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

2. 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. Looks like a real signal Probably noise, or not very useful; a hint of a signal

3. What is signal and noise?

4. 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 Rev. Sci. Inst., 1966, 37, 93-102. Mostly Signal Mostly Noise signal Std. Dev.

5. Where does noise come from? Chemical noise Temperature, pressure, humidity, fumes, etc. Instrumental noise

6. Detector and post-detector noise Thermal (Johnson) noise Shot noise Flicker (1/f) noise Environmental noise Popcorn (burst) noise Microphonic noise

7. 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. rms = root-mean-square noise voltage k = Boltzmans constant T = temperature R = resistance of element ( ) f = bandwith (Hz) = 1/(3t r ) t r = rise time

8. 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 298K 77K lowers thermal noise by factor of ~2 rms = root-mean-square noise voltage k = Boltzmans constant T = temperature R = resistance of element ( ) f = bandwith (Hz) = 1/(3t r ) t r = rise time N 2 (l): bp=77K

9. 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 i rms = root-mean-square noise current = average direct current e = electron charge f = bandwidth (Hz)

10. 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 noisemore red (low frequency) components

11. Environmental noise Comes from the surroundings Biggest source is antenna effect of instrument cabling J. Chem. Educ., 1968, 45, A533-542.

12. Noise contributions in different frequency regimes Frequency independent Supposedly 1/f mostly at low frequencies Occurs at discrete frequencies

13. 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

14. 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

15. 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).

16. 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

17. Lock-in amplifiers continued Synchronous demodulation Converts AC signal to DC signal synchronous with chopperfollows reference Low-pass filtering Back converts high frequency DC signal to return filtered, low frequency output.

18. 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.

19. Ensemble averaging continued 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 4 2 16 4 64 8

20. 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.

21. Moving average smooth Similar to a boxcar average, but changes in time AverageStandard Deviation S/NRelative S/N Original100.26.016.61 4 point100.23.033.42.0 16 point100.21.567.14.0 100 point 100.20.61609.7

22. Downside of moving average smoothing

23. 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