Squared telecom parameters

Question

I observe that some telecommunication parameters are squared like noise power is $σ^2=-95\text{ dBm}$, and the amplifier coefficient $η^2=20\text{ dB}$. What is the meaning of the square? Can I remove it with mathematical calculations?

Answer 1

It's common in physics and engineering to find that the energy of or the power transferred in some system is proportional to the square of some "signal" parameter, for example the voltage across a resistor, the pressure of a sound wave in a material, or the displacement of a mass connected to a spring.

Presumably the parameters you're looking at, $\sigma$ and $\eta$, are chosen to be proportional to some signal parameter, and then the power associated with them naturally is related to the square of these parameters.

Without knowing exactly what $\sigma$ and $\eta$ refer to (these symbols are used for many different things in different engineering fields), it's hard to be more specific than that.

Answer 2

1.- standard deviation

Without any further clarification from the question originator, and understandly assumint that the supplied values are related to signals, then it's reasonable to consider that σ stands for standard deviation.

Defition of standard deviation :

https://www.math.net/standard-deviation#:~:text=Standard%20deviation%20Standard%20deviation%20is%20a%20statistical%20measure,values%20tend%20to%20be%20close%20to%20the%20mean.

mean variance standard deviation covariance all concisely explained here :

https://www.mun.ca/biology/scarr/Mean_&_Variance.html

2.- Signals with null DC

In Communications, wireless modulated signals, radiated received by antennas, over-the-air sinals, as well as spectral noise, have null DC values.

Radiated Electromagnetic fields do not have DC.

Indeed noise and signals can and are often biased, on circuit boards there are plenty of examples to be found.

These are conducted, not radiated, signals.

3.- dB dBm clarification

In the context of this question, since the question originator starts equating squared σ and η to dBm and dB values, such practice is often found in literature for signals and noise with null DC component.

dBm and dB are often the short for dBmW and dBW that it's assumed this way there's no way to confuse them with dBmV and dBV.

Experts advise, in case of any doubt, when mixing power W and Volts V in same text, to use the whole expressions dBmW instead of dBm and dBW instead of dBW when referring to signal power and the equivalent with Volts when signal amplitude levels referred.

4.- Signal to Power

So σ is commonly assigned to signal deviation and η to noise deviation.

So when both signal s and noise n have null mean values, then taking the absolute values abs(s) abs(n) both fall most of the time, statistically speaking, below certain levels.

This is, most of the times one measures s and n the readings of s and n below certain abs() level, now squared are equivalently proportional to power level but not equal.

The proportionality factor between the square of the values and power is the pervasive impedance.

All types of matter have (electrical) impedance.

5.- Power measurements require known impedance(s)

Circuit imedances are measured in Ohms = Ω (Omega, last character of Greek alphabet, as in Rev22:13).

RF and microwave systems are mostly built with the following impedances: 75Ω for VHF some UHF, and 50Ω system impedances, alothough there are plenty of 50Ω for UHF systems.

So the averaged power, in a wide sense (rms? avg? another question),

is squared amplitude over impedance

However, accademia often obviate system impedances and tacitally assume 1Ω allowing a lot more theory to be conveyed in the same text.

When one reads -95dBm for a sigma square probably means a weak signal, because dBm is another short for dBmW.

30dBmW = 0 dBW means 1mW = 1e-3W

6.- Disambiguation circuit impedance and wave impedance

Matter is characterised by different values of impedance.

Signals impedances in general, are often referred as

Z = R +1j*X,

R resistance and X reactance

Z (Ω) directly relates Voltage (Volts) and Current (Amperes).

However electromagnetic waves, contrary to acoustic waves, travel without need of matter presence.

Then there are books that use η in a particular way as electromagnetic wave impedance also in Ω Ohm :

wave impedance is the ratio between |E| electric field and |H| magnetic field

electric field units [V/m]

mangetic field units [A/m]

B is not magnetic field, but magnetic flux.

magnetic flux as H * A , A : area, square feet, square meters . B is not H/A

https://www.vedantu.com/physics/magnetic-flux

B units : Tesla [T]

 1 T = 1Wb/sqm =1N/(C * (m/s)) = 1N/(A * m)

Weber [Wb] : 1 Wb = 1 V·s = 1 T·m2 = 1 J/A = 10e8 Mx 

Maxwell [Mx]

sqm square meter, same as m2

Air / vacuum (wave, aka intrinsic) impedance is the wave impedance of mobile phone signals encounter when propagating through atmosphere : about 377Ω.

η as wave impedance is often also found in literature as Z0.

when Z0 values depart from standard values as the ones shown above it may referred with variable names like Z1 Z2 .. for instance in impedance transformers.