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### 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

Looks like a real signal

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

Signal-to-Noise Ratio (S/N)Mostly

Signal

signal

Std. Dev.

Mostly

Noise

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.

nrms = root-mean-square noise voltage

k = Boltzman’s constant

T = 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

N2(l): bp=77K

nrms = root-mean-square noise voltage

k = Boltzman’s constant

T = temperature

R = resistance of element (W)

Df = bandwith (Hz) = 1/(3tr)

tr = rise time

10.5 e-/s

10 e-/s

Shot noise- Arises from statistical fluctuations in quantized behaviors
- Electrons crossing junctions or surfaces
- Independent of frequency
- Example: current

irms = root-mean-square noise current

I = average direct current

e = electron charge

Df = 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

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

Enhancing signal-to-noiseGrounding 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
- Thermal and shot noise
- High pass filter removes low frequency noise
- Drift and flicker noise
- Narrow-band electronic filters

High freq removed.

Low freq preserved

(passed).

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 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

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

Ensemble averagingcontinued

- Signal-to-noise improves with increasing number of data sets

N = rms noise

n = number of replicate scans

i = number of replicate scans

in other data set

# Scans, n Relative S/N

1 1

4 2

16 4

64 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

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

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