# Power spectral density generated by AWG

I'm having trouble interpreting the results of my spectrum analyzer, concerning noise generated with an AWG. So, step 1. The AWG generates gaussian noise with mean 0 and variance A at a sampling rate of 1.2GSamples. I will look at the noise in the 50 - 150 MHz band, so it should still be produced relatively well there. For A I chose for example 0.2, with a Vpp of 4.5, which means that the entire time trace of 3 million points should be produced by the AWG without too much trouble; no clipping or anything.

Now, I then hook up the AWG to the spectrum analyzer, which has a 50 Ohm input. I pick a 100MHz center frequency, 100 MHz span, and set my RBW to 1 MHz. I take about 1000 points. Now, the amplitudes are in dBm so I convert them to watts and divide by the RBW (so Watts per Hertz), and go back to dBm/Hz. This should be a proper power spectral density; moreover the inbuilt function of the spectrum analyzer that calculates the PSD gets me the same values.

However, this does not work out with the theory. In theory, the spectral density of white noise with variance A can be calculated through the autocorrelation function:

$$\begin{array} RR_{xx}[\tau] &=& E\left[ x[t] x[t+\tau] \right]\\ &=& \left \{ \begin{array} EE \left[ x[t]^2 \right], {\rm if\ }\tau=0 \\ 0, {\rm otherwise} \end{array} \right. \\ &=& A \delta[\tau] \end{array}$$ where the last delta is the Kronecker delta. By the Wiener Khinchin theorem the power spectral density is then simply given by the Fourier transform of the above, which is rather nicely equal to just A.

Linking this to my measured power spectral density, I divide by the 50 Ohm resistance and go to the dBm/Hz units. Great; except the numbers I get are completely different. From the above considerations I expect a power spectral density of around -24 dBm/Hz, and I measure -78 dBm/Hz. Clearly something is terribly off, but I simply do not know what.

So this is my question; what am I doing wrong to measure such a different power spectral density?

As for the amplitude calibration of the spectrum analyzer, I checked by using a signal generator at 54 MHz with a power of -50 and -60 dBm; it reports these values quite nicely; that is probably not the issue.

• The difference between -24 and -78 is not a rounding error, interestingly it's suspiciously close to a factor of 500. That's the sort of error you'd get when you're off by a factor of 2 (easy to do) and by a factor of 1000 (MHz for kHz somewhere, or watts for mW? (I'll bet it's W <-> mW)). If those factors don't leap out at you, then try checking the sum another way. What is the total power the AWG is producing, over what total BW, so the expected spectral density is ?? – Neil_UK Jun 24 '16 at 14:16
• @Neil_UK Thanks for your comment! You helped me figure out the problem. The error can be found by looking at how I defined the spectral density; it is a voltage squared, in the end. Divide this by R and you have a power, but not a power per Hertz. I had to divide this by the bandwidth of the AWG, like you mentioned. Not trivial for anything but white noise, but it works out perfectly! So I guess I can either delete the question or just leave it open. Thanks a lot. – user129412 Jun 24 '16 at 16:19