01. Decibels and the link budget

A satellite link spans a transmit power of tens of watts and a received power of a fraction of a picowatt. That is a ratio of order $10^{13}$ . Working with those numbers directly is error-prone, so radio engineering works in logarithms.

The definition

A decibel expresses a power ratio:

X_{\mathrm{dB}} = 10\log_{10}\left(\frac{P_1}{P_2}\right)

Because logarithms turn multiplication into addition, a chain of gains and losses becomes a column of numbers you add up. That is the entire reason the convention exists.

Ratio	Decibels
1	0 dB
2	3.01 dB
10	10 dB
100	20 dB
1000	30 dB
0.5	$-3.01$ dB

Two useful habits: doubling is close to 3 dB, and every factor of ten is exactly 10 dB. Together they let you sanity-check a budget without a calculator.

The factor is 10 for power ratios and 20 for amplitude or field ratios, because power goes as amplitude squared. Free-space path loss uses 20 because it is written in terms of distance and wavelength, not power. Mixing the two is the most common decibel error.

Absolute units

A bare decibel is a ratio and cannot express an absolute quantity. Suffixed units fix a reference:

Unit	Reference	Used for
dBW	1 watt	Transmit power, EIRP
dBm	1 milliwatt	Received power. $0$ dBm $= -30$ dBW
dBi	An isotropic radiator	Antenna gain
dB/K	1 kelvin	Receive figure of merit, G/T
dBHz	1 hertz	Bandwidth, and C/N0

Adding two absolute quantities is meaningless. Adding a ratio to an absolute quantity is how a budget works: dBW plus dBi gives dBW.

Building the budget

Start with what the transmitter radiates in the direction of interest:

\mathrm{EIRP} = P_{\mathrm{tx}} + G_{\mathrm{tx}} - L_{\mathrm{feed}}

Subtract the spreading loss over the slant range:

L_{\mathrm{fs}} = 20\log_{10}\left(\frac{4\pi d}{\lambda}\right)

Add what the receiver recovers, expressed as a single figure of merit, and subtract Boltzmann’s constant to land in carrier-to-noise density:

\frac{C}{N_0} = \mathrm{EIRP} - L_{\mathrm{fs}} - L_{\mathrm{atm}} + \frac{G}{T} - k

with $k = -228.6$ dBW/K/Hz.

$C/N_0$ has units of dBHz because it is a power ratio per unit bandwidth. To compare against a demodulator threshold quoted as $E_b/N_0$ , divide by the data rate:

\frac{E_b}{N_0} = \frac{C}{N_0} - 10\log_{10}(R_b)

where $R_b$ is the bit rate in bits per second.

Margin

Margin is what is left after the demodulator takes its share:

M = \frac{E_b}{N_0}\bigg|_{\mathrm{received}} - \frac{E_b}{N_0}\bigg|_{\mathrm{required}}

A positive margin means the link closes. The question that matters is not whether it closes, but under what conditions it was evaluated: at what elevation, and at what availability. A budget with 6 dB of margin at zenith in clear sky can be several decibels negative at the elevation mask in rain.

Worked check

A satellite at 550 km, seen at zenith, at 20 GHz.

Wavelength is $\lambda = c/f = 3\times10^8 / 2\times10^{10} = 0.015$ m.

L_{\mathrm{fs}} = 20\log_{10}\left(\frac{4\pi \times 5.5\times10^{5}}{0.015}\right) \approx 173\ \mathrm{dB}

At the same frequency and 10 degrees elevation the slant range is roughly 2.5 times greater, adding $20\log_{10}(2.5) \approx 8$ dB. That 8 dB is the difference between a budget that works across a pass and one that works only overhead.

Next steps

Link budgets for how the tool assembles these terms and quotes margin at an availability target.

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