Page 213 - DSP Integrated Circuits

198 Chapter 5 Finite Word Length Effects

Figure 5.11 Quantization in a WDF of allpass section with all poles on the imaginary axis

5.5 SCALING OF SIGNAL LEVELS

As just discussed, parasitic oscillations can be started in fixed-point arithmetic by
an overflow of the signal range. Unfortunately, for most filter structures, the para-
sitic oscillation persists even when the cause of the overflow vanishes. Special pre-
cautions must therefore be taken to assure that the filter does not sustain a
permanent oscillation. Measures must also be taken to prevent overflow from
occurring too often, since overflows cause large distortion. Typically, an overflow
should not occur more frequently than once every 10*> samples in speech applica-
tions. Of course, the duration and perceived effect of the overflow must also be
taken into account. However, it is difficult to give precise guidelines for overflow
probability since one overflow may increase the probability of repeated overflows.
The probability of overflow can be reduced by reducing signal levels inside the
filter. This can be accomplished by inserting so-called scaling multipliers that
affect signal levels inside the filter only. The scaling multiplier must not affect the
transfer function. On the other hand, the signal levels should not be too low,
because the SNR (signal-to-noise ratio) would then become poor, since the noise
level is fixed for fixed-point arithmetic.
Scaling is not required in floating-point arithmetic since the exponent is
adjusted so that the mantissa always represents the signal value with full precision.
An important advantage of using two's-complement representation for nega-
tive numbers is that temporary overflows in repeated additions can be accepted if
the final sum is within the proper signal range. However, the incident signal to a
multiplier with a noninteger coefficient must not overflow, since that would cause
large errors. For example, the only critical overflow node in the signal-flow graph
shown in Figure 5.12 is the input to the multiplier with the noninteger (1.375)
coefficient. The multiplication by 2 can be regarded as an addition, and delay ele-
ments do not effect the signal values. In other types of number systems, the scal-
ing restrictions are more severe.