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Basically my question is:

enter image description here

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?

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    \$\begingroup\$ Good question. At very low frequencies, this looks like integration of offset or drift which certainly would have no particular limit nearing DC (long term integration.) But there are different mechanisms operating when you get near long term drift areas and they don't all "look like" 1/f mechanisms. So I think a good answer here would provide an intriguing sense of understanding in these regions, which I frankly lack. As I said, good question. Hopefully, a good answer will bring out the salient factors getting very near DC. \$\endgroup\$ – jonk Mar 29 '18 at 22:22
  • \$\begingroup\$ It seems to me that you are implying that to have a finite area the curve needs to decay, as if there were an infinite f interval from say 1 Hz to 0 Hz. This is not the case, there is only a 1 Hz interval there. The 'extension' to 10^-infinity is just a mathematical artifact of the logarithmic scale. Also, as Jasen said, the limit f->0 is an idealization too. The lowest frequency attained so far is 1/(age of the universe). \$\endgroup\$ – Sredni Vashtar Mar 29 '18 at 22:46
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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 :)

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    \$\begingroup\$ You are right, but I was referring exclusively to the 1/f noise that appears in ranges like the one shown in the plot. Also the Big Bang is like a delta so its spectrum might be flat haha \$\endgroup\$ – user171780 Mar 30 '18 at 12:44
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\$ Does ~f ~go \to 1/\infty ?\$ unknown , unproven but close to it

\$\dfrac{1V}{\sqrt{Hz}} @10^{-14} Hz\$ equals.... wait for it

= 31,709.8 centuries .. now that's a little flicker but which century?

Is this the probability of gamma wave hitting electrons out of orbit?

In audio it is called "Pink Noise" and it exists everywhere in nature.

The true cause is not known, but it exists for as long as you measure it even the last 60 years, as has been done.

What scientists in China do know is that , the origin of 1/f noise is the interaction between the system and random effect.

In dust particle sizes , we see the same histogram of qty vs size if we equate the frequency of occurence of dust particles in a unit volume. How small can they go? only particle physicists can answer this and they keep finding small particles with more energy required to find them.

1 M.Keshner,1/f noise,proceedings of the IEEE, 70(1982), pg212-218
[2] B.Mendlebrot and R.Voss, Noise in physical System and 1/f Noise,
Elsevier Science, 1983, ch. Why is Fractal and when should noises by scaling?, pg31-39
[3] R.F.Voss and J.Clarke, 1/f Noise in music and speech, Nature, 258(1975), pg31-38
[4] B.B.Manderbrot, Some noise with 1/f spectrum, abridge between direct current and white noise, IEEE Transaction on Information Theory, IT-13(1967), pg289-298 [5] B.B.Manderbrot and J.W.V.Ness, Fractional Browinian motions, fractional noises and application, Siam Review, 10(1968), pg422-437
[6] V.Solo, Intrinsic random functions and the paradox of 1/f noises, SIAM Journal of applied Mathematics, 52(1992), pg270-291
[7] X.C.Zhu and Y.Yao, The low frequency noise of HgCdTe photoconductors, Infrared Research, 8(1989)5,pg375-380. (in Chinese)
[8] M.K.Yu, F.S.Liu, 1/f noise theory of 1/f noise, Physics Acta, 32(1983)5, pg593-606, (in Chinese)
[9] J.Clark and G.Hawking, Phys. Rev. B14(1974)2862
[10] J.Kurkijarvi, Phys. Rev. B6(1972)832
[11] 高安秀树,分数结,地震出版社,1994,pg63-65
[12] Xu Shenglong , 1/f noise exploration , Technical Acoustics , 1997 , pg63-67
[13] Xu Shenglong , Statistical dynamics of 1/f noise, Infrared Technology, 25(2003), pg63-67
[14] Xu Shenglong,Re-study of statistical dynamics of 1/f noise,China Measurement Technology,33(2007), pg79-83
[15] W u Peijun , The Low Frequence 1/f Voltage Noise of the Ti Film Microbridge, CHINESE JOURNAL OF LOW TEMPERATURE PHYSICS,16(1994), pg350-353

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Having read the Journal of Solid State Circuits for decades, wherein the various causes of noise of all forms is a crucial discussion for phase-locked-loop performance, I'll provide some recollections from an ATT or IBM presentation at the annual ISSCC (conference) approximately 2005.

There are various trapped-charges on the crystal surface and also buried inside the crystal at various non-ideal non-cubic "dislocations" where various perfect regions meet at imperfect atomic patterns.

These trapped-charges have relaxation times, from the microseconds to the seconds (and perhaps longer). Thus as individual electrons escape from these tiny storage locations, we see tiny impulses. Finite bandwidth measuring systems, or our circuits, round off these impulses into "noise".

And as signal polarities reverse, charges move back into these charge-traps, again in the form of tiny impulses.

Apparently there are more charge-traps for the very long duration relaxation times, and we get more power at the lower frequencies.

Cleaner silicon surfaces reduce the 1/F noise.

And silicon boules (the huge almost-pure 12" by 24" beasts provided by zone refiners) with fewer internal dislocations reduce the 1/F noise.

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It's the red line. Not the green.

I like to think of 1/f noise as thermal noise, and heat moving around different parts of a silicon die (or transistor). If you've ever watched glowing embers in a fire, it could be analogous to those temperature fluctuations, but on a different scale (at least that's how I think of 1/f noise).

There is no way to really know here is what AOE (Art of electronics 3rd edition by Horowitz and Hill) says:

You often hear talk about low-frequency noise power conforming to a “1/f law,” as if there’s some statutory requirement involved. You might at first think that this cannot possibly be true, because (you say to yourself) a 1/f power spectrum cannot continue forever, since it would imply unlimited noise amplitude. If you waited long enough, the input offset voltage (or input current, in this case) would become unbounded. In fact, the popular mythology of a low frequency noise catastrophe (to which your thinking would have fallen victim) is quite without merit: even if the noise power density continues as 1/f all the way down to zero frequency, its total noise power (i.e., the integral of noise power density) diverges only logarithmically, given that \$ \int f^{−1}df = \log f \$. To put some numbers to it, the total noise power in a pure 1/ f spectrum between 1 microhertz and 10 Hz is only 3.5 times greater than that between 0.1 Hz and 10 Hz; going down another six decades (to 10−12 Hz), the corresponding ratio grows only to 6.5. Put another way, the 1/ f total noise power, going all the way down to a frequency that is the reciprocal of 32,000 years (when Neanderthals still roamed the planet, and there were no op-amps), is just six times greater than that of the usual datasheet 0.1–10 Hz “low-frequency noise.” So much for catastrophes. To find out whether the low-frequency noise of real opamps continues to conform to a 1/ f spectrum, we measured the current noise spectrum of an LT1012 op-amp all the way down to 0.5 millihertz,130 with the result of Figure 8.107. As we remarked above, this op-amp is unusual in that its current noise density in rises faster than the usual 1/√f (pink noise) for a decade around 1Hz; but even so it settles back to the canonical pink noise, and ultimately becomes something closer to “pale white” ( f −1/4 or slower). You could conclude that this demonstrates the unphysical nature of 1/ f behavior all the way down to zero. But there’s another possible explanation, namely that this opamp is afflicted with some mild burst noise. That would be consistent with the “faster-than-pink” slope around 1Hz (recall the burst-noise spectrum in Figure 8.6), and it would also lead you to incorrectly attribute a “slower-than-pink” slope at the low frequency end of the spectrum in Figure 8.107.

enter image description here
Source: Art of electronics
enter image description here
Source: Art of electronics

The most interesting graph to me is 8.106, which shows a times series of a low noise amp with different filtering. The largest amplitude noise is 100Hz-1kHz, and then 0.1-1Hz. If this graph were continued to 0.01-0.1Hz it would probably not increase much (and that test was not ran because it would have taken too long or the filter would have been difficult to build. But do a thought experiment, take the 0.1Hz-1Hz and stack it from end to end a few times. The amplitude would not increase but you just increased the time, so if you were to do an FFT, you would not see the amplitude increase and at some point it comes back to DC which would be a value around zero. Why zero? because that is where the average value of the noise is.

enter image description here

In my line of work, I have ran FFTs on the months scale (I don't have any on hand) but they do flatten off and don't go up forever.

A second thing to note is you will have many other sources of noise on the half hour to days scale, you are entering the relm of temperature noise. Air conditioners, the diurnal cycle, weather and pressure start to effect low level measurements.

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