CHAPTER 15 -- MEASUREMENT, SIGNAL, AND NOISE
Experimental physics relies on detecting signals in the presence of noise.
15.1 SIGNAL-TO-NOISE RATIO (SNR)
SNR is the ratio of the power of the desired signal to the power of the background noise.
SNR = P_signal / P_noise
Often expressed in decibels (dB):
SNR_dB = 10 * log10(P_signal / P_noise)
For voltage amplitudes:
SNR = A_signal / A_noise
SNR_dB = 20 * log10(A_signal / A_noise)
Improving SNR:
- Signal Averaging: Summing N repeated measurements increases the signal by N, but noise (uncorrelated) increases by sqrt(N). Thus, SNR improves by sqrt(N).
- Bandwidth Reduction: Filtering out frequencies where the signal is not present reduces noise power.
15.2 DETECTION LIMITS
The detection limit is the smallest signal that can be reliably distinguished from noise (typically SNR > 3).
- Factors limiting detection:
- Thermal Noise (Johnson-Nyquist): V_noise = sqrt(4 * k_B * T * R * Delta_f). Generated by thermal agitation of electrons in conductors.
- Shot Noise: Due to the discrete nature of charge carriers (photons or electrons).
15.3 SPECTROSCOPY
Spectroscopy is the study of the interaction between matter and electromagnetic radiation as a function of wavelength or frequency.
- Absorption Spectroscopy: Measuring dip in intensity as light passes through a sample.
- Emission Spectroscopy: Measuring light emitted by excited atoms.
15.4 LINE BROADENING
Spectral lines are not infinitely sharp; they have a finite width.
Mechanisms:
1. Natural Broadening: Due to the uncertainty principle (Delta_E * Delta_t >= h_bar/2). Short-lived states have broad energy uncertainties.
2. Doppler Broadening: Thermal motion of atoms causes Doppler shifts, widening the observed line distribution.
3. Pressure/Collisional Broadening: Collisions interrupt the emission process, effectively shortening the lifetime of the state.
4. Inhomogeneous Broadening: Variations in the local environment (e.g., magnetic field inhomogeneities in NMR) cause different parts of the sample to resonate at slightly different frequencies.
