On the Wikipedia page for 1/f noise (at the bottom of the page) it suggests the noise can be reduced if the signal of interest is at DC. DC signals suffer from significant 1/f noise, so one method of removing this is to modulate the signal at some higher frequency and use a lock-in-amplifier to detect the now modulated signal synchronously. However, to my understanding, lock-in-amplifiers then convert the modulated signal back into a DC signal, surely again limited by the 1/f noise as before, so how is this method helpful for DC signals?
Experimental Physics – How to Remove $1/f$ Noise with Lock-In Amplifiers
electrical engineeringelectronicsexperimental-physicsnoisesignal processing
Related Solutions
From your description of the experiment (please correct me if my assumptions are wrong), it sounds like your apparatus consists of the application of a controlled stress to the sample (and the sensor), and the resulting strain in the sensor is measured. Whenever the stress applied by your apparatus changes, it will take some time for the system to settle to it's new equilibrium. It could be that sampling at $1 Hz$ is allowing plenty of time for equilibration, but sampling at higher frequencies you are recording the oscillations of the system as it has not yet settled.
One way to test would be to run the experiment without changing the applied force, just recording the strain at various sampling rates, and looking to see if the noise spectrum still depends on the sample rate in the way you describe. If it does, then the noise is a result of the frequency dependence of the electronics. If it does not, then the noise is resulting from the physical behavior of the sample
There's no magic solution. Your options are (in something like an order of preference):
- Prevent the 50 Hz from entering your signal in the first place by (a) making your setup less sensitive to the 50 Hz field, and (b) removing sources of the 50 Hz field.
- Filter out the 50 Hz with a notch filter.
- Subtract the 50 Hz signal by using a magnetometer (i.e. coil of wire) to measure the ambient 50 Hz field and experimentally determine the transfer function (i.e. phase) from this measurement to the 50 Hz line in your signal. (This sort of feedforward is always finicky since it will break if the transfer function changes.) This is a special case of Wiener filtering.
- Tolerate the 50 Hz.
50/60 Hz problems will be quite difficult to diagnose in a forum like this, since finding the problem usually involves quite a bit of fiddling around with the experimental setup.
Basic steps towards mitigation include:
- Make sure you are using shielded coaxial cables for single-ended signals, and twisted pair for differential signals.
- Seek out and destroy any ground loops! Sometimes this means connecting the shield of a coaxial cable only on one end. (For instance, you are using a differential amplifier to subtract the two halves of the photodiode: is there a ground-loop there? Can you shorten those cables, or twist them together?) The first thing to do is to simply move cables around while watching the 50 Hz line on a spectrum analyzer. Sometimes you can find the offending cable quite easily in this manner.
- Turn off the room lights, and/or make sure your experiment is shielded from line-powered lights.
- Move AC-powered equipment away from your experiment, especially anything involving CRTs, motors, or switching power supplies. You could use a simple magnetometer (coil of wire hooked to high-impedance differential input of spectrum analyzer) to hunt for the worst sources of 50 Hz field.
- Verify that the 50 Hz is not on the laser light itself, and that the 50 Hz field is not itself moving your mirrors (for instance, if they are suspended and magnetic).
Best Answer
Say the original signal is $y$ with noise spectrum $s(f)$. Let $A$ be the gain of an amplifier, and let $z(f)$ be the noise introduced by the amplifier at the output. A d.c. amplifier of gain $A_{dc}$ would produce at its output approximately $$ A_{dc} (y + s(0)) + z(0). $$ There is more than one way to modulate the signal so as to use a lock-in amplifier. One way is to chop the signal, so that it appears and disappears altogether at the chosen frequency and phase. In this case the output is approximately $$ A\left( y + s(f)\right) + z(0). $$ Another method is to modulate some parameter $x$, in which case the output is something like $$ A\left( \frac{dy}{dx} + s(f)\right) + z(0). $$ Broadly speaking, the d.c. amplifier amplifies the d.c. noise whereas the lock-on amplifier amplifies the noise at the chosen frequency $f$. So if the latter is smaller (which it almost always is if you pick $f$ sensibly) then the lock-in method is superior. In practice you would only use the amplifier when $z(0)$ does not dominate $A s(f)$.
(Thanks to Jamie1989 for pointing out an omission in the first version of this answer.)