CHEM 524 -- Course Outline (Part 10)—2005 modification

VII. Signal to noise considerations (Text - Chap. 5)

A. Noise definitions:

• Average signal of several measurements: E = S E_i/n
• Standard deviation (rms excursion from mean): E_rms = s_E = [S (E_i-E)²/(n-1)]^1/2
• random -- non-correlated to other aspects of measurement
• fundamental -- intrinsic to detection method
• chemical -- errors in sampling

1. Types

• white (Gaussian), ubiquitous
• pink (1/f), -- diode detectors (e.g. MCT) show this, often roll off at ~1KHz
• interference (at f) –could be many things, e.g. line frequency, radio stations, neighbors etc.
• flicker (~signal) – typically chemical or due to instrument stability issues

1. evaluate by understanding noise power spectrum –particular to every experiment/instrument
2. use to design modulation or detection scheme – choose optimal frequency to operate

2. Amplitude transfer function (book: Table 5-1, Fig. 5-4)

• mathemeatical representation of device efficiency as function of frequency: H(f) = E_out/E_in
• band pass: Df =_-/^- |H(f)|²df -- frequency range with attenuation < 3dB or H(f) > 0.707
• effect on noise: s_E² = _-/^- P(e)|H(f)|²df where P(e) -- density spectrum -- _-/^- = integral, sorry!
• white noise: s_E² = P(e)Df
• time constant -- low pass at f = 0 -- DC/stability trade off
• wait 5t to make measurement, rule of thumb: t < 1/10 measurement time
• integrating circuit effectively faster, can improve S/N in same time and can reject interfering signals

B. Quantum/shot noise -- square root dependence on signal level

due to random photon field and random probability of emission of at interface

• PMT: photon noise: f = n_P, s_P = n_P^1/2 , S/N = f /s_P = n^1/2,
• cathode: (S/N)_n = n_C/s_C = K n_P/(K n_P)^1/2 = (Kn_P)^1/2 , in current: i_C/s_C = (i_Ct/e) ^1/2
• anode: s_E = [2eDf(1+a)mGE]^1/2, m - multiplier, G - gain in V/A, E - signal (R^.f),
• a - multiplier add on noise [vary 0.1-0.5, good PMT ~ (d-1)^-1, d - gain per dynode]

C. Others

1. Flicker, due to sample or blank vary and especially source or temperature fluctuations that impact the signal level, level ~ light signal: s_F ~ E_S
2. Dark current (e.g. field emission dynode or amplifier output level) -- excess noise, additive
3. Quantization noise (finite digital resolution) -- s_q = q/12 , if s >q/2 – q = quantization level (if less then limited by readout resolution, s_q=q/2)
4. Thermal (Johnson) noise - (thermal fluctuation of e^- in resistor) s_J = (4kTRDf)^1/2 –
1. cooling, narrowing band pass help,
2. lowering R also, but usually costs signal (in volts)
5. Uncorrelated sources, sum the noise: (read Section 5.4, 5.5)

D. Bottom line -- understand Figures relating S/N and E (fig. 5.6), A/s_A vs A (fig. 5.7)

1. Emission—different noise sources approach the ideal shot-noise limit

a. Shot noise limit: S/N = i_s/[K(i_s+i_d)]^1/2—K=2eDf(1+a)—improve reduce Df, dark current, i_d

b. Signal limit: S/N = [i_s/K]^1/2—improve with more S

c. Flicker limit: S/N = z^-1—becomes constant at high S

2. Absorption—ratioing signals makes more complex: s_A = 0.43s_T/T from A = -0.43 ln T

a. (S/N)^-1 = s_A/A = -s_T/TlnT --new form for plots, lower is better in this view

b. 0%T limiting conditions—dark or amplifier or readout limited—min 0.43 A

reduce dark noise, IR this dominates—cool detector

c. Shot noise limited—min 0.87 A –reduce bandwidth, increase light level

d. Flicker—since constant, improves with absorbance, but not real, since losing light

E. Enhance S/N

1. Filtering ---time domain

1. average e.g. multiplex -- time avg. idea, integrate signals in each channel) - multiple (n) scan average, increase S/N = n^1/2
2. time constant—attenuate the high frequency components to enhance the DC

2. Filter -- frequency domain (Df select signal) –

a. best: fully digitize signal, FT to frequencies,

b. multiply by H(f), back transform

3. Adjust levels –

a. shot (raise to flicker limit),

b. dark (cool detector),

c. flicker (adjust instrument, e.g. Double beam -- counter drift, long time changes – measure signal and blank simultaneously)

4. Photon counting -- best for low light level -- (S/N)_PC/(S/N)_i = [f_d(1+a)]^1/2, f_d discriminator coeff., (1+a) term gives 5-25% improvement
5. Modulation -- demodulate with lock-in, boxcar, or correlation –

a. Modulation can be major advantage when dark noise and 1/f noise limited—additive noise

b. all discriminate against noise which is broad band and no time correlation to signal (except flicker) - (Fig. 5-9)