Also known as: Friis transmission equation, Friis formula, Friis equation
The Friis transmission equation predicts the power a receiving antenna extracts from a transmitting antenna across an unobstructed free-space path: in its ideal form P_r / P_t = G_t · G_r · (λ / 4πd)², where G_t and G_r are the antenna gains, λ is the wavelength, and d is the separation.1 Published by Harald T. Friis in 1946, it is the foundation of every free-space link budget and the origin of the free-space path loss term.
How it works
The formula is built from two physical ideas. First, an isotropic source spreads its power over the surface of an expanding sphere, so the power density at distance d falls as 1/(4πd²) — the inverse-square law. A transmit antenna with gain G_t concentrates energy toward the receiver, multiplying the density in that direction. Second, the receive antenna captures power in proportion to its effective aperture, and aperture relates to gain by A_e = G_r·λ²/4π. Multiplying the arriving power density by the effective aperture yields:
P_r = P_t · G_t · G_r · (λ / 4πd)²
Two consequences deserve emphasis. The inverse-square in distance means doubling range costs 6 dB of received power. The wavelength-squared factor means that, for antennas of fixed gain, a longer wavelength (lower frequency) delivers more received power — the low band travels “better” not because space treats it differently but because a fixed-gain antenna has a larger physical aperture at longer wavelengths. Expressed in decibels, the (λ/4πd)² term is exactly the negative of free-space path loss.
In practice
Friis is an idealization: it assumes a clear line of sight, far-field distances, matched polarization, impedance-matched and lossless antennas, and no obstruction or multipath. Real links add correction terms for feedline loss, polarization mismatch, atmospheric absorption, and fading — all handled as extra dB in a link budget. Even so, the bare equation sets the ceiling: no terrestrial path beats free space, so Friis is the optimistic bound against which real measurements are compared.
Relevance to SDR
For a receive-only SDR, Friis quantifies the arriving power from a known transmitter. Given the transmitter’s EIRP (which already folds in P_t·G_t), the receive antenna gain, the frequency, and the distance, the equation predicts P_r — the input level that your front end must lift above its noise floor to decode. It also explains recurring field observations: why the same trunking system is easier to hear on its VHF control channel than an equivalent 800 MHz one with identical antennas, and why every 6 dB of arriving margin corresponds to a doubling of usable range.
GopherTrunk does not evaluate the Friis equation in its decode chain, but the equation is the right mental model for predicting whether a distant site delivers enough signal to lock, and how much receive gain would recover a marginal link.
Sources
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Friis transmission equation — Wikipedia, derivation from aperture and gain, wavelength dependence, and idealizing assumptions. ↩