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Microgravity

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**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 [itex]P_S[/itex] divided by the noise power [itex]P_N[/itex](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: [itex]SNR_{dB} = 10 \log_{10} SNR[/itex].

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,*

**the ratio by [itex]10 \log(BW)[/itex]."**__decrease__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

**(?!) that value. To make things more confused, the resulting "S/N" is**

*adds***in dB (nor in dB/Hz). Right after the first quote, the guide continues: (italics mine)**

*not**"For example, if the analyzer is -70 dB/Hz but you have a channel bandwidth of 30 kHz:*

[itex]S/N = -70 \ dB/Hz + 10 \log(30 \ kHz) = -25.2 \ dB/30 \ kHz[/itex] "

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! :)