Noise figure, stacked: why your LNA goes where it goes

Every radio receiver has the same job. A weak signal arrives at the antenna, some amount of noise arrives with it, and the receiver tries to hand the demodulator something with a decent signal-to-noise ratio. The receiver itself is an obstacle to this project because every component in it adds noise of its own. How much depends on the components and on the order they're in.

If you've ever heard someone say "the noise figure of the first stage is everything" or "the LNA has to go at the antenna," they were talking about the Friis formula. It's a small piece of engineering math that, once it clicks, makes most of the weird choices in receiver design look obvious.

Below, the same territory in four scenes. Start with where noise comes from, then where noise figure as a number comes from, then what happens when you chain stages together, and finally why the LNA ends up at the antenna and not near the radio.

Where the noise floor comes from

Resistors hiss. Electrons in anything above absolute zero rattle around, and some of that rattling shows up as voltage across any resistor in the circuit. In a receiver, the resistor is effectively the antenna (or the load it connects to), and the available noise power works out to a specific, unavoidable number:

P_noise = k * T * B

where k is Boltzmann's constant, T is the temperature in kelvin, and B is the bandwidth your receiver looks at. At 290 K (roughly room temperature), that works out to approximately −174 dBm per hertz of bandwidth. Wider filter, wider noise. Hotter resistor, hotter noise.

Move the sliders. Widen the bandwidth from 1 kHz to 1 MHz and the noise floor rises by 30 dB. Heat the front end from 290 K to a hypothetical 2900 K and the floor rises by another 10. A signal at, say, −110 dBm sits comfortably above a narrow-bandwidth floor and disappears into the noise at wider ones, without the signal itself changing at all.

This is the part of the receiver you can never quite beat. You can shape the filter, pick a lower temperature, or move the measurement further down the chain, but whatever noise the antenna delivers is the floor you have to work with.

A single stage: what noise figure actually measures

Put a signal and some noise into an amplifier. The amplifier has some gain, which multiplies both by the same factor, leaving the signal-to-noise ratio unchanged. So if gain were the only thing an amplifier did, noise figures wouldn't need to exist.

Real amplifiers, however, add their own noise at the output. You can model this as an extra noise source sitting at the input of an ideal amplifier, and the noise figure is just a compact way to say how much extra. Formally:

NF (dB) = 10 * log₁₀ ( SNR_in / SNR_out )

which in practice means "this stage made my SNR worse by this many decibels." A 0 dB noise figure is a perfect amplifier that adds no noise. A 3 dB NF is an amplifier that doubles the noise power at the output relative to what was at the input. Real low-noise amplifiers at HF and low VHF get down to a few tenths of a dB. At microwave they might be 0.5 to 1.5 dB if you paid well.

Cranking the gain slider up doesn't change the SNR degradation, which is the point: gain is free, noise figure isn't. Cranking the NF slider is the one that actually matters for the output SNR, and the damage is identical regardless of what the input signal level was.

Friis: why chain order matters

Now chain two amplifiers together. The first one degrades the SNR by its NF. The second one degrees the SNR by its own NF, but applied to the signal and noise that the first stage has already amplified. If the first stage has high gain, the noise it produced is much larger than the second stage's own noise, so the second stage's NF barely matters.

That's the whole idea behind the Friis formula:

F_total = F₁ + (F₂ − 1) / G₁ + (F₃ − 1) / (G₁ * G₂) + ...

where F is the linear noise factor (F = 10^(NF/10)) and G is the linear gain. Everything is in linear units because you can't add decibels, only multiply them.

The crucial thing is the G₁ in the denominator of every term after the first. If G₁ is large, those later terms shrink to nothing. If G₁ is small, or worse, negative (an attenuator has G < 1), those terms blow up.

Try a few chains. Start with "LNA -> mixer -> IF amp" and watch the contribution bar. Almost all the output noise is being contributed by the LNA, which is a little paradoxical until you remember that the LNA is the first thing touching the signal, and everything downstream is looking at a signal that's already 18 or 20 dB stronger than what the antenna is providing.

Now move the LNA to the second position and put the mixer first. The mixer is a terrible first stage because it loses 7 dB of gain (it's essentially an attenuator) and has a noise figure of 7 dB on its own. With the mixer first, the LNA's nice 1.5 dB noise figure is divided by a gain of 0.2 (a conversion loss of 7 dB in linear terms), which makes its contribution much larger than it should be. The total NF climbs dramatically.

Receivers almost always end up in the same left-to-right order because Friis leaves very little room to disagree.

The LNA has to go at the antenna

One of the most common mistakes in a ham station is running the LNA at the radio instead of up at the antenna. Intuitively it seems fine. An amplifier amplifies regardless of where it is, and putting it near the radio means the DC power and the control are easier. Friis has opinions.

Both chains use the same LNA and the same feedline loss. Putting the LNA before the feedline makes the feedline loss nearly invisible because it's followed by the LNA's large gain, and the noise it adds is small compared to the noise already present. Putting the LNA after the feedline means the feedline loss is in front of the amplifier, which is equivalent to having the first stage of your receiver be a 4 dB attenuator whose own noise figure is also 4 dB (loss and noise figure of a passive component are always equal).

Even a good LNA cannot recover the signal that the feedline loss has already corrupted. Friis is not a law you can argue with, and the difference in total system noise figure between "at the antenna" and "at the radio" can easily be 3 to 6 dB depending on how long the run is. That's the difference between hearing a weak DX station and never knowing it was there.

For microwave paths, the effect is worse because coaxial feedline loss goes up steeply with frequency. At 10 GHz, a few meters of LMR-400 eats a huge fraction of the signal before it reaches the radio. This is why every remote 10 GHz beacon setup you'll ever see has the LNA mounted in a weatherproof enclosure right at the dish, with a low-loss waveguide or short coax feed to the LNA input, and a standard coax run after it carrying the amplified signal back to the shack. The LNA's job is to get in front of the feedline loss before it happens.

Practical rules

Once you've spent a while with the Friis formula, a few rules of thumb fall out naturally:

• The first active stage should have the lowest noise figure you can afford, and it should live as close to the antenna as possible.
• Everything before that first stage is loss, which adds directly to the system noise figure. Keep it short, keep it low-loss, and resist the temptation to add "just one more connector."
• Once the signal has been amplified by ~20 dB, later noise figures barely matter. You can use cheap, high-NF parts after the LNA and suffer essentially nothing.
• An attenuator of L dB has a noise figure of exactly L dB. A 6 dB pad in front of your LNA costs you 6 dB of system noise figure. Do not put one there.
• The total noise figure of the cascade is an upper bound on how good any link budget involving that receiver can ever be. No amount of antenna gain will help if the receiver itself is destroying the SNR.

The Friis formula is one of those pieces of math that looks abstract until you start moving blocks around in the scenes above, at which point it stops feeling abstract and starts feeling like an old acquaintance who shows up every time you design a receiver and reminds you, politely, that no, the LNA really does have to go at the antenna.