Representation of Audio Signals 441common use. We recognize that the old coinage system in the United Kingdom used the
base of 12 for pennies, as, indeed, the old way of marking distance still uses the unit of
12 inches to a foot.
The binary counting scale has many useful properties. Counting in the base of 2 means
that there can only be two unique digits, 1 and 0. Thus each column must represent a
power of 2 (2 2 1 , 4 2 2 , 8 2 3 , 16 2 4 , and so on) and, by convention, we use a 1
to mark the presence of a power of 2 in a given column. We can represent any number
by adding up an appropriate collection of powers of 2 and, if you try it, remember that
20 is equal to 1. We refer to each symbol as a bit (actually a contraction of the words
binary digit). The bit that appears in the units column is referred to as the least signifi cant
bit ( LSB), and the bit position that carries the most weight is referred to as the most
signifi cant bit (MSB).
Binary arithmetic is relatively easy to perform since the result of any arithmetic operation
on a single bit can only be either 1 or 0.
We have two small puzzles at this stage. The fi rst concerns how we represent numbers
that are smaller than unity and the second is how negative numbers are represented. In
the everyday decimal (base of 10) system we have adopted the convention that numbers
which appear to the right of the decimal point indicate successively smaller values. This
is in exactly the opposite way in which numbers appearing to the left of the decimal
point indicated the presence of increasing powers of 10. Thus successive columns
represent 0.1 1/10 1 0 ^1 , 0.01 1/100 1 0 ^2 , 0.001 1/1000 1 0 ^3 , and so on.
We follow the same idea for binary numbers and thus the successive columns represent
0.5 1/2 2 ^1 , 0.25 1/4 2 ^2 , 0.125 1/8 2 ^3 , and so on.
One of the most useful properties of binary numbers is the ease with which arithmetic
operations can be carried out by simple binary logic. For this to be viable there has to be
a way of including some sign in the number itself since we have only the two symbols 0
and 1. Here are two ways it can be done. We can add a 1 at the beginning of the number
to indicate that it was negative or we can use a more fl exible technique known as two’s
complement. Here the positive numbers appear as we would expect but the negative
numbering is formed by subtracting the value of the intended negative number from
the largest possible positive number incremented by 1. Table 15.1 shows both of these
approaches. The use of a sign bit is only possible because we will arrange that we will use
the same numbering and marking convention. We will thus know the size of the largest