LabView Unearths Tiny Signals buried in a Noise - (flip) Kromer's

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24 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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LabView Rescues Tiny Signals from a Sea of Noise

Philip
F
Kromer

and
Roger Bengtson

Department of Physics
,
University of Texas at Austin

Category

R&D / Lab Automation

Products Used


LabView 6.0.2; PCI
-
MIO
-
16E
-
4

The Challenge

Measure an extremely small signal obscured by noise thousands of times greater in magnitude, at a minimum of added
expense
and

hardware, i
n order to determine the resistance of a high
-
temperature superconductor.

The Solution

Use the signal
-
processing capabilities of LabView
to implement a

technique known as lock
-
in amplification. Compared to
hardware

lock
-
in amplifiers, the LabView approach
yields excellent price/performance, increased functionality, superior
flexibility, and the ability to inspect the signal at all stages of processing
1
.

Measuring Small Signals is Difficult

An experiment
in our lab
determines

the resistance of a high
-
temper
ature superconductor
2

by direct measurement:
we
apply a
known current across the sample, and measure its re
sistive voltage drop. Th
is signal

varies from
~
1 mV
while
normally
conducting
to

<
1
0

n
V

while
superconducting.

Even under carefully controlled condit
ions, there are intractable sources of
noise, due to fundamental physical processes, that
obscure this signal
3
. What is worse, a significant
portion of this noise has a “1/
f

spectrum:” its intensity
increases at low frequencies and is worst at DC,
precisel
y where
the

signal

naturally resides
. Finally,
intransigent sources of measurement error, such as
offset drift, thermoelectric voltages, and common
-
mode
error,
act to
corrupt the signal
.

Figure
1

shows

a ty
pical
input amplitude spectrum

(note the log scale
)
. One can
see the DC offset,
the
60 Hz interference, and the
mixture of 1/
f

and
broadband noise.

A
verag
ing

over a long time

reduces
the noise

by
narrowing the
bandwidth
and
effectively trading
aw
ay
response
time

for improved noise rejection. However,
no a
mount of averaging can distinguish
our slowly
varying signal from
DC and low
-
frequency
components
of
the
noise and error sources.

The Lock
-
in Technique Provides an Answer…

Instead, we
will
use lock
-
in amplification
4

to recover the signal
.
Rather than apply

a constant (DC) current, we
generate a
purely
sinusoidal

(AC)

reference

signal

and apply

a scaled current
)
2
cos(
ref
ref
ref
t
f
I
i



across the sample
.

The resistive
voltage drop
has
the same frequency and phase as the applied current:


)
2
cos(
)
2
cos(
ref
samp
ref
ref
samp
samp
t
f
R
I
t
f
V
v




.

However,
our actual
input
contain
s

not only

the desired
v
samp
, but also
the
undesirable offset,
interference
,

and noise

(see
Figure 1
)
.
We can represent t
he noise and interference a sum of randomly varying signals at all frequencies and pha
ses
:







n
)
2
cos(
n
n
n
offs
samp
in
f
t
f
V
V
v
v


.

To
select

only the interesting
(matching the reference
in
phase and frequency) part
of the input signal, we
use

a simple trick
from trigonometry. Recall the
cosine
sum rules:


)
sin(
)
sin(
)
cos(
)
cos(
)
cos(
b
a
b
a
b
a





)
sin(
)
sin(
)
cos(
)
cos(
)
cos(
b
a
b
a
b
a



.

Add

and re
arrang
e; the product of two cosines
yields
a cosine at the difference
frequency
plus
a cosine at the sum

frequency
:




)
cos(
)
cos(
2
1
)
cos(
)
cos(
b
a
b
a
b
a




.


Figure 1: Log Amplitude versus Frequency (for F
ref
=1122 Hz)

M
ultiply
ing

the noisy input signal
in
v

by
)
2
cos(
2
ref
t
f



copies

the input
,

shifted
up a
nd down by
ref
f
:









n
)
2
cos(
)
2
cos(
2
)
2
cos(
2
)
2
cos(
)
2
cos(
2
2
n
n
n
ref
ref
offs
ref
ref
samp
in
ref
mult
f
t
f
V
t
f
t
f
V
t
f
t
f
V
v
u
v



















n
)
2
]
cos([
)
2
cos(
)
2
2
cos(
n
ref
n
n
ref
offs
ref
samp
samp
f
t
f
f
V
t
f
V
t
f
V
V




,

T
he voltage drop has the same frequency as
the
reference,
so
their difference term has zero frequency (DC).
Therefore,
filter
out
all frequencies below a cutoff

freque
ncy
ref
filt
f
f

:








filt
n
)
2
]
cos([
n
ref
n
n
samp
lock
f
f
t
f
f
V
V
V


.

This is

the
desired voltage drop, with
a
small
remnant

of
noise
,
those components indistinguishable in frequency
and

phase
from the
reference.

… And LabView provides a Solution

L
ock
-
in amplification is

traditionally accomplished
with
expensive, monolithic hardware
.
Our LabView
implementation
achieve
s

reasonable performance at
greatly reduced price
,

with superior
flexibility
5
.

Besides

a
National Instruments
PCI
-
MIO
-
16E
-
4

data acquisition
board
6
,
t
he only

additional hardware
devices are

an input
amplifier

and

a
voltage
-
controlled current source, both
shop
-
built
7
.

Our LabView
program i
nstructs the
DAQ
board

to
generate

a
waveform
continuously
at the
arbitrarily
chosen

reference frequency

of 1122 Hz
.

This
re
ference

drives
a voltage
-
controlled
current source
connected
across the superconductor sample.
The
superconductor’s
bulk
resistance
causes a
proportional voltage drop, which
is
passed to
a high
-
gain
,

low
-
noise differential amplifier
.

T
he DAQ board
continuo
usly acquires

the
amplified voltage drop,
the voltage
from
a thermocouple on
the sample,
and other
signals of interest.
As each buffer is acquired, our program processes the signals
and
performs preliminary data analysis
.
The
input signal i
s

lock
ed
-
in



mu
ltipl
ied

by
the unit reference
, scale
d
, and filter
ed


giving
the
sample voltage

and t
he nominal
bulk resistance (sample voltage over applied current)
.
Since the temperature changes slowly and needs no special attention,
we
take

the average over each buffe
r (LabView automagically compensates and scales the thermocouple data).

The program interactively graphs the various waveforms and their Fourier transforms

(see
Figure 3
)
.
T
his ability to observe
the
signal as it proceeds through
the lock
-
in
is
a distinc
t advantage
of the
so
ftware
approach
. Furthermore, the plots display
the natural quantities of interest
:

resistance, time, temperature, etc
.;

no subsequent data analysis is required.

The

key to the extraordinary stability and accuracy of our
device
is th
a
t, once acquired, all signal processing is
accomplished
digitally
.

Since the reference

signal

is
internally
calculated
,
its
accuracy

is

limited only by the
floating
-
point resolution of the computer.
Reading and
amplifying the

input signal
introduces
a vari
ety of
physical and measurement

artifacts,
but once safely
within the computer
no further degradation

occurs
.

There are subtle signal
-
processing pitfalls that must be
avoided, such as
synchronization (phase error)
,
aliasing

(undersampling)
and

windowing ar
tifacts (discontinuities
at the buffer edges)
5
. We
align

acquisition
with
the
waveform generation trigg
er to
minim
iz
e

phas
e error, and
use continuous acquisition

to

prevent

window
ing

artifacts.
Continuous acqu
isition
allows reference frequencies up to
several kilohertz on our modest PC with a 250 kS/s DAQ
board. One may instead read non
-
continuous chunks
containing an exact number of waveforms
,
each
synchronous with the
waveform generation. This allows
referenc
e frequencies up to about ten percent of the
maximum sampling
frequency
.


Figure 2: Block Diagram for Lock
-
In Detector



Figure 3: LabView Front Panel

Results

Figure
4

shows a data set recorded as the sample
warmed up
from approximately 80 to 200 K using
an
applied current
of
10 mA at
1122

Hz. The input
was sampled at 72 kHz,
then
lock
ed
-
in

and
filter
ed

at 0.5 Hz.

The inset graph shows the initial portion of the
transition on an expanded
y
-
axis.

One can clearly
observe the shape and details of the transition near
115 K.
The transition is not sharp
,
a general
characteristic of high
-
t
ransition temperature
superconductors. Above the transition, we can see
that the resistance increases linearly with
temperature. Below the transition, we find
a
superconducting
resistance of

0 ± 0.02 microohms.

Conclusions

Using LabView, we have implemente
d a versatile,
low
-
cost digital lock
-
in amplifier
8
. The
device

shows negligible offset drift

and

is robust against
noise and interference


yet
it
requires
minimal
hardware

and may
be customized for
each task.
I
t
is capable of
10 nV sensitivity,
a
quality
factor of
Q

= Δ
f

/

f

= 10
5

or more
, and
noise rejection of ~
120 dB (
can extract
signals
from
noise up to ~10
6

times greater in
amplitude
).




1

For implementation details, circuit diagrams, and source code, please

see
http://w
ww.ph.utexas.edu/~phy453/lockin/

or
contact t
he authors:

Philip (Flip) Kromer (
flip@physics.utexas.edu
)
or
Roger Bengtson (
bengtson@physics.utexas.edu
).

2

Superconductivity:



G.C. Brown, J.O. Rasure, and W.A. Morrison,
American Journal of Physics
. 5
7(12), 1142
-
1144 (1989).



M.J. Pechan and J.A. Horvath,
American Journal of Physics
.
58(7), 642
-
644 (1990).



Semiconductor
kits are
available from
Colorado Superconductor
, 1623 Hillside Drive, Fort Collins, CO 80524.


3

Sources of, and defenses against, noise:



"Signal Enhancement"
(
http://www.srsys.com/html/a
pplicationnotes.html
, or
p.225

of their
catalog
).

Stanford Research
Systems, Sunnyvale, CA, 1999
.
A summary of fundamental noise sources.



S.J.
Shah,

Field Wiring and Noise Considerations
,”

National Instruments
,

Austin, TX, 1994
;

see
http://digital.natinst.com/appnotes.nsf/web/index
, #25
.



Low Level Measurements Handbook
, ed. J. Yeager and M.A.
Hrusch
-
Tupta.
Keithley Instruments
, Cleveland, OH, 1998.
An excellent introduction to precision measurement, and
freely available
up
on request
.



P. Horowitz and W. Hill, The Art of Electronics. Cambridge University Press, New York, 1980.

4

Lock
-
in detection:



M. Stachel, "
The Lock
-
in Amplifier: Exploring Noise Reduction and Phase
,
"

(
http://www.lockin.de/
)
.
An excellent web
-
based introduction to lock
-
in detection, complete with Java simulations.



P. Temple,
American Journal of Physics

43(9),
p
801

(1975).



"
About Lock
-
in Amplifiers
"

(
http://www.srsys.com/html/applicationnotes.html
).

Stanford Research Systems
, Sunnyvale,
CA, 1999. A functional de
scription of lock
-
in amplifiers
.



Lock
-
in Applications Anthology, ed. Douglas Malchow.
EG&G Princeton Applied Research
, Princeton, NJ, 1985. A
freely available guide to applic
ations of the lock
-
in analyzer.



D.W. Preston and E.R. Dietz, The Art of Experimental Physics
, pp 367
-
375
. Jo
hn Wiley & Sons, New York, 1991
.

5

Data Acquisition:



Data Acquisition Handbook
, ed. J. Yeager and M.A. Hrusch
-
Tupta.
Keithley Instruments
, Cleveland, OH, 1998. An
excellent introduction to data acquisition, and
freely available on request
.

6

We use
the

PCI
-
MIO
-
16E
-
4

(NI 6040E) multifunction I/O board; it has 16 12
-
bit, 250 kS/s analog inputs; two 12
-
bit, 1
MS/s analog outputs; and two 24
-
bit counters.
National Instruments
, 11500 N. MoPa
c Expressway, Austin, TX 78759

7

Our
front
-
end amplifier

is based on
Texas Instruments
'
INA114

precision instrumentation amplifier. Other suitable de
vices
include
Analog Devices
'
AD624

and Texas Instruments'
OPA111
.
A c
ircuit
d
iagram
is
available
on our website.

8

Our source code is freely available under the Gnu Public License; download at
http://www.ph.utexas.edu/~phy453/lockin/
.

0
2
4
6
8
10
12
14
16
18
50
70
90
110
130
150
170
190
Temperature (K)
Resistance (milliohms)
Liquid Nitrogen
Temperature 77 K
Onset of
superconductivity
Room Temperature
300K
Zero-point Resistance
-0.04
0
0.04
0.08
80
85
90
95
100
105
110
Normal conductivity:
graph is linear

Figure 4: Resistance versus Temperature for a High
-
Temperature Superconductor