Analytical figures of merit, noise, and S/N ratio Chemistry 243.

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

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

R = resistance of element (W)Df = bandwith (Hz) = 1/(3tr)tr = rise time

Thermal (Johnson) noise continued

Dependent upon bandwidth (Df) 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

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

R = resistance of element (W)Df = 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

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11 e-/s

2rmsi Ie f irms = root-mean-square noise currentI = average direct currente = electron chargeDf = 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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Sn

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