Page 222 - DSP Integrated Circuits

5.6 Round-Off Noise 207

In the second step, the next critical node, as seen from the input, is measured
using the new set of coefficients in the first section. We get

an
a
The coefficients ao2> !2> d «22 are then divided by || ^5 ||oo so that the output
of the second section and the input to the third section become properly scaled.
The new set of coefficients are

By successively measuring the norms to the critical nodes using the scaled
coefficients in the preceding sections we get

Note that it is necessary to measure the Loo-norms and scale the sections one
at a time, from the input to the output of the filter, since the Loo-norms depend on
the scaling of the previous sections. This is not necessary if we use L2-norms for
scaling.

Comparing the coefficients obtained in Examples 5.6 and 5.7 we observe that
a narrow-band input signal presents a more difficult case than a wide-band signal
in the sense that the former may produce larger signal levels inside the filter. This
give rise to the smaller numerator coefficients.
The two scaling methods just discussed are the ones used in practice. How-
ever, they lead only to a reasonable scaling of the filter. In practice, simulations
with actual input signals should be performed to optimize the dynamic range.

5.6 ROUND-OFF NOISE

In most digital filter implementations only a few types of arithmetic operations—
addition, subtraction, and multiplication by constant coefficients—are used. The
advent of VLSI has fostered the use of more complex basic operations such as vec-
tor-multiplication.
In practice, only fixed-point arithmetic is used in application-specific ICs since
the hardware is much simpler and faster compared with floating-point arithmetic.
Also, the quantization procedure needed to suppress parasitic oscillations in wave
digital filters is more complicated for floating-point arithmetic. Moreover, the analy-
sis of the influence of round-off errors is very complicated [28,32,351, because quan-
tization of products as well as addition and subtraction causes errors that depend on
the signal values. Errors in fixed-point arithmetic are with few exceptions indepen-
dent of the signal values and can therefore be analyzed independently. Lastly, the