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.

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Looks like a real signalProbably noise, or not very useful;

a hint of a signal

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

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xSN s

SN RSD

Rev. Sci. Inst., 1966, 37, 93-102.

MostlySignal

MostlyNoise

signal

Std. Dev.

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.

4rms kTR f

rms = root-mean-square noise voltagek = Boltzman’s constantT = temperature

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

white noise Can be reduced by narrowing bandwidth

Slows instrument response time More time required for measurement

Reduced by lowering T Common to cool detectors

298K77K lowers thermal noise by factor of ~2

4rms kTR f

rms = root-mean-square noise voltagek = Boltzman’s constantT = temperature

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

10.5 e-/s

10 e-/s

11 e-/s

2rmsi Ie f irms = root-mean-square noise currentI = average direct currente = electron chargef = 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 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.jpghttp://farm2.static.flickr.com/1227/578199978_17e8133c7c_o.jpg

Analog filtering Low pass filter removes

high frequency noise Thermal and shot noise

High pass filter removes low frequency noise Drift and flicker noise

Narrow-band electronic filters

Example of low-pass filter

High freq removed.Low freq preserved

(passed).

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 amplifierscontinued 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

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J. Chem. Educ., 1979, 56, 148-153.

Ensemble averagingcontinued Signal-to-noise improves with increasing

number of data sets

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S SnN N

N = rms noise n = number of replicate scansi = number of replicate scans in other data set

# Scans, n Relative S/N

1 1 4 216 464 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

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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

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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