5.3.1 Any discussion of sampling rate and signal
aliasing, must include a discussion of the frequency content of the
signal being measured. In the case of free-fall lifeboats, the signal
being measured is the acceleration force time-history. The acceleration
force time-history, which is the variation of the acceleration force
with time, can be decomposed into a combination of sine and cosine
curves of varying amplitudes and frequencies. This concept is represented
mathematically as:
The term h(t) is the amplitude of a resultant time-history
at time t. The time-history is T seconds in duration. The quantities
aj and bj are the amplitudes of the cosine and
sine curves, respectively, associated with frequency f
j. n different frequencies, and associated amplitudes, are included
in the analysis.
5.3.2 To obtain a more intuitive understanding
of the significance of this equation, consider the curves presented
in Figure 5.3. Shown on this figure are two sine curves and one cosine
curve of different amplitudes and frequencies. The 2 hz and 20 hz
signals are sine curves with amplitudes of 1.5 and 0.5, respectively.
The 6 hz signal is a cosine curve with an amplitude of unity. The
amplitude of the combined signal at any particular time is equal to
the sum of the amplitudes of the three other curves at that same time.
For this particular example, then, the amplitude of the combined signal
at any time is:
When considering Equation 5.2, recall that the frequency
in radians is equal to 2π times the frequency in Hertz.
Figure 5.3 A Signal that is a Combination of Three Sinusoids
5.3.3 By using principles from calculus of complex
variables, Equation 5.1 can be reduced to a form in which frequency
and its associated amplitude are more readily apparent, namely:
where
The term Ai is the amplitude of the resultant
sinusoid and θi is the phase angle. With the equation
presented in this form, there is a single amplitude associated with
each frequency. For the example presented in Figure 5.3, a plot of
the frequency content versus amplitude is shown in Figure 5.4. The
frequency content of the combined curve is 2, 6, and 20 hz .
Figure 5.4 Frequency Content of Example Problem
5.3.4 Selection of a data sampling rate requires
knowledge of the highest frequency that is of significance in the
analysis to be performed as well as the magnitude of other frequencies
with significant amplitudes present in the system being measured.
Let us first deal with the highest frequency that is important in
the analysis. If the sampling rate is not rapid enough an aliased
signal (a false signal) such as that shown in Figure 5.5 will be returned.
In Figure 5.5 the actual signal is a sine curve with an amplitude
of 1.5 and a frequency of 15 hz. This sine curve was "sampled" every
60 milliseconds; these data points are indicated by the solid boxes.
By sampling at this slow rate, an apparent signal with a frequency
of 1 ⅔ hz and an amplitude of 1.5 was obtained. The apparent
signal is significantly different than the actual signal and as such
is probably of very little value. In this particular case, the actual
signal increases in an opposite direction from the apparent signal.
Starting at time t=0, the actual signal initially increases positively
whereas the apparent signal initially increases negatively.
Figure 5.5 Apparent Signal from a Signal that Was Sampled at too Slow a
Rate
5.3.5 Clearly, the sampling rate must be rapid
enough to properly describe a signal oscillating at the highest important
frequency. Although the general shape of a sinusoid can be described
with as few as two data points, more data points provide a more reliable
description of the shape. As shown in Figure 5.6, four data points
provide a reasonable description of the shape of a sine (or cosine)
curve. More data points will describe the curve better but five points
generally provide an adequate description. A good rule of thumb"
often used in experimental measurement is that a signal should be
sampled at a rate which will enable a sinusoid with a frequency five
times greater than that of importance to be adequately described.
As such, the minimum sampling rate is generally 20 times the highest
important frequency. If, for example, 20 hz is the highest frequency
to be considered, the data should be sampled 400 times per second
(4 samples per cycle times 20 cycles per second times 5).
Figure 5.6 Minimum Number of Data Points Required to Describe a Sinusoid
5.3.6 One problem does however, arise in experimental
measurement of mechanical or structural systems. Very often high frequency
vibration is present. In free-fall lifeboat systems, such vibration
occurrs when the lifeboat slides along the launch ramp and again when
it impacts the water. If the amplitude of the vibration is significant,
the sampled signal can be an alias of the true signal (small amplitude,
high frequency vibration is not a concern). An alias signal can be
observed in Figure 5.7. In this example, the important frequency is
2 hz with an amplitude of 2.0. This is the "true' signal" shown on
the figure. There was 15 hz unoise" with an amplitude of 1.5 that
was superimposed over the true signal. The resulting signal is the
actual signal shown on Figure 5.1; the actual signal is that which
will be measured even though it contains the unwanted high frequency
data. If the actual signal is sampled enough (about 20 times the highest
frequency) the unwanted frequencies can be later removed through filtering.
If, however, the signal is not sampled rapidly enough to properly
describe these high frequencies, the data will be aliased, the high
frequency data is falsely translated into the low frequency data.
The apparent curve in Figure 5.7 was obtained by sampling the data
at 16 ⅔ hz; such a rate is adequate for the true signal but
is not adequate for the high frequency noise. As such, the apparent
signal resembles the true signal but the amplitude is different; the
apparent signal is erroneous and may lead to improper conclusions
about the behavior of the system. A signal should therefore be sampled
at a rate quick enough to describe large amplitude, high frequency
vibration that may be present. After a signal has been sampled, there
is little that can be done to remove the aliased power (Press, et.
al., 1988). If this results in an excessively high sampling rate,
anti-aliasing filters can be used to limit high frequencies in the
signal before it is sampled.
Figure 5.7 An Alias of a True Signal
5.3.7 Experience with free-fall lifeboats has
indicated that sampling rates in the order of 600800 hz are
adequate to provide a reliable acceleration force time-histories that
has negligible aliasing. Such rates have been used on both GRP and
aluminum boats with free-fall heights as high as 30 meters. Care should
be taken, however, to properly mount the accelerometers on rigid parts
of the boat. If the accelerometers were placed in the middle of a
large flat panel, for instance, this sampling is probably not rapid
enough.