Page 50 - Prosig Catalogue 2005

SOFTWARE PRODUCTS
ALIASING, ORDERS & WAGON WHEELS

spectrum. When converting to synchronous sampling again it is essential to
Applying the Shannon Theorem directly we have quite simply the result determine the tacho edge accurately. The next step is to interpolate the
that with synchronous sampling where we have a sample rate of N points amplitude. This in theory can be done precisely by using a {(sin x) /x}
basis but it requires signals and integrations of infinite length. In practice
Training & Support Incidentally, if we Fourier analyze over an exact number of revolutions, Lagrange based method.
, is given by
per revolution then our highest order to avoid aliasing, O
max
a reasonable estimate can be achieved with a relatively small number of
points by using the most appropriate interpolation technique, such as a
Synchronous sampling avoids these complications.
say P revs, then our order spacing is 1/P orders.
Appendix - Aliasing Demonstrations
Spatial Sampling
One of the classical demonstrations of aliasing is the so called “wagon
wheel effect” where in films as the stage coach goes faster the wheels
Just for completeness, the same applies if we do spatial sampling by
measuring at equally spaced distance increments. The corresponding
a visual demonstration of the wagon wheel effect then the link below is
domain is wave number. Thus if we sample a road surface at L points per
excellent.
metre then our highest wave number to avoid aliasing, w
max , is given by appear to go slower ad then to go backwards. If you would like to see
Initially the spokes rotate anticlockwise then slow and begin to rotate
clockwise. What is particularly nice in this presentation is that there are
Condition Monitoring to sample at 100K samples/second. So if we are dealing with a shaft allow manual control of the wheel speed.
other ‘markers’ at a smaller angular spacing that remain in the “non-
Why use synchronous sampling?
aliased” region. Thus one can see parts of the wheel rotating and other
With high speed data acquisition systems it is quite usual to be able
parts stationary, most impressive and convincing. There are sliders that
speed of say 6000 rpm, which is 100 revs/sec, then the highest order is
See “Wagon-wheel effect” from Michael’s “Optical Illusions & Visual
[100000/ (2x100)] = 500. With synchronous sampling we would need an
Phenomena” (http://www.michaelbach.de/ot/mot_wagonWheel/index.
encoder with 1000 points/rev to achieve the same level. Often we are not
html)
interested in such high orders and there is rarely any high order content.
Another common form of illustrating aliasing is to show a high frequency
That means we can use lower time based sample rates or lower points per
revolution. So what is the advantage?
see the blue curve but rather we only see the data at the * points, which
The essential point is that when we transform a synchronously sampled
have been aliased to a much lower frequency.
signal we are taken directly into the order domain. If we have sampled sine wave sampled at too low a rate. If we sample too slowly we do not
data covering exactly B revolutions then our order spacing will be (1/B)
orders. Further the measurements at each order will be exact as they are
precisely centred at each order.
If we have a time based signal there are two approaches. One way is to
Software sampled data by software. Both of these methods require the tachometer
use waterfall analysis and the other is to convert the data to synchronously
signal to be captured at the same time. The accuracy depends upon
precisely locating the tacho edges. The problem is illustrated below.
Suppose the blue coloured signal is the actual tacho signal and the green
one is the digitally sampled tacho. Now the * on the measured signal
(green) are the actual data points. The actual tacho goes between a low
of zero and a high of unity. So suppose we set the tacho crossing level at
0.6 and use the first data point that occurs at or above the threshold level
on a rising edge as determining the actual tacho crossing point. This will For those of a mathematical leaning then we may find the apparent digital
directly lead to time jitter, adding and subtracting from each period. This frequency from the following formula:
will appear as noise and possibly as false frequencies.
Clearly the software could improve the situation, for example in DATS the where f is the sample rate, f is the actual frequency, f is the ‘digital’
Hardware is at its smallest.
s
d
a
frequency, and K is the integer, starting from zero, such that
We may write this as
For example consider a sample rate of 500 samples per second and what
happens to various frequencies.
Actual Frequency f Hz Minimising value K Alias frequency f Hz
a
d
180 0 1 1 180
System Packages the time location of the crossing point. Note how the pattern repeats. 20
220
280
380
120
software uses an interpolation process to determine a better estimate of
480
1
80
580
1
With waterfalls the next step is to determine a speed curve and then to
1
180
680
carry out Fourier analysis at appropriate speed steps. The orders are then
220
2
780
extracted from those frequency points which are the closest to the actual
order being extracted. This is yet another approximation. An improved
120
2
880
amplitude estimate is to determine the rms value over a short interval
980
2
20
in the frequency domain. As one can see from the very description of
the stages there exists considerable room for errors in the final order
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