Analytical figures of merit,
noise, and S/N ratio
Chemistry 243
Noise
A signal is only of analytical value if it can be
definitively attributed to the species/system of interest
in the presence of noise.

Probably noise, or not very useful;
a hint of a signal
2.5
2.5
2
2
signal
signal
Looks like a real signal
1.5
1
1.5
1
0.5
0.5
0
20
40
60
data point
80
100
0
20
40
60
data point
80
100
What is signal and noise?
Signal-to-Noise Ratio (S/N)

Signal-to-noise ratio (S/N) is a measure of the
quality of an instrumental measurement


Ratio of the mean of the analyte signal to the
standard deviation of the noise signal
High value of S/N : easier to distinguish analyte
signal from the noise signal
Mostly
Signal
signal
S
N
S
N

x
Std. Dev.
s

1
Mostly
Noise
R SD
Rev. Sci. Inst., 1966, 37, 93-102.
Where does noise come from?

Chemical noise


Temperature, pressure, humidity, fumes, etc.
Instrumental noise
Detector and post-detector
noise






Thermal (Johnson) noise
Shot noise
Flicker (1/f) noise
Environmental noise
Popcorn (burst) noise
Microphonic noise
Thermal (Johnson) noise


Random motions of charge carriers
(electrons or holes) that accompany thermal
motions of solid lattice of atoms.
Lead to thermal current fluctuations that
create voltage fluctuations in the presence of
a resistive element

Resistor, capacitor, etc.
 rm s 
rms = root-mean-square noise voltage
k = Boltzman’s constant
T = temperature
4 kTR  f
R = resistance of element (W)
f = bandwith (Hz) = 1/(3tr)
tr = rise time
Thermal (Johnson) noise
continued

Dependent upon bandwidth (f) but not f itself


Can be reduced by narrowing bandwidth



white noise
Slows instrument response time
More time required for measurement
Reduced by lowering T

Common to cool detectors

298K77K lowers thermal noise by factor of ~2
 rm s 
rms = root-mean-square noise voltage
k = Boltzman’s constant
T = temperature
4 kTR  f
R = resistance of element (W)
f = bandwith (Hz) = 1/(3tr)
tr = rise time
N2(l): bp=77K
Shot noise

Arises from statistical fluctuations in
quantized behaviors



Electrons crossing junctions or surfaces
Independent of frequency
Example: current
11 e-/s
10.5 e-/s
10 e-/s
irm s 
2 Ie  f
irms = root-mean-square noise current
I = average direct current
e = electron charge
f = bandwidth (Hz)
Flicker (1/f) noise


Magnitude is inversely proportional to the
frequency of the signal
Significant at frequencies lower than 100 Hz


Long-term drift
Origin is not well understood

Dependent upon materials and device shape


Metallic resistors have 10-fold less flicker noise than
carbon-based resistors.
Referred to as “pink” noise—more red (low
frequency) components
Environmental noise


Comes from the surroundings
Biggest source is “antenna” effect of
instrument cabling
J. Chem. Educ., 1968, 45, A533-542.
Noise contributions in
different frequency regimes
Frequency
independent
Supposedly 1/f—
mostly at low
frequencies
Occurs at discrete
frequencies
Enhancing signal-to-noise

Hardware methods




Grounding and shielding
Difference and
Instrumentation
Amplifiers
Analog Filtering
Lock-In Amplifiers
 Modulation and
Synchronous
Demodulation

Software methods

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Ensemble averaging
Boxcar averaging
Digital filtering
Correlation methods
Grounding and shielding

Surround circuits (most critical conductors)
with conducting material that is connected to
ground


Noise will be picked up by shield and not by circuit
Faraday cage
http://www.autom8.com/images_product/table_farady_benchtop.jpg
http://farm2.static.flickr.com/1227/578199978_17e8133c7c_o.jpg
Analog filtering

Low pass filter removes
high frequency noise


High pass filter
removes low frequency
noise


Thermal and shot noise
High freq removed.
Low freq preserved
(passed).
Drift and flicker noise
Narrow-band electronic
filters
Example of low-pass filter
Lock-in amplifiers

Modulation

Translate low frequency signal
(prone to 1/f noise) to a high
frequency signal which can
amplified and then filtered to
remove 1/f noise
Mechanical chopper
Lock-in amplifiers
continued

Synchronous demodulation


Converts AC signal to DC signal synchronous with
chopper—follows reference
Low-pass filtering

Back converts high frequency DC signal to return filtered,
low frequency output.
Ensemble averaging to
increase S/N

Averaging multiple
data sets taken in
succession

Divide sum of data
sets by number of
data sets
n
S
Sx 
i
i 1
n
J. Chem. Educ., 1979, 56, 148-153.
Ensemble averaging
continued

Signal-to-noise improves with increasing
number of data sets
S N 

n

n S
N

i
N = rms noise
n = number of replicate scans
i = number of replicate scans
in other data set
# Scans, n
1
4
16
64
Relative S/N
1
2
4
8
Boxcar averaging






Smoothing irregularities and increasing S/N
Assumes signal varies slowly in time
Multiple points are averaged to give a single
value
Often performed in real time
Detail is lost and utility limited for rapidly
changing samples
Boxcar integrators commonly used in fast
(pico- to microsecond) measurements using
pulsed lasers.
Moving average smooth
Similar to a boxcar average, but changes in time
signal
(arb units)

0
200
400
600
data point
800
1000
Average
Standard
Deviation
S/N
Relative
S/N
Original
100.2
6.0
16.6
1
4 point
100.2
3.0
33.4
2.0
16 point
100.2
1.5
67.1
4.0
100
point
100.2
0.6
160
9.7
Downside of moving average
smoothing
140
130
signal
(arb units)
120
110
100
90
80
70
60
50
0
200
600
400
data point
800
1000
Digital filtering

Fourier transform


Convert data from time- to frequency-domain,
manipulate to remove higher frequency noise
components, regenerate time-domain signal
Polynomial data smoothing


Moving average smooth
Least-squares polynomial smoothing