From June 2023, John Engelbrecht has a big electronics project going, recording 1/f noise originating in a dual-JFET, monolithic, Linear Integrated Systems integrated circuit.  (This part is a low-noise part, don't think it has special, high-noise output.)  The recording is targeted for a year-long record, in 3-hour pieces of data.  The reason for such a long record is that 1/f noise is supposed to get bigger (wider voltage swings) with time, without limit.  This may be true, but the voltage changes very gradually and takes a very long time to get anywhere.  1/f noise is widely described as being mysterious.  https://www.researchgate.net/publication/340937331_Origins_of_1f_Noise_in_Electronic_Materials_and_Devices_ A_Historical_Perspective     

Edoardo Milotti's 2001 1/f noise: a pedagogical review is a great paper, and shows 1/f noise is strange ways:  ocean tides, music, etc.

1/f noise in electronic circuits:  in resistors, and in semiconductors.  In the latter, "traps" are said to be the source.  The voltage of 1/f noise is in the microvolt range, so to see it you have to amplify a lot.  And to see 1/f noise for hours or longer, your amplifier has to be DC coupled, which is much harder than AC coupling.  But instrumentation amplifiers and chopper amplifiers are up to the task.

This web page starts at the nine-month point in the year-long recording.  I have a four-page document giving a briefing, Mar 23 2024 John Engelbrecht San Antonio partial results of 9-month 1_f recording LSK489 v2.odt.  Below are excerpts.

Above is a typical frequency spectrum of 1/f noise, in this case from my Tektronix DPO 2002B recording for 200 seconds, then Fourier-series analyzed by Python code.  Frequencies of the harmonics are indicated across the top of the chart, starting at 0.005Hz.  The harmonic voltages are discrete voltages, each at a frequency, and the virtual dots are connected by lines to make them more visible.  The negative slope is apparent, as 1/f noise always has a negative slope.  The dropoff at 4 Hz is due to an anti-aliasing filter, a 6-pole low-pass filter.

1/f Noise from an LSK489 (Linear Integrated Systems) Monolithic Dual-JFET Diff Amp, Amplified by Another 24900, DC Coupled Overall

How Mysterious is 1/f Noise?

John Engelbrecht San Antonio March 23 2024 johnenge@earthlink.net

An interim report at the nine-month point of a one-year recording of 1/f noise, with power continuously applied. The silicon traps are uninterrupted in whatever they do to create 1/f noise. Overall DC gain 334,000, no negative feedback. file: Mar 23 2024 John Engelbrecht San Antonio partial results of 9-month 1_f recording LSK489 v2.pdf

Results at the nine-month point:

1. The straight-line, negative slope of 1/f noise spectrum is easy to see, evident at just 50 harmonics and 7 bits of resolution. https://scistatcalc.blogspot.com/2013/12/fft-calculator.html is easy to process on 100 samples that are appropriately low-pass filtered to exclude aliasing.

2. On the other hand, time-domain 1/f noise is hard to see on a digital panel meter. A skeptical observer wonders when it is going to “take off” since 1/f noise is supposed to be unbounded in time. Charting is needed, then a comparison to white noise (high-pass filtered, above 200Hz) makes the 1/f noise evident in the time domain.

3. 1/f noise looks very bounded with an amplifier-chain gain of 334,000, during a year, for monolithic-pair, JFET first stage, staying within a ±25V band at the output. Qualifications:   oven keeps first and second stages within 0.2ºC band, 0.1% thin-film resistors, stable supplies, no drop-outs of supplies, DC coupling. Ten years can be expected to stay within ±35V band. 1000 years probably stays within ±50V.  Longer periods are invalidated by corrosion, if not by room-temperature diffusion of the P-N dopants in the silicon semiconductor.       The slowness of 1/f noise growth with time is similar to the sum of 1/n from n=1 to infinity, the "harmonic series."  At n=5, the sum is above 5.  At n=1000, the sum is only 7.48, and at n=1000000, it has grown only to 14.4.  Mathematicians say it is unbounded or diverging, but it is hard to see that the sum could get to 1000.

4. Even an op amp shows 1/f noise, when operated open-loop. Demo can be set up in an hour on solderless breadboard.  But the op amp's output is quite temperature sensitive.

5. Popcorn noise, probably from BJTs, contaminates my circuit 10% of the time, but at popcorn frequency < 2Hz. But popcorn noise does not detract from the negative, straight-line slope of 1/f harmonics.

Objectives in a later report:

1. Prove or disprove straight-line slope of 1/f noise on log-log chart down to 0.032μHz

2. Analyze 789,000 harmonics of Fourier Series. FFT is not up to this, instead use the sine-cosine coefficients integration method with Python.

3. Interpret the concept that 1/f noise is unbounded, interpret “per root Hz” when frequencies are in the millihertz, use of DAC to adapt offset voltage on month-to-month basis, note the mistaking of instrumentation “drift” for 1/f noise

4. Review literature about 1/f noise being stationary or non-stationary

5. Suitability of Raspberry Pi for one year of data recording, oven for temperature stability, derivation of temperature compensation, popcorn noise but not in LSK489. Two R Pis, each with its own I2C ADC, can do the job.

The Lowest Harmonics are Plagued by Noise

Though one wishes to see a constant slope all the way to the lowest harmonics, those low harmonics are victims of randomness and are unlikely to cooperate when one draws a straight line. The lowest harmonics are prisoners of randomness.

What do twenty runs of data reveal about the lowest harmonics?

Use Python to run 20 different spectrums on 20 parts out of a 125000-capture dataset by DSO, each part being a selection of 125000/20 = 6250 samples. Extract only the lowest 30 harmonics on each. Collect those 30 harmonics for each of the 20 runs, and chart in spreadsheet. See if there is a clustering that makes drawing lines obvious.

Answer: there is no clustering of slopes, just various negative slopes. And four of the Fourier series have positive slope at the lowest harmonics, not negative slope!

This dooms any long-term (months to a year) recording of 1/f noise, followed by Fourier analysis to get the spectrum, indeterminate about whether there is a consistent 1/f slope that extends to the first harmonic, which would be 0.03μHz for a year. There is a good chance that my ultimate Fourier series, covering a year of data taking, will not display a nice, straight, 1/f line below the 20th harmonic. Such is randomness.

Potential Publication in IEEE Instrumentation and Measurement Society

The IEEE Instrumentation and Measurement Society accepts informal papers about practical subjects for its monthly publication.  I would like to offer results of my 1/f noise study, and I think it would need co-authoring with a person who has an advanced degree, who can contribute on the math and statistics side of this endeavor, especially about making a year of data periodic so that the Fourier series even exists.  If a co-author is a professor, I acknowledge that a professor would need an article that makes an advance on the art, and the article would need to be peer reviewed.  I am open to that.

I seek such a co-author.

Details:  see below, "Theory and Statistics."

John Engelbrecht IEEE Life Senior Member    BSEE UT Austin 1972    johnenge@earthlink.net 512 773 9266    12311 Culebra Rd APT 6104 San Antonio TX 78253    March 23 2024

The Theory and Statistics... 

...that John Engelbrecht needs help with (need a coauthor) for an IEEE paper, possible title "Practical 1/f Noise Recording for a Year."

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reference Wikipedia  pink noise:  There is no known lower bound to background pink noise in electronics. Measurements made down to 10−6 Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour.

reference Wikipedia  Flicker_noise

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Milotti 2002  1/f noise: a pedagogical review

https://www.researchgate.net/publication/2167452_1f_Noise_A_pedagogical_review/link/02e7e51501765e6c9f000000/download?_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6InB1YmxpY2F0aW9uIiwicGFnZSI6InB1YmxpY2F0aW9uIn19

Milotti's conclusion:  Do we have by now an "explanation" of the apparent universality of flicker noises? Do we understand 1/f noise? My impression is that there is no real mystery behind 1/f noise, that there is no real universality and that in most cases the observed 1/f noises have been explained by beautiful and mostly ad hoc models.

John Engelbrecht Mar 29 2024    But Milotti mentions electron "traps" only with electron-tube cathodes and resistors, not with bulk semiconductor crystal defects, the main source for JFET 1/f noise.  Bulk-semiconductor traps, being atom-sized, have more potential to be mysterious.  (But are resistor traps also atom-sized?)

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John Engelbrecht does get the negative-slope, straight line through Fourier harmonics, but only when using log-log chart.  But commercial semiconductor 1/f noise chart on a data sheet is often with linear vertical scale, nV/root Hz.  Are these at odds with each other?

1/f noise power is supposed to approximately increase by 10 when harmonic freq is reduced to 1/10.  Does that mean harmonic VOLTAGE increases by 10 when the frequency goes down by 100?  (For the reason that power varies by voltage squared.)  

Confusion about "per root Hz."  When frequencies are above 50 Hz, I can believe "per root Hz," but below 1Hz or below 1mHz, what does that mean?  I think my constant sampling period (sample freq 27.5 samples per second) makes "per root Hz" transparent.  (I may naively chart days, weeks, and months of 20.4-second averages of samples at 27.5Hz sample freq.)

But how to convert my 20.4-second averages to nV/root Hz?  So that I come out with an apples-to-apples comparison to nV/root Hz spec from the LSK489 datasheet:  en, Noise Voltage, 3.5 nV/√Hz at VDS = 15V, ID = 2.0mA, f = 10Hz, NBW = 1Hz.     (Thinking out loud:  A solution may be to record some noise at exactly 10Hz, with 1Hz bandwidth [9.5Hz to 10.5Hz], and that offers a calibration point.  But can I build an op-amp bandpass filter with such a narrow bandwidth?  I built a 6-pole 4Hz low-pass filter, so maybe I can, but the LPF has a lot of ringing.  Is it better to take existing Fourier harmonics, both the sine and cosine, and throw away all but those falling within 9.5Hz to 10.5Hz, and rebuild the time domain?)

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John Engelbrecht

What is the importance of POWER, as in Wikipedia pink noise, "In pink noise, there is equal energy per octave of frequency."   Or "The power spectrum of pink noise is 1/f  only for one-dimensional signals."

An electrical engineer thinking about power thinks in terms of volts-squared/ohms, P=V^2/R.

If I amplified my 1/f noise (from my gain-334000 amplifier) with a voltage-gain-of-one, class A, DC-coupled amplifier and applied it to a one-ohm resistor, it would be delivering -1.5V or +2.3V at various instants, and that would cause power of several watts in the one-ohm load.  

Is this the same sense of POWER that Wikipedia mentions?  

I have a DC-offset control on my gain-334000 amplifier that offsets the input-offset voltage of the LSK489 diff amp.  (Vio is in the low-millivolt range.)  The offset control has been constant for nine months of 1/f recording.  If I were to change this offset, the output would make a step change and the one-ohm resistor power would change.  Surely this is a different sense of POWER than Wikipedia mentions.

John needs understanding of the power that Wikipedia talks about.

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Stationarity of 1/f noise can be wide-sense or strict-sense, says Wikipedia on Stationary Process.  The math gets intense.  John needs help interpreting the math, to say whether real, 1/f (not simulated) noise for a year is either strict-sense stationary, wide-sense stationary, or non-stationary.   

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John Engelbrecht knows that the lowest harmonics of 1/f noise rarely fall along a straight, negative-slope line, while harmonics above 50 have a neat fit.  The reason is that low harmonics are themselves noisy, and there is no way to average the variation by repeating the measurement--who wants to wait 10 years for 10 repetitions of a one-year 1/f measurement, for the sake of averaging the lowest harmonics?

Is there a better way to say "low harmonics are themselves noisy"?  It seems too hand-wavy.

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John Engelbrecht has a vague idea that a Fourier series on real data works best when the resolution of voltage measurement tracks the number of harmonics.  Example:  16-bit ADC has resolution of 65536 levels.  If 65536 samples are taken, Fourier harmonics are valid.  If the sampling goes as long as a million samples, but the volt resolution remains at 16 bits, is there a consequence of a noise floor that creeps into the Fourier series?  John thinks so.  (Noise floor is always seen with RF spectrum analyzers.)

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When one uses the SciStatCalc FFT calculator, it delivers a Fourier spectrum that has the same number of coefficients as the pasted input sample count.  The output is mirrored around the middle.  John E has been discarding all coefficients past the midpoint, and the low half charts as a reasonable 1/f spectrum, as plotted on log-log chart.  

Is it normal for an FFT spectrum to be so mirrored?  Is it normal for a Fourier spectrum to have half the coefficients of the number of input samples, even when the Fourier analysis is by the integration method with sin and cos?  (As John is using for sample size over 200.)