Friis noise formula

Harald Friis’s cascade formula answers a practical question: you have an antenna, a run of coax, an LNA, maybe a filter, and an SDR. How noisy is the whole receiver?

Refer every stage’s noise back to the input of the chain. The first stage’s noise temperature counts in full. The second stage’s is divided by the first stage’s gain, because by the time the signal reaches it, it has already been amplified by that much. The third is divided by the gain of the first two, and so on. Add them up and you have the receiver noise temperature T_rx.

Two things fall straight out of it. First, the earliest stage dominates. Second, a lossy stage (gain less than one) near the front makes everything after it worse, because it divides the signal before the noise of later stages is added.

Worked example: same parts, two orders

Example values: an LNA with 1.0 dB noise figure (75.1 K) and 20 dB gain; 3 dB of coax at room temperature (288.6 K, G = 0.501); an SDR with 6 dB noise figure (864.5 K).

LNA first, at the feed:

• LNA: 75.1 K
• coax: 288.6 / 100 = 2.9 K
• SDR: 864.5 / (100 × 0.501) = 17.2 K
• T_rx = 95.2 K

Coax first, LNA indoors:

• coax: 288.6 K
• LNA: 75.1 / 0.501 = 149.8 K
• SDR: 864.5 / (0.501 × 100) = 17.2 K
• T_rx = 455.7 K

Identical parts, and moving the LNA to the other end of the cable makes the receiver nearly five times noisier. That is the whole argument for a masthead LNA, and LAB-011 runs it with real cable lengths.

Using it

Convert every noise figure to kelvin and every gain to a linear ratio first. The receive-chain noise calculator does this for any order of stages and shows each stage’s contribution, which is the useful part: it tells you which component to upgrade next. T_rx then feeds into the system temperature.