Unusual SNR in the guide of a Spectrum Analyzer

[Microgravity]

If I posted in the wrong section, please feel free to move the thread in the correct one. Since this is my first post in the forum, I have yet to get a good grasp of it. With that said, let's begin!

Prologue: SNR (Signal to Noise Ratio), sometimes indicated with S/N, is a very important parameter, especially in my branch of engineering. Depending on the context, it may have slightly different definitions. The one to which I will refer from now on is, simply stated, the power of the useful signal P_S divided by the noise power P_N(both of them calculated in the same bandwidth). Defined this way, SNR is a dimensionless quantity and can be expressed in dB (decibels). More precisely, the formula used is: SNR_{dB} = 10 \log_{10} SNR.

No problem should arise from what I wrote, since everything is pretty much very well-defined. Well, not until I actually read the guide of an analog Spectrum Analyzer. A guide, I must specify, aimed to introduce basic measurements and practical application of theory to students with the instrument. Playing (for a moment) devil's advocate, I must say that the Spectrum Analyzer is a bit old - it doesn't even have an USB interface and the only way to save data is by using a floppy disk (!). With all that said, let's state the actual problem: derive the definition of SNR used by the guide and recognize and correct all the errors (if any) made in the process of applying it; a.k.a. finding an unusual error(s) with an unusual definition.

The Guide: Firstly, the guide doesn't use the definition I stated above of SNR. In fact, it doesn't explicitly state which definition it uses. The only hint to that is the fact that the S/N is measured in dB/Hz (?!). In the "basic measurement" of S/N, after explaining the setup, the guide states: (emphasis and italics mine)

"Read the signal-to-noise in dB/Hz, that is with the noise value determined for a 1-Hz noise bandwidth. If you wish the noise value for a different bandwidth, decrease the ratio by 10 \log(BW)."

If you are thinking "what is BW?", good question! :) It is not defined, but it is safe to assume it simply means "bandwidth". Apart from that, I am not particularly satisfied with this pseudo-explanation, but I am going, for a moment, to accept it. The critical word is "decrease" - in fact, I do wish to calculate the S/N in a different bandwidth. The irony is that, after the very next sentence (which I will state in a moment), there is an example where it adds (?!) that value. To make things more confused, the resulting "S/N" is not in dB (nor in dB/Hz). Right after the first quote, the guide continues: (italics mine)

"For example, if the analyzer is -70 dB/Hz but you have a channel bandwidth of 30 kHz:
S/N = -70 \ dB/Hz + 10 \log(30 \ kHz) = -25.2 \ dB/30 \ kHz "

An interesting result. But (as all good tales) the story doesn't end here! In fact, in the beginning, there was a very simple error (at least, in my humble opinion), making all the following calculations useless. The S/N showed (both in an image in the guide and by the instrument in a laboratory experience) was negative (in dB, i.e. S/N<1), when very simple calculations and a very simple laboratory experience with known signals and artificial noise showed it should have been positive (in dB). An error in the sign - which reverses everything was said and adds another problem in the already confused situation.

Regrettably, I don't (and will not) have access to the Spectrum Analyzer, thus no test can be performed on it. However, I think that, with all the information provided, it is possible to reasonably deduce:
a) The definition of S/N the guide "wanted to use" (but did not actually use);
b) The error(s) the guide made in the quoted sentences and in the example;
c) The problems with "dB/Hz" (and the like) definitions of S/N.

I already have answers for these questions, but I wanted to share my experience with everyone here, since I would like to know your opinions on the matter.

For the answers: Please, avoid comments such as "throw away that guide!" - I already did! - or "get a better Spectrum Analyzer" (I'm not in charge, thus I cannot). Please, refrain from such line of thought (and similar ones); only approach the topic if you are genuinely interested: be critic, but also very polite. Thanks! :)

 

[mfb]

I think the guide uses the wrong signs for the signal to noise ratio, but the other things look consistent.

A larger bandwidth gives more noise, with the same signal strength you get a worse SNR. The SNR value gets decreased, consistent with 70 -> 25.2 for 1Hz -> 30kHz (with the switched sign).

 

[Averagesupernova]

There are a lot of specs on a spectrum analyzer that can be quite confusing. Are you sure that resolution bandwidth is not being referred to? Do you know how a spectrum analyzer works? If you do/did, I think some things would make more sense to you. Maybe I am misinterpreting what you are saying?
-
The nuts and bolts of a spectrum analyzer is as follows: A local oscillator sweeps and upconverts the incoming signal to an intermediate frequency that is above the highest measureable frequency that the spectrum analyzer can measure. This IF is often downconverted to one or more IFs and then detected. This IF(s) have an adjustable bandwidth and this WILL affect the S/N ratio. The trace on the screen is synchronized with the sweeping of the LO. Take a carrier that is very very weak and feed into the instrument. Set the scan width wide enough and you may not see this signal. It is below the noise level. This is because typically when the scan width widens the bandwidth of the IF needs to widen as well otherwise the instrument cannot scan fast enough to be user friendly. You will wait 10 seconds for the trace to go across the screen. So, if you zoom in and set the scan width narrow enough or change the bandwidth of the IF (resolution bandwidth) narrower the noise level drops and the signal shows up. There is also video bandwidth, video averaging and other settings that can help the instrument see weak signals. Hope this helped.

 

[Microgravity]

Thanks for the fast reply!

First of all, I know how a spectrum analyzer works and I am very familiar with super-heterodyne principles, including things such as: IF, LO, RBW (Resolution BandWidth), VBW (Video BandWidth), etc. I will address the RBW issue later. I think mfb pointed out the correct way: there is a sign error. This was exactly my first thought.

However, it doesn't end here (sadly). It is sufficient to take the example (with the correct sign):
70 \ dB/Hz - 10\log_{10}(30 \ kHz) = 25.2 \ dB/30 \ kHz

As the formula in my opening post, there are a number of problems with this.

1. In order to use the dB-scale, logarithms can only be taken on dimensionless quantities. "dB" can only be used with a pure number: \log_{10}(30 \ kHz) is a conceptual mistake. With that said, nothing prevents you to use a reference value in the division, but you must explicitly write so! The guide should have read: (e.g.) \log_{10} \frac{30 \ kHz}{1 \ Hz}.
2. The true meaning of "dB/Hz" must be clarified (this is very important!). What definition of S/N is used? This is the true question.
3. The "apple and oranges" problem. You start with dB/Hz and end up with dB/30 kHz, thus making useless the whole point of "bandwidth conversion". Strictly speaking, you are not calculating SNR anymore.
4. The numerical problem. The numbers are off-scale: adding 30 kHz worsens SNR by several orders of magnitude?! This is suspect, to say the least.

Now, back on RBW. No, the guide doesn't refer to it in those sentences. I am fairly confident that "BW" doesn't refer to RBW (the example, in fact, speaks of "channel bandwidth"), since RBW is a well-defined acronym in the guide. The example was not meant to understand how RBW affects SNR (which is another topic entirely), but how the "bandwidth conversion formula" should work.

In any case, it couldn't have played a role in the particular experiment we did. Not only we used a "strong" sinusoidal signal (well-above noise level), but we put RBW to the minimum and also set a reasonable (not too large) Span. The signal wasn't below the noise floor (not even close to it), but clearly distinguished from it. In addition, we also manipulated VBW to obtain an even better display. Conclusion: effects tied to RBW (and VBW) were essentially sorted out.

In conclusion: yes, specs of Spectrum Analyzer are quite confusing; nonetheless, this case is quite strange and asks for a deeper explanation.