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208 Chapter 5 Finite Word Length Effects

high dynamic range provided by floating-point arithmetic is not really needed in
good filter algorithms, since filter structures with low coefficient sensitivity also uti-
lize the available number range efficiently. We will therefore limit the discussion to
fixed-point arithmetic.
Using fixed-point arithmetic with suitably adjusted signal levels, additions
and multiplications generally do not cause overflow errors. However, the products
must be quantized using rounding or truncation. Errors, which are often called
rounding errors even if truncation is used, appear as round-off noise at the output
of the filter. The character of this noise depends on many factors—for example,
type of nonlinearity, filter structure, type of arithmetic, representation of negative
numbers, and properties of the input signal.
A simple linear model of the quantization
operation in fixed-point arithmetic can be used if
the signal varies over several quantization levels,
from sample to sample, in an irregular way. A sim-
ple model is shown in Figure 5.16. The quantiza-
tion of a product

is modeled by an additive error Figure 5.16 Linear noise
model for
quantization
where e(n) is a stochastic process.
Normally, e(ri) can be assumed to be white noise and independent of the sig-
nal. The density function for the errors is often approximated by a rectangular
function. However, the density function is a discrete function if both the signal
value and the coefficient value are binary. The difference is only significant if only
a few bits are discarded by the quantization. The average value and variance for
the noise source are

cj-2 = Q^/l.2,, Q is the data quantization step, and Q c is the coefficient quantization
step. For long coefficient word lengths—the average value is close to zero for round-
ing and Q/2 for truncation. Correction of the average value and variance is only nec-
essary for short coefficient word lengths, for example, for the scaling coefficients.