Chapter 4
Time Quantization and Sampling
Sampling may be one of the most confusing issues involved in data
acquisition.
All digital instruments sample, including data acquisition boards.
It's one of the things that make them digital. It also makes digitalinstrument
measurements harder to interpret and more prone to error than analog
measurements. Digital measurements are not inherently less accurate—I
dealt with that in Chapter 3—but digital measurements are more
complicated, so there are more ways of messing them up.
The most obvious specification having to do with digitalinstrument
sampling is the sampling rate, and the most talked about rule regarding
sampling rate is Nyquist's Theorem. Most people interpret Nyquist's
Theorem to mean that a digital instrument cannot measure a signal
whose frequency is equal to or greater than one half the sampling
rate. (One half the sampling rate is popularly known as the Nyquist
frequency.) Thus, if the sampling rate is 50 kS/s (kilosamples per
second), you run into problems when the signal frequency gets up
into the neighborhood of 25 kHz.
What Nyquist's Theorem actually states is that you cannot accurately
reconstruct from sampled data the waveform of a signal whose frequency
is equal to or greater than one half the sampling rate. People then,
in an effort to figure out what Nyquist's Theorem means to Joe Research
Scientist using a data acquisition system, assume that if you can't
reconstruct it, you can't measure it.
That ain't necessarily so.
Nyquist's Theorem addresses the problem of aliasing, not measurement
accuracy. To understand what Nyquist's theorem means for your particular
data acquisition situation, you have to think about what effect
aliasing has on the particular measurement you're making.
Imagine a signal made up of a 40 kHz fundamental with one overtone
(80 kHz) whose amplitude equals that of the fundamental. This is
not the sort of signal you run across every day, but it is a very
instructive one nonetheless.
To get a baseline, let's look at the signal through the eyes of
a really fast data acquisition system running at 4 MS/s. Figure
4.1a shows that we get a nice, clean, well characterized waveform.
Figure 4.1b shows the actual Fourier transform calculated the hard
way using an actual, honesttogosh Fourier integral evaluated numerically.
Click image to see full size
Figure 4.1: Highspeed sampling (4 MS/s in this case) collects
a data set that faithfully represents a waveform (a) and can be
Fourier transformed into a good representation of the signal's spectrum
(b).
The wavetrain shows a nice, clean waveshape. It is exactly what
you would expect to see for a wellsampled waveform displayed by,
say, a digital oscilloscope, and it is very similar to what you
would expect to see on an analog oscilloscope. Note that the data
record includes 1,000 samples and ten periods of the fundamental
waveform. These will both become important facts later on.
Spectral Analysis
The Fourier transform displays the signal's spectrum. In this case,
although the waveform is very nicely sampled, we don't see a very
clean spectrum. That is because we have only 1,000 points in the
waveform record, and although 1,000 points gives you a pretty good
looking waveform display, it really isn't very many to use to get
a Fourier transform.
Also, note that the fundamental fits rather neatly into the data
record—there are exactly 100 samples in each waveform period
and exactly ten periods in the record. I've even contrived to have
the waveform start at exactly zero phase. Without these nice coincidences
(which, by the way, would be very difficult to achieve in a real
dataacquisition situation), the spectrum would look even messier.
Generally, the longer the waveform record (i.e., the more periods
in the record) and the better the wavform fits into the sampling
window (i.e., the closer to an integer number of periods) the nicer
the spectrum will appear.
The above paragraph should tell you that long data buffers in DAQ
boards are not just there for convenience. Since the Windows environment
(Windows 3.x/95/NT) is notorious for getting around to accepting
data when it feels like it, rather than when the data is ready,
you have to be concerned about latency, which is the lag between
putting the data out from the DAQ board and when it gets stored
nice and safe in the computer's RAM. If you're gonna use that data
in a Fourier analysis, don't expect it to work if you use more than
one buffer's worth.
Plan on starting with an empty buffer on the DAQ card, filling
it in one go, then loading it into memory. Don't expect to do a
Fourier transform on 10,000 data points obtained by filling a 1,000point
buffer ten times. Latencyinduced phase errors will wipe you out!
If your DAQcard buffer carries 1,000 points, that's how long your
data set for the Fourier transform is going to be, and there's nothing
you can do about it. (Well there is, but time stamping every individual
sample with subnanosecond resolution is not usually an option!)
Back to the spectrum in Figure 4.1. For all its flaws, the essential
features show up pretty well: we see two spectral lines at the appropriate
frequencies (40 and 80 kHz) and they have approximately the right
amplitude (both roughly equal to one). The amplitudes are a little
bit low because the record's shortness spreads the spectral lines
out, spilling some of their power into adjacent frequency bins.
When we reduce the sampling rate to 200 kS/s (Figure 4.2), the
spectrum gets smeared out more because there are now only five samples
tracing each period of the fundamental. Smearing out the spectral
lines reduces the peak amplitude. In other words, the system's spectral
resolution drops because it can't discern the waveshape as clearly.
Again, lower resolution leads to a lower peakamplitude measurement
because some of the total spectral power gets shifted into adjacent
frequency bins.
Click image to see full size
Figure 4.2: Marginal sampling (200 kS/s) makes for a less faithful
representation of the spectrum, but does not lead to aliasing.
Sampling at 200 kS/s still gives a Nyquist frequency of 100 kHz,
which is greater than the highest frequency component in the signal.
Figure 4.3 shows what happens when we drop the sampling rate still
further—to 100 kS/s, which gives a Nyquist frequency of 50
kHz. The Nyquist frequency is now below the highest frequency in
the signal. Aliasing occurs, producing ghosts at 20 kHz and 60 kHz.
Note that the aliasing does not affect the apparent strength of
the real signal components. Furthermore, note that there's no way
to tell which spectral components are real and which are aliases.
Click image to see full size
Figure 4.3: Undersampling (100 kS/s) causes aliasing. Ghost frequencies
appear at 20 kHz and 60 kHz.
So, undersampling of a signal will mess up spectral measurements.
Oversampling with a limited record length can also mess up the spectrum,
as Figure 4.4 shows. To get Figure 4.4, I used a sampling rate fast
enough to put all 1,000 data points onto the same wave period. In
other words, the entire record is one waveperiod long. Needless
to say, that makes working out the frequencies of the components
pretty dicey for the Fourier transform. That vagueness translates
into poor resolution and badly smearedout lines. The peaks, however,
are still right on the money as far as frequency is concerned. And,
of course, aliasing is not a problem!
Click image to see full size
Figure 4.4: Extreme oversampling (40 MS/s) to reduce the waveform
record to one fundamental period long (a) produces a clean spectrum,
but with very poor resolution (b).
NonSpectral Analysis
Okay, undersampling ruins spectral measurements. But, frequency
measurements are not exactly the bread and butter of data acquisition.
Suppose, for example, what you really want are the maximum, minimum,
average (of absolutes—the average of signed values is zero
for nonDC signal components) and RMS values from the timedomain
waveform. Table 4.1 shows these values for several different sampling
rates. Clearly, Nyquist's theorem has nothing to do with measurements
of these values. The values measured with a sampling rate of 125
kS/s are identical to the 2 MS/s values, despite the fact that 125
kS/s is undersampling by Nyquist's criterion. The 125 kS/s values
are more accurate even than the 200 kS/s values, although Nyquist's
Theorem says the latter should be fine.
Table 4.1: Signallevel values detected using various sampling
rates
Sample Rate
2 MS/s
200 kS/s
125 kS/s
100 kS/s
75 kS/s
50 kS/s
Maximum
1.125
1.000
1.125
1.000
1.121
1.000
Minimum
1.990
1.760
1.990
1.760
1.973
1.760
Average
0.826
0.815
0.826
0.815
0.830
0.815
RMS
0.999
0.999
0.999
0.999
1.001
0.999
Something else is going on here.
The something else that's going on here is that, for nonspectralanalysis
measurements, Nyquist's theorem doesn't say a whole lot. For those
measurements, what you want is a lot of data points randomly scattered
throughout the waveform. It doesn't make a bit of difference how
often you take a data point, so long as they are randomized and
you have a lot of them.
Again, the ability to take a lot of data points (buffer size) is
at least as important as being able to take them rapidly (sampling
rate).
You can get good randomization simply by not having your sample
rate make a simple ratio with your fundamental frequency. 125 kS/s
is a good match with 40 kHz because the ratio (3.125) is nonintegral
to three significant figures. You should get even better mixing
with a sample rate of, say, 123,456 kHz. I tried it, and got a match
(to the third decimal place) with all the 2 MS/s numbers except
the minimum value. The nicely randomizing sample rate found at least
one data point at 2.000, which the (unrandomized) 2 MS/s data set
missed.
So, unless you're looking at the shape of a repetitive waveform
or doing spectral analysis, Nyquist's theorem is not a good guide!
Transient Waveforms
If spectral analysis is not exactly the bread and butter of data
acquisition, the same can truly be said of repetitive waveforms.
Although lots of people use data acquisition systems for capturing
repetitive waveforms, DAQ was invented for capturing transients,
and that is still where its greatest strength lies.
A transient waveform is not just a single occurrence of a repetitive
waveform. In a repetitive waveform the spectral bandwidth is moreorless
limited. That is, the amplitudes of the harmonics asymtotically
approach zero. Thus, you can usually identify a finite frequency
band and a definite cutoff frequency for a repetitive waveform and
say that everything beyond that is of no interest. You can then
(at least in principle) get a sampling rate high enough to keep
your Nyquist frequency above the cutoff frequency.
As Figure 5 shows, the spectrum for even a simple step transient
has infinite bandwidth. That is, harmonics may asymtotically approach
something, but it isn't necessarily zero. Of course, real transients
can't have infinite risetimes (or, in the case of this negative
step, fall times) because real electronics can't react instantly.
Real step functions have somewhat rounded steps (caused by poor
highfrequency response) or overshoots (caused by poor lowfrequency
response) or even ringing (caused by resonance effects).
Click image to see full size
Figure 5: Transients have essentially unlimited spectral bandwidth.
All of this doesn't change the fact that an ideal step function
has an infinite bandwidth and you ain't gonna capture it anywhere
near perfectly with a real data acquisition system because you can't
have an infinite sampling rate. The best you can do is to sample
as fast as you can, then chop the bandwidth off using an antialiasing
filter with a cutoff below the Nyquist frequency. If you're lucky,
your DAQ system's cutoff will be high enough to pass the important
components in your transient.
Then, of course, you have to be honest and report what you did:
what your cutoff frequency was, what the sampling rate was, etc.
Being honest and admitting all this presupposes you had your thinking
cap on and determined these things in the first place!
TimeDivision Multiplexing
Now that I've cleared up what Nyquist's Theorem means to data acquisition
mavens, it's time to point out that a fourchannel, 300 kS/s data
acquisition board does not necessarily give you an Nyquist frequency
of 150 kHz! You see, most data acquisition boards have several input
channels, but only one analogtodigital convertor (ADC). To funnel
those n dataacquisition channels through that one ADC, they use
timedivision multiplexing (TDM).
Suppose you have that fourchannel DAQ board with one ADC sampling
at 300 kS/s. You start the acquisition at time zero and, for the
first sampling interval, the ADC is connected to channel 1. For
the second sampling interval (which starts 3.333 microseconds later),
the board's signalrouting electronics connects the ADC input to
channel 2. The third sample (starting at 6.667 microseconds) comes
from channel 3 and the fourth from channel 4. For the fifth sample,
the electronics reconnects the ADC to channel 1, and so forth.
That's called scanning. While the ADC is zipping along at 300 kS/s,
scanning drops the actual sample rate for each of the input channels
(which is what really counts) to only 75 kS/s! To get a real sampling
rate of 300 kS/s, you have to run the board as a onechannel DAQ
board.
That is why most multichannel boards let you select certain channels
for use and leave others out. It's not just a case of simply not
hooking a signal into the channel n input. You have to tell the
board not to scan through the unused channels, otherwise your true
sampling rate suffers badly.
Until recently, nearly all DAQ boards multiplexed through a single
ADC. Since the vendor couldn't predict how many signals you'd actually
need for your application, they'd publish the ADC sampling rate
in their specifications. It would be up to the customer (in other
words you ) to figure out the sampling rate you'd actually get—by
dividing the published scanning rate by the number of channels you
plan to actually use.
Recently, in an effort to keep real sampling rates up with real
user needs (especially for boards with 8, 16 or more input channels),
manufacturers have begun installing additional ADCs. The reason
they can do it now, whereas they couldn't do it before, is improved
semiconductor integration. Cramming more electronics into each chip
means they can now cram more ADCs into one chip. That means more
ADCs can fit on a DAQ board along with all the associated electronics.
Having two ADCs doubles the real sampling rate per channel. Four
quadruples it, and so forth.
Of course, when a vendor tells you that they have a fourchannel,
300 kS/s board with two ADCs, you now have to ask whether they have
two 300 kS/s ADCs or two 150 kS/s ADCs. You also have to ask if
all the ADCs are available to all of the channels.
In other words, if you connect signals to channels 1 and 2, do
you have two channels operating at 150 kS/s (by fully utilizing
both ADCs), or do you still only have 75 kS/s because you've stupidly
(or by necessity, due to some constraint in the application) loaded
both channels into one ADC while the other one sits idle? You may
have the potential to get faster sampling, but to get it for real
you need to apply a little more brain power as well.
Chapter 1  Chapter
2  Chapter 3  Chapter 4
