Noise and Bandwidth

Nu uh, no how, Nyquist didn't say that. You have invented a whole new category of misuse of the Nyquist-Shannon sampling theorem which is not covered therein, but I suggest you read this anyway:

Get or make a true RMS meter, set the bandwidth, and...

Most oscilloscopes sample at a much higher bandwidth than the sampling rate. This is often the most correct thing to do (see my paper above), and when it's not, you're expected to be smart enough to gin up an anti- aliasing filter.

When you sample a broadband noise signal with an ideal sampler (meaning that the sampling process takes the value of the signal at an instant), then the total power of the sampled signal will, with few exceptions, be the total power of the unsampled signal. This holds regardless of sample rate. (The exceptions happen if that broadband signal has concentrations of energy at the sampling rate or multiples thereof -- if the signal has no tones in it, then the above holds true).

When you sample a broadband noise signal with a sampler whose bandwidth is greater than the signal's bandwidth (as you are doing with your O-scope), then the total power of the sampled signal is equal to the total power of the unsampled signal -- regardless of sampling rate.

So what you're really finding out is that sample bandwidth >= analog bandwidth, regardless of the sample rate.

--

Tim Wescott 
Wescott Design Services 
http://www.wescottdesign.com

take the 1.5Vrms divide by 10 for the gain then divide by sqrt(100MHz) [remember the result is per rtHz, NOT per rtradian, or something.

I got 15nVrms/rtHz how did you get 5.8??

Just to be clear, that was to be read with heavy sarcasm... :^)

Ah, nice article. And good formatting.

Or, as your article correctly states, not. After all, how else does it claim 350MHz and 150MS/s on the faceplate? :)

I was looking for a more analytical explanation. I'd do it myself, but it sounds like a pain e.g. doing Parseval's theorem integrated over a continuous frequency band. On the upside, it is nice to know it does what it's supposed to (mix together all frequencies, from Fs to light).

Now, I wonder if I can convince someone that I can measure true RMS of a noise signal with one single instantaneous measurement*...

(*Insert more sarcasm here. Obviously, a measurement has the usual sqrt(1/N) accuracy factor if nothing else, since statistics are like thermodynamics, you can't escape it. To use "like" lightly...)

Tim

--
Seven Transistor Labs 
Electrical Engineering Consultation 
Website: http://seventransistorlabs.com

On Thu, 20 Nov 2014 15:14:56 -0600, "Tim Williams" Gave us:

One must utilize a one bit quantum computer to analyze such single moment events. :-)

They, like useable fusion reaction systems, are only ten years away from perfection.

Once we get that computer though, we will not need the reactor, since the computer will be able to create and control a tiny matter/anti-matter collision, negating our needs for anything else. :-)

And that was how the big unbang got started.

"Hey, I'll give you a Nickel for every hole you can find in that incredible theory sweater you are wearing..."

I still like how GPS signals can get picked and read, all the way down right next to the noise floor.

Hmm, don't know where I pulled that number from. Must've mixed them up in my mind. The amp is actually 0.6mV RMS out of a 0.27mV RMS instrument baseline, or 0.54mV from the amp, or 54uV RMS input referred, or

5.3nV/rtHz.

It also measures 0.32mV RMS through an LNA, but the LNA's BW is less, so this isn't as useful. Accounting for the BW difference, it's about right.

Tim

--
Seven Transistor Labs 
Electrical Engineering Consultation 
Website: http://seventransistorlabs.com

I don't have time to be really correct in my math, but the outline of the proof goes like this:

First, I'm using "power" to mean "signal amplitude squared", i.e., I'm blithely ignoring impedance and voltage (or current, or velocity) units.

The average power of the signal is the integral of the signal's instantaneous energy over time, divided by the time period of the integral

-- so far so good. We get the RMS voltage by squaring the voltage, averaging, and taking the square root.

Now, if you ideally sample the signal's power (which is the same as sampling the signal and squaring), that sample will have a probability distribution (I think it's a Rayleigh distribution for Gaussian noise -- you can check on that). The mean of that probability distribution is exactly equal to the signal power (really! Honest! You can believe me -- after you go check your statistics book).

So if take a whole bunch of samples of the signal and average them, you will have the average power of the signal -- you don't even have to sample the signal evenly; you just need to get enough samples so that you're not just sampling one particular peak or trough.

Once you've verified that the distribution you're working with really _is_ the Rayleigh distribution you can also find it's variance; that'll give you a clue as to the minimum number of samples you need to take, although you'll need more if you're looking at a slowly-settling signal.

Well, you _can_, without sarcasm. You just get a measurement whose variance is high enough to make it useless. But as a power measurement you get a number whose expected value is correct (although I'm 99.44% sure that it is no longer right when you take the square root).

--

Tim Wescott 
Wescott Design Services 
http://www.wescottdesign.com

Yes, the noise wil be measured over the analog bandwidth of the front end of the scope. That will usually be much less than the Nyquist frequency Fs/2. It's common for a 100 MHz scope to sample at 1 GHz. So the 100 MHz is the measurement BW.

Assuming it really is 100 MHz.

If the sample set is too short in duration to capture low frequency noise, you should see a lot of variation in the reported RMS if you do multiple measurements. Sometimes you'll catch a high level, sometimes not.

--

John Larkin         Highland Technology, Inc 
picosecond timing   precision measurement  

jlarkin att highlandtechnology dott com 
http://www.highlandtechnology.com

+1

I'm a big fan of using two-section RC lowpass filters with HP 3400A or

3403A true-RMS meters. Well into the HF range, you can get very accurate cutoff frequencies this way, because the components are so good.

Also IME most 'scopes, and most spectrum analyzers, don't do a good job of measuring noise. For instance, your garden variety analogue spectrum analyzer will show a noise floor 2.5 dB below the true RMS value. (See HP/Agilent/Keysight/FuzzyNuts app note AN150.) You have to calibrate the daylights out of your system to be able to measure noise accurately.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs 
Principal Consultant 
ElectroOptical Innovations LLC 
Optics, Electro-optics, Photonics, Analog Electronics 

160 North State Road #203 
Briarcliff Manor NY 10510 

hobbs at electrooptical dot net 
http://electrooptical.net

You are presuming there is no antialiasing filter.

Yeah, if you just sample without a front end filter, noise above the Nyquist point gets folded back. But that kind of sampling is not Kosher.

Is that 2.5dB 'above' true rms value?

Can you provide exact URL to that paper?

Even the FFT of random noise in LTspice appears as 3dB larger than true rms, even with tones being EXACTLY rms.

I have no solution except to assume peak of a bin is the rms value, then after 'removing' all the peaks, to find the overall noise density function = sqrt(mean(vf.*conj(vf)))/sqrt(BW)/sqrt(2)

Without that sqrt(2) energy appears larger. Not just in 'pure' math solving, but same in LTspice. As an experiment, find the 'average' of noise energy of a random noise source and it will be around 3dB too high.

A VERY good 1Vrms gaussian white noise source in LTspice [very flat, does NOT have an apparent 2^10 repetition buried in there] is to make a current source drive with the following charactristic: I={sqrt(12/5)*(rand(time/dt)+rand(tstop/dt+1+2*int(time/dt))+rand(3*(tstop/dt+1)+3*int(time/dt)) +rand(6*(tstop/dt+1)+4*int(time/dt))+rand(10*(tstop/dt+1)+5*int(time/dt))-2.5) } that drives a 1 ohm resistor then look at the Vrms and the FFT 'average' noise floor. See. 3dB too high. [don't forget, no filter, 1 sample, and increase to 1,000,000] it reads about -47dB and should read closer to

-50dB. [sqrt(1/BW)]

If you add a tone current source, the tone's peak is EXACTLY Vrms.

note the Vrms is slightly less than 1Vrms, due to an artifact in the way LTspice calculates rms by 'averaging between values'

NoiseVoltageSource.asc

Version 4 SHEET 1 4876 2620 WIRE -448 -160 -576 -160 WIRE -288 -160 -448 -160 WIRE -256 -160 -288 -160 WIRE -576 -144 -576 -160 WIRE -448 -144 -448 -160 WIRE -576 -48 -576 -64 WIRE -448 -48 -448 -64 WIRE -448 -48 -576 -48 WIRE -448 -16 -448 -48 FLAG -448 -16 0 FLAG -288 -160 out SYMBOL res -592 -160 R0 SYMATTR InstName R1 SYMATTR Value 1 SYMBOL bi2 -448 -144 R0 WINDOW 3 24 28 Invisible 2 SYMATTR Value I={sqrt(12/5)*(rand(time/dt)+rand(tstop/dt+1+2*int(time/dt))+rand(3*(tstop/dt+1)+3*int(time/dt)) +rand(6*(tstop/dt+1)+4*int(time/dt))+rand(10*(tstop/dt+1)+5*int(time/dt))-2.5) } SYMATTR InstName Bnoise1 TEXT -1352 -64 Left 2 !.tran 0 {tstop} {tstart} {dt/10}\n.param Voff=0 sz=0.5\n.save V(out) TEXT -1352 -184 Left 2 !.options plotwinsize=0 TEXT -1352 -144 Left 2 !.param dt=5uS N=200000 \n.param tstop={N*dt} tstart={0} Fstop={1/2/dt} Fstart={1/N/dt} TEXT -1000 -152 Left 2 ;Valid range is 1Hz to 100kHz TEXT -368 -24 Left 2 ;1Vrms gaussian white noise generator

page 53 in particular.

Probably the same calculation is used, with the same results.

Tim

--
Seven Transistor Labs 
Electrical Engineering Consultation 
Website: http://seventransistorlabs.com

Geez, what's wrong with all you doubters?

My TDS460 does it just fine, and SDR receivers do it just fine.

The stated scope BW is 350MHz, and drops off gracefully above there. On the downside, it's a TDS460... on the upside, the performance is still classic Tek quality. It's no TDS2102 or whatever, with still-shitty menus but even worse interface (try panning) and ruined step response.

Of course, I can't capture single event transients in ET sampling, but I don't need to do that very often.

Tim

--
Seven Transistor Labs 
Electrical Engineering Consultation 
Website: http://seventransistorlabs.com

Very good point. However, there are a great number of applications that involve sampling, but are not Jewish. So you still have to actually analyze what the aliasing will do, and decide whether you really need an anti-aliasing filter.

--

Tim Wescott 
Wescott Design Services 
http://www.wescottdesign.com

The noise bandwidth is defined by the analog bandwidth of the whole system, amps/filters/sampler. It doesn't depend on the sample rate.

I measure GHz-bandwidth noise using an equivalent-time sampling scope that free-run triggers at about 100 KHz. I can add a coax lowpass filter to restrict the measurement bandwidth. It doesn't matter when you sample, as long as you do the math on a lot of samples. HP used to sell a 1 GHz BW AC analog voltmeter that basically used an oscilloscope sampler circuit in the front end. The sampler was triggered randomly to avoid aliasing against periodic inputs.

--

John Larkin         Highland Technology, Inc 
picosecond timing   precision measurement  

jlarkin att highlandtechnology dott com 
http://www.highlandtechnology.com

Thanks, Tim. Got a copy.

I looked at the 'core' calculation. Bet to a mathematician the explanation is obvious. I gave up and live with it.

For an even more hand-wavey explanation, consider the detector is peak reading (at least, the analog one is), so you have your sqrt(2) for the sine. Then something about statistics I think. Probably the same for the analytical / FFT case, since that's usually done in terms of peak sine components, but RMS noise is... lumpier.

(oops, premature posting)

Tim

--
Seven Transistor Labs 
Electrical Engineering Consultation 
Website: http://seventransistorlabs.com

On Fri, 21 Nov 2014 13:52:29 -0600, "Tim Williams" Gave us:

It's like... fractal, man...

OK... OK... It's like CHAOS, man... :-) Better?

Who needs that one bit Quantum Computer? When any old good (well structured observance) guess will do.

WE *are* that computer, and we just don't know it. Look at the savants.

"Badges... We don' need no stinking badges!!!"

On Fri, 21 Nov 2014 12:12:40 -0800, DecadentLinuxUserNumeroUno Gave us:

And as to those huge bags of data to process into information...

Think of this trek...

"It's full of stars..." --David Bowman, 2001, A Space Odyssey