Digital storage oscilloscopes 133
purpose display and can be generally recommended. Where the
waveform under investigation is known to be smooth and
generally of a sinusoidal shape, sine interpolation provides a good
representation with as few as three or even only 2.5 samples per
cycle. However, it should not in general be used for pulse
waveforms, as here it can introduce ringing on the display which
is not present on the actual waveform if the pulse risetime is less
than about two or three sample periods, see Figure 7.11.
Having mentioned perceptual aliasing above, perhaps a word
should be said about true aliasing, although this is really more an
unfortunate result of inappropriate control settings on the
acquisition- rather than on the display - process. The topic has
already been mentioned in Chapter 6, see Figure 6.9 and
associated text. A theorem due to Nyquist states that to define a
sine wave, a sampling system must take more than two samples
per cycle. It is often stated that at least two samples per cycle are
necessary, but this is not quite correct. Exactly two samples per
cycle (usually known as the 'Nyquist rate') suffice if you happen
to know that they coincide with the peaks of the waveform, but
not otherwise, since then although you will know the frequency
of the sine wave, you have no knowledge of its amplitude. And
if the samples happen to occur at the zero crossings of the
waveform, you would not even know it was there. However,
with
more than
two samples per cycle - in principle 2.1 samples
would be fine - the position of the samples relative to the sine
wave will gradually drift through all possible phases, so that the
peak amplitude will be accurately defined.
As we have seen in Figure 7.9, a good sine interpolator can
manage very well with as few as 2.5 samples per cycle, always
assuming of course that the waveform being acquired is indeed a
sine wave. For non-sinusoidal waveforms, a sine interpolator is
inappropriate (except in the case of certain instruments which can
suitably preprocess the waveform before passing it to the sine
interpolator). For non-sinusoidal waves, accurate definition of the
waveform requires that the sampling rate should exceed twice the
frequency of the highest harmonic of significant amplitude. If
frequency components at more than half the sampling rate are
present, they will appear as 'aliased' frequencies lower than half