2}\,d w$ Note that in general, each ${H}_{i}(w)$ , and thus the variance at the output due to each quantizer, is different; for example, the system as seen by a quantizer at theinput to the first delay state in the Direct-Form II IIR filter structure to the output, call it ${n}_{4}$ , is with a transfer function ${H}_{4}(z)=\frac{z^{-2}}{1+{a}_{1}z^{(-1)}+{a}_{2}z^{-2}}$ which can be evaluated at $z=e^{iw}$ to obtain the frequency response.

A general approach to find ${H}_{i}(w)$ is to write state equations for the equivalent structure as seen by ${n}_{i}$ , and to determine the transfer function according to $H(z)=CzI-A^{(-1)}B+d$ .

The above figure illustrates the quantization points in a typical implementation of a Direct-Form II IIRsecond-order section. What is the total variance of the output error due to all of thequantizers in the system?

By making the assumption that each ${Q}_{i}$ represents a noise source that is white, independent of the other sources, and additive,

the variance at the output is the sum of the variances atthe output due to each noise source: ${}_{y}^{2}=\sum_{i=1}^{4} {}_{y,i}^{2}$ The variance due to each noise source at the outputcan be determined from $\frac{1}{2\pi }\int_{-\pi }^{\pi } \left|{H}_{i}(w)\right|^{2}{S}_{{n}_{i}}(w)\,d w$ ; note that ${S}_{{n}_{i}}(w)={}_{{n}_{i}}^{2}$ by our assumptions, and ${H}_{i}(w)$ is the transfer function from the noise source to the output .

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