Note · RF circuits
Noise figure, explained with one receiver chain
Every receiver datasheet quotes a noise figure, and every RF course derives the Friis formula. What usually gets lost is the intuition: why one stage gets almost all the attention, and why a short piece of cable in the wrong place can cost as much as a bad amplifier. This note works through one example chain to show both.
What noise figure measures
Noise factor F is how much a block degrades the signal-to-noise ratio, measured with a source at the standard temperature T0 = 290 K:
A perfect, noiseless block has F = 1, or 0 dB. A block with NF = 3 dB adds as much noise as the source already had. Note that F is a ratio of powers and NF is the same thing in decibels. The math below uses F, and the datasheets use NF, so you convert back and forth constantly.
The Friis formula
For blocks in cascade, each with noise factor Fi and available gain Gi (both as linear ratios):
Read it as: each stage's added noise is divided by all the gain in front of it. Once the signal has been amplified, later stages have a much harder time hurting it.
One receiver chain, worked through
Take a simple three-stage receiver: a low-noise amplifier (LNA), a mixer, and an IF amplifier.
| Stage | NF | Gain | F (linear) | Term in eq. (2) |
|---|---|---|---|---|
| LNA | 1.5 dB | 20 dB | 1.413 | 1.413 |
| Mixer | 10 dB | 8 dB | 10.0 | 9.0 / 100 = 0.090 |
| IF amplifier | 4 dB | 30 dB | 2.512 | 1.512 / 631 = 0.002 |
| Total | 1.78 dB | 1.505 |
The mixer has a 10 dB noise figure, which looks terrible on its own. Behind 20 dB of LNA gain it adds only about 0.3 dB to the total. The IF amplifier barely registers. The receiver's noise figure is essentially the LNA's noise figure plus a small penalty.
Where a cable belongs
A passive, lossy block at T0 has a noise factor equal to its loss: a 2 dB loss means NF = 2 dB. Put 2 dB of cable or filter loss before the LNA and it adds directly to the total, giving 3.78 dB. Put the same loss after the LNA and it is divided by 20 dB of gain, adding under 0.02 dB. This is why antennas often have the LNA mounted right at the feed, and why the input filter's loss is fought over so hard.
From noise figure to sensitivity
The thermal noise floor at room temperature is −174 dBm/Hz. A receiver's sensitivity, the weakest signal it can detect, follows directly:
For our chain with a 20 MHz bandwidth (73 dB) and a demodulator that needs 10 dB SNR: Pmin = −174 + 73 + 1.78 + 10 ≈ −89 dBm. Every dB saved in the LNA, or in the loss in front of it, is a dB of sensitivity.
Takeaways
Spend the design effort on the first stage and on anything lossy in front of it. Give the first stage enough gain that later stages stop mattering, but not so much that it compresses on strong signals. Linearity is the other side of this tradeoff, and a good topic for another note.
Further reading
- H. T. Friis, “Noise figures of radio receivers,” Proc. IRE, vol. 32, no. 7, pp. 419–422, 1944.
- B. Razavi, RF Microelectronics, 2nd ed., Prentice Hall, 2011.
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