Multiplexing and Sampling Theory
THE ECONOMY OF MULTIPLEXING
Sampled-Data Systems

An ideal data acquisition system uses a single ADC foreach measurement channel. In this way, all data arecaptured in parallel and events in each channel can becompared in real time. But using a multiplexer, Figure
3.01, that switches among the inputs of multiple channels and drives a single ADC can substantially reducethe cost of a system. This approach is used in so-calledsampled-data systems. The higher the sample rate, thecloser the system mimics the ideal data acquisition
system. But only a few specialized data acquisition
systems require sample rates of extraordinary speed.
Most applications can cope with the more modest
sample rates typically offered by mainstream dataacquisition systems.

Solid State Switches vs. Relays

A multiplexer is an array of solid-state switches orelectromechanical relays connected to several input
channels. Although both approaches are used in a wide
variety of applications, neither one is perfect; each typecomes with various advantages and disadvantages.
Electromechanical relays, for example, are relatively
slow, about 1,000 samples/sec or less for the fastest reed
relays, but they can handle large input voltages, andsome can isolate voltages of several kV. A relay’s size andMultiplexers
Analoginputs
Mux

PGA

Gain
control lines
Mux
control lines

ADC

Fig. 3.01. The multiplexer (MUX) is a fast switchthat sequentially scans numerous input-signal channels and directs them in a preprogrammed manner toa single ADC for digitizing. This approach saves thecost of using an ADC for each channel.

Figure 3.01

contact type determine its current carrying capacity. For
instance, laboratory instrument relays typically switch
up to 3A, while industrial applications use larger relays
to switch higher currents, often 5 to 10A.
Solid-state switches, on the other hand, are much fasterthan relays and can reach sampling rates of several MHz.
However, these devices can’t handle inputs higher than
25V, and they are not well suited for isolated applications. Moreover, solid-state devices are typically limited
to handling currents of only one mA or less.
Another characteristic that varies between mechanical relays and solid-state switches is called ON resistance. An ideal mechanical switch or relay contact
pair has zero ON resistance. But real devices suchas common reed-relay contacts are 0.010 W or less,a quality analog switch can be 10 to 100 W, and ananalog multiplexer can be 100 to 2,500 W per channel.
The ON resistance adds directly to the signal source
impedance and can affect the system’s measurement
accuracy if not compensated.
Analog switching devices have another undesirable
characteristic called charge injection. This means thata small portion of input-gate drive voltage is coupled tothe analog input signal and manifests as a spike in the
output signal. This glitch produces measurement errors
and can be seen riding on the input signal when thesource impedance is too high. A compensating circuit
can minimize the effects of charge injection, but themost effective method is to keep source impedance as
low as possible to prevent it from developing in thefirst place.
Channel-to-channel cross talk is another non-ideal
characteristic of analog switching networks, especiallyintegrated circuit multiplexers. Cross talk develops whenthe voltage applied to any one channel affects the accuracy of the reading in another channel. Conditions are
optimum for cross talk when signals of relatively highfrequency and high magnitude such as 4 to 5V signals
are connected to one channel while 100 mV signals
are connected to an adjacent channel. High frequency
multiplexing also exacerbates cross talk because thesignals couple through a small capacitance between
switch channels. Low source impedance minimizes the
cross talk and eliminates the charge injection.

1

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Scan Sequencer
All channels within a scan group aremeasured at a fixed scan rate

Scan groupt

Programmable
Constant
scan rate
Channel1 #2
Gain2 x1
Unipolar or Bipolar3 Uni

#4
x8
Uni

#7
x8
Bi

#2
x2
Uni

D4

#18 5
x100
Bi

#19
x10
Bi

#16
x1000
Uni

t

1. Unipolar or bipolar operation can be programmed for each channel dynamically by the sequencer
2. Gain can be programmed for each channel dynamically by the sequencer
3. Channels can be sampled dynamically by the sequencer
4. Expansion channels are sampled at the same rate as on-board channels

Fig. 3.02. This is an example of a 512 location scan sequencer operating in a 100 kHz data acquisition system. The sequencer canprogram each channel dynamically for unipolar or bipolar operation. The sequencer also can program the gain for each channel on
the fly and change the order in which the channels are scanned.

Speed

Figure 3.02

A data acquisition system with a software-selectable range canmeasure different ranges on different channels (but at a relatively
slow rate) with a command to change the gain between samples.
But the technique has two problems. First, it is relatively slow.
That is, issuing a software command to change the gain of aprogrammable-gain amplifier (PGA) can take tens or hundreds of
ms, lowering the system’s sample rate to several Hz. Second, thespeed of this sequence is often indeterminate due to variationsin PC instruction cycle times. Cycling through the sequence
continuously generates samples with an uneven (and unknown)
spacing in time. This complicates time-series analysis and makes
FFT analysis impossible because its algorithm requires evenly
spaced samples.

Multiplexing reduces the rate at which data can be acquired
from an individual channel because of the time-sharing strategy between channels. For example, an ADC that can sample
a single channel at 100 kHz is limited to a 12.5 kHz/channel
sampling rate when measuring eight channels.
Unfortunately, multiplexing can introduce yet other problems.
For instance, the multiplexer’s high source impedance can
combine with stray capacitance to increase settling time andgenerate cross talk between channels. Multiplexer impedance
itself also can degrade signals. A solid-state multiplexer with an
impedance of tens or hundreds of ohms is worse than a relay
with a typical resistance of 0.010 W or less.

A better implementation hosts a sequencer that sets the maximum
acquisition rate and controls both channel selection and associated amplifier gain at random. For example, one widely used dataacquisition system running at 100 kHz and 1 MHz uses software
selectable channel gain and sequencing. (See Figure 3.02.) The
100 kHz system provides a 512 location scan sequencer that lets
operators use software to select each channel and its input amplifier gain for both the built-in and expansion channels. Each scangroup can be repeated immediately or at programmable intervals.
The sequencer circuitry overcomes a drastic reduction in the scanrate for expansion channels, a major limitation encountered withmany plug-in data acquisition boards.

In spite of these negative issues, the advantages of multiplexing
outweigh its disadvantages, and it has become a widely usedtechnique to minimize cost without compromising performance.
Because the measurement errors are known and specified, they
can be compensated at each stage of the data acquisition system
to ensure high accuracy at the output.

Sequence vs. Software-Selectable Ranges
Most data acquisition systems accommodate a variety of input
ranges, although the manner in which they do so varies considerably. Some data acquisition systems allow the input range
to be switched or jumper selected on the circuit board. Others
provide software-selectable gain. This is more convenient, but
a distinction should be made between data acquisition systems
whose channels must all have the same gain and other systems
that can sequentially select the input range for each channel.
The more useful system accepts different input ranges on different channels, especially when measuring signals from different
transducers. For example, thermocouples and strain gages require
input ranges in tens of mV and use special signal conditioners,
while other sensors might output several volts.

All channels are scanned, including expansion channels, at
100 kHz, (10 µs per channel). (See Figure 3.03.) Digital inputs
also can be scanned using the same scan sequence employed
for analog inputs, enabling the time correlation of acquired
digital data to acquired analog data. Such systems permit eachscan group (containing up to 512 channel/gain combinations)
to be repeated immediately or programmed to intervals of up
to 12 hours. Within each scan group, consecutive channels aremeasured at a fixed 10 µs per channel rate.

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Multiplexer With
Sequence-Selectable Gain

Buffers
InputsAnaloginputs16
to1
Mux

Expansionchanneladdressing

4

x1, x2, x4, & x8
PGA

2

Expansion
gain addressing

Hardwaresequencer

BufferMux

BufferCLK

ADC

Triggering
circuitryFig. 3.04. Buffering signals ahead of the multiplexer
increases accuracy, especially with high-impedance sources orfast multiplexers

Fig. 3.03. A multiplexer with sequence-selectable gain lets
the data system operate at its maximum acquisition rate.

FUNDAMENTAL CONCEPTS

PGA

BufferTo dataacquisition
system2

4

BufferFigure 3.04
Sample and Hold

Sampling Rates

Ch 1

When ADCs convert an analog voltage to a digital representation, they sample the value of the measured variable several
times per second. Steady or slowly changing dc voltages might
require sampling rates of only several Hz, but measuring various
ac and sine waves is different. Enough samples/sec must be madeto ensure that the wave being measured in the continuous-time
domain can be faithfully reproduced in both the continuous-time
domain and the discrete-time domain.

Ch 2
Ch 3
Ch 4

Source Impedance

IA

LPF

S/H

IA

LPF

S/H

IA

LPF

S/H

IA

LPF

S/H

Mux

To dataacquisition
systemChanneladdresslines

Sample enable

Switched bias resistors(per channel)

Most signal sources have impedances of less than 1.5 kW, so sucha maximum source impedance is usually not a problem. However,faster multiplexer rates require lower source impedances. For example, a 1 MHz multiplexer in a 12-bit system requires a source
impedance less than 1.0 k W. When the source impedance exceeds
this value, buffering is necessary to improve accuracy. A buffer
is an amplifier with a high input impedance and extremely low
output impedance. (See Figure 3.04.) A buffer on each channel
Figure ensures
3.05 higher
located between the transducer and the multiplexer
accuracies by preventing the multiplexer’s stray capacitance fromdischarging through the impedance of the transducer.

Fig. 3.05. A four-channel multiplexer with low-pass filtersand simultaneous sample-and-hold circuits ensures that all
the samples are taken within a few ns of each other.

simultaneous sample and hold implementation, all channels are
sampled within 100 ns of each other.
Figure 3.05 shows a common scheme for SS&H. Each input
signal passes through an instrumentation amplifier (IA), a lowpass filter, and into a sample-and-hold buffer (S/H). When thesample enable line goes high, each S/H samples its input signal
and holds it while the multiplexer switches through the readings. This scheme ensures that all the samples are taken within
50 ns of each other, even with up to 256 simultaneous channels
connected to a single instrument.

Sample and Hold ADCs
Time Skew

A multiplexed ADC measurement introduces a time skew amongchannels, because each channel is sampled at a different time.
Some applications can’t tolerate this effect. But a sample and holdcircuit placed on each input ahead of the multiplexer remedies
time-skew problems. In a simultaneous sample and hold circuit(SS&H) each channel is equipped with a buffer that samples thesignal at the beginning of the scan sequence. The buffer output
holds the sampled value while the multiplexer switches throughall channels, and the ADC digitizes the frozen signals. In a goodNyquist Theorem

Transforming a signal from the time domain to the frequency
domain requires the application of the Nyquist theorem. The
Nyquist sampling theorem states that if a signal only contains

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Acceptable Nyquist Sampling Rates

1

1

0.8

0.8
0.6

0.4

Amplitude (V)Amplitude (V)

0.6

0.2
0
-0.2
-0.4
-0.6

0.2
0
-0.2
-0.4
-0.6

-0.8
-1

0.4

-0.8
0

100

200

300

400

500

-1

600

Frequency (Hz)Original Sine Wave
Sampled less than 1 time per cycle

100

200

300

400

500

Frequency (Hz)600

Original Sine Wave
Sampled about 4 times per cycle

Fig. 3.06. When sampling an ac signal at less than twotimes the Nyquist frequency, the original waveform cannot be
reproduced faithfully.

Fig. 3.07. When ac inputs are sampled more than twice the
Nyquist frequency of the sine wave, the frequency content of
the signal is preserved, and all the Fourier components of theperiodic waveform are recovered.

frequencies less than cutoff frequency, fcFigure, all the3.07
information(2.08) in
the signal can be captured by sampling it at a minimum frequencyof 2fc. This means that capturing a signal with a maximum frequency component of fmax requires that it must be sampled at
2fmax or higher. However, common practice dictates that whileworking in the frequency domain, the sampling rate must be
set more than twice and preferably between five and ten times
the signal’s highest frequency component. Waveforms viewed
in the time domain are usually sampled 10 times the frequency
being measured to faithfully reproduce the original signal andretain accuracy of the signal’s highest frequency components.

Fourier Transforms
1
0.9
0.8
0.7

Volts

9)

Inadequate Nyquist Sampling Rates

0.6
0.5
0.4
0.3

Aliasing and Fourier Transforms

0.2
0.1

When input signals are sampled at less than the Nyquist rate, ambiguous signals that are much lower in frequency than the signal
being sampled can appear in the time domain. This phenomenon
is called aliasing. For example, Figure 3.06 shows a 1 kHz sine wave
sampled at 800 Hz. The reconstructed or transformed frequency ofthe sampled wave is much too low and not a true representationof the original. If the 1 kHz signal were sampled at 1,333 Hz, a
333 Hz alias signal would appear. Figure 3.07, on the other hand,
Figure3.08(2.10)
shows the signals when sampling the same
1 kHz
wave
at morethan twice the input frequency or 5 kHz. The sampled wave now
appears closer to the correct frequency.

0

0

500

1000

Frequency [Hz]500 Hz square wave
4 kHz sample rate
No filters

1500

2000

Fig. 3.08. The Fourier transform of a 500 Hz square wave issampled at 4 kHz with no filtering.

is a frequency spectrum display of the sampled data. It shows
how much energy at a given frequency is in a particular signal. For the purpose of illustration, assume that the example
processes only frequencies under 2 kHz. Ideally, a Fourier transform of a 500 Hz square wave contains one peak at 500 Hz, thefundamental frequency, and another at 1,500 Hz, the third
harmonic, which is 1/3 the height of the fundamental. Figure
3.08, however, shows how higher frequency peaks are aliased
into the Fourier transform’s low-frequency range. The low-passfilter with cutoff at 2 kHz shown in Figure 3.09 removes mostof the aliased peaks.

Conversely, input frequencies of half or more of the sampling
rate will also generate aliases. To prevent these aliases, a lowpass, anti-aliasing filter is used to remove all components ofthese input signals. The filter is usually an analog circuit placed
between the signal input terminals and the ADC. Although thefilter eliminates the aliases, it also prevents any other signals
from passing through that are above the stop band of the filter,
whether they were wanted or not. In other words, when selecting a data acquisition system, make certain that the per channel
sampling frequency is more than twice the highest frequency
intended to be measured.

When the sampling rate increases to four times the highest
frequency wanted, the Fourier transform in the range of interest
looks even better. Although a small peak remains at 1 kHz, it is

Another example of aliasing is shown in Figure 3.08 for a squarewave after passing through a Fourier transform. A Fourier ­transform

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1.4

1.6

1.2

1.4
1.2

0.8

Volts

Volts

1

0.6

0.8

0.4

0.2
0

1

0.6

0.4

0.2
0

500

1000

Frequency [Hz]

1500

0

2000

0

500

1000

Frequency [Hz]

1500

2000

500 Hz square wave
8 kHz sample rate
2 kHz low-pass filter

500 Hz square wave
4 kHz sample rate
2 kHz low-pass filter

Fig. 3.09. The Fourier Transform of a 500 Hz square wave
is sampled at 4 kHz with a low-pass filter
cutoffat 2
kHz.
Figure3.10(2.12)

Fig. 3.10. The Fourier Transform of a 500 Hz square wave
is sampled at 8 kHz with a low-pass filter cutoff at 2 kHz.

most likely the result of an imperfect square wave rather thanan effect of aliasing. (See Figure 3.10).

Fourier Transforms
1.2

Discrete Fourier Transform

1

When ac signals pass through a time-invariant, linear system,
their amplitude and phase components can change but their frequencies remain intact. This is the process that occurs when thecontinuous time domain ac signal passes through the ADC to thediscrete time domain. Sometimes, more useful information can beobtained from the sampled data by analyzing them in the discrete
time domain with a Fourier Series rather than reconstructing theoriginal signal in the time domain.

Amplitude

1)

Fourier Transforms

Fourier Transforms

0.6
0.4
0.2
0
450 460 470 480 490 500 510 520 530 540 550

Frequency [Hz]The sampled data pass through a Fourier transform function tocull out the fundamental and harmonic frequency information.
The amplitude of the signal is displayed in the vertical axis, andthe frequencies measured are plotted on the horizontal axis.

Windowing

0.8

Fourier transform of a sine wave using a window function
Fourier transform of a sine wave with no window function

Fig. 3.11. A Fourier Transform is shown with and without awindow function.

Figure 3.11 (2.13)

Real-time measurements are taken over finite time intervals. In
contrast, Fourier transforms are defined over infinite time intervals, so limiting the transform to a discrete time interval produces
sampled data that are only approximations. Consequently, theresolution of the Fourier transform is limited to approximately 1/T
where T is the finite time interval over which the measurement
was made. Fourier transform resolution can only be improvedby sampling for a longer interval.

spurious oscillations at the expense of a slight loss in triggering resolution. Many possible window functions may be used,
including Hanning, Hamming, Blackman, Rectangular, andBartlett. (See Figures 3.12, 3.13, and 3.14.)

Fast Fourier Transforms

The Fast Fourier Transform (FFT) is so common today that FFT
has become an imprecise synonym for Fourier transforms ingeneral. The FFT is a digital algorithm for computing Fouriertransforms of data discretely sampled at a constant interval.
n
The FFT’s simplest implementation requires 2 samples. Other
implementations accept other special numbers of samples. If thedata set to be transformed has a different number of samples thanrequired by the FFT algorithm, it is often padded with zeros tomeet the required number. Sometimes the results are inaccurate,
but most often they are tolerable.

A finite time interval used for a Fourier Transform also generates spurious oscillations in the transform’s display as shown in
Figure 3.11. From a mathematical viewpoint, the signal that’s
instantaneously turned on at the beginning of the measurement
and then suddenly turned off at the end of the measurement
produces the spurious oscillations. These spurious oscillations
are usually eliminated with a function called windowing, thatis, multiplying the sampled data by a weighting function. A
window function that rises gradually from zero decreases the5
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Measurement With
Window Function

1

1

0.5

0.5

Amplitude2
1.5

Amplitude

2
1.5

0

2.5
2
1.5

0

-0.5

-0.5
-1

-1

-1.5

-1.5

-2

-2

1

0.5
0

Fig. 3.13. In this case, the frequency information is preserved
when multiplied by a windowing
function. It minimizes the effect ofirregularities at the beginning andend of the sample segment.

Fig. 3.12. The frequency information obtained from a measurement
can be incorrect when the waveform
contains abrupt beginning or ending points.Figure 3.13 (2.15)

Common Window
Function

Amplitude

Measurement Without
Window Function

HammingHanning
Blackman

Fig. 3.14. The three most common types of window functions
include Hamming, Hanning, andBlackman.

Figure 3.14 (2.16)
of a similar number of points. This is becoming less of an issue,
however, as the speed of modern computers increases.

Digital Filtering
1.546

Digital vs. Analog Filtering

Digital filtering requires three steps. First, the digital signal is subjected
to a Fourier transform. Then the signal amplitude in the frequency
domain is multiplied by the desired frequency response. Finally thetransform signal is inverse Fourier transformed back into the time
domain. Figure 3.15 shows the effect of digital filtering on a noisy
signal. The solid line represents the unfiltered signal while the twodashed lines show the affect of different digital filters. Digital filters
have the advantage of being easily tailored to any frequency response
without introducing phase error. However, one disadvantage is thata digital filter cannot be used for anti-aliasing.

1.544

Volts

1.542
1.54
1.538
1.536
1.534

0

0.05

0.1

0.15

Time [seconds]

0.2

0.25

Unfiltered signal
50 Hz low-pass digital filter
5 Hz low-pass digital filter

In contrast to digital filters, analog filters can be used for antialiasing. But changing the frequency response curves is moredifficult because all analog filters introduce some phase error.

Fig. 3.15. The solid line is the noisy, unfiltered signal, andthe two dashed lines show the same signal at the output of
the 5 and 50-Hz low-pass digital filters.

Settling Time

Source impedance and stray capacitance influence the multiplexer input settling time. Their effect can be predicted witha simple equation:

Standard Fourier Transforms
A Standard Fourier Transform (SFT) can be used in applications
where the number of samples cannot be arranged to fall on one of
the special numbers required by an FFT, or where it cannot tolerate
the inaccuracies introduced by padding with zeros. The SFT is also
suitable where the data are not sampled at evenly spaced intervals
or where sample points are missing. Finally, the SFT can be used
to provide more closely spaced points in the frequency domain
than can be obtained with an FFT. In an FFT, adjacent points are
separated by 1/T, the inverse of the time interval over which themeasurement was made.

EQN. 3.01 Time Constant

T = RC,
Where:
T = time constant, s
R = source impedance, W
C = stray capacitance, F
For example, determine the maximum tolerable source impedance
for a 100 kHz multiplexer. The time between measurements on adjacent channels in the scan sequence is 10 µs. During time T = RC,
the voltage error decays by a factor of 2.718, or one time constant,
T. But a time interval 10 times longer (10 T) is required to reduce
the error to 0.005%. Consequently, a fixed time of 10 µs between

Many standard numerical integration techniques exist for computing SFTs from sampled data. Any other technique selected forthe problem at hand probably will be much slower than an FFT

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For a 16-bit data acquisition system with 100 pF input capacitance,(Cin) and a multiplexer resistance (Rmux) of 100 W, the maximum
external resistance is:scans (Tscan) and an error of 0.005% requires that T = 1 µs. But ina typical multiplexed data acquisition system, this is insufficient
settling time, so the data would continue to be in error. The relationship can be explained as follows. Most 100 kHz converter’s sample
and hold circuits are set to acquire the signal at about 80% of thesample window (10 µs) to allow 8 µs of settling time. Subtracting
this from the scan time results in a sampling time of:

EQN. 3.04 External ResistanceR ext =

EQN. 3.02 Settling Time

T ext
(C in )(l n 2 16 ) – R mux

Rext = 4,670 W

Tsample = Tscan – Tsettle
Tsample = 10 µs – 8 µs
Tsample = 2 µs

The simplified examples above do not include effects due tomultiplexer charge injection or inductive reactance in the measurement wiring. In actual practice, the practical upper limit on
source resistance is between 1.5k and 2 kW.

In a typical 16-bit data acquisition system, the internal settling
time (Tint) may be 6 µs. The external settling time may then becomputed as follows:
EQN. 3.03 External Settling TimeT ext =

Equation 3.04
T settle 2 – T int 2

Text = 5.29 µs

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