Page 215 - DSP Integrated Circuits

200 Chapter 5 Finite Word Length Effects

The magnitude of the scaled input signal to the multiplier will be equal to, or
less than, the magnitude of the input signal of the filter. The input to the multi-
plier will never overflow if the input to the filter does not overflow. This scaling pol-
icy is therefore called safe scaling. The safe scaling method is generally too
pessimistic since it uses the available signal range inefficiently. The safe scaling
method is suitable for short FIR filters because the probability for overflow is high
for a filter with a short impulse response.
It is sometimes argued that parasitic oscillations caused by overflow can not
occur if the filter is scaled according to the safe scaling criterion. However, this is
not correct since abnormal signal values can occur due to malfunctioning
hardware—for example, in the memory (delay) elements due to external distur-
bances (e.g., ionic radiation and disturbances from the supply lines). The likeli-
hood of temporary malfunction of the hardware increases as the size of the
transistors is scaled down. In the future it will therefore be necessary to use filter
structures that are guaranteed to suppress all types of parasitic oscillations and
return the filter to normal operation.

EXAMPLE 5.5
Scale the signal levels in an FIR filter using the safe scaling criterion. The filter is
realized using the direct form, linear-phase structure shown in Figure 4.6. The
original impulse response is

First, the input to the multipliers that have noninteger coefficients must be
scaled. Hence, the input signal must be divided by 2 because the preceding addi-
tions effectively double the signal level. This operation may in practice be accom-
plished by either a shift operation or by simply extending the word length by one
bit. Second, the output of the filter must be scaled such that the output uses the
available signal range effectively. For the unsealed filter we have

where the factor 2 comes from the scaling at the input of the filter. The new, scaled
coefficients are obtained by dividing by S.

We get