Cascaded Noise Figure (Friis)

Harald T. Friis at Bell Labs published his now-famous cascade noise-figure formula in his 1944 paper "Noise Figures of Radio Receivers". Friis worked on receiver design for the first WWII radar systems and trans-Atlantic telegraph carriers, where extracting signals from the noise floor was life-or-death. His formula captures the key insight that earlier stages in a receiver contribute far more to total noise than later stages — the first stage\'s gain divides the noise contribution of every subsequent stage.

The consequence: receivers are designed "LNA first" — a low-noise amplifier with the lowest achievable NF and enough gain to mask later stages\' noise sits at the antenna input. Cryogenically cooled LNAs (NF < 0.2 dB) power satellite ground stations, deep-space network receivers, and radio telescopes. The Arecibo radio telescope\'s LNA chain had a system noise temperature of ~30 K, equivalent to NF ≈ 0.4 dB — approaching the thermodynamic limit set by the cosmic microwave background.

Modern RF front-end chips (RFMD, Qorvo, Analog Devices) integrate LNA + mixer + IF amp on one die, optimizing the Friis cascade together. Software-defined radio platforms (HackRF, USRP) quote a single-chip NF of 6–10 dB — modest by dedicated-LNA standards but perfectly adequate for consumer and hobbyist use. The Friis formula remains the primary design tool for analyzing these chains.

Enter noise figure (dB) and gain (dB) for each of three cascaded stages. The tool converts each NF to linear noise factor F, applies Friis\' cascade formula, and returns total NF, total noise factor F, total gain, and the percent contribution of the first stage\'s noise to the total.

Design rule emerges naturally: minimize F₁ (use a quiet LNA first) and maximize G₁ (enough first-stage gain to crush subsequent stages\' noise). Lossy passive elements (filters, attenuators, long cables) contribute their loss directly to NF with no gain to compensate — minimize them before the LNA. All math runs client-side; no values leave your browser.