We get SNR in two ways: the amplitude of applied force and the integration time. So we are limited in two ways: stability of the lock to applied forces and time of locklosses / calibration fluctuations.
At the sites, you probably know that we blow our spectrum out of the water with the calibration lines, with SNRs of about 100 on the scale of about 10 seconds. For us this might be impossible, since we aren't as quiet.
If we want 1% calibration on our sweeps, we'll need 0.01 = Uncertainty = sqrt( (1 - COH^2)/(2 * Navg * COH^2) ), where COH is the coherence of the transfer function measurement and Navg is the number of measurements at a specific frequency. This equation comes from Bendat and Piersol, and is subject to a bunch of assumptions which may not be true for us (particularly, that the plant is stationary in time).
If we let Navg = 10, then COH ~ 0.999.
Coherence = Gxy^2/(Gxx * Gyy), where x(t) and y(t) are the input signal and output signal of the transfer function measurement, Gxx and Gyy are the spectral densities of x and y, and Gxy is the cross-spectral density.
Usually SNR = P_signal / P_noise, but for us SNR = A_signal / A_noise.
Eric Q and Evan H helped me find the relationship between Coherence and SNR:
P = Pn + Pc, Pn = P * (1 - Coh), Pc = P * Coh
==> SNR = sqrt( Pc / Pn ) = sqrt( Coh / 1 - Coh )
From Coh ~ 0.999, SNR ~ 30.
Question for Craig: What does the SNR of our lines have to be? IF we're only trying to calibrate the actuator in the audio band over long time scales, it seems we could get by with more frequency noise. Assuming we want a 1% calibration at 50-500 Hz, what is the requirement on the frequency noise PSD curve?