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does not have a code value at muting/blanking level and as a result the code value is not proportional to the signal
voltage.
In studying the transfer function it is better to avoid complicating matters with the aperture effect of the DAC. For
this reason it is assumed here that output samples are of negligible duration. Then impulses from the DAC can be
compared with the original analog waveform and the difference will be impulses representing the quantizing error
waveform. As can be seen in Figure 2.27 , the quantizing error waveform can be1 2 Q to + 1 2 Q with equal probability.
Note, however, that white noise in analog circuits generally has Gaussian amplitude distribution, shown in (d).
thought of as an unwanted signal which the quantizing process adds to the perfect original. As the transfer function
is non-linear, ideal quantizing can cause distortion. As a result practical digital audio devices use non-ideal
quantizers to achieve linearity. The quantizing error of an ideal quantizer is a complex function, and it has been
researched in great depth. [ 12 ]
Figure 2.27: In (a) an arbitrary signal is represented to finite accuracy by PAM needles whose peaks are at the
centre of the quantizing intervals. The errors caused can be thought of as an unwanted signal (b) added to the
original. In (c) the amplitude of a quantizing error needle will be from -
As the magnitude of the quantizing error is limited, its effect can be minimized by making the signal larger. This will
require more quantizing intervals and more bits to express them. The number of quantizing intervals multiplied by
their size gives the quantizing range of the convertor. A signal outside the range will be clipped. Clearly if clipping is
avoided, the larger the signal, the less will be the effect of the quantizing error.
Consider first the case where the input signal exercises the whole quantizing range and has a complex waveform.
In audio this might be orchestral music; in video a bright, detailed contrasty scene. In these cases successive
samples will have widely varying numerical values and the quantizing error on a given sample will be independent
of that on others. In this case the size of the quantizing error will be distributed with equal probability between the
limits.
Figure 2.27 (c) shows the resultant uniform probability density. In this case the unwanted signal added by
quantizing is an additive broadband noise uncorrelated with the signal, and it is appropriate in this case to call it
quantizing noise. This is not quite the same as thermal noise which has a Gaussian probability shown in Figure
2.27 (d). The subjective difference is slight.
Treatments which then assume that quantizing error is always noise give results which are at variance with reality.
Such approaches only work if the probability density of the quantizing error is uniform. Unfortunately at low levels,
and particularly with pure or simple waveforms, this is simply not true. At low levels, quantizing error ceases to be
random, and becomes a function of the input waveform and the quantizing structure. Once an unwanted signal
becomes a deterministic function of the wanted signal, it has to be classed as a distortion rather than a noise. We
predicted a distortion because of the non-linearity or staircase nature of the transfer function. With a large signal,
there are so many steps involved that we must stand well back, and a staircase with many steps appears to be a
slope. With a small signal there are few steps and they can no longer be ignored.
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