464 Chapter 15
is thus a rather poor way of handling the restricted output word width. A slightly better
approach is to round up the output by adding a fraction to each output number just prior to
truncation. If we added 00000110, then shifted four places and truncated the output would
become 01010 ( 1010), which, although not absolutely accurate, is actually closer to the
true answer of 9.75. This approach moves the statistical value of the error from 0 to 1
toward / 0.5 of the value of the LSB, but the error signal that this represents is still
very highly correlated to the required signal. This close relationship between noise and
signal produces an audibly distinct noise that is unpleasant to listen to.
An advantage is gained when the fraction that is added prior to truncation is not fi xed but
random. The audibility of the result is dependent on the way in which the random number
is derived. At fi rst sight it does seem daft to add a random signal (which is obviously a
form of noise) to a signal that we wish to retain as clean as possible. Thus the probability
and spectral density characteristics of the added noise are important. A recommended
approach commonly used is to add a random signal that has a triangular probability
density function (TPDF) ( Figure 15.14 ). Where there is suffi cient reserve of processing
power it is possible to fi lter the noise before adding it in. This spectral shaping is used
to modify the spectrum of the resulting noise (which you must recall is an error signal)
such that it is biased to those parts of the audio spectrum where it is least audible. The
mathematics of this process are beyond this text.
A special problem exists where gain controls are emulated by multiplication. A digital
audio mixing desk will usually have its signal levels controlled by digitizing the position
of a physical analogue fader (certainly not the only way by which to do this, incidentally).
Movement of the fader results in a stepwise change of the multiplier value used.
(a)x t
Figure 15.14(a) : The amplitude distribution characteristics of noise can be described in terms
of the amplitude probability distribution characteristic. A square wave of level 0 or 5 V can
be described as having a rectangular probability distribution function (RPDF). In the case of
the 5-bit example, which we are using, the RPDF wave form can be considered to have a value
of0.12 or 0.12 [(meaning 0.5 or0.5), equal chances of being positive or negative].