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252 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
where each bit b i represents either a0ora1.This representation will
help us to understand the advantages and disadvantages of each signed
format and to develop simple MATLAB functions. We will then extend
these concepts to fractional real numbers for both fixed-point and floatingpoint
arithmetic.
6.6.1 FIXED-POINT SIGNED INTEGER ARITHMETIC
In this arithmetic, positive numbers are coded using their binary representation.
For example, using 3 bits, we can represent numbers from 0 to
7as
0 1 2 3 4 5 6 7
-+----+----+----+----+----+----+----+-
000 001 010 011 100 101 110 111
Thus, with 8 bits the numbers represented can be 0 to 255, with 10 bits
we can represent the numbers from 0 to 1023, and with 16 bits the range
covered is 0 to 65535. For negative numbers, the following three formats
are used: sign-magnitude, one’s-complement, and two’s-complement.
Sign-magnitude format In this format, positive numbers are represented
using bits as before. However, the leftmost bit (also known as the
most-significant bit or MSB) is used as the sign bit (0 is +, and 1 is −),
and the remaining bits hold the absolute magnitude of the number as
shown here:
Sign Bit
-+ Absolute Magnitude
+---+----------------------+
| | |
+---+----------------------+
This system has thus two different codes for 0, one for the positive 0, the
other one for the negative 0. For example, using 3 bits, we can represent
numbers from −3 to3as
-3 -2 -1 -0 0 1 2 3
-+----+----+----+----+----+----+----+-
111 110 101 100 000 001 010 011
Thus, 8 bits cover the interval [−127, +127], while 16 bits cover [−32, 767,
+32, 767]. If we use B bits in the sign-magnitude format, then we can
represent integers from −(2 B−1 − 1) to +(2 B−1 − 1) only.
This format has two drawbacks. First, there are two representations
for 0. Second, the arithmetic using sign-magnitude format requires one
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