UploadFile_6417

Representation of Numbers 251
new filter Ĥ(z) and its output ŷ(n) are as close as possible to the original
filter H(z) and the original output y(n).
Since the quantization operation is a nonlinear operation, the overall
analysis that takes into account all three effects described above is very
difficult and tedious. Therefore, we will study each of these effects separately
as though it were the only one acting at the time. This makes the
analysis easier and the results more interpretable.
We begin by discussing the number representation in a computer—
more accurately, a central processing unit (CPU). This leads to the process
of number quantization and the resulting error characterization. We
then analyze the effects of filter coefficient quantization on digital filter
frequency responses. The effects of multiplication and addition quantization
(collectively known as arithmetic round-off errors) on filter output
are discussed in Chapter 10.
6.6 REPRESENTATION OF NUMBERS
In computers, numbers (real-valued or complex-valued, integers or fractions)
are represented using binary digits (bits), which take the value of
either a 0 or a 1. The finite word-length arithmetic needed for processing
these numbers is implemented using two different approaches, depending
on the ease of implementation and the accuracy as well as dynamic range
needed in processing. The fixed-point arithmetic is easy to implement but
has only a fixed dynamic range and accuracy (i.e., very large numbers or
very small numbers). The floating-point arithmetic, on the other hand, has
a wide dynamic range and a variable accuracy (relative to the magnitude
of a number) but is more complicated to implement and analyze.
Since a computer can operate only on a binary variable (e.g., a 1 or
a 0), positive numbers can straightforwardly be represented using binary
numbers. The problem arises as to how to represent the negative numbers.
There are three different formats used in each of these arithmetics:
sign-magnitude format, one’s-complement format, and two’s-complement
format. In discussing and analyzing these representations, we will mostly
consider a binary number system containing bits. However, this discussion
and analysis is also valid for any radix numbering system—for example,
the hexadecimal, octal, or decimal system.
In the following discussion, we will first begin with fixed-point signed
integer arithmetic. A B-bit binary representation of an integer x is given
by 1
x ≡ b B−1 b B−2 ... b 0 = b B−1 × 2 B−1 + b B−2 × 2 B−2 + ···+ b 0 × 2 0 (6.28)
1 Here the letter b is used to represent a binary bit. It is also used for filter coefficients
{b k }. Its use in the text should be clear from the context.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.