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

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?

• Most (if not all) lock-in amplifier do the maths using sin and cos so I'm not sure how that affects your question. Jan 27, 2021 at 15:51
• Essentially, low pass filtering IS time domain averaging. You use averaging to reduce short-term fluctuations to see long term trends. That is precisely the same as reducing high frequency content. The simple un-weighted average, (sum of N samples / N) is a rectangular impulse response, and will have a sin(x)/x frequency response (aka sinc filter).
– user16324
Jan 27, 2021 at 16:33
• I followed the link you provided, scrolled down to the white paper on the principles of operation, and figure 5 therein seems to address your question. The white paper is quite good: I downloaded it for careful reading later.
– Ed V
Jan 27, 2021 at 18:03