4.3 ANALOG SIGNAL
Figure 4.2 contras ts analog and digital forms of data representation. An analog signal is a time-varying
quantity s(t), which if transformed into digital form (e.g. by an analog-to-digital converter ADC), each
point of s(t) is then represented by a sequence of pulses of constant duration and amplitude. Note that
measurement of a continuous analog quantity requires a certai n finite time interval,
s
, and the observed
value of s(t) is a time average over the interval
s
:
s
s
¼
1
s
Z
s
0
sðtÞdt (4.1)
Obviously, the recorded time average value
s
s
will depend on the measurement time
s
and on how
rapidly the analog signal changes in time, which can be characterized by the signal’s highest and
lowest frequencies, f
H
and f
L
respectively. The difference
D
f ¼ f
H
– f
L
is called bandwidth. In the
T
A
External stimulus
FIGURE 4.1
A generic model of a minimal sensor configuration: transducer (T) and a downstream device used for signal
conditioning, e.g. amplifier (A)
BOX 4.1 DIGITAL AND ANALOG DEVICES
The binary switches, discussed in Chapter 3, are used to represent and process information in digital form: they have
two states (known as binary states), which are marked as the digits 0 and 1 (see Box 3.1). As was discussed in
Chapter 3, in digital operation, the barriers in the switches (e.g. the FET) are abruptly set from high to low values and
vice versa.
The barrier height can also be decreased gradually to a value somewhere between its maximum and minimum
values. Devices (e.g. the FET) operating in the continuous mode are called analog, while devices operating in the
discrete (e.g. binary) mode are called digital. Most sensors are analog devices.
In many microelectronic systems, both analog and digital parts co-exist. To enable interaction between the two,
signal converters are used: analog-to-digital (ADC) and digital-to-analog converters (DAC).
4.3 Analog signal 93
discussion below it is always assumed that f
L
¼ 0 (Hz). The highest frequency (and therefore the
bandwidth) is related to the signal’s time duration, t
s
,as
Df ¼ f
H
w
1
t
s
(4.2)
Note that in theory, for a signal of finite time duration, f
H
/ N. In practice however, the highest
frequency of a signal is understood to be that frequency below which the main (e.g. 90%) part of the
signal’s energy is located. For such a ‘relaxed’ definition, formula (4.2) holds for most practical cases.
As can be seen from Figure 4.2, for reliable analog recording, the recording interval,
s
, (sampling
time) must be much less than the signal duration: s << t
s
or f
s
>> f
H
.
The fundamental limit on the minimum sampling time is given by the Nyquist-Shannon-
Kolmogorov theorem:
s
min
1
2Df
w
t
s
2
(4.3)
0
1
2
3
4
5
6
101100
t
s
011 010
001
A/D
analog-to-digital converter
FIGURE 4.2
Examples of analog (top) and corresponding digital (bottom) signals. The digital signals shown in the figure
represent several sample points of the analog signal, generated by an analog-to-digital converter. Recording/
conversion of each point of the analog signal requires certain sampling time
s
which should typically be much
smaller than the signal duration t
s
94 CHAPTER 4 Sensors at the micro-scale
To illustrate (4.3), consider strategies for detecting a si ngle rectangular pulse of duration t
s
(Fig. 4.3) by using a repetiti on of rectangular sampling pulses of duration
s
. Two things need to
happen for a successful detection: (i) the sampling signal needs to ‘catch’ the measured signal,
i.e. there must be a time overlap between the two, and (ii) the recorded time average value
s
s
of
the detected signal should be reasonably close to the actual value, s
m
. In the following, we
require that in a ‘mi nimal successful measurement,’ the measured value should be no less than
a half o f the actual value, i.e.
s
s
s
m
2
. First let
s
>> t
s
(Fig. 4.3b). In this case, the probability the
sampling and the measured signals overlap is close to 1. However, the measured time average
value of the signal over interval
s
, according to (4 .1) is s
s
/0. For more accurate time average
measurements,
s
should be reduced, e.g.
s
~t
s
(Fig. 4.3c). However, if the repetition interval is
larger than t
s
, there is a non-zero probability that the s ampling will miss the measured signal. If
s
< t
s
, the overlap probability is always 1 and it is easy to show that if
s
¼ t
s
/2, then s
s
s
m
/2
(Fig. 4.3d).
t
s
0
s
m
(a)
s 0
t
s
s
m
0
>> t
s
(b)
s =0…s
m
t
s
s
m
0
~ t
s
(c)
2
m
s
s
s
m
0
t
s
~ t
s
/2
(d)
FIGURE 4.3
Illustration to the sampling theorem: different strategies for detection of a single rectangular pulse (see text for
discussion)
4.3 Analog signal 95
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