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

Overview

A link budget is a comprehensive accounting of all the gains and losses in a radio frequency (RF) communication link from transmitter to receiver. It answers the fundamental question: "Will my signal survive the journey through space?"

The link budget determines whether a spacecraft can successfully communicate with Earth (or another spacecraft) by calculating if the received signal power exceds the noise floor by enough margin to decode the data reliably.

In space, we can't just "turn up the volume" — power is precious, antennas have size limits, and physics (inverse square law) is unforgiving. The link budget is our survival calculator.

Core Components of a Link Budget

1. EIRP — Effective Isotropic Radiated Power

Why This Concept? Real antennas don't radiate equally in all directions — they focus power like a flashlight focuses light. EIRP combines transmitter power and antenna gain into one number that represents the "effective" radiated power in the best direction.

Physical meaning of antenna gain: If power were radiated isotropically, intensity at distance $r$ would be: $I_{\text{iso}} = \frac{P_t}{4\pi r^2}$

With a directional antenna of gain $G_t$, the intensity in the main beam direction becomes: $I_{\text{actual}} = \frac{P_t \cdot G_t}{4\pi r^2}$

We define EIRP such that: $I_{\text{actual}} = \frac{\text{EIRP}}{4\pi r^2}$

Equating these: $\text{EIRP} = P_t \cdot G_t \quad \text{(watts)}$

In decibels (the practical form): $\text{EIRP (dBW)} = P_t\text{(dBW)} + G_t\text{(dBi)}$

where dBW = decibels relative to 1 watt, and dBi = decibels relative to isotropic.

Step 1 — Convert power to dBW: $P_t = 10 \text{ W} = 10 \log_{10}(10) = 10 \text{ dBW}$ Why this step? We work in dB because gains and losses multiply (add in dB), making calculations manageable.

Step 2 — Calculate EIRP: $\text{EIRP} = 10\text{ dBW} + 25\text{ dBi} = 35 \text{ dBW}$

Physical meaning: This system radiates as much power in its main beam as a 3162 W isotropic transmitter ($10^{3.5} \approx 3162$ W).

2. Path Loss — The Fundamental Range Penalty

Why does this happen? As a spherical wavefront expands, the same total power spreads over an increasingly large area ($4\pi r^2$). Power per unit area (intensity) decreases as $1/r^2$.

Step 1 — Power density at distance $r$:

Starting with EIRP, the power density (watts per square meter) at distance $r$ is: $S = \frac{\text{EIRP}}{4\pi r^2}$

Why $4\pi r^2$? That's the surface area of a sphere at radius $r$ — the power spreads over this entire area.

Step 2 — Received power by antenna:

A receiving antenna with effective aperture $A_e$ (square meters) collects: $P_r = S \cdot A_e = \frac{\text{EIRP} \cdot A_e}{4\pi r^2}$

The effective aperture relates to gain by: $A_e = \frac{G_r \lambda^2}{4\pi}$

where $\lambda$ is wavelength and $G_r$ is receiver antenna gain.

Why this relation? It comes from antenna theory — a bigger antenna or higher gain means larger effective collecting area.

Step 3 — Substitute aperture: $P_r = \frac{\text{EIRP} \cdot G_r \lambda^2}{4\pi \cdot 4\pi r^2} = \text{EIRP} \cdot G_r \cdot \frac{\lambda^2}{(4\pi r)^2}$

Step 4 — Define path loss $L_p$:

Path loss is the ratio of transmitted to received power (in the absence of antenna gains): $L_p = \left(\frac{4\pi r}{\lambda}\right)^2$

In decibels: $L_p\text{(dB)} = 20\log_{10}\left(\frac{4\pi r}{\lambda}\right) = 20\log_{10}(4\pi) + 20\log_{10}(r) - 20\log_{10}(\lambda)$

Using $\lambda = c/f$ where $c = 3 \times 10^8$ m/s: $L_p\text{(dB)} = 20\log_{10}(r) + 20\log_{10}(f) + 20\log_{10}\left(\frac{4\pi}{c}\right)$

The standard form (r in km, f in MHz): $L_p\text{(dB)} = 32.45 + 20\log_{10}(r_{\text{km}}) + 20\log_{10}(f_{\text{MHz}})$

Step 1 — Convert frequency: $f = 8.4 \text{ GHz} = 8400 \text{ MHz}$

Step 2 — Apply formula: $L_p = 32.45 + 20\log_{10}(2.5 \times 10^8) + 20\log_{10}(8400)$

Calculate each term:

• $20\log_{10}(2.5 \times 10^8) = 20(8 + 0.398) = 167.96$ dB
• $20\log_{10}(8400) = 20(3.924) = 78.48$ dB

$L_p = 32.45 + 167.96 + 78.48 = 278.89 \text{ dB}$

Why so huge? Space is vast! This ~279 dB loss means the signal is attenuated by a factor of $10^{27.9}$ — nearly a billion times weaker.

3. G/T — Figure of Merit for Receivers

$\frac{G}{T}\text{(dB/K)} = G_r\text{(dBi)} - 10\log_{10}(T_{\text{sys}})$

Why this combination?

• High $G_r$ → collect more signal✓
• Low $T_{\text{sys}}$ → less noise competing with signal ✓

Both improve signal-to-noise ratio (SNR). Combining them into one metric lets us compare receiver systems directly.

The received carrier-to-noise ratio $C/N_0$ (in dB-Hz) is: $\frac{C}{N_0} = \text{EIRP} - L_p + \frac{G}{T} - 10\log_{10}(k)$

where $k = 1.38 \times 10^{-23}$ J/K is Boltzmann's constant.

Derivation sketch:

• Received carrier power: $C = \text{EIRP} - L_p + G_r$
• Noise power density: $N_0 = kT_{\text{sys}}$
• Ratio: $C/N_0 = (\text{EIRP} - L_p + G_r)/(kT_{\text{sys}})$
• In dB: separate $G_r$ and $T_{\text{sys}}$ → the $G/T$ term emerges naturally

Physical meaning: G/T captures "how well does my receiver grab signal versus generating noise?"

$\frac{G}{T} = 63 - 10\log_{10}(25) = 63 - 10(1.398) = 63 - 13.98 = 49.02 \text{ dB/K}$

Why this is excellent: Deep space ground stations achieve40-55 dB/K. Higher G/T means we can detect weaker signals from farther spacecraft.

4. Eb/N0 — Energy Per Bit to Noise Density Ratio

Why Eb/N0 instead of SNR?

• Eb/N0 normalizes for data rate: a slow link (fewer bits per second) can tolerate lower signal power than a fast link for the same error rate
• Eb/N0 connects directly to information theory — Shannon's limit depends on Eb/N0

Starting with carrier-to-noise density: $\frac{C}{N_0} \text{ (ratio, not dB)}$

Received carrier power $C$ is distributed across data rate $R_b$ (bits/second): $E_b = \frac{C}{R_b}$

Why? Each bit gets its "share" of the total power. Faster data rate = less energy per bit.

Therefore: $\frac{E_b}{N_0} = \frac{C}{N_0 \cdot R_b}$

In decibels: $\frac{E_b}{N_0}\text{(dB)} = \frac{C}{N_0}\text{(dB-Hz)} - 10\log_{10}(R_b)$

Complete scenario: Mars orbiter downlink

• Transmitter power: $P_t = 15$ W = 11.76 dBW
• Transmit antenna gain: $G_t = 22$ dBi
• EIRP: $11.76 + 22 = 33.76$ dBW
• Path loss (from earlier): $L_p = 278.89$ dB
• Receive G/T: $49.02$ dB/K (DSN 34-m)
• Boltzmann constant: $k = 1.38 \times 10^{-23}$ = -228.6 dBW/K/Hz
• Data rate: $R_b = 2$ Mbps = $10\log_{10}(2 \times 10^6) = 63$ dB-Hz

Step 1 — Calculate C/N0: $\frac{C}{N_0} = \text{EIRP} - L_p + \frac{G}{T} - (-228.6)$ $= 33.76 - 278.89 + 49.02 + 228.6 = 32.49 \text{ dB-Hz}$

Why add 228.6? In dB, subtracting a negative is adding. This is the "$-k$" term.

Step 2 — Calculate Eb/N0: $\frac{E_b}{N_0} = 32.49 - 63 = -30.51 \text{ dB}$

Wait, negative?! This is a problem! Let's check required Eb/N0...

Step 3 — Compare to requirement:

For uncoded BPSK at BER = $10^{-5}$: $E_b/N_0$ required≈ 9.6 dB

For turbo-coded system (rate 1/2): $E_b/N_0$ required ≈ 1-2 dB

Our -30.51 dB is way below requirements!

What went wrong? Let's recalculate with realistic spacecraft EIRP of ~60 dBW (100W + 40 dBi high-gain antenna):

$\frac{C}{N_0} = 60 - 278.89 + 49.02 + 228.6 = 58.73 \text{ dB-Hz}$ $\frac{E_b}{N_0} = 58.73 - 63 = -4.27 \text{ dB}$

Still negative! Need to reduce data rate or use powerful coding. At100 kbps (50 dB-Hz): $\frac{E_b}{N_0} = 58.73 - 50 = 8.73 \text{ dB}$

Now we're in the ballpark for coded systems. This illustrates the fundamental tradeoff: distance, data rate, and power are locked in battle.

Mistake 1: Mixing dB and linear units ❌ Wrong: $P_r = P_t\text{(dBW)} \times G_t\text{(dBi)} / L_p\text{(dB)}$ ✓ Right: $P_r\text{(dBW)} = P_t\text{(dBW)} + G_t\text{(dBi)} - L_p\text{(dB)}$

Why this feels right: In linear units, we multiply gains and divide losses. But in dB, these become addition and subtraction.

The fix: In dB-land, multiplication → addition, division → subtraction, powers → multiplication.

Mistake 2: Forgetting the sign of losses ❌ Wrong: $C/N_0 = \text{EIRP} + L_p + G/T$ ✓ Right: $C/N_0 = \text{EIRP} - L_p + G/T$

Why this feels right: Losses are big positive numbers (278dB), so adding them makes the budget "bigger."

The fix: Losses subtract from the budget. They're called losses for a reason! Think of your bank account: withdrawals are negative.

Mistake 3: Confusing C/N0 and Eb/N0

Why this feels right: Both are "signal to noise" ratios with similar formulas.

Key difference:

• $C/N_0$ is power per Hz (doesn't depend on data rate)
• $E_b/N_0$ is energy per bit (explicitly depends on data rate)
• $E_b/N_0 = C/N_0 - 10\log_{10}(R_b)$ connects them

Mistake 4: Ignoring implementation losses

Real systems have losses that eat into the budget:

• Pointing loss (antenna not perfectly aimed): 0.5-2 dB
• Polarization mismatch: 0.5-3 dB
• Atmospheric absorption (ground stations): 0.5-2 dB
• Hardware losses (cables, filters): 1-2 dB

The fix: Include a "loss budget" section. Margins of 3+ dB ensure the link closes even with imperfections.

Complete Link Budget Equation

Putting it all together:

$\boxed{\frac{E_b}{N_0}\text{(dB)} = P_t\text{(dBW)} + G_t\text{(dBi)} - L_p\text{(dB)} + G_r\text{(dBi)} - 10\log_{10}(T_{\text{sys}}) +228.6 - 10\log_{10}(R_b) - L_{\text{misc}}\text{(dB)}}$

Or more compactly: $\boxed{\frac{E_b}{N_0} = \text{EIRP} - L_p + \frac{G}{T} + 228.6 - 10\log_{10}(R_b) - L_{\text{misc}}}$

Practical Design Insights

The Iron Triangle of Space Communication:

1. Distance (path loss): $+20$ dB per10× distance
2. Data rate: $+10$ dB per 10× rate
3. Transmit power/antenna: Limited by mass, power budget, deployability

You can't have it all. Mars missions choose:

• Critical telemetry: Low rate (kbps), closes at any geometry
• Science data: High rate (Mbps), requires favorable geometry + ground station availability
• Deep space (Jupiter+): Every dB matters, use gravitational assists to get closer

Why G/T is better than SNR for system comparison:

• G/T is a property of the receiver alone
• SNR depends on the transmitter too
• Ground stations quote G/T; missions can plug into link budget directly

Why Eb/N0 is universal:

• Shannon limit: $\frac{E_b}{N_0} \geq \frac{2^{R/W}-1}{R/W} \approx -1.59$ dB at spectral efficiency $R/W → 0$
• Modulation schemes (BPSK, QPSK,8PSK) each have known $E_b/N_0$ vs BER curves
• Coding (Reed-Solomon, turbo, LDPC) reduces required $E_b/N_0$ by 5-8 dB

EIRP is how loud your friendells AND whether they're using a megaphone. A megaphone doesn't make their voice louder, but it focuses the sound toward you instead of wasting it in all directions. EIRP combines both into one number.

Path Loss is why sounds get quieter far away. The sound energy spreads out into a bigger and bigger sphere. By the time it reaches you, you're only catching a tiny fraction. In space, radio waves do the same thing — the farther you are, the weaker the signal, following a specific math rule (twice as far = one-quarter the strength).

G/T is how good your ears are (G) versus how much background noise there is (T). Big ears (big antenna) catch more signal. Quiet background (low noise) means you can hear whispers. G/T combines both.

Eb/N0 is like asking: "Does each word have enough volume to hear over the crowd noise?" If your friend talks really fast, each word is shorter and harder to catch. If they talk slowly, each word is clearer even if they're quieter overall.

Connections

• Free-Space Path Loss Derivation — electromagnetic wave propagation
• Antenna Gain and Effective Aperture — why bigger dishes help
• Noise Temperature and Noise Figure — understanding receiver noise
• Shannon-Hartley Theorem — theoretical limit on data rate
• Modulation Schemes (BPSK, QPSK) — achieving spectral efficiency
• Forward Error Correction — coding gain in link budgets
• Deep Space Network (DSN) — NASA's ground station infrastructure
• Fris Transmission Equation — alternative link budget formulation

#flashcards/physics

What is EIRP and how is it calculated? :: EIRP (Effective Isotropic Radiated Power) is the power that an isotropic antenna would need to radiate to produce the same power density as a directional antenna in its main beam. EIRP = Pt × Gt (linear) or EIRP(dBW) = Pt(dBW) + Gt(dBi) in decibels.

Concept Map

Hinglish (regional understanding)