Lock-in Amplifier
Lock-in amplifier: demodulation gain, ENBW, noise rejection, and SNR improvement.
ENBW
0.2500 Hz
Detuning Δf = 0.000 Hz
Output X (rms)
0.707 µV
Output noise 5.00 nV rms
SNR in → out
0.224 → 141
SNR Improvement
×632.5
√(B_in/ENBW) = ×632.5
X = g · (A/√2) · cos φ · |H(Δf)|, noise = e_n · √ENBW (g = 1 sine, 2√2/π square reference)
A ±1 square reference also detects odd harmonics: g_k = 2√2/(kπ) at Δf_k = |f_s − k·f_r|, k = 3, 5, … At f_s = 2m·f_r the harmonics 2m ∓ j beat at the same |Δf| and add: √(g_a² + g_b² + 2g_a·g_b·cos 2φ) · |H(Δf)|
RC cascade: ENBW = 1/4τ, 1/8τ, 3/32τ, 5/64τ (N = 1–4) | Butterworth: ENBW = (1/4Nτ)/sin(π/2N)
SNR improvement = g · |cos φ| · |H(Δf)| · √(B_in / ENBW)
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Ideal multiplier with a unit-power sine or square reference, white input noise and an order-N output filter (cascaded RC or Butterworth). A square reference also detects odd harmonics k·f_r with gain 2√2/(kπ); the strongest harmonic is shown, and at f_s = 2m·f_r the two harmonics that beat at the same |Δf| are added as phasors. A detuned signal is shown by its beat envelope. The square-reference noise penalty assumes the input noise bandwidth B_in ≫ f_r (many odd harmonics in band). Reference phase noise, dynamic reserve, harmonic pickup and 1/f noise are left out.
- Scofield J. H. (1994). Frequency-domain description of a lock-in amplifier. American Journal of Physics 62(2), 129–133. Demodulation, output filter and noise bandwidth. doi:10.1119/1.17629
- Stanford Research Systems (1993). SR830 DSP Lock-In Amplifier manual. ENBW 1/4T, 1/8T, 3/32T and 5/64T for 6–24 dB/oct output filters. thinksrs.com