Link budget — path loss, EIRP, G - T, Eb - N0: Visual walkthrough

This is the whole link budget built in pictures, starting from nothing but a transmitter switched on in empty space. By the last figure you will be able to draw the entire chain — power out, power spreading, power collected, noise fighting back, and the final "did the bits survive?" number — on a napkin.

We define every symbol the instant it appears. If you have never seen a decibel or a logarithm, you will still be able to follow line one.

Step 1 — A transmitter alone in space: what "power spreading" means

WHAT. We switch on a radio transmitter. It pumps out a number of watts we call $P_{t}$ — the transmit power. A watt is just "joules of energy per second"; think of it as how fast energy leaves the antenna.

WHY. Before any gains, losses, or antennas, we must picture the raw fact of radio: energy leaves a point and spreads outward as a growing sphere. Everything else in the budget is a correction to this one picture.

PICTURE. In the figure, the red dot is the transmitter. The energy it sends out this second lives on the surface of an expanding ball. At distance $r$ (metres from the transmitter) that ball's surface has area

area of a sphere of radius r 4 π r^{2}

Figure — Link budget — path loss, EIRP, G - T, Eb - N0

The intensity — power landing on each square metre — is the total power divided by that area:

I_{iso} = \frac{P _{t} watts sent out}{m ^{2} of sphere 4 π r ^{2}} [\frac{W}{m ^{2}}]

The subscript "iso" means isotropic — radiating equally in all directions, like a bare light bulb. Real antennas do better, which is Step 2.

Step 2 — Aiming the power: gain and EIRP

WHAT. We replace the bare bulb with a directional antenna — a dish that focuses the same watts into a narrow beam, like a flashlight reflector focuses the same bulb.

WHY. We can't make more watts (power is precious on a spacecraft), so instead we aim the watts we have. We need one number that captures "how much brighter is the beam than a bare bulb would be."

PICTURE. The figure shows two intensity patterns from the same $P_{t}$ : a round isotropic glow (thin black) and a focused red beam. The red beam is far more intense along its axis.

We call the focusing factor the transmit gain $G_{t}$ — a plain dimensionless ratio (no units):

G_{t} = \frac{I _{beam} intensity on axis}{bare-bulb intensity I _{iso}}

A gain of $300$ means "the beam is $300 \times$ brighter on-axis than a bare bulb of the same power." Where does $G_{t}$ come from physically? A bigger dish → tighter beam → higher gain — see Antenna Gain and Effective Aperture.

Now the clever bookkeeping trick. Instead of tracking $P_{t}$ and $G_{t}$ separately, define one number, the Effective Isotropic Radiated Power (EIRP) — the wattage a bare bulb would need to match our beam:

EIRP = real watts P_{t} \times focusing factor G_{t} [W]

So the on-axis intensity is simply the isotropic formula with EIRP in place of $P_{t}$ :

S = \frac{EIRP}{4 π r ^{2}}

Step 3 — Decibels: turning multiply into add (so we can survive Step 6)

WHAT. We switch our units from raw ratios to decibels (dB).

WHY. In the next steps numbers get monstrous — signals weaken by factors of $1 0^{27}$ . Multiplying such numbers by hand is hopeless. A logarithm turns multiplication into addition, so the whole budget becomes a tidy sum: add the gains, subtract the losses.

PICTURE. The figure is a number line. On top, the linear scale crams huge numbers into an unusable smear. Below, the same values on a dB (log) scale are evenly, readably spaced.

The rule. For any power ratio $x$ :

x_{dB} = 10 lo g_{10} (x)

Here $lo g_{10} (x)$ answers the question "10 to what power gives $x$ ?" We chose base $10$ (not base $e$ ) purely because engineers count in powers of ten. Two facts you'll reuse constantly:

lo g_{10} (a \cdot b) = lo g_{10} a + lo g_{10} b lo g_{10} (a^{2}) = 2 lo g_{10} a

The first is why multiply becomes add. The second is why an $r^{2}$ path loss carries a factor of $20$ (that's $10 \times 2$ ).

Two named reference units:

dBW = dB relative to $1$ watt → $P_{dBW} = 10 lo g_{10} (P /1 W)$ .
dBi = dB relative to isotropic → $G_{dBi} = 10 lo g_{10} (G_{t})$ .

So Step 2's boxed result becomes an addition:

EIRP (dBW) = power in dB P_{t} (dBW) + gain in dB G_{t} (dBi)

Step 4 — The receiver catches a raindrop: effective aperture

WHAT. A receiving dish far away intercepts a patch of the expanding wavefront and funnels it into the electronics.

WHY. The transmitter's power is now smeared over a sphere millions of km wide. Our dish only catches the sliver that lands on its mouth. We need to know how big a bite the dish takes.

PICTURE. The figure shows the far wavefront (nearly flat by now) sweeping past, and the red dish scooping out one small circular area labelled $A_{e}$ .

The dish's effective aperture $A_{e}$ (in m²) is the area of wavefront it effectively collects. Power caught = intensity $\times$ area:

P_{r} = W/m^{2} arriving S \times m^{2} collected A_{e} = \frac{EIRP \cdot A _{e}}{4 π r ^{2}}

Antenna theory links aperture to receive gain $G_{r}$ and wavelength $λ$ (the length of one wave, in metres):

A_{e} = \frac{G _{r} receive gain λ ^{2} wave size squared}{4 π}

Why does $λ$ appear? A dish focuses a wave well only when the dish is many wavelengths across; smaller wavelengths (higher frequency) let the same physical dish grab more — full story in Antenna Gain and Effective Aperture. Substituting:

P_{r} = EIRP \cdot G_{r} \cdot the geometry factor \frac{λ ^{2}}{( 4 π r ) ^{2}}

Step 5 — Isolating the villain: free-space path loss $L_{p}$

WHAT. We separate the pure distance-and-frequency penalty from the antenna gains, and name it path loss $L_{p}$ .

WHY. Gains ( $G_{t}$ , $G_{r}$ ) are things we design. The distance penalty is fixed by physics and geometry — Mars is where Mars is. Isolating it lets us see clearly what we're fighting.

PICTURE. The figure is the same expanding sphere as Step 1, but now annotated as a shrinking bar of "surviving power" against a log distance axis — the red curve plunging as $r$ grows.

Define $L_{p}$ as the reciprocal of that geometry factor (so it's a number $> 1$ that we divide by):

L_{p} = (\frac{4 π r}{λ})^{2}

Read it term by term: $4 π r$ is the sphere's spreading, $λ$ in the denominator is the wavelength's helping hand, and the outer square is the inverse-square law showing up. In dB — and here the $lo g (a^{2}) = 2 lo g a$ rule turns the square into a factor of $20$ :

L_{p} (dB) = 20 lo g_{10} (\frac{4 π r}{λ}) = farther \to worse 20 lo g_{10} r + higher freq \to worse 20 lo g_{10} f + constant 20 lo g_{10} \frac{4 π}{c}

using $λ = c / f$ with $c = 3 \times 1 0^{8}$ m/s. Folding the constant and converting to convenient units ( $r$ in km, $f$ in MHz) gives the working form used across the Friis world:

L_{p} (dB) = 32.45 + 20 lo g_{10} (r_{km}) + 20 lo g_{10} (f_{MHz})

Step 6 — The noise floor: G/T and the birth of $C / N_{0}$

WHAT. We put the whole chain together to get the carrier-to-noise-density ratio $C / N_{0}$ — signal versus noise.

WHY. A faint signal is useless if noise is louder. Every warm object (the electronics, the sky, the ground) radiates random RF noise. Detecting the message means the carrier power $C$ must stand above this hiss.

PICTURE. The figure shows two bars: the tall received carrier $C$ after all the adds and subtracts, and a fuzzy red band — the noise density $N_{0}$ . The gap between them is the budget.

Received carrier, in dB, is a running total — start loud, add focusing, subtract distance, add collecting:

C = loud start EIRP - distance L_{p} + dish grabs G_{r}

The noise. Any receiver's noise is described by a system noise temperature $T_{sys}$ (in kelvin) — a made-up temperature whose thermal hiss equals the real noise (see Noise Temperature and Noise Figure). The noise power in each hertz of bandwidth is

N_{0} = k T_{sys}, k = 1.38 \times 1 0^{- 23} J/K = - 228.6 dBW/K/Hz

where $k$ is Boltzmann's constant — nature's conversion from temperature to power-per-hertz. Divide $C$ by $N_{0}$ , and the receiver's two knobs — gain $G_{r}$ (want big) and $T_{sys}$ (want small) — merge into one figure of merit, the G/T ratio:

\frac{G}{T} (dB/K) = G_{r} (dBi) - 10 lo g_{10} (T_{sys})

Putting it all together (subtracting a negative $k$ = adding $228.6$ ):

\frac{C}{N _{0}} (dB-Hz) = EIRP - L_{p} + \frac{G}{T} - 10 lo g_{10} (k)

Step 7 — Per-bit accounting: $E_{b} / N_{0}$ , the number that decides success

WHAT. We convert the raw signal-to-noise into energy per bit over noise density, $E_{b} / N_{0}$ — the metric that actually predicts errors.

WHY. $C / N_{0}$ doesn't know how fast we're sending. Cram the same power into twice as many bits per second and each bit gets half the energy. Bit errors depend on energy per bit, so we must divide out the data rate.

PICTURE. The figure shows the fixed carrier power being sliced into $R_{b}$ bit-boxes: few big boxes (slow, robust) vs. many thin boxes (fast, fragile). The red slice is "one bit's energy."

Each bit's share of the carrier, over the data rate $R_{b}$ (bits per second):

E_{b} = \frac{C}{R _{b}} ⟹ \frac{E _{b}}{N _{0}} = \frac{C}{N _{0} R _{b}}

In dB — one clean subtraction:

\frac{E _{b}}{N _{0}} (dB) = \frac{C}{N _{0}} (dB-Hz) - 10 lo g_{10} (R_{b})

This is the payoff number. Compare it to the required $E_{b} / N_{0}$ for your modulation (helped by Forward Error Correction); the difference is your link margin. Its ultimate floor is set by the Shannon-Hartley Theorem.

The one-picture summary

Every step is one arrow in a single left-to-right ledger: start loud, focus, cross the void, collect, out-shout the noise, split across bits, compare to the threshold.

Steps 1-3 EIRP - Steps 4-5 L_{p} + Step 6 \frac{G}{T} - 10 lo g k - Step 7 10 lo g R_{b} = \frac{E _{b}}{N _{0}}

Recall Feynman retelling — the whole walkthrough in plain words

A transmitter switches on and its energy races outward as a growing bubble; spread over the bubble's skin, it thins as one-over-distance-squared (Step 1). We aim it with a dish so the beam is many times brighter than a bare bulb — that focus times the real watts is EIRP (Step 2). To add instead of multiply through the coming giant numbers, we switch to decibels, where multiplying becomes adding and squares become factors of twenty (Step 3). Far away, our own dish scoops one small patch of the thinned bubble — its effective aperture — and that patch, plus wavelength, sets how much we catch (Step 4). Stripping out the antennas leaves the pure distance-and-frequency penalty, path loss, a monstrous ~279 dB for Mars (Step 5). Against the faint survivor stands the receiver's noise, summarised by G/T; dividing signal by noise-per-hertz gives C/N₀, the gap between voice and hiss (Step 6). Finally we ask how fast we're talking: split the carrier across the bits per second to get energy per bit, and E_b/N_0 tells us whether each bit will be read correctly (Step 7). Add the gains, subtract the losses, subtract the rate — one honest sum decides if the signal survives the journey.

Quick self-check

Path loss is

20 lo g

of range, not

10 lo g

— why?

Because power falls as

r^{2}

, and

lo g (r^{2}) = 2 lo g r

, so the dB factor is

10 \times 2 = 20

If you double the data rate, what happens to

E_{b} / N_{0}

It drops by

10 lo g_{10} (2) \approx 3

dB — each bit gets half the energy.

Why merge

G_{r}

and

T_{sys}

into G/T?

Because signal-to-noise depends on gain over noise; one figure of merit lets you compare whole receivers at a glance.