The “lock-in” detection technique can extract, from a possibly noisy waveform, the amplitude of a signal that is synchronous with a known reference signal. This paper examines the effects of input noise on the output of a lock-in amplifier. We present quantitative predictions for the root-mean-square size of the resulting fluctuations and for the spectral density of the noise at the output of a lock-in amplifier. Our results show how a lock-in amplifier can be used to measure the spectral density of noise in the case of a noise-only input signal. Some implications of the theory, familiar and surprising, are tested against experimental data.
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For example, the voltage density of Johnson noise of a 4-kΩ resistor at room temperature is .
The data in this paper were taken using an SR530 dual-phase lock-in amplifier from Stanford Research Systems. Users of lock-ins which use square-wave demodulation will experience numerical differences such as 2/π compared to the derivations in this paper.
Our noise waveform was the Johnson noise of a 100-kΩ resistor at ambient temperature, amplified by voltage gain 600 and then filtered using 100-Hz high-pass and 10-kHz low-pass filters of two-pole Butterworth topology. The voltage noise density of the output in the vicinity of 1 kHz is predicted to be 24 μV/, and this was experimentally confirmed by separate analog and digital methods.
For this and other investigations in this paper, we chose to use only a one-pole filter at the lock-in's output, to conform with the model implicit in Eq. (19).
The rms fluctuations were otherwise measured just as in Sec. V A.
The Fourier spectra were obtained using an SR770 FFT spectrum analyzer from Stanford Research Systems. For all the spectra shown, the frequency span was fixed at 48.75 Hz, and the spectra obtained are lists of 400 Fourier magnitude coefficients obtained at 122-mHz spacing. The spectra were acquired using flat-top windowing for best estimation of the peak heights of spectral lines, and the spectra shown are equally weighted averages of 128 non-overlapping acquisitions.
