Audio Engineering

Representation of Audio Signals 463

The second technique is called dither , which will be covered fully later. Consider, for
the present, that the output word width is simply cut (in the example given earlier such
as to produce a 5-bit answer). The need to handle the large numbers that result from
multiplication without overfl ow means that when small values are multiplied they are
likely to lie outside the range of values that can be expressed by the chosen word width.
In the example given earlier, if we wish to accommodate the most signifi cant digits of
the second multiplication (156) as possible in a 5-bit word, then we shall need to lose
the information contained in the lower four binary places. This can be accomplished
by shifting the word four places (thus effectively dividing the result by 16) to the right
and losing the less signifi cant bits. In this example the result becomes 01001, which is
equivalent to decimal 9. This is clearly only approximately equal to 156/16.

When this crude process is carried out on a sequence of numbers representing an audio
analogue signal, the error results in an unacceptable increase in the signal-to-noise ratio.
This loss of accuracy becomes extreme when we apply the same adjustment to the lesser
product of 3 5  15 since, after shifting four binary places, the result is zero. Truncation

Carry

Carry in

Carry

Out Out

A

AB B

Data

A

Bits

B Carry in Out Carry out

0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1

0 1 1 0 1 0 0 1

0 0 0 1 0 1 1 1

Figure 15.13 : A 2-bit full adder needs to be able to handle a carry bit from an adder

handling lower order bits and similarly provide a carry bit. A large adder based on this circuit

would suffer from the ripple through of the carry bit as the fi nal sum would not be stable

until this had settled. Faster adding circuitry uses look-ahead carry circuitry.