noisenoise-spectral-density
Basically my question is:
I doubt noise density goes to infinity because we can reach the limit f→0 in any DC circuit in contrast to the limit f→∞ (which is an idealization because all circuit behaves as a low pass for enough f).
If noise density is limited, at which f, and how does it decay?
The problem in your reasoning is that you do not show the complete path of the signal. More specific the impedance level of the signal.
You are right in that you cannot have both a high impedance and a low noise. If you want low noise you must keep the impedance low. Simple as that.
In the two circuits you have drawn it is unclear what the impedance is of the source which you use to feed a signal to your amplifier. Assuming that the AC coupling capacitors are large and that this source impedance is low (for example: 50 ohms) then the noise will be low !
Why ? Because the noise generated by the 100 Mohm DC bias resistors will be shorted by the AC coupling capacitors and that low source impedance. So in this situation the effective signal impedance (at a certain frequency) is much lower than 100 Mohm. Resulting in a low noise.
If the 50 ohms source impedance was not there that noise current would multiply by the 100 Mohm of the resistor itself resulting in a high noise level.
You can do calculations on this more easily by considering the noise current generated by the 100 Mohm resistors. That current will be multiplied by the signal source impedance (for example, 50 ohms) resulting in a small noise voltage !
So the circuit on the right is no better than your circuit on the left. Carefully read how they measured that low noise and try to figure out what the impedance level of the input signal was. I guarantee you that they will have used a source impedance such that the noise of the 100 Mohm DC biasing resistors can be neglected (a very low source impedance, they might even have shorted/grounded the inputs !). In that circuit the noise of the FETs should be dominant as these should determine the lowest possible noise level (at least in a properly designed amplifier).
Another way of posing this question; most ΣΔ ADCs appear to specify a peak-to-peak input-referred noise for a single measurement under various conditions (e.g filter mode, data rate, PGA gain, Vref, etc.). Given a known input noise spectrum as described above, how do I compute a noise amplitude to compare with the ADC input referred noise to know whether I can resolve my input noise in a given configuration?
The easiest way to do this is to call the 1/f noise flat
Source: https://www.edn.com/electronics-blogs/the-signal/4408242/1-f-Noise-the-flickering-candle-
Since we already know how to handle white noise sources, it's easier to work with them. For example with this amplifier I would draw a line at 10e-6 from 0.1 to 10Hz
There isn't a way (currently) to mathematically sample a 1/f noise distribution mathematically. Why? because 1/f noise is more like a random walk, and the noise values now depend on the previous noise values. A close approximation is to generate a white noise distribution and then filter it with a low pass filter.
There are ways to simulate 1/f noise, and you could match up the amplitude, as described in this paper: 1/f noise: a pedagogical review.
Figure 25: 1/f noise generated with the algorithm described in section 10, amplitude vs. time(both linear scales and arbitrary units); starting from top, = 0, 1, 1.5 and 2.
Best Answer
At lower frequencies less common events become part of the signal,on scales of seconds hearbeats and footsteps on a scale of weeks there are electrical storms, on a scale of months there are seasonal effects, on a scale of years earthquakes etc...
At \$2.3\times10^{−18}Hz\$ one must include the big bang :)