16.1 Scaling

Overflow is clearly a serious problem, since the errors it introduces are very large. As we shall see, it is also responsiblefor large-scale limit cycles, which cannot be tolerated. One way to prevent overflow, or to render it acceptably unlikely, is to scale the input to a filter such that overflow cannot (or is sufficiently unlikely to) occur.

In a fixed-point system, the range of the input signal is limited by the fractional fixed-point number representation to $\left|x(n)\right|\le 1$ . If we scale the input by multiplying it by a value $$ , $0< < 1$ , then $\left|x(n)\right|\le$ .

Another option is to incorporate the scaling directly into the filter coefficients.

Fir filter scaling

What value of $$ is required so that the output of an FIR filter cannot overflow( $\forall n\colon \left|y(n)\right|\le 1$ , $\forall n\colon \left|x(n)\right|\le 1$ )? $$\left|y(n)\right|=\left|\sum_{k=0}^{M-1} h(k)x(n-k)\right|\le \sum_{k=0}^{M-1} \left|h(k)\right|\left|\right|\left|x(n-k)\right|\le \sum_{k=0}^{M-1} \left|h(k)\right|\times 1$$ $$$$ $$< \sum_{k=0}^{M-1} \left|h(k)\right|$$ Alternatively, we can incorporate the scaling directly into the filter, and require that $$\sum_{k=0}^{M-1} \left|h(k)\right|< 1$$ to prevent overflow.

Iir filter scaling

To prevent the output from overflowing in an IIR filter, the condition above still holds:( $M$∞ ) $$\left|y(n)\right|< \sum$$∞ 0 h k so an initial scaling factor $< \frac{1}{\sum }$∞ 0 h k can be used, or the filter itself can be scaled.

However, it is also necessary to prevent the states from overflowing, and to prevent overflow at any point in the signal flow graph where the arithmetic hardware wouldthereby produce errors. To prevent the states from overflowing, we determine the transfer function from the input to all states $i$ , and scale the filter such that $\forall i\colon \sum$∞ 0 h i k 1

Although this method of scaling guarantees no overflows, it is often too conservative. Note that a worst-case signal is $x(n)=\mathrm{sign}(h(-n))$ ; this input may be extremely unlikely. In the relatively common situation in which the input is expected tobe mainly a single-frequency sinusoid of unknown frequency and amplitude less than 1, a scaling condition of $$\forall w\colon \left|H(w)\right|\le 1$$ is sufficient to guarantee no overflow. This scaling condition is often used. If there are several potential overflowlocations $i$ in the digital filter structure, the scaling conditions are $$\forall i, w\colon \left|H_i(w)\right|\le 1$$ where $H_i(w)$ is the frequency response from the input to location $i$ in the filter.

Even this condition may be excessively conservative, for example if the input is more-or-less random, or if occasionaloverflow can be tolerated. In practice, experimentation and simulation are often the bestways to optimize the scaling factors in a given application.

For filters implemented in the cascade form, rather than scaling for the entire filter at the beginning, (whichintroduces lots of quantization of the input) the filter is usually scaled so that each stage is just preventedfrom overflowing. This is best in terms of reducing the quantization noise. The scaling factors are incorporatedeither into the previous or the next stage, whichever is most convenient.

Some heurisitc rules for grouping poles and zeros in a cascade implementation are:

• Order the poles in terms of decreasing radius. Take the pole pair closest to the unit circle and group it withthe zero pair closest to that pole pair (to minimize the gain in that section). Keep doing this with all remainingpoles and zeros.
• Order the section with those with highest gain ( $(\left|H_i(w)\right|)$ ) in the middle, and those with lower gain on the ends.

Leland B. Jackson has an excellent intuitive discussion of finite-precision problems in digitalfilters. The book by Roberts and Mullis is one of the most thorough in terms of detail.