Noise Figure and the Low-Noise Amplifier

The hiss that never sleeps

Press your ear to a seashell and you hear a soft roar — not the ocean, just the random rush of ambient sound your ear can't quite silence. Every electronic receiver has its own seashell roar. Even with the antenna disconnected and the circuit sitting in a dark, perfectly shielded box, a resistor's terminals jitter with a tiny random voltage. The charge carriers inside are not still; they rattle around in proportion to temperature, and that thermal agitation shows up as a faint, hissing noise voltage. This is thermal noise (also called Johnson–Nyquist noise), and it is the floor beneath every radio signal on Earth.

The remarkable thing is how universal it is. The available noise power from any resistor at temperature T, over a bandwidth B, is simply P = kTB — it doesn't depend on the resistor's value, only on temperature and how much bandwidth you listen to. Here k is Boltzmann's constant. At room temperature (290 K, the standard reference), this works out to about −174 dBm per hertz of bandwidth. Memorise that number; it is the bedrock of RF engineering. Open up a 1 MHz channel and the noise floor rises to −174 + 10·log₁₀(10⁶) = −114 dBm. There is no clever circuit, no cooling short of liquid helium, that lets you hear a signal buried below that floor.

Signal-to-noise: the only ratio that matters

A signal is only useful if you can tell it apart from the noise. That contest is captured by the signal-to-noise ratio (SNR): the ratio of signal power to noise power, almost always quoted in decibels. An SNR of 20 dB means the signal is 100 times stronger than the noise — crystal clear. An SNR of 0 dB means signal and noise are equal, and most demodulators are already in serious trouble. Push below that and the bits start flipping. In communications work this same idea is written as SNR, and it ties directly to how many bits per second a channel can carry: Shannon's capacity theorem says the maximum data rate grows with log₂(1 + SNR). More noise, fewer bits. It's that blunt.

Here is the cruel part. When a signal flows through any real amplifier or mixer, it doesn't come out cleaner — it comes out worse. The block amplifies your signal, yes, but it also amplifies the noise that came in *and* sprinkles in fresh noise of its own. So the SNR at the output is always lower than the SNR at the input. The signal got louder; the noise got louder faster. Every active stage you add charges a tax in SNR, and you can never get a refund.

Noise figure: the SNR tax, measured

If every stage degrades SNR, we need a number that says by how much. That number is the noise factor F: the ratio of the SNR at the input to the SNR at the output, measured under a standard 290 K source. Express it in decibels and you get the noise figure (NF = 10·log₁₀ F). A perfect, noiseless amplifier would have F = 1, NF = 0 dB — it amplifies signal and noise equally and adds nothing. A real amplifier with NF = 3 dB has cut your SNR in half; the stage added as much noise as the source itself delivered.

Definition (under a 290 K source):

        SNR_in        F  = noise factor   (linear)
  F  = ------         NF = 10·log10(F)    (dB)
        SNR_out

Quick feel for the numbers:
  F = 1     -> NF = 0   dB   (ideal, impossible)
  F = 1.26  -> NF = 1   dB   (excellent LNA)
  F = 2.00  -> NF = 3   dB   (SNR cut in half)
  F = 10    -> NF = 10  dB   (a passive mixer, say)

Equivalent 'noise temperature' view:
  F = 1 + Te/T0 ,  T0 = 290 K
  NF = 1 dB  <->  Te ~=  75 K   (radio-astronomy land)
  NF = 3 dB  <->  Te = 290 K

Two subtleties worth carrying with you. First, noise figure is always referenced to a 290 K source temperature — that's why a cryogenically cooled radio telescope amplifier is rated in *noise temperature* (kelvin) instead, because once you go far below 290 K the dB scale stops being intuitive. Second, a purely passive, lossy component — a cable, an attenuator, a filter — has a noise figure equal to its loss. A 2 dB filter in front of your receiver doesn't just throw away 2 dB of signal; it adds 2 dB of noise figure to the whole chain. That single fact will haunt the next section.

Friis: why the first stage rules them all

A receiver is never one block — it's a chain: amplifier, then mixer, then filter, then more amplifiers, each with its own gain and its own noise figure. In 1944, Harald Friis worked out exactly how these stack up, and the result is one of the most consequential formulas in all of electronics. It says the total noise factor is dominated by the first stage, because every later stage's noise contribution is *divided down* by all the gain that came before it.

Friis cascade formula (noise factors are LINEAR, not dB):

            F2 - 1     F3 - 1     F4 - 1
  Ftot = F1 + ------  + -------- + ----------- + ...
               G1        G1·G2      G1·G2·G3

Notice:
  * F1 enters with FULL weight  -> stage 1 dominates
  * F2 is shrunk by G1
  * F3 is shrunk by G1·G2  (often already negligible)

=> Put a high-gain, low-noise stage FIRST and the rest
   of the chain almost stops mattering.

Read that formula like a story. The first stage's noise, F₁, lands on the total with its full, undiminished weight. The second stage's *extra* noise (F₂ − 1) is divided by the first stage's gain G₁. The third stage's extra noise is divided by G₁·G₂ — by now a huge number — so it's almost invisible. If your first amplifier has 20 dB of gain (a factor of 100), it shrinks everything downstream by 100×. This is the whole reason RF receivers are built the way they are: get a clean, high-gain amplifier in first, and you've effectively bought immunity from the noise of every block behind it.

The low-noise amplifier: first, fast, and quiet

Friis hands the RF designer a clear marching order: make the first active block the cleanest, highest-gain stage you can build. That block is the low-noise amplifier (LNA) — the unsung gatekeeper sitting right behind the antenna of every phone, GPS receiver, Wi-Fi radio, satellite dish, and radio telescope on the planet. Its entire reason for existing is to lift the faint incoming signal well above the noise floor of everything downstream, while adding almost no noise of its own. A modern silicon-germanium or GaAs LNA might deliver 15–20 dB of gain with a noise figure under 1 dB; a cryogenic radio-astronomy LNA can reach a few tenths of a kelvin of added noise temperature.

Designing an LNA is where this rung shakes hands with the previous one. Recall from S-parameters that to transfer maximum power into an amplifier you want a *conjugate impedance match* at the input. But here's the catch that defines the whole craft: the source impedance that gives an amplifier its lowest noise figure is generally *not* the same impedance that gives it maximum gain or a perfect input match. The transistor has one magic source impedance for noise (call it Γ_opt) and a different one for power transfer. You cannot have both at once.

So the LNA designer performs a careful compromise — a *noise match* that lands close to Γ_opt for the best noise figure, traded off against enough of a gain/input match to keep the stage stable and the return loss acceptable. The classic trick is the inductively degenerated cascode: a small inductor in the transistor's source creates a real input resistance out of thin air (with no resistor, which would add noise), letting the designer hit both a 50 Ω input match and near-optimum noise simultaneously. This tension between noise match and gain match — and the related fight against distortion, captured by the third-order intercept point — is exactly the trade-off the next rung explores.

A worked example: the LNA earns its keep

Numbers make Friis vivid. Imagine a GPS receiver where, after the antenna, the signal must pass through a lossy cable and filter (call it 3 dB of loss, so F = 2) before reaching a mediocre mixer (NF = 10 dB, F = 10, gain near unity) and finally a stack of baseband amplifiers (NF = 15 dB, F ≈ 31). Let's compute the total noise figure two ways: without an LNA, then with a good LNA bolted on right after the antenna.

WITHOUT an LNA  (lossy front end goes straight to mixer)
  Stage 1: cable+filter   F1 = 2    G1 = 0.5  (-3 dB)
  Stage 2: mixer          F2 = 10   G2 = 1
  Stage 3: baseband amps  F3 = 31

  Ftot = 2 + (10-1)/0.5 + (31-1)/(0.5*1)
       = 2 + 18 + 60 = 80
  NF   = 10*log10(80)  ~= 19.0 dB   <-- terrible

WITH a good LNA placed FIRST, before the lossy cable
  Stage 0: LNA            F0 = 1.26 (NF 1.0 dB)  G0 = 100 (20 dB)
  Stage 1: cable+filter   F1 = 2    G1 = 0.5
  Stage 2: mixer          F2 = 10
  Stage 3: baseband amps  F3 = 31

  Ftot = 1.26 + (2-1)/100 + (10-1)/(100*0.5)
              + (31-1)/(100*0.5*1)
       = 1.26 + 0.01 + 0.18 + 0.60 = 2.05
  NF   = 10*log10(2.05) ~= 3.1 dB   <-- 16 dB better!

Sixteen decibels. That is the LNA earning its keep. Watch *why* it works in the formula: with the LNA first, its own 1 dB noise figure sets the floor (F₀ = 1.26 lands with full weight), and everything behind it gets divided by the LNA's gain of 100. The lossy cable that wrecked the no-LNA design now contributes a measly 0.01 to the total. The receiver's sensitivity — the weakest signal it can still decode at a usable SNR — just improved by 16 dB, which in free space translates to roughly six times the range, or a signal forty times weaker that you can still hear. A GPS satellite is 20,000 km away broadcasting at the power of a faint lightbulb; without that quiet first amplifier, your phone would simply never find it.

1. Convert every block's noise figure from dB to a linear noise factor: F = 10^(NF/10).
2. Convert every gain from dB to linear too (a loss is a gain less than 1).
3. Apply Friis: add F₁, then each later (Fₙ − 1) divided by the product of all preceding gains.
4. Convert the total noise factor back to dB: NF = 10·log₁₀(F_total).
5. Sanity check: the system NF should sit just above the first stage's NF — if it doesn't, your first stage lacks gain.