# Quantifying white noise with a spectrum analyzer

The setup of my problem is as follows. I have an experiment in which I want to apply a noisy voltage with a white (flat) spectral density over a large frequency range, say 0 to 7 GHz for example. Not only do I want it to be white, I want to be able to tune its amplitude A. I plan to do so by using an AWG and generating Gaussian noise with zero mean and a variance of square root A; one can derive that for gaussian noise with a vanishing mean

$$S(f) = \sigma^2 = A$$

Moreover, I have access to an AWG with a sample rate of up to 50 GSamples, so the frequency range should be attainable as well.

One problem I now have however, is that I need to quantify how well the amplitude I set on the AWG actually corresponds to what is produced. This is because I do not actually generate a new time trace with different amplitudes, I simply produce 1 trace (I can make it 16 Gpoints long) and I change the peak to peak voltage it places the points into. That makes it a bit unintuitive to see what V_pp corresponds to what A.

So my idea was to use a spectrum analyzer and simply measure the spectral density that is being output for a specific V_pp. It'll allow me to verify that the spectral density is actually flat as well. So this is what I started doing, but I am running into trouble due to my limited understanding of how to operate the spectrum analyzer.

The problem is that I do not know how to choose the resolution bandwidth and the video bandwidth in such a way that the resulting spectrum is as accurate as possible. What I mean by that is that, by changing these bandwidths, the actual amplitude of the spectral density measured by the spectrum analyzer also changes, quite considerably. So how do I decide which values make the most sense?

From what I read about the spectrum analyzer, the resolution bandwidth allows for the discrimination of signals with closely spaced frequency components. So as my noise is supposed to be white, should I make this bandwidth as small as possible, to discriminate as many frequency components as possible?

Similarly, the video bandwidth determines the capability to discriminate between two different power levels, as far as I understand. Should this then also be set as low as possible?

It seems obvious to me that my interpretation of the above cannot be correct, as in that case most scenarios would simply boil down to setting these bandwidths to low values. There has to be something more nuisances that I am missing, and this is what I am asking your help for.

As an aside, I will be applying the voltage to a resistor to produce a current, which will be used to produce a flux, which will be sent through squid loops in a Josephson junction. So I suppose there is also the question of if the noise the spectrum analyzer sees (both in the frequency domain and the amplitude domain) is trivially related, or if there are filters and such which complicate things entirely. However, I think at this stage this is not too important. Knowing exactly what signal I sent in is a good starting point before thinking about what arrives.

• So maybe I'm lost in the biblical length of this question but, in short, is your question this "how do I use a spectrum analyser to prove the flatness of white noise". – Andy aka May 26 '16 at 12:48
• I'm sorry for that, I'll try to make it a bit shorter. You're right that it looks like a massive wall of text. My question is not so much how I prove the flatness, but how do I use a spectrum analyzer to accurately determine the amplitude of the spectral density. What I mean is that the different settings of the spectrum analyzer change the amplitude you measure, while (the way I understand it anyway) the AWG actually outputs noise with a specific spectral density amplitude. Or, how do I know if the white noise is at -6dBm or -4 dBm when the settings of the analyzer change the amplitude – user129412 May 26 '16 at 12:52
• power = U²/R. U²=\sigma². R= spectrum analyzers impedance. Is Ohm's law what you're asking for? – Marcus Müller May 26 '16 at 12:54
• No, my question is regarding the choice of the resolution bandwidth and the video bandwidth, and how they change the power reported by the spectrum analyzer. Going from voltage to power is not the issue, knowing if the power reported can be trusted is. In the end I have a parameter in my system that is linearly proportional to the aforementioned amplitude A. – user129412 May 26 '16 at 12:55

When measuring noise you must always specify the bandwidth in which you measured it !

As you lower the Resolution Bandwith (RBW), the noise level you see on the Spectrum Analyzer (SA) will go down as well. This makes sense as there is less power present in a smaller frequency band (and more frequencies outside the RBW are suppressed).

For flat noise, -100 dBm in 1 MHz is the same as -80 dBm in 10 MHz or -120 dBm in 0.1 MHz.

The Video Bandwidth (VBW) provides a sort of averaging on the image, leave it on "auto" for the moment.

What model SA are you using ? Maybe it has a noise measuring option ?

• I see, that might be the core of my misunderstanding. I'm definitely not well versed in these systems and I am still finding my way, so I apologies for the probably trivial concepts I might struggle with. In any case, let me state my issue as follows then. Given that my noise is white up to the point I want it to be (the spectrum is flat up to half the sampling rate and then drops off), how do I then use my spectrum analyzer to determine the amplitude of the spectral density? If I understand correctly what you wrote, I can only say this if I also know the bandwidth of what the noise acts on? – user129412 May 26 '16 at 13:06
• To maybe expand on why I don't fully understand this, the way I see the spectral density is just as the fourier transform of the autocorrelation of the noise, which to me has a well determined value, regardless of any bandwidth. – user129412 May 26 '16 at 13:08
• Oh, and the analyzer is an old one, the Agilent E4407B Spectrum Analyzer – user129412 May 26 '16 at 13:16
• Oh, that is not an old one, it has a color display ! It will do fine for your purposes. If it read "HP" instead of Agilent and had a green monochrome display, then it would be old (but still very usable). – Bimpelrekkie May 26 '16 at 13:22
• For measuring I would use a frequency which is significantly lower that the sampling rate of the AWG, like Fs/10 and then choose a RBW of Fs/100. So if Fsample is 100 MHz, on the SA: Fcentre = 10 MHz, Span = 10 MHz and then choose RBW = 1 MHz. To check if the noise is flat only change Fcentre, for example to 20 MHz, 50 MHz and see if you get the same nosie power level. – Bimpelrekkie May 26 '16 at 13:33