Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 42

and 0 is not acceptable in the mathematical sense. Sign magnitude form is
abbreviated to SM.

2.7.1.2 Ones Complement Form

In ones complement form ð 1sC Þ the same numbers f 0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 g are

designated as fþ 0 ; 0 g . This method is relatively easy

to implement but still has the problem of a non-unique zero.

; þ 1 ; þ 2 ; þ 3 ; 3 ; 2 ; 1

2.7.1.3 Twos Complement Form

Twos complement form (2sC) is the most popular numbering and is practised widely
in all machines. In this scheme the positive zero and negative zero are removed and
the numbers f 0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 g are mapped as f 0 ; 1 ; 2 ; 3 ; 4 ; 3 ; 2 ; 1 g:

2.7.1.4 The Three Forms in a Nutshell

Table 2.1 illustrates the three representations in a nutshell. Sign magnitude is very
popular in A/D converters and it is almost a standard practice in all arithmetics to
use a twos complement number system.

Table 2.1 The three representations in a nutshell

b x k c SM 1sC 2sC

000 þ 0 þ 0 0

001 þ 1 þ 1 1

010 þ 2 þ 2 2

011 þ 3 þ 3 3

100 0 3 4

101 1 2 3

110 2 1 2

111 3 0 1

2.7.2 Fixed-Point Numbers
Similar to the imaginary decimal point, we have a binary point that can be
positioned anywhere in the binary number b n (2.36) and can be written as

b n ¼ x n : x n 1 ... x i ... x 3 x 2 x 1 :

2 n 1 ; all other things
remain the same. It could be SM , 1sC or 2sC . The above form is called the
normalised form. It essentially imposes a restriction that it is permitted to have only
one bit to the left of the imaginary binary point

Now the value of b n takes the form of a rational number b n =