Hardware and Software for Digital Signal Processors - Digital Signal Processing: Fundamentals and Applications - page 413

EXAMPLE 9.1
Given
a.

2 ð1Þ

b. 2 ð3Þ

operate each expression using 2's complement.
Solution:
a.

010
001
010
000
þ 000
00010
The 2's complement of 00010 ¼ 11110. Removing two extended sign bits 1 gives 110. The answer is 110 (2),
which is within the system.
b. 010
011
010
010
þ 100
00110
The 2's complement of 00110 ¼ 11010. Removing two extended sign bits leaves 010. Since the binary number
010 is 2, which is not (6) as what we expect, overflow occurs; that is, the result of the multiplication (6) is out of
our dynamic range (4 to 3).

Let us design a system treating all the decimal values as fractional numbers, so that we obtain the
fractional binary 2's complement system shown in Table 9.2 .
To become familiar with the fractional binary 2's complement system, let us convert a positive
fraction number 3

1
4 in decimals to their 2's complements. Since

4 and a negative fraction number

3
4 ¼ 0 2 0

þ 1 2 1 þ 1 2 2

Table 9.2 A 3-Bit 2's Complement System Using Fractional Representation

Decimal Number Decimal Fraction Two's Complement

3 3/4 0.11

2 2/4 0.10

1 1/4 0.01

0 0 0.00

1 1/4 1.11

2 2/4 1.10

3 3/4 1.01

4 4/4 ¼ 1 1.00