How exactly does the low-pass filtering in a lock-in amplifier correspond to a time-averaging?

Question

In this white paper of Zurich instruments two ways of explaining a lock-in amplifier are shown: low-pass filtering and time-averaging. Whereas the explanation with time-averaging can easily be done with complex numbers, the low-pass filtering seems to be more complicated (since it involves Fourier analysis to get from the time to the frequency domain?): https://www.zhinst.com/others/en/lock-amplifiers

A larger bandwidth in the frequency domain seems to correspond to a smaller time interval for time-averaging in the time domain. Therefore, there is a trade-off between good time resolution and high SNR. The signal buried in noise can vary over time, therefore it is important to set the right bandwidth. Most modern lock-in amplifiers work digitally by applying a low-pass filter, right? But couldn't they also perform the mathematical operation in the time domain (multiplying the sin and cos / complex numbers) ?

My question is if somebody can explain this relationship between the time and frequency domain more exactly. I couldn't really find any information about that. In the white paper it is also not explictly explained. And what exactly does the lock-in amplifier do when it applies a digital low-pass filter?

Answer 1

Time- and frequency-domain representations of lock-in amplifiers are equivalent. The signal processing algorithms to implement a lock-in amplifier are carried out in the time domain. However, its functionality becomes more intuitive when described in the frequency domain.

A lock-in amplifier receives a periodic signal at a known frequency and outputs the amplitude and phase of that signal with respect to a reference at the same frequency. This is done by a dual-phase (0° and 90°) demodulator as shown in the figure below. Each branch of the demodulator includes a mixer followed by a low-pass filter (LPF).

When mixing two signals with the same frequency ω, you generate two components at frequency difference and frequency sum i.e., 0 and 2ω. The LPF is supposed to remove the 2ω component from the signal and only allow the DC component to pass through. Therefore, the filter bandwidth must be much less than 2ω in order to effectively suppress the undesired frequency component. Moreover, the LPF rejects the out-of-band noise to improve the signal-to-noise ratio (SNR) of the measurement. The lower the bandwidth is, the higher the SNR. In a nutshell, the LPF of a lock-in amplifier has two tasks: 1) suppressing the unwanted frequency components and, 2) rejecting the out-of-band noise.

In the time-domain, a low-pass filter is represented and implemented by a weighted integrator in which the signal is multiplied by a weight function and then the outcome is integrated over the time continuously. The weight function is determined by the filter type. For instance, a sinc filter has a uniform weight function which results in a simple moving averager; while, for a Butterworth filter, the weight function is exponential and thus the integrator is like an exponential moving averager. The time-constant associated with the exponential weight function is inversely proportional to the filter bandwidth as shown in Table 1 of the White Paper. Standard digital lock-in amplifiers use the Butterworth filter which can be efficiently implemented by IIR filters in the time domain. The temporal response of the Butterworth filter of several orders is provided here.

In summary, the digital signal processing algorithms used in lock-in amplifiers including mixers and low-pass filters are implemented in the time domain, while their functionality is better explained, interpreted, and understood by the users in the frequency domain.