Audio Engineering

Representation of Audio Signals 461

processing, one approach being to set the result of an overfl owing sum to the appropriate
largest positive or negative number. The process of adding two sequences of numbers
that represent two audio waveforms is identical to that of mixing the two waveforms in
the analogue domain. Thus when the addition process results in overfl ow the effect is
identical to the resulting mixed analogue waveform being clipped.

We see here the effect of word length on the resolution of the signal and, in general, when a
binary word containing n bits is added to a larger binary word comprising m bits the resulting
word length will require m  1 bits in order to be represented without the effects of overfl ow. We
can recognize the equivalent of this in the analogue domain where we know that the addition of
a signal with a peak-peak amplitude of 3 V to one of 7 V must result in a signal whose peak-peak
value is 10 V. Don’t be confused about the rms value of the resulting signal, which will be

37

282

269

(^22) 

.

. Vrms,

assuming uncorrelated sinusoidal signals.

A binary adding circuit is readily constructed from the simple gates referred to
earlier, and Figure 15.13 shows a 2-bit full adder. More logic is needed to be able to
accommodate wider binary words and to handle the overfl ow (and underfl ow) exceptions.

If addition is the equivalent of analogue mixing, then multiplication will be the equivalent
of amplitude or gain change. Binary multiplication is simplifi ed by only having 1 and 0
available since 1 1  1 and 1 0  0.

Since each bit position represents a power of 2, then shifting the pattern of bits one place
to the left (and fi lling in the vacant space with a 0) is identical to multiplication by 2. The
opposite is, of course, true of division. The process can be appreciated by an example:

Decimal Binary

3 00011

    5 00101

 15 00011

(Continued)