The Impact of Finite Wordlength Implementation - Digital Signal Processing: An Introduction with MATLAB and Applications - page 188

compact representations. This compactness is achieved at the expense of an
induced error (quantization noise), which is expressed as

e q ð n Þ¼ Q ½ x ð n Þ x ð n Þ; ð 4 : 5 Þ

where Q[x(n)] represents the quantized value of x(n).
The quantization error depends on the representation format (in particular,
fixed-point or floating-point) and on the method of quantization being used, spe-
cifically, whether truncation or rounding is used [ 1 ].

4.3.1 Quantization of Fixed-Point Numbers

A common practice in DSP is to represent data in a digital machine either as a
fixed-point fraction or as a floating-point binary number with the mantissa as
fraction. In each of these formats, three different forms can be used to represent a
negative number.
As indicated in Fig. 4.1 , the fixed-point format assumes that a number is rep-
resented with b ? 1 bits, with the most significant bit (MSB) conveying the sign of
the number and the remainder of the bits constituting a binary fraction. That is, the
binary point is just to the right to the sign bit. With this format the LSB is
equivalent to 2 -b and represents the quantization step-size.
Several different variants of fixed-point representation are commonly
used. These variants include 1's complement, 2's complement, sign-magnitude,
and offset binary (see, for example, Mitra [ 1 ]). When one converts from one
format to another, it is frequently necessary to quantize. This quantization is
usually performed with a simple rounding or truncation, as discussed more fully
below.

4.3.1.1 The Rounding Method

In rounding, the number is quantized to the nearest quantization level. Assum-
ing b bits are used to represent the number's magnitude, then the quantization step
is 2 -b . After rounding, therefore, the maximum rounding error is 2 -b /2. Conse-
quently, the range of the rounding error e qr is

1

2 ð 2 b 2 k Þ \e qr 1 2 ð 2 b 2 k Þ:

ð 4 : 6 Þ

Figure 4.3 a shows the transfer characteristics of the rounding quantization
method, where D ¼ 2 b is the quantization step. Notice that the rounding error is
independent of the format being used to represent the negative fractions. This is so
because the rounding operation is decided by the magnitude of the number.