CHEM 524 -- Course Outline (Part 10)

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

• white (Gaussian),
• pink (1/f),
• interference (at f) and
• flicker (~signal)

1. evaluate by understanding noise power spectrum
2. use to design modulation or detection scheme

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

• 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
• whilte 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

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

• 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]

1. Flicker, due to sample or blank vary (eg. source fluct.) ~ light signal:
2. Dark current (e.g. field emission dynode or amplifier output level) -- excess noise
3. Quantization noise (finite digital resolution) -- s_q = q/2 , q- quantization level
4. Thermal (Johnson) noise - (thermal fluctuation of e^- in resistor) s_J = (4kTRDf)^1/2 -- cooling, narrowing band pass help, 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. Filtering ---time domain (avg or use time constant, eg. multiplex -- time avg idea, integrate signals in each channel) - multiple (n) scan average, increase S/N = n^1/2
2. Filter -- frequency domain ( select signal) -- best: fully digitize signal, FT to frequencies, multiply by H(f), back transform
3. Adjust levels - shot (raise to flicker limit), dark (cool detector), flicker (adjust instrument, eg. Double beam -- counter drift -- signal and blank simultaneous)
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 -- all discriminate against noise which is broad band and no time correlation to signal (except flicker) - (Fig. 5-9)