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## Eseo-emc.fr

Below you find a summary of my findings on calculations with dimensional decibels.

Although this should be quite straightforward and commonly known, I was not ableto find a concise calculation rule anywhere on the web. As such, I decided to try andwrite one myself.

At the end, in ‘Examples’, you find my consequent opinion on a measure for phase
noise, the carrier-to-noise-density-ratio.

If you find something better or if you disagree, please do not hesitate to contact me
at Questions, comments and offences are highly appreciated.

**Introduction**
Although not standarthere are a lot of specialisations on the basic decibel incommon use. These specialisations can be divided in two classes.

The first class of special decibels are the dimensionless decibels. These decibels
are indeed the logarithm of a power ratio. The specialisations differ in the explicitreference power used. Examples are dBi (relative to the gain of an isotropic antenna),dBc (relative to the carrier power) and dBr (relative to something defined in the context).

As the dimensionless decibels are logarithms of a power ratio, they are logarithms of adimensionless number.

The second class of special decibels are dimensional decibels. These decibels are the
logarithm of a dimensional quantity. If possible this quantity is unambiguously pro-portional to power. Examples are dBW (logarithm of a quantity in W), dBm (logarithmof a quantity in mW) and dBV (logarithm of a quantity in V2).

**Algebraic Conventions**
In this document I will use square brackets to obtain the unit of a quantity, so [P] = W.

I will use round brackets to indicate the unit of a quantity, so P(dBW) = 10 log (P(W)).

(The upright typeface shows that dBW and W are the units of a quantity P, instead ofthe arguments of a function P, which would be typeset in italic.)
1The SI (Syst`eme international d’unit´es) only specifies the neper, bel and decibel for logarithmic ratios.

When determining the unit of dimensional decibels, we have to distinguish between
the signs of numbers in decibels and operators to add and subtract. In this document, Iwill do this by always prepending decibel numbers with a sign. All pluses and minusesin excess of the signs must, therefore, be operators. For example:

**Dimensional Decibels**
Looking at dimensions, we can only sensibly take the logarithm of a dimensionlessquantity. To make the argument of the logarithm dimensionless we have to divide thequantity by its unit. We say that the reference of the resulting decibel is this unit. Wedenote this by appending the symbol of the unit directly to the dB symbol (no spaces).

For example:
Here we see that the argument of the logarithm is dimensionless, which is sensible. Theunit ‘dBW’ reminds us that the logarithm was taken from a number, relative to 1 W.

Some common exceptions to this convention are enumerated in
If the unit contains a fraction, I will use −1 and no spacing. If the decibel number is
a per something number, I will use / and spacing. Examples:
Pnoise(dBW) = N0(dBWHz−1) + bandwidth(dBHz);
loss(dB) = specific loss(dB /m) · length(m).

If this notation becomes illegible, we may use round brackets to group the referenceafter the decibel, for example dB(W Hz−1 m−2).

Table 1: Dimensional decibelswith a notation different thandBreference.

**Calculation Rule**
To determine we the decibel reference of a sum of decibel terms, we take the product ofthe references:
In words: we multiply the references of the terms added, while we divide by thereferences of the terms subtracted.

Whenever we encounter a subtracted reference power (e.g. Pcarrier), we may infor-
matively add the corresponding identifier to the reference of the result (e.g. ’c’).

**Examples**
Below, I will illustrate the calculation rule by means of common calculations and errors.

**Distinction Between Operators and Signs**
We will calculate the received surface power density S from the received power Preceivedand the antenna’s effective area Aeff. Note what happens when we simplify the equationand thereby confuse operators and signs:
S(dBWsm−1) = Preceived(dBW) − Aeff(dBsm)
= −60 dBW − −40 dBsm = −20 dBWsm−1
Numerically, both answers are correct, but the units after simplification are wrong.

**dB’s Really Indicate Ratios**
The fact that dB’s indicate a power ratio can be shown by looking at the units:
If the subtracted power is the carrier, we may informatively add ’c’ to the unit:
[Psignal(dBW) − Pcarrier(dBW)] = dBc.

**Phase Noise**
The carrier-to-noise-ratio CNR in transceiver design is defined as follows (with theconsequent unit of dB:
[Pcarrier(dBW) − Pnoise(dBW)] = dB.

Sometimes, this measure is used when describing phase noise. Pcarrier then is the powerof the wanted, mathematical sine. Pnoise is the the total power in the physical, ‘smearedout’ power density spectrum due to phase noise. By above definition, the CNR will beunity, or 0 dB; the phase noise does not attenuate, but rather, spread the power overa broader frequency spectrum. Consequently, the CNR in this definition is not veryuseful in describing phase noise.

More often, the carrier-to-noise-density-ratio C/N0 is used (and this is what people
mean when talking about CNR in the context of phase noise):
Another measure is the phase noise L, which is the phase noise measured or in-
terpolated over a bandwidth of 1 Hz at ∆ f from the carrier, with respect to the carrierpower. The phase noise power is measured over a known bandwidth, and as such, it isa measure for its power density. The reference thus should be:
From the definitions we see that C/N0 and L are reciprocal. Numerically (in decibels),
they will only differ in sign, but the references are mutually reciprocal (dBHz vs. dBHz−1or dBcHz−1).

Source: http://eseo-emc.fr/wp-content/uploads/2012/07/Dimensional-Decibel-Calculation-Rule.pdf

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