The Short Answer
Noise figure (NF) is the number of decibels by which a component or a receiver raises the noise above what a perfect, noiseless one would produce at the reference temperature of 290 K. A 3 dB noise figure means the stage doubles the noise power passing through it.
In a chain of stages, noise figures do not add. Each stage adds its noise to a signal that has already been amplified by everything in front of it, so the noise of a late stage matters far less than the noise of an early one. The Friis cascade formula puts that into arithmetic. Working in linear noise factor F and linear gain G rather than decibels:
F_total = F₁ + (F₂ − 1) / G₁ + (F₃ − 1) / (G₁ · G₂) + (F₄ − 1) / (G₁ · G₂ · G₃) + …
Every stage after the first is divided by the gain ahead of it. Put 20 dB of low noise gain at the front and everything behind it is divided by 100, which is why a good LNA at the antenna hides the noise of the whole chain. The corollary is the expensive one. A passive loss has a noise figure equal to its loss, so 3 dB of feeder in front of the amplifier adds 3 dB to the system noise figure, decibel for decibel, because there is no gain ahead of it to divide it down.
Where you put the LNA is therefore not a packaging decision. It is worth more decibels than almost anything else you can buy.
The quantity comes in two forms, and using the wrong one is the first place a calculation goes wrong.
Noise factor (F) is the linear ratio of the signal to noise ratio at the input to the signal to noise ratio at the output. A stage that halves the SNR has F = 2. It is a plain ratio, and it is the form you must use inside the cascade formula.
Noise figure (NF) is that same ratio expressed in decibels. It is the form quoted on every datasheet, because decibels are easier to read and easier to compare:
NF = 10 · log₁₀(F) and F = 10^(NF / 10)
Keep both in view, because the datasheet speaks one language and the formula speaks the other. There is a third form, the equivalent noise temperature, which becomes indispensable on satellite paths but only muddies things before the cascade is clear, so it waits until later in this article.
Harald Friis gave his name to two different equations, and confusing them is common enough to be worth heading off. The Friis transmission equation relates received power to transmitted power across a free space path, and it is the backbone of a link budget. The Friis cascade formula, published in 1944 in Noise Figures of Radio Receivers, relates the noise factor of a chain to the noise factors and gains of its stages. This article is about the second one. The Friis transmission calculator handles the first.
The cascade formula reads:
F₂ − 1 F₃ − 1 F₄ − 1
F_total = F₁ + -------- + --------- + -------------- + ...
G₁ G₁ · G₂ G₁ · G₂ · G₃
Two rules govern its use, and breaking either one produces a wrong answer that still looks plausible.
First, F and G must be linear ratios, never decibels. Convert every datasheet figure with F = 10^(NF/10) and G = 10^(Gain/10) before you start, and convert back at the end with NF = 10 · log₁₀(F). Adding noise figures in decibels is the most common error in receiver design, and it always overstates the result.
Second, the order of the stages is part of the answer. The same components in a different sequence give a different system noise figure. That is not a quirk of the maths, it is the whole point of it.
Why the First Stage Dominates
Look at what the denominators do. The second stage is divided by G₁. The third is divided by G₁ · G₂. By the fourth stage, with a 20 dB amplifier and a 10 dB amplifier in front of it, the divisor is 1000, and a genuinely noisy stage contributes almost nothing.
The physical reason is straightforward. Noise added at the front of the chain gets amplified by every stage that follows it, exactly as the wanted signal does. Noise added at the back is amplified by nothing. By the time the signal reaches the demodulator it is large, and the demodulator’s own noise is small beside it. Whatever damage was done at the antenna, though, has been faithfully amplified all the way through, and no amount of later gain will undo it. Gain lifts the signal and the noise together, so it can never repair a signal to noise ratio that has already been spoiled.
This is why the first active device gets specified to a tenth of a decibel while the receiver behind it is allowed 5 or 6 dB without anyone losing sleep.
Here is the fact that decides tower design.
Any passive lossy component at ambient temperature has a noise figure equal to its insertion loss. A 3 dB feeder run has a 3 dB noise figure. A 1.5 dB bandpass filter has a 1.5 dB noise figure. A 0.3 dB connector has a 0.3 dB noise figure. Its gain, of course, is the negative of the same number.
The reason is that the attenuator does two things at once. It attenuates the signal, and it attenuates the noise arriving with it, but then it replaces that lost noise with thermal noise of its own, because it is a warm resistive object and warm resistive objects generate noise. The noise coming out is unchanged while the signal coming out is smaller, so the signal to noise ratio has fallen by exactly the loss.
Put that component first in the chain and it lands on the system twice over. Its noise figure has nothing ahead of it to divide it down, and its gain is less than one, so it multiplies up the noise of everything behind it. In the cascade formula, a G₁ of 0.5 does not divide the second term, it doubles it. The feeder will cost you its loss on the transmit side no matter what. On the receive side, you get to choose whether it costs you that loss again in noise figure.
Where to Put the LNA: A Worked Four Stage Chain
Take a UHF base station receive chain with four components. The components are fixed, and so are their prices. The only decision is the order.
| Component | Noise figure | Gain |
|---|
| Receive bandpass filter | 1.5 dB (its loss) | −1.5 dB |
| Low noise amplifier | 1.0 dB | +20 dB |
| Feeder and jumpers | 3.0 dB (its loss) | −3.0 dB |
| Receiver front end | 6.0 dB | (last stage) |
The two candidate chains differ only in where the tower ends and the shelter begins.
Option A: LNA in the shelter Option B: LNA at the masthead
Antenna Antenna
│ │
Feeder −3.0 dB ◄ loss first Filter −1.5 dB
│ │
Filter −1.5 dB LNA +20 dB ◄ gain first
│ │
LNA +20 dB Feeder −3.0 dB
│ │
Receiver NF 6 dB Receiver NF 6 dB
System NF = 5.6 dB System NF = 2.7 dB
Option A: the LNA in the equipment shelter
The amplifier goes on the rack, where it is warm, dry and easy to reach. The signal comes down the tower first.
F₁ = 1.995 G₁ = 0.501 (feeder, 3.0 dB)
F₂ = 1.413 G₂ = 0.708 (filter, 1.5 dB)
F₃ = 1.259 G₃ = 100 (LNA, 1.0 dB NF, 20 dB gain)
F₄ = 3.981 (receiver, 6.0 dB)
F_total = 1.995 + 0.413/0.501 + 0.259/0.355 + 2.981/35.5
= 1.995 + 0.823 + 0.730 + 0.084
= 3.632
NF_total = 10 · log₁₀(3.632) = 5.6 dB
The filter and the amplifier go up the tower, immediately behind the antenna. The feeder now sits after 20 dB of gain.
F₁ = 1.413 G₁ = 0.708 (filter, 1.5 dB)
F₂ = 1.259 G₂ = 100 (LNA, 1.0 dB NF, 20 dB gain)
F₃ = 1.995 G₃ = 0.501 (feeder, 3.0 dB)
F₄ = 3.981 (receiver, 6.0 dB)
F_total = 1.413 + 0.259/0.708 + 0.995/70.8 + 2.981/35.5
= 1.413 + 0.366 + 0.014 + 0.084
= 1.876
NF_total = 10 · log₁₀(1.876) = 2.7 dB
The same four components, in a different order, give 5.6 dB or 2.7 dB. Moving the amplifier up the tower is worth 2.9 dB of system noise figure, and it costs nothing but the mounting bracket and the bias tee.
Those 2.9 dB are 2.9 dB of receiver sensitivity, which is 2.9 dB straight into the link budget. In free space that is about a 40 per cent increase in range. Over real terrain, where the path loss exponent is nearer 3.5 than 2, it is closer to 20 per cent, which is still more coverage than most antenna upgrades will buy you.
Which Stage Is Actually Costing You
The cascade formula does more than produce a total. Because the terms stay separate, each one tells you what that stage contributed, and the terms move around dramatically when the order changes. Reading them on the excess noise, F − 1, since the leading 1 is the source noise rather than anything a component added:
| Stage | Contribution to excess noise, Option A | Option B |
|---|
| Feeder (3.0 dB) | 37.8% | 1.6% |
| Filter (1.5 dB) | 31.3% | 47.1% |
| LNA (1.0 dB) | 27.7% | 41.7% |
| Receiver (6.0 dB) | 3.2% | 9.6% |
The number to take away is what happens to the filter. Once the LNA is at the masthead, the filter ahead of it becomes the dominant term, at 47 per cent of the excess noise, while the feeder that dominated Option A has fallen to under 2 per cent. The design problem has moved. In a well built masthead chain the money is no longer in the amplifier, it is in the insertion loss of whatever sits between the antenna and the amplifier, and every connector, jumper and arrestor in that short run is being charged at full price.
The receiver, meanwhile, contributes under 10 per cent even in the good configuration, so specifying a quieter radio would barely move the total. That is the practical use of the breakdown. It stops you spending money on the stage that is not the problem.
How Much LNA Gain Is Enough?
If gain divides down the noise behind it, the obvious move is to use as much as possible. The formula says otherwise, and quite firmly.
Take Option B above and raise the LNA gain from 20 dB to 30 dB. The feeder and receiver terms shrink by a factor of ten, and the total noise factor falls from 1.876 to 1.788, so the noise figure improves from 2.73 dB to 2.52 dB. Ten extra decibels of gain bought two tenths of a decibel.
The reason is that the chain has a floor. With infinite gain, every term after the LNA vanishes and the noise factor tends to F₁ + (F₂ − 1) / G₁, which for this chain is 1.778, or 2.50 dB. The filter and the LNA set that floor between them, and no amount of gain can push the system below it. At 20 dB of gain you are already within a quarter of a decibel of the limit, and at 30 dB you are within two hundredths.
The working rule is this. Once the LNA gain exceeds the losses that follow it by about 10 to 15 dB, the noise figure has essentially converged, and further gain is buying nothing. What further gain does buy, reliably, is trouble at the other end of the dynamic range.
Gain Is Not Free: The Dynamic Range Squeeze
Every decibel of gain in front of a stage improves that stage’s noise contribution and worsens its linearity contribution, and the symmetry is exact. The cascade formula for third order intercept, working in linear power, runs the other way:
1 / IIP3_total = 1 / IIP3₁ + G₁ / IIP3₂ + (G₁ · G₂) / IIP3₃ + …
The gain that appears in the denominator of the noise cascade appears in the numerator of the intercept cascade. Gain early in the chain divides down the noise of what follows and multiplies up its intermodulation. You cannot have one without the other.
This is the dynamic range squeeze, and it is why a receiver design is a negotiation rather than an optimisation. Push the noise figure down with a big masthead amplifier and you drive the mixer behind it closer to compression, so a strong nearby transmitter that was harmless before now generates third order products inside your passband. On a shared communications site, where a paging transmitter or a co-located base station may be delivering a great deal of energy into your antenna, that is not a theoretical risk. It is the usual outcome.
Two habits keep you out of it. Filter before you amplify, which is why the bandpass filter sits ahead of the LNA in the worked example above even though it costs 1.5 dB of noise figure to put it there. And use the least gain that gets the job done, because the decibels beyond convergence are pure dynamic range given away for nothing. A quieter receiver that overloads whenever the site is busy is not a better receiver.
Noise Temperature: When Decibels Hide the Difference
With the cascade settled, the third form of the quantity is worth picking up. Equivalent noise temperature (Te) recasts a stage’s noise as the physical temperature a resistor would need to be at to generate it, referred to the input:
Te = T₀ · (F − 1), where T₀ = 290 K
This is not an affectation, it is the working language of satellite and radio astronomy design, and the reason is visible in the table below. A 0.5 dB and a 1.0 dB LNA look almost interchangeable in decibels, half a decibel apart on a datasheet. In noise temperature they are 35 K and 75 K, better than a factor of two. When you are hunting the last few kelvin on an earth station, decibels compress precisely the difference you are paying for, and noise temperatures have the further convenience of adding directly.
| Noise figure (dB) | Noise factor F | Noise temperature Te (K) |
|---|
| 0 | 1.00 | 0 |
| 0.2 | 1.05 | 14 |
| 0.5 | 1.12 | 35 |
| 1.0 | 1.26 | 75 |
| 1.5 | 1.41 | 120 |
| 2.0 | 1.58 | 170 |
| 3.0 | 2.00 | 289 |
| 6.0 | 3.98 | 865 |
| 10.0 | 10.0 | 2610 |
The 3 dB row is the one to memorise. A 3 dB noise figure is a noise temperature of very nearly 290 K, which is simply the statement that the stage adds as much noise as the reference source itself and so doubles the total. The noise figure converter moves between all three forms, and the same noise temperature rolls up into the G/T figure of merit, the antenna gain divided by the system noise temperature, which is how an earth station compresses its entire receive performance into a single number.
| Component | Typical noise figure |
|---|
| Cryogenically cooled LNA | 0.1 to 0.3 dB |
| Masthead LNA, VHF and UHF | 0.5 to 1.5 dB |
| Satellite LNB | 0.6 to 1.2 dB |
| Base station or good SDR front end | 2 to 4 dB |
| Handheld or consumer receiver front end | 5 to 8 dB |
| Passive diode mixer | roughly equal to its conversion loss, 6 to 8 dB |
| Filter, duplexer, circulator | equal to its insertion loss, 0.5 to 3 dB |
| Coaxial feeder | equal to its loss at the operating frequency |
| Connector, jumper, arrestor | equal to its loss, 0.1 to 0.5 dB each |
These are engineering ballparks for orientation, not specifications. Design against the datasheet for the parts you are actually buying, and note that the feeder row is the one people forget to look up, because cable loss climbs with frequency and the figure at 400 MHz is not the figure at 2 GHz.
Measuring It: The Y Factor Method
Design is one thing and the bench is another. The standard way to measure the noise figure of a real device is the Y factor method, and it needs a calibrated noise source with a known excess noise ratio (ENR), typically about 5 dB or 15 dB.
The source is switched between its hot state, where it injects a known excess noise, and its cold state, where it is simply a matched load at ambient temperature. You measure the output noise power in each state, and Y is the ratio of the two:
Y = N_hot / N_cold
With the cold state at the reference temperature, the noise figure follows directly:
NF (dB) = ENR (dB) − 10 · log₁₀(Y − 1)
A worked instance: a 15.2 dB ENR source produces a measured Y of 13.4 dB, which is a linear ratio of 21.9. Then NF = 15.2 − 10 · log₁₀(20.9) = 15.2 − 13.2 = 2.0 dB.
Two cautions are worth carrying to the bench. The measurement includes everything between the source and the analyser, so the noise figure of the analyser itself has to be calibrated out first, or you will measure the test set rather than the device. And a device with very low gain gives a Y factor close to unity, where the logarithm becomes brutally sensitive to small errors, so low gain stages are the hardest ones to measure well.
There is a point at which the pursuit becomes a waste of money, and it arrives sooner than most specifications admit.
The noise figure only sets the receiver’s internal noise. What the receiver actually has to contend with is the total noise at the antenna terminals, and below roughly a few hundred megahertz the noise arriving from outside, atmospheric noise, galactic noise and above all man made noise from switch mode supplies, solar inverters, powerlines and industrial plant, frequently exceeds anything the front end generates on its own. When the antenna is delivering more noise than the receiver adds, the system is externally noise limited, and a 0.5 dB LNA performs identically to a 1.5 dB one. The floor is being set outside the building.
On a congested site, interference sets the floor rather than thermal noise at all, and the honest fix is filtering, siting and coordination, not a quieter amplifier. This is the receiver side of why RF interference keeps getting worse, and it is why an HF or low VHF receiver has never needed the exquisite front end that a satellite downlink demands.
The test is simple enough to run. Disconnect the antenna and terminate the input, and note the noise level. Reconnect the antenna and note it again. If the level rises appreciably, the environment is louder than the receiver and your decibels are to be won by filtering and siting. If it barely moves, the receiver is setting its own floor and a better LNA will genuinely pay. Do that measurement before you write the LNA specification, not after.
- Adding noise figures in decibels. They do not add. Convert to linear noise factor, run the Friis cascade, convert back. Adding in dB always overstates the result.
- Putting the feeder before the amplifier. Loss ahead of the first gain stage lands on the system noise figure at full value and multiplies up everything behind it. This is the 2.9 dB in the worked example, given away for free.
- Forgetting the small losses in front of the LNA. The jumper, the lightning arrestor and two connectors can total half a decibel, and in a masthead chain that half decibel is charged at full price. Specifying an LNA to 0.3 dB and then feeding it through a 0.5 dB run of odds and ends is not a design, it is a rounding error with a purchase order attached.
- Assuming more gain is better. Beyond about 10 to 15 dB more than the losses that follow it, extra gain does nothing for the noise figure and costs you dynamic range and intermodulation performance decibel for decibel.
- Amplifying before filtering. An LNA with an unfiltered input amplifies every strong signal on the site, not just the wanted one, and the mixer behind it pays the bill.
- Chasing noise figure on an externally noise limited band. Below a few hundred megahertz, and on any congested site, the environment often sets the floor. Measure before you specify.
- Mixing up the two Friis formulas. One is the free space transmission equation and one is the cascade noise formula. They share a name and nothing else.
Frequently Asked Questions
What is noise figure in simple terms? Noise figure is how much noise a component or receiver adds on top of a perfect, noiseless one, expressed in decibels. A 3 dB noise figure means the stage doubles the noise power passing through it. A 0 dB noise figure would be an ideal device that adds nothing, which does not exist.
What is the difference between noise figure and noise factor? They are the same quantity in different units. Noise factor F is the linear ratio, and noise figure NF is that ratio in decibels, so NF = 10 · log₁₀(F). Datasheets quote the decibel form, but the Friis cascade formula requires the linear form, so you must convert before you calculate.
How do you calculate the noise figure of a cascade? Use the Friis cascade formula, F_total = F₁ + (F₂ − 1)/G₁ + (F₃ − 1)/(G₁ · G₂) + …, with every noise factor and gain expressed as a linear ratio rather than in decibels. Each stage is divided by the total gain in front of it, then convert the result back with NF = 10 · log₁₀(F_total).
Why does the first stage dominate the noise figure? Because noise added at the front is amplified by every stage that follows it, while noise added later is not. In the cascade formula this shows up as each subsequent stage being divided by the gain ahead of it, so a high gain, low noise first stage divides everything behind it down towards insignificance.
What is the noise figure of a passive component like a cable or a filter? It equals its insertion loss. A 3 dB cable has a 3 dB noise figure, a 1.5 dB filter has a 1.5 dB noise figure. The component attenuates the signal but replaces the attenuated noise with its own thermal noise, so the signal to noise ratio falls by exactly the loss.
Where should the LNA go, at the antenna or in the shelter? At the antenna, in almost every case. Any feeder loss ahead of the amplifier adds directly to the system noise figure, so mounting the LNA at the masthead recovers that loss. In the worked example above the same four components give 5.6 dB in the shelter and 2.7 dB at the masthead.
Does more LNA gain always mean a lower noise figure? No. Once the LNA gain exceeds the losses that follow it by roughly 10 to 15 dB, the system noise figure has converged on a floor set by the LNA and whatever sits in front of it, and further gain improves it by only hundredths of a decibel while degrading intermodulation performance and dynamic range one for one.
Why a Spreadsheet Stops Being Enough
A single cell can hold the Friis formula, and for a three stage chain that is genuinely all you need. Real front ends outgrow it quickly, because the noise figure is never the only thing being decided.
- The chain is longer than you think. Antenna, jumper, arrestor, filter, LNA, feeder, second filter, splitter, receiver. Ten or fifteen stages is ordinary, and a splitter feeding several receivers branches the chain rather than extending it.
- Noise figure and intercept point have to be solved together. The gain that helps one hurts the other, and optimising either alone gives an answer that fails on site.
- Cable loss is frequency dependent. The same feeder is a different stage at 150 MHz and at 2 GHz, so a chain that was fine on one band is not automatically fine on another.
- The order is a variable, not an input. The useful question is usually which arrangement of a fixed set of parts gives the best result, and that means recalculating the cascade for every permutation.
- The result has to reach the link budget. A system noise figure is only interesting once it becomes a noise floor, then a sensitivity, then a fade margin on a real path.
Keeping those in step by hand is where the errors get in, and they are the quiet kind that survive review because every individual number looks right.
Build it in noIM₃
The noise figure converter moves between noise figure, noise factor and noise temperature, runs the Friis cascade stage by stage and handles the Y factor measurement side. The noise floor calculator takes the system noise figure you have just derived and turns it into a noise floor and a receiver sensitivity, with a cascade builder for multi stage front ends. The SNR calculator checks the resulting margin against the modulation threshold, and the link budget calculator carries the whole thing through path loss to a received level and a fade margin. For the transmit side of Friis’s other equation, the Friis transmission calculator is the one you want.
Key Takeaway
Noise figure measures the noise a stage adds, and in a chain those figures never simply add. The Friis cascade formula divides every stage by the gain ahead of it, so the first stage sets the system noise figure and everything after it is progressively discounted. Because a passive component has a noise figure equal to its loss, any feeder, filter or connector placed before the first amplifier is charged at full price, which is why the same four components in the worked example give 5.6 dB with the LNA in the shelter and 2.7 dB with the LNA at the masthead. Put low noise gain as close to the antenna as you can, filter before you amplify, use the least gain that gets you to the floor, and check whether the band is externally noise limited before you spend anything at all on the last half decibel.
References and Further Reading
The methods and figures here draw on the standard receiver design and noise measurement literature. These are pointers to the primary sources rather than a claim that every figure is quoted verbatim, and the worked examples are our own. Component figures vary between manufacturers, so design against the datasheet for the parts you are specifying, and check the current, in force version of any standard before relying on it.
- H. T. Friis, Noise Figures of Radio Receivers, Proceedings of the IRE, vol. 32, no. 7, July 1944, the paper that introduced the cascade formula.
- IEEE and the earlier IRE standards on noise definitions, which fix the 290 K reference temperature that noise figure is defined against.
- Keysight (formerly Agilent and Hewlett Packard) Application Note 57-1, Fundamentals of RF and Microwave Noise Figure Measurements, and Application Note 57-2 on noise figure measurement accuracy, for the Y factor method and its error budget.
- Rohde and Schwarz application notes on noise figure measurement, for the instrumentation and calibration side.
- D. M. Pozar, Microwave Engineering, for the derivation of the noise figure of a lossy passive network and the cascade relations.
- ITU-R Recommendation P.372, Radio Noise, for the external and man made noise levels that determine when a system is externally noise limited rather than receiver limited.
- TIA TSB-88, Wireless Communications Systems Performance in Noise and Interference Limited Situations, for land mobile receive system planning and site noise.
