Analytical figures of merit, noise, and S/N ratio

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

3. What is signal and noise?

4. 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.

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. 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

8. 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 • 298K77K 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

9. 11 e-/s 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)

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” noise—more 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. 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-noise

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 High freq removed. Low freq preserved (passed). Example of low-pass filter

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 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.

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 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

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

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