White noise standard

[TimFox]

0.1 to 10 Hz:  It is conventional to measure 1/f pink noise per decade of frequency (each decade contributes an equal amount of noise power to the white thermal-noise spectrum).

[Kalvin]

<snip>
I could write a test sketch (it's programmed in the Arduino environment using Teensyduino), but unfortunately I don't have a real oscilloscope or frequency analyzer (nor access to one right now) to analyze and quantify the output.

You can use Matlab or GNU Octave for getting the spectrum. Just pass the sequence through a zero-order hold, and compute spectrum. If you want, you can apply some filtering to the zero-order output.

Alternatively, you can record the samples into an audio file, and use some software for analysis. Even LTSpice can be used for reading audio data and performing analysis.

[Kalvin]

Hello everyone!

What can I use as a white noise standard to verify my low-noise amplifier? This amplifier is inteded to measure the white noise. But how can I check it? I say about cheap standard. I do the amplifier for myself, it's not my job, so I would want to have noise source say 10 uV rms for as low cost as possible. Of course, I can take 1G resistor, it give me just ~10 uV rms in range 0.1..10 Hz. But I don like such impedance source: 1 G. My amplifier input impedance must be higher several times.

Sorry if I ask a question that had been discussed before. I used Google search but could find anything worth.

You can measure your amplifier's frequency response by sweeping a sine wave. No need to use white noise source for that.

[MrYakimovYA]

You can measure your amplifier's frequency response by sweeping a sine wave. No need to use white noise source for that.

Yes, I can. But it's very interesting to me to watch reals white noise on oscilloscope or measure it with DMM.

[RoGeorge]

One of my forever postponed projects:
- get noise from a random process in the physical world, for example a PN junction
- pass the signal through a comparator to get a digital stream of random 0/1
- pass the stream through a von Neumann debiasing (or extractor) to get a mathematically guaranteed white noise distribution:  https://mathoverflow.net/questions/152107/proof-of-von-neumanns-debiasing-algorithm

Now that we have (discrete) ideal white noise, we can generate arbitrary shapes of probability distributions (https://en.wikipedia.org/wiki/Inverse_transform_sampling), and get Gaussian noise, pink noise, etc. 

This is an animated example from Wikipedia (getting Gaussian noise from white noise), just that the above way would make possible to get custom shaped noise of guaranteed distributions starting from the physical randomness of an unknown distribution (physical randomness as in not a pseudo-random).


Image from:  https://en.wikipedia.org/wiki/Inverse_transform_sampling

[Nominal Animal]

<snip>
I could write a test sketch (it's programmed in the Arduino environment using Teensyduino), but unfortunately I don't have a real oscilloscope or frequency analyzer (nor access to one right now) to analyze and quantify the output.

You can use Matlab or GNU Octave for getting the spectrum. Just pass the sequence through a zero-order hold, and compute spectrum. If you want, you can apply some filtering to the zero-order output.

As I already explained, numerically the sequence is essentially indistinguishable from true uniform random numbers (i.e., white noise).  If it wasn't, it wouldn't pass BigCrush.  (It isn't cryptographically secure, though; so given a sequence, there may be ways to predict what the next number is.)

A DFT of any subsequence will have a (pseudo-)random spectrum, but it –– every bin (except the zeroth/DC one) in the DFT –– will and do tend to the same average and the same standard deviation, depending only on the size of the DFT and the scaling of the samples.  (I have verified this already, of course.)

I just do not have the equipment to measure the real-world output (including DAC properties, DMA clock jitter), to quantify it as a white noise source.

[David Hess]

You can measure your amplifier's frequency response by sweeping a sine wave. No need to use white noise source for that.

Yes, I can. But it's very interesting to me to watch reals white noise on oscilloscope or measure it with DMM.

I measure the noise on my analog oscilloscope using the tangental method to check the calibration of my DMM.

https://www.eevblog.com/forum/beginners/measuring-amplifier-noise-with-scope/
https://youtu.be/5Rk8I5BT2KU

[Kalvin]

<snip>
I just do not have the equipment to measure the real-world output (including DAC properties, DMA clock jitter), to quantify it as a white noise source.

As you have already analyzed the numerical sequence of the random number generator, and it is producing expected spectrum (white noise in this particular case), then it is a white noise generator. The other factors (like DAC properties and DMA jitter) are just implementation dependent imperfections. These imperfections just should not be significant at the frequency range of interest.

For audio frequencies it is quite easy to produce sufficiently good white noise using digital LFSR-techniques, as these imperfections will affect the high frequency components. By increasing the output sample rate will push these imperfections higher in the frequency spectrum, until they will be filtered out (you should have some kind of reconstruction low-pass filter at the output of DAC). As the required sample rate and frequency spectrum goes up, the practical limitations of digital techniques will start to appear. At RF frequencies semiconductor junction noise will be typically used as a noise source.

[Nominal Animal]

you should have some kind of reconstruction low-pass filter at the output of DAC

Exactly.  I know that in theory this should be almost perfect white noise source up to ~ 50kHz or more, but not being able to quantify it exactly makes me hesitant to make any claims about it.  Also, to make it useful for real-world measurements, there should be a high-pass filter at very low frequencies (10 Hz, perhaps).

Some time ago I did some careful examination of simple pulse density modulation (which is often used in direct digital synthesis), using an adder.  If you restrict the output dynamic range, for example 8-bit to 16..240 (down to 87.5% of the dynamic range available, essentially as if it was just 7.8-bit and not full 8 bit), there will be at least one transition every 16 clocks, pushing the quantization noise quite high in the spectrum, making it easier to filter out.  (In other words, that when using PDM, you can push the quantization noise to higher frequencies by using clock frequency higher than you need, restricting to a smaller dynamic range that does not include 0% or 100%.  If your output stage has symmetric rise and fall times, this yields a very linear output after a low-pass filter.)

This, too, is something that looks good and valid on paper, but I think one really needs to test and measure it in real life before relying on it.

(I know it sounds strange coming from someone who does software and is not an EE, but at the core, I'm a physicist.  Even when I do simulations or simulators, the first question after the first results always is Does this make any sense?)

[Kalvin]

Also, to make it useful for real-world measurements, there should be a high-pass filter at very low frequencies (10 Hz, perhaps).

This can be done in digital domain, too. A simple IIR-type DC-removal filter (with selectable corner frequency) is very easy to implement as part of LFSR sequence generation. Requires only few arithmetic operations.

https://www.embedded.com/dsp-tricks-dc-removal/

This kind of IIR-filter will affect the phase somewhat at the lower frequencies. Probably this won't be an issue in practice, but can be checked with Matlab / GNU Octave by running the LFSR-sequence into a FIR-filter and IIR-filter (with identical corner frequencies), and comparing the output spectrums.

[Kalvin]

(I know it sounds strange coming from someone who does software and is not an EE, but at the core, I'm a physicist.  Even when I do simulations or simulators, the first question after the first results always is Does this make any sense?)

That is called real-life wisdom. 

[Nominal Animal]

Also, to make it useful for real-world measurements, there should be a high-pass filter at very low frequencies (10 Hz, perhaps).

This can be done in digital domain, too. A simple IIR-type DC-removal filter (with selectable corner frequency) is very easy to implement as part of LFSR sequence generation. Requires only few arithmetic operations.

https://www.embedded.com/dsp-tricks-dc-removal/

This kind of IIR-filter will affect the phase somewhat at the lower frequencies. Probably this won't be an issue in practice, but can be checked with Matlab / GNU Octave by running the LFSR-sequence into a FIR-filter and IIR-filter (with identical corner frequencies), and comparing the output spectrums.

Yup; Teensy LC is 32-bit Cortex-M0+ with a single-cycle 32×32=32 multiply operation, so the DAC being just 12-bit, there is ample precision to do some filtering without worrying about affecting the output.

As an aside, for FIR filters, I actually made an example standalone HTML page (you can save it locally and use it in your browser) for this here.  Just put the coefficients, say -2 1 1, into the top bar (that reads FIR Filter Coefficients when empty), and it'll show you its frequency (magnitude) and phase response.  I intended it as an example of how we could use the very well optimized JavaScript engines in common browsers, to implement local tools using HTML+CSS+JS, run locally in our browsers even without any network connection; and not as an actual FIR filter analysis tool, though.