Compute the total noise figure of a 3-stage RF receiver chain in real time with the Friis formula. Change the gain and noise figure of each stage to reveal the total NF, equivalent noise temperature and per-stage noise contributions, and understand intuitively why the low-noise amplifier (LNA) sits first.
Parameters
Stage 1 gain G1
dB
Power gain of the first stage (LNA)
Stage 1 noise figure NF1
dB
First-stage NF — almost sets the total NF
Stage 2 gain G2
dB
Stage 2 noise figure NF2
dB
Stage 3 gain G3
dB
A negative value represents mixer / filter loss
Stage 3 noise figure NF3
dB
Results
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Total noise figure NF_total (dB)
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Total gain (dB)
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Equiv. noise temperature Te (K)
—
Stage 1 contribution (linear F)
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Stage 2 contribution (linear)
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Is stage 1 dominant?
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RF receiver chain — signal and noise flow
Signal (blue) and noise (red) wavepackets flow left to right through three stage blocks. Both the signal level and the noise floor rise at each stage, and the cumulative NF after each stage is shown.
The Friis cascade noise formula. Each later stage's noise contribution is divided by the (product of) preceding gains, so the first-stage noise factor dominates the overall performance.
$$NF=10\log_{10}F,\qquad T_e=(F-1)\,T_0$$
Noise figure NF [dB] and equivalent noise temperature Te [K]. Both are alternative expressions of the noise factor F and convert into one another. Reference temperature T₀ = 290 K.
Here the gains G and noise factors F are all linear ratios (not dB), obtained with F = 10^(NF/10) and G = 10^(G_dB/10). Reference temperature T₀ = 290 K.
What is Cascade Noise Figure?
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"Noise figure" measures how much noise a receiver adds, right? Can't I just add it up stage by stage?
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Good question — and no, it is not a simple sum. A receiver is a "cascade": after the antenna come an amplifier, a mixer, a filter, and so on, chained together. Each stage amplifies the signal but also adds its own noise. The key fact is that the noise of a later stage is evaluated together with a signal that has already been strongly amplified by the earlier stages. So the later-stage noise is "diluted" by the earlier-stage gain. The Friis formula captures this exactly.
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Diluted, I see. The formula is F_total = F1 + (F2−1)/G1 + (F3−1)/(G1·G2) — so the division is the "dilution" part.
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Exactly. The second stage's noise contribution (F2−1) is divided by the first-stage gain G1, and the third stage's by G1·G2. Try raising the first-stage gain G1 from 15 dB to 30 dB on the left. You will see the total NF stick right against the first-stage NF1. Conversely, drop G1 to 0 dB and the later-stage noise leaks straight through, making the total NF much worse.
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So the first stage is hugely important? Is it pointless to work hard on a low-noise second stage?
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That is precisely "first-stage dominance". That is why an RF receiver always puts a low-noise amplifier — an LNA, with a low NF and high gain — right at the front. With the default values, raise the second-stage NF2 from 4 dB up to 12 dB. The total NF barely moves, doesn't it? When the first-stage LNA has enough gain, the whole receiver shrugs off a noisy second stage. The practical conclusion: spend your money and effort on the first stage.
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The bar chart below shows the stage-1 contribution towering over the others — that's what "dominant" means. There is also an equivalent noise temperature shown. What is that?
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The equivalent noise temperature Te restates the noise factor in terms of temperature: Te = (F−1)·T0, with T0 = 290 K. For ultra-low-noise systems where the NF is below 1 dB — satellite-comms or radio-astronomy receivers — it is easier to see differences in Te (in kelvin) than in dB. The gap between NF = 0.5 dB and 0.3 dB looks tiny in dB, but in Te it is 35 K versus 21 K, a clear difference. Both carry the same information, and you can always switch with F = 1 + Te/T0.
Frequently Asked Questions
The total noise factor of cascaded stages is given by the Friis formula F_total = F1 + (F2−1)/G1 + (F3−1)/(G1·G2). F is the linear noise factor and G is the linear gain; convert from noise figure NF [dB] with F = 10^(NF/10) and from gain G [dB] with g = 10^(G/10). The total noise figure is NF_total = 10·log10(F_total) [dB]. The key feature of the formula is that each later stage's noise contribution is divided by the gain of all preceding stages.
In the Friis formula the second stage's noise contribution is (F2−1)/G1 and the third stage's is (F3−1)/(G1·G2) — each divided by the product of the preceding gains. So the larger the first-stage gain G1, the more the later-stage noise is diluted, and the total NF ends up almost equal to the first-stage NF1. That is why the first stage must be an LNA with a low noise figure and high gain. When the LNA gain is high enough, a noisy second stage barely degrades the overall performance.
The equivalent noise temperature Te expresses the noise factor F in terms of temperature, defined as Te = (F−1)·T0, where T0 is the reference temperature 290 K. Noise figure NF [dB] is common in commercial radio, while Te [K] is used for extremely low-noise systems such as satellite communications and radio astronomy. The two are different expressions of the same information and convert with F = 1 + Te/T0. For example, NF = 1.5 dB corresponds to F ≈ 1.41 and Te ≈ 120 K.
The most effective change is lowering the first-stage noise figure NF1. Since the total NF is almost equal to the first stage, dropping NF1 by 0.5 dB drops the total NF by roughly 0.5 dB. Next, raising the first-stage gain G1 shrinks the later-stage contribution (F2−1)/G1 and suppresses the influence of the following stages. Conversely, placing a lossy element (negative gain) such as a cable or filter before the first stage adds its loss directly to the total NF, so it should be avoided.
Real-World Applications
Cellular and Wi-Fi receiver front ends: In a smartphone or a Wi-Fi router, an LNA sits immediately behind the antenna, followed by a mixer, IF amplifier and filters. The receive sensitivity is set almost entirely by the noise figure of that first LNA, so manufacturers spend their lowest-NF device there. Move the first-stage NF1 in this tool and you see the total NF shift by almost the same amount — exactly why LNA selection is the most critical choice.
Satellite communications and GPS receivers: Signals from geostationary or navigation satellites are extremely weak, so these receivers describe their performance with equivalent noise temperature. A low-noise block downconverter (LNB) is an ultra-low-noise design with Te on the order of tens of kelvin and is mounted right at the antenna. Any cable between the antenna and the LNB adds its loss straight to the noise temperature, so the LNB is connected directly to the antenna feed.
Radio astronomy and radar: A radio-telescope receiver may even cryogenically cool its first stage to push Te as low as possible. By the Friis formula, if the first stage has a low Te and enough gain, the noise of the following mixer and spectrometer is diluted to a negligible level. The "cumulative noise figure" chart in this tool visualizes exactly this — the NF is almost set by the first stage and barely climbs afterwards.
Test instrument and spectrum-analyzer design: The receiver in a spectrum analyzer or network analyzer is also multi-stage, and the displayed average noise level (DANL) is set by the total NF. When adding an internal or external preamplifier, the Friis formula estimates how far the total NF improves. It also explains why a low-gain preamplifier brings little benefit.
Common Misconceptions and Pitfalls
The biggest misconception is that "noise figure is in dB, so just add it stage by stage". The Friis formula works with the linear noise factor F and linear gain G, not the noise figure in dB. NF = 3 dB means F = 2, NF = 6 dB means F ≈ 4 — you put the F values into the formula and only at the end convert back with NF_total = 10·log10(F_total). Adding or dividing values while they are still in dB gives a completely wrong answer. That is exactly why this tool performs the 10^(NF/10) conversion internally.
Next, the careless belief that "a lossy element before the first stage does no real harm". A cable, connector or band filter between the antenna and the LNA is a "stage" with negative gain. An element with loss L dB has a noise figure of L dB, and when it becomes the first stage, the contribution of the following LNA is divided by that loss and worsens. The total NF ends up roughly "loss + the original NF". A 1 dB cable loss is a direct 1 dB hit to receive sensitivity. That is why the LNA is placed as close to the antenna as possible.
Finally, the assumption that "a good total NF means the receiver is finished". The Friis formula deals only with small-signal noise performance; distortion under strong inputs (IP3, the 1 dB compression point) and dynamic range are separate issues. Pushing the first-stage gain too high makes the later stages saturate and intermodulate under strong signals. In practice, you set the gain distribution of each stage by trading off lowering the NF against preserving linearity. Only when both noise and distortion are considered do you have a sound receiver design.