Noise lab

Thermal (Johnson–Nyquist) noise. The open-circuit voltage density of a resistance R at absolute temperature T is √(4kTR) (k = 1.380649 × 10⁻²³ J/K), flat in frequency, Gaussian, independent of any current. It is the fluctuation–dissipation theorem at work: whatever dissipates must also fluctuate (Nyquist, Phys. Rev. 32, 1928). Into a matched load it delivers kT per hertz, −174 dBm/Hz at 290 K. The generator draws independent Gaussian samples with σ = √(4kTR·fs/2).

Shot noise. A dc current of independent carriers crossing a barrier has a density √(2qI). The generator counts Poisson-distributed electrons per sample (mean I/(q·fs)), so at low counts the histogram is skewed and discrete, as it really is.

Flicker and brown noise are synthesised in the frequency domain: complex Gaussian coefficients in every FFT bin scaled to the density (4kTR·fc/f for 1/f, 4kTR·(f₀/f)² for brown), inverse transformed. The record is Gaussian, has exactly the target spectrum on average, and loops without a click. The measured slope comes from a least-squares fit to a Welch (Hann, 50% overlap) estimate.

Popcorn noise is a two-state random telegraph signal (switching rate r each way) on thermal noise; its autocorrelation is e^−2r·τ and its spectrum a Lorentzian (Machlup, J. Appl. Phys. 25, 1954). Sums of such traps with spread time constants give 1/f (McWhorter, 1957).

Adding noise. Uncorrelated sources add in power; with correlation ρ, v = √(Σvᵢ² + 2ρΣvᵢvⱼ). The equivalent noise bandwidth of an nth-order Butterworth filter is f₃·(π/2n)/sin(π/2n) — 1.571, 1.111, 1.047, 1.026 for orders 1 to 4 — and of n cascaded identical RC poles fp·√π·Γ(n − ½)/(2Γ(n)) — 1.571, 1.220, 1.155, 1.128 times their −3 dB bandwidth. A density e·√(1 + fc/f) integrates to e·√(fc·ln(f₂/f₁) + f₂ − f₁) (the closed form in opamp-real). Gaussian noise is outside ±3.3σ 0.1% of the time, hence peak-to-peak ≈ 6.6 × rms.

Signal chain. Every noise generator — the source’s 4kT·Rs (or a photodiode’s shot noise and shunt resistance), the preamp’s e_n (split into its white and 1/f parts) and i_n·|Z_s| (with Z_s = Rs
jωL for a pickup), a transimpedance amplifier’s 4kT/Rf and e_n·|1/Z_f + 1/Z_s| (the "e_nC" noise that rises with frequency), later stages' e_n — is carried to the output by the gain after it (single-pole preamp and gain-stage bandwidths, Butterworth filters) and integrated numerically over the band. With an ADC the band stops at Nyquist; analog noise above Nyquist aliases in, spread over 0…fs/2; the ADC adds white noise of rms (full scale/2√2)·10^−SNR/20 over 0…fs/2 (6.02N + 1.76 dB if no SNR is given). The op-amp figures are the catalogue of opamp-real, the same model the circuit simulator's noise analysis uses. The model ignores feedback-network resistors, op-amp gain-bandwidth limits and the preamp’s input impedance loading the source; for those, simulate the stage.

Friis. F = F₁ + (F₂ − 1)/G₁ + (F₃ − 1)/(G₁G₂) + …, Te = T₀(F − 1), T₀ = 290 K (Friis, Proc. IRE 32, 1944) — the same formula as the decibels tool’s chain. Passive losses have NF equal to their loss at 290 K. The noise floor is k·(T_ant + Te)·B.

Averaging and lock-in. Averaging N sweeps cuts uncorrelated noise by √N. The lock-in multiplies by 2·sin and 2·cos of the reference and low-passes (or averages): the error of X is 2σ·√(ENBW/fs) for white noise of rms σ sampled at fs, with ENBW = 1/(4τ) for one RC pole, 1/(8τ) for two, fs/2N for an average of N samples. Oversampling uses converters-lab's simulation. The chopper demo modulates the input with a ±1 square wave around an amplifier with offset and 1/f noise, demodulates and low-passes.

Shielding and differential signalling is a qualitative model with round numbers — about 40 dB for a grounded shield against electric fields, about 34 dB for twisting against magnetic loops, common-mode rejection limited by the CMRR or the imbalance — meant to show which fix addresses which coupling, not to predict a real installation (see Ott, Electromagnetic Compatibility Engineering, and Morrison, Grounding and Shielding).