Ferkans — Interactive Telecom Tutor

From Transmitter to Receiver

Every wireless link begins with an electromagnetic wave leaving a transmit antenna and ends with what remains of that wave arriving at a receive antenna. Between these two points, the wave interacts with the environment — it spreads, reflects, diffracts, scatters, and is absorbed. Understanding these mechanisms is essential for predicting coverage, capacity, and reliability.

Definition:
Maxwell's Equations (Communications Perspective)

In a source-free, linear, isotropic medium, Maxwell's equations in phasor form are:

$\nabla \times \mathbf{E} = -j\omega\mu\mathbf{H}, \qquad \nabla \times \mathbf{H} = j\omega\varepsilon\mathbf{E}$

These yield the wave equation for plane waves propagating in the $z$ -direction:

$\mathbf{E}(z, t) = E_0\,e^{j(\omega t - \beta z)}\,\hat{\mathbf{x}}$

where $\beta = \omega\sqrt{\mu\varepsilon} = 2\pi/\lambda$ is the phase constant, $\lambda = c/f$ is the wavelength, and $c = 1/\sqrt{\mu_0\varepsilon_0} \approx 3 \times 10^8$ m/s in free space.

For communications, we rarely solve Maxwell's equations directly. Instead, we use their consequences: the Friis equation, reflection coefficients, and diffraction models.

Definition:
Isotropic Radiator and Antenna Gain

An isotropic radiator radiates equally in all directions. At distance $d$ , the power flux density is

$S = \frac{P_t}{4\pi d^2} \quad \text{(W/m}^2\text{)}$

A real antenna concentrates power in certain directions. The antenna gain $G$ is the ratio of power density in the direction of maximum radiation to that of an isotropic radiator:

$S_{\max} = \frac{P_t G_t}{4\pi d^2}.$

The effective aperture relates gain to the physical collecting area:

$A_e = \frac{\lambda^{2}}{4\pi} G_r.$

Theorem: Friis Transmission Equation

For a clear line-of-sight link with separation $d \gg \lambda$ , the received power is

$P_r = P_t G_t G_r \left(\frac{\lambda}{4\pi d}\right)^2.$

In decibels:

$P_r\,\text{(dBm)} = P_t\,\text{(dBm)} + G_t\,\text{(dBi)} + G_r\,\text{(dBi)} - PL_{\text{fs}}\,\text{(dB)}$

where the free-space path loss is

$PL_{\text{fs}}(d) = 20\log_{10}\!\left(\frac{4\pi d}{\lambda}\right) \text{ dB}.$

The $1/d^2$ dependence arises from the spreading of power over a sphere of area $4\pi d^2$ . The $\lambda^{2}$ term reflects the effective aperture of the receive antenna — higher frequencies have smaller apertures and thus collect less power for a given gain.

Proof

Derivation

The received power is the power flux density times the effective aperture:

$P_r = S \cdot A_e = \frac{P_t G_t}{4\pi d^2} \cdot \frac{\lambda^{2} G_r}{4\pi} = P_t G_t G_r \left(\frac{\lambda}{4\pi d}\right)^2$ . $\blacksquare$

Example: Link Budget Calculation

A 2.4 GHz Wi-Fi access point transmits 20 dBm with antenna gain $G_t = 5$ dBi. A laptop at $d = 50$ m has $G_r = 2$ dBi. Find the received power in free space.

Solution

Wavelength

$\lambda = c/f = 3 \times 10^8 / 2.4 \times 10^9 = 0.125$ m.

Free-space path loss

$PL_{\text{fs}} = 20\log_{10}\!\left(\frac{4\pi \times 50}{0.125}\right) = 20\log_{10}(5027) = 74.0$ dB.

Received power

$P_r = 20 + 5 + 2 - 74.0 = -47.0$ dBm.

This is well above typical Wi-Fi sensitivity ( $\approx -75$ dBm), so the link has $\approx 28$ dB of margin. $\blacksquare$

Common Mistake: Free-Space Path Loss Is NOT Absorption

Mistake:

Saying "free space absorbs the signal" or "path loss means the energy is lost."

Correction:

Free-space path loss is a geometric effect: power spreads over an ever-larger sphere. No energy is absorbed — the total power crossing any enclosing sphere is still $P_t$ . What decreases is the power density at the receiver, and hence the power captured by the finite-area antenna.

Key Takeaway

Free-space path loss follows the inverse-square law: doubling the distance costs 6 dB, and doubling the frequency also costs 6 dB. The Friis equation is the starting point for every link budget — all other propagation effects are modelled as deviations from it.

Historical Note: Harald T. Friis and the Transmission Equation

Harald T. Friis derived his famous equation in 1946 at Bell Labs. It was one of the first rigorous analyses of radio link performance and remains the foundation of every link budget calculation. Friis also introduced the concept of noise figure, which we encounter in receiver design (Chapter 7).

Definition:
Effective Isotropic Radiated Power (EIRP)

The EIRP combines transmit power and antenna gain into a single figure of merit:

$\text{EIRP} = P_t \cdot G_t \qquad \text{or} \qquad \text{EIRP (dBm)} = P_t\,\text{(dBm)} + G_t\,\text{(dBi)}.$

Regulatory limits are typically expressed as maximum EIRP. For example, the FCC limits unlicensed operation at 2.4 GHz to 36 dBm EIRP (4 W).

⚠️Engineering Note

Practical Link Budget Considerations

The Friis equation gives the ideal free-space received power. A real link budget must additionally account for:

Cable and connector losses: typically 1–3 dB at the transmitter and receiver
Body loss: 3–5 dB when a mobile device is held near the body
Polarisation mismatch: up to 3 dB for linearly polarised antennas with random orientation
Atmospheric absorption: negligible below 10 GHz, but 0.1–0.5 dB/km at 28 GHz and up to 15 dB/km at 60 GHz (oxygen absorption)
Rain attenuation: significant above 10 GHz (e.g., 7 dB/km at 28 GHz in heavy rain)
Implementation margin: 2–3 dB for real-world impairments not captured by the model

These losses are additive in dB and can easily consume 10–15 dB of the link margin.

Practical Constraints

•
Cable loss increases with frequency: 0.1 dB/m at 900 MHz, 0.3 dB/m at 28 GHz for standard coax
•
Atmospheric absorption at 60 GHz limits outdoor range to ~200 m
•
Rain attenuation above 10 GHz requires additional fade margin in tropical climates

Quick Check

By how many dB does free-space path loss increase when the distance doubles?

3 dB

6 dB

10 dB

20 dB

Correction:

6 dB

Correct. $PL \propto d^2$ , so doubling $d$ increases $PL$ by $10\log_{10}(4) = 6$ dB.

Path Loss

The ratio of transmitted to received power (in dB): $PL = P_t/P_r$ (linear) or $PL = P_t(\text{dBm}) - P_r(\text{dBm})$ (dB).

EIRP

Effective Isotropic Radiated Power: $\text{EIRP} = P_t G_t$ . The power that an isotropic antenna would need to produce the same peak power density.

Antenna Gain

The ratio of radiation intensity in a given direction to that of an isotropic radiator. Related to effective aperture by $G = 4\pi A_e / \lambda^{2}$ .

Electromagnetic Wave Propagation Basics