A topic for an
undergraduate
laboratory experiment
in signal processing.
Shlomo Engelberg
and Yona Bendelac
©COREL TEXTURES
M
easuring physical constants by examining
suitable types of noise has a long history. In
1918, Shottky described two types of noise:
thermal noise and shot noise. By the 1920s,
measurements of thermal noise had been used to measure
the Boltzmann constant, KB , and measurements of shot noise
had been used to measure the charge on the electron, e0 .
Since the 1920s, many people have designed experiments
to measure physical constants using noise. (See [1] for an
example of such an experiment and for additional information about the theory behind the experiments.) In addition to
their early use in determining the value of physical constants, noise experiments have become part of undergraduate laboratories at several universities. (Reference [1]
describes such a laboratory experiment.)
Our experiments differ from previous experiments in that
a computer does much of the data gathering and data processing. Once an undergraduate student understands the
elements of the theory of thermal and shot noise, a little bit
about the Windows commands that interface with the
SoundBlaster, and something of filter design using MATLAB, the student will be able to design the apparatus and
implement the measurements described in this article. (The
December 2003
thermal noise experiment is particularly simple to design
and implement.) Before describing the experiments, we
briefly discuss thermal and shot noise.
Thermal Noise
Thermal noise is variously known as thermal, Johnson, or
Nyquist noise. It is noise produced in a resistor because the
electrons in the current that flows through the resistor are
subject to random thermal motions in addition to any
motion imposed upon them by an applied electric field.
Because of the nature of thermal noise, thermal noise has a
constant power spectral density, i.e., mean square voltage
per unit frequency, up to very high frequencies. Noise
whose power spectral density is constant is called white
noise. In 1928, Harry Nyquist [2] found that the power spectral density of thermal noise is
Sthermal ( f) = 4KB TR
where KB is the Boltzmann constant, T is the absolute temperature in kelvin, and R is the resistance (in ohms) of the
resistor producing the thermal noise. This formula is known
as the Nyquist formula. From the formula, we see that the
IEEE Instrumentation & Measurement Magazine
1094-6969/03/$17.00©2003IEEE
49
mean square voltage associated with the thermal noise in a
frequency band !f wide that is produced by a resistor of
resistance R held at temperature T is
2
V thermal
−rms = 4KB TR!f.
Shot Noise
Under certain conditions, the electrons that make up a constant current arrive in a random fashion. Under these conditions, in which N electrons pass a point each second, the
current does not have precisely one electron pass a given
point each 1/N s. On average, one electron passes every
1/N s; in any given time interval, more electrons or fewer
electrons than expected will arrive.
When measuring a constant current, these variations are
seen as noise that “rides” on the constant current; this noise
is called shot noise. You can understand where the name
comes from if you think of the sound that shot falling on a
tin roof would make. If the electrons arrive in a random
fashion, then the noise power per unit frequency due to shot
noise measured across a resistor with resistance R is
Our Goal
The Boltzmann constant and the charge on the electron—KB
and e0 , respectively—are known to very high accuracy, and
we are not trying to improve man’s knowledge of the constants. The Boltzmann constant, in units that match the units
picked for the other terms, is
KB = 1.3897 × 10−23 JK−1 .
The charge on the electron is
e0 = 1.6022 × 10−19 C .
We present a simple, largely computerized, way of measuring the Boltzmann constant, and we explain how our
method can be used to simplify the measurement of the
charge on the electron.
Instrumentation Considerations
Both shot and thermal noise are white noise out to relatively
high frequencies. The spectral width, however, is not the
same for both types of noise. Thermal noise is more broadband than shot noise. As our goal is to use thermal and shot
Sshot ( f) = 2e0 Vavg R
noise with a minimum of fuss, we filter out the high- and
low-frequency portions of the noise and only make use of a
where e0 is the charge on the electron (measured in relatively narrow band of frequencies.
coulombs) and Vavg is the dc voltage (measured in volts)
Working at high frequencies is expensive and tricky. We
across a resistor of resistance R (measured in ohms). This want to perform our experiments as simply as possible, and
formula is a variant of the Shottky formula for shot noise.
this is a sufficient reason to filter out the high frequencies.
Why filter out the low frequencies? One reason is that in the case of shot noise there is a
(relatively) large dc current flowing too, and
Noise Source
Filter
RMS Voltage
we do not want it (the signal) corrupting our
noise measurements. However, even when
measuring thermal noise—which need not
Fig. 1. The experimental setup.
have a dc component—it is important to filter
out the low-frequency portion of the noise.
In addition to shot noise and thermal noise, there is
R3 = 10K
another type of noise that is associated with many devices.
This type of noise is called flicker, pink, or “one over f ”
noise because its power spectral density looks like
−
LF 355
+
R2 = 100 Ω
R1 = 100K
Fig. 2. The circuit used to produce thermal noise.
50
S1/ f ( f ) ∝ 1/f α
where α is close to one. Flicker noise contributes most at low
frequencies; we work at higher frequencies.
The Experiment: Thermal
Noise and the Boltzmann Constant
When using either thermal or shot noise to measure physical
constants, a system whose general form is given in Figure 1
is used. There is a noise source (in the case of thermal noise,
a resistor with some additional circuitry), a filter used to
remove the high and low frequencies, and then there is an
element that is capable of measuring the mean square (or
rms) voltage at the output of the filter.
IEEE Instrumentation & Measurement Magazine
December 2003
To measure the Boltzmann constant, we used a simple experimental setup. First we connected an “input” resistor, R1, and
two other resistors, R2 and R3, as shown in Figure 2. A picture of
the system as we implemented it is given in Figure 3. The output
of the noise source is amplified thermal noise. Rather than using
analog filters, this noise is fed to a computer’s SoundBlaster
through the microphone input. Using the Windows Sound
Recorder utility, we sample the noise and store the samples in a
.wav file. Opening MATLAB, we use the “wavread” command
to convert the .wav file into a MATLAB vector. Finally, we use
MATLAB’s filter design tools to design and implement a bandpass filter that passes frequencies from 800 Hz to 4 kHz. We pass
the vector of measurements through this filter, and we measure
the mean square value of the output of the voltage. From this
measurement, we deduce the value of KB .
Let us consider the various stages of the measurement and
what they contribute to the final measured value. The analog
first stage amplifies the noise by the amount A. Next, the
SoundBlaster amplifies the signal. Let us call its amplification
B. The digital band-pass filter then filters the sampled signal.
This filter removes high- and low-frequency components and
amplifies the frequencies in its passband by a factor C. Thus,
the mean square voltage at the output is approximately
Thus, we did not need to worry much about high-frequency
signals passing into our system and being aliased down to
low frequencies by the sampling operation. We also found
that the SoundBlaster provides an amplification of about 30
to the signal throughout the frequencies of interest to us.
The next component was a digital filter. Using MATLAB,
we designed a digital filter that did almost precisely what
we wanted. The gain of the filter is approximately zero outside of the passband and is almost constant in the passband.
MATLAB makes it easy to make sure that the passband is
almost exactly where one wants it.
Other Noise Sources
There will be many sources of noise when performing any
experiment. As long as the noise sources are all independent of
one another, if they have different sources then the noise powers combine. In the case of our measurement, we have thermal
noise and other sources of noise. We are only interested in the
thermal noise and would like to avoid the other sources.
2
= 4KB RTA2 B2 C2 ( fmax − fmin ).
V rms
As we know or can measure all the values here except for KB ,
and we assume a value for room temperature, T, that contributes an error of the same order of magnitude as the other
errors we observe, we can use a measurement of the rms value
of the output voltage to compute the value of the constant. We
did this using a more precise measurement of the filter’s effect
and achieved an accuracy of slightly better than 5%. See Figure
4 for a typical run of our MATLAB program with actual data.
Designing the Experiment:
Avoiding Bad Parameters
To understand why the different stages are designed as they are,
we must understand what we are trying to avoid. Suppose a
resistor was attached to a multimeter that measures rms voltage.
What voltage would be measured? It is hard to say. The rms
voltage due to thermal noise depends on the bandwidth over
which the noise measurement is done, and with this type of
setup the bandwidth is being fixed by the (generally unknown)
bandwidth of the measuring device. In our experiment, we
want to avoid problems of this type; we want to set all necessary
bandwidths in a predictable and reproducible fashion.
Let us consider the components of our setup. The analog portion is an op-amp amplifier. If we do not want to have to deal
with the frequency-dependent portion of the transfer function of
the amplifier, then we must make sure that the transfer function
is constant in the region of interest to us. In this case, this means
making sure that the gain-bandwidth product is large enough to
handle the gain we use and the bandwidth we desire.
The next component in the setup is the SoundBlaster. The
SoundBlaster we used has antialiasing filters at its input.
December 2003
Fig. 3. The actual noise source used in the determination of the Boltzmann
constant.
Fig. 4. The input to and output of the MATLAB program that calculates the
value of the Boltzmann constant. Note that the program requires as input one
file with data taken while the resistor is in place and one file taken while a
short circuit replaces the resistor. (Note that R here is the resistor referred to
as R1 in the text.)
IEEE Instrumentation & Measurement Magazine
51
When we measure the output of our circuit, there are two
contributions to the measured value
2
2
2
= V thermal
V meas−rms
−rms +V other−noise−rms .
To remove the second part, R1 can be shorted out and the
2
remaining noise can be measured. The result is V other−noise−rms
almost precisely. (Actually, there are two types of noise contributed by R1 . One is the thermal noise that we are trying to measure,
and the other is connected to unwanted currents flowing through
the op-amp. To make this experiment work well, one must use a
low-noise op-amp. We use the LF355 [3].) Subtracting the mean
square noise that is measured when R1 is shorted out from the
measured value of the noise when the resistor is in the circuit,
2
2
, one finds V thermal
from V meas−rms
−rms to reasonably high accuracy. It is this value that we used in calculating the Boltzmann constant. A sample run of the MATLAB program we used is given in
Figure 4. Note that one of the inputs is a measurement taken with
the noise-producing resistor replaced by a short circuit.
The Second Experiment: Shot Noise
and the Charge of the Electron
Using almost the same setup, we determined the charge on the
electron. We had to add some parts, however, and the measurement required some amount of work on the part of the
operator. Our results for the charge of the electron were about
as good as for Boltzmann’s constant, but the method is not that
much simpler than the other methods that have been used.
The Shottky formula holds when the electrons in a dc current arrive in a totally random way. It is interesting that in
many currents the electrons do not arrive randomly [4]. To
produce a “constant” current in which the electrons arrive
randomly, we used the setup described in [5]. A small incandescent bulb generates a random stream of photons, and a
vacuum photodiode is used to convert this stream of randomly arriving photons into a stream of randomly arriving electrons. The output of the vacuum photodiode is fed to the
input of a simple feedback amplifier. This voltage is then pro-
cessed by an analog system in [5]. We feed the output of the
amplifier into the SoundBlaster and then use the system that
we have already described to calculate e0 . The schematics of
the noise detector that we used are given in Figure 5. (They
are one part of the circuit used in [5].) Note that the Shottky
formula involves the dc voltage across the resistor, Vavg . The
voltage must be measured manually. Once this value has been
entered into the computer, it can calculate the charge on the
electron. Using our system, we achieved an accuracy of better
than 5% in our measurement of the charge on the electron.
Summary
Using well-known formulas for thermal and shot noise, we
designed experiments for measuring two physical constants:
the Boltzmann constant and the charge on the electron. We
were able to perform the experiments in a particularly simple
fashion by making use of widely available computer-based
measurement and processing tools. In particular, the measurement of the Boltzmann constant only required one operational amplifier and three resistors. Our measurements
achieved reasonable accuracies, considering the nature of the
instruments with which we performed the measurements.
The experiments are well suited to use in a laboratory setting,
and they allow one to see some of the connections between
random noise, physics, and signal processing.
Acknowledgments
We acknowledge with thanks many helpful and pleasant
discussions with Beni Goldberg.
References
[1] MIT Department of Physics, “Johnson noise and shot noise: The
determination of the Boltzmann constant, absolute zero temperature
and the charge of the electron.” Available: http://www.mit.edu/
afs/athena/course/8/8.13/WWW/JLExperiments/JLExp_43.pdf
[2] J.B. Johnson, “Electronic noise: The first two decades,” IEEE
Spectr., vol. 8, pp. 42–46, Feb. 1971.
[3] National Semiconductor, “LF155/LF156/LF256/LF257/LF355/
LF356/LF357 JFET input operational amplifiers.” Available:
–65V
51k
0.01mF
160V
http://cache.national.com/ds/LF/LF155.pdf
51k
0.33mF
+ 63V
110nF
+
[4] M. de Jong, “Sub-Poissonian shot noise,” Phys. World, vol. 9, no.
15V
Zener
Diodes
+
8, pp. 22, Aug. 1996.
[5] D.R. Spiegel and R.J. Helmer, “Shot-noise measurements of the
electron charge: An undergraduate experiment,” Amer. J. Physics,
1p39
150k 2%
Ro +15V
–
+
150k
2%
150k
Input Test
–15V
vol. 63, no. 6, pp. 554–560, 1995.
330K 1%
0.3uF10K1%
4.7k
DC Out
10k
+15V
–
+
–15V
To Sound
Blaster
Fig. 5. The schematic of the shot noise detector circuit. The point marked
“input test” is used to input a test signal [5]. At the point marked “dc out,” the
dc value of the measured voltage is available. The element labeled “1p39” is
the vacuum photodiode that was used; the two op-amps are 741. The
incandescent bulb is not shown in the schematic. (This circuit is part of the
setup described in [5] and appears with permission.)
52
Shlomo Engelberg received his B.E.E. in 1988, his M.E.E. in 1990,
his M.S. (in mathematics) in 1991, and his Ph.D. (in mathematics) in 1994. He teaches electronics engineering at the Jerusalem
College of Technology-Machon Lev. His interests include signal
processing, control theory, and applied mathematics.
Yona Bendelac is currently completing his bachelor’s degree in
electronics engineering at the Jerusalem College of TechnologyMachon Lev. He is an academic reservist in the Israeli army. His
interests include signal processing and computer programming.
IEEE Instrumentation & Measurement Magazine
December 2003