What is the Friis transmission equation?

The Friis transmission equation gives the power received by one antenna from another in free space: P_r/P_t = G_tG_r(λ/4πR)², published by Harald T. Friis of Bell Labs in 1946.^[1] In decibels, P_r = P_t + G_t + G_r − FSPL, with FSPL = 20·log₁₀(4πR/λ).^[2]

What assumptions does Friis make?

• Both antennas are in each other’s far field (R ≥ 2D²/λ and R ≫ λ).
• Free space: no ground reflection, obstruction or atmospheric absorption.
• Antennas are impedance- and polarization-matched; otherwise multiply by (1−|Γ_t|²)(1−|Γ_r|²) and the polarization loss factor.^[3]
• Gains are evaluated in the directions toward each other.

How do you use Friis in a worked example?

A 2.45 GHz link with P_t = 10 dBm, G_t = G_r = 2 dBi, R = 100 m: FSPL = 32.45 + 20·log₁₀(0.1) + 20·log₁₀(2450) ≈ 80.2 dB, so P_r ≈ 10 + 2 + 2 − 80.2 = −66.2 dBm. Real indoor links lose more to walls and multipath; see link budget.

Why does path loss grow with frequency?

The λ² term reflects the shrinking effective aperture of a fixed-gain receive antenna, not any loss in space. Friis noted that with fixed aperture areas A_t and A_r, P_r/P_t = A_tA_r/(λ²R²), which improves with frequency.^[1]

How is Friis extended to radar?

For monostatic radar with gain G and target radar cross section σ, the radar range equation is P_r = P_tG²λ²σ/((4π)³R⁴), so received power falls as R⁻⁴.^[4] See radar antennas.

Frequently asked questions

See also

• Link budget
• Free-space path loss
• Effective aperture
• Gain

References

Online sources accessed October 21, 2026; printed works cited by edition. See all sources.

1. “A Note on a Simple Transmission Formula”. H. T. Friis, Proceedings of the IRE, vol. 34, no. 5, pp. 254-256. May 1946.
2. “Recommendation ITU-R P.525: Calculation of free-space attenuation”. ITU-R. current edition.
3. “Antenna Theory: Analysis and Design, 4th ed.”. C. A. Balanis, Wiley. 2016.
4. “Introduction to Radar Systems, 3rd ed.”. M. I. Skolnik, McGraw-Hill. 2001.