What is the Friis noise formula?
F_total = F1 + (F2 − 1) ÷ G1 + (F3 − 1) ÷ (G1 × G2) + …, with every noise factor F and gain G as a linear power ratio. Convert the result back with NF = 10 log10(F).
FREE TOOL
Cascaded noise figure calculator
Up to six stages go into this noise figure calculator, each with its gain and NF, and the Friis formula gives the total NF and gain of the chain. See how much each stage adds, along with the noise temperature, noise floor and receiver sensitivity.
Examples
dB
A loss is a negative gain: a 2 dB filter is −2.
dB
Passive part at 290 K: same as its loss. Blank pair = unused stage.
dB
dB
dB
dB
dB
dB
dB
dB
dB
dB
Hz
Optional: gives the noise floor and sensitivity.
dB
At the demodulator. Blank is treated as 0 dB.
K
Optional: antenna temperature. Blank uses 290 K.
Cascaded noise figure
1.946 dB
- Total gain
- 41 dB
- Noise factor F
- 1.565 linear
- Equivalent noise temperature Te
- 163.9 K
- Noise floor at the input
- -112.03 dBm
- Sensitivity
- -102.03 dBm with 10 dB SNR
Friis assumes matched stages, available power gains and noise figures measured with a 290 K source. Enter a loss as a negative gain. For a passive part at 290 K the noise figure equals its loss.
The floor uses kT0 = −173.98 dBm/Hz at 290 K, the exact value behind the rounded −174 dBm/Hz.
What each stage adds
| Stage | Gain | NF | Gain ahead | Adds to F | Share of added noise | Cascade NF |
|---|---|---|---|---|---|---|
| 1 | 20.0 dB | 1.0 dB | 0.0 dB | 1.26 | 45.8 % | 1.00 dB |
| 2 | −2.0 dB | 2.0 dB | 20.0 dB | 0.00585 | 1.0 % | 1.02 dB |
| 3 | −7.0 dB | 7.0 dB | 18.0 dB | 0.0636 | 11.3 % | 1.23 dB |
| 4 | 30.0 dB | 6.0 dB | 11.0 dB | 0.237 | 41.9 % | 1.95 dB |
How to calculate noise figure in a cascade
Convert each stage to linear values, then apply the Friis formula: F = F1 + (F2 − 1) ÷ G1 + (F3 − 1) ÷ (G1 × G2) and so on. Two LNAs with 1 dB and 3 dB noise figures and 12 dB and 15 dB of gain, followed by an ADC with a 20 dB noise figure, give F = 1.519, which is 1.82 dB. The first stage dominates because later noise is divided by all the gain ahead of it.
Noise figure formula for cascaded stages
Noise figure measures how much a stage degrades the signal-to-noise ratio. In a chain, each stage’s added noise is reduced by the gain that comes before it.
F = F1 + (F2 − 1) ÷ G1 + (F3 − 1) ÷ (G1 × G2) + …
- F
- noise factor of the stage or chain, linear
- G
- power gain of the stage, linear
- NF
- noise figure in dB
NF = 10 log10(F) F = 10^(NF ÷ 10) G = 10^(gain ÷ 10)
Noise temperature expresses the same quantity against the 290 K reference, and it cascades the same way:
Te = T0 × (F − 1) Te = T1 + T2 ÷ G1 + T3 ÷ (G1 × G2) + …
- Te
- equivalent input noise temperature in K
- T0
- reference temperature, 290 K
The noise floor and sensitivity follow from the total noise figure:
Sensitivity (dBm) = 10 log10(k T0 × 1000) + 10 log10(BW) + NF + SNR
- k
- Boltzmann constant, 1.380649 × 10⁻²³ J/K
- BW
- noise bandwidth in Hz
- SNR
- signal-to-noise ratio the demodulator needs, in dB
The first term is −173.98 dBm/Hz, usually rounded to −174 dBm/Hz. If the source is not at 290 K, as with an antenna looking at cold sky, the floor is k (Ta + Te) BW instead.
Cascade noise figure worked examples
Example 1: two LNAs and an ADC, from TI SLYT839
LNA1 is 1 dB NF and 12 dB gain, LNA2 is 3 dB and 15 dB, and the ADC is 20 dB with 0 dB gain. The terms are 1.259 + 0.0628 + 0.1975, which sum to F = 1.519, or 1.82 dB. TI publishes 1.8 dB for this chain. The noise temperature is 150.6 K.
Example 2: where the filter goes
A chain of a 20 dB LNA with 1 dB NF, a 2 dB filter, a 7 dB loss mixer and a 30 dB IF amplifier with 6 dB NF has a noise figure of 1.95 dB. Put the filter ahead of the LNA and it becomes 3.61 dB, which is the 1.66 dB cost of moving the loss forward. These stage values are illustrative.
Example 3: sensitivity of that receiver
With the LNA first and a 1 MHz noise bandwidth, the noise floor is −173.98 + 60 + 1.95 = −112.0 dBm. A demodulator that needs 10 dB of SNR sets the sensitivity at −102.0 dBm.
Noise figure to noise temperature table, with noise factor
Te = 290 × (F − 1).
| NF (dB) | F (linear) | Te (K) |
|---|---|---|
| 0.5 | 1.122 | 35.39 |
| 1.0 | 1.259 | 75.09 |
| 1.5 | 1.413 | 119.6 |
| 2.0 | 1.585 | 169.6 |
| 3.0 | 1.995 | 288.6 |
| 4.0 | 2.512 | 438.4 |
| 5.0 | 3.162 | 627.1 |
| 6.0 | 3.981 | 864.5 |
| 8.0 | 6.31 | 1540 |
| 10.0 | 10 | 2610 |
| 15.0 | 31.62 | 8881 |
| 20.0 | 100 | 28710 |
Noise floor by bandwidth
kT0 × BW × F at 290 K, in dBm, before any required SNR is added.
| Bandwidth | NF 0 dB | NF 3 dB | NF 6 dB | NF 10 dB |
|---|---|---|---|---|
| 1 Hz | −174.0 dBm | −171.0 dBm | −168.0 dBm | −164.0 dBm |
| 1 kHz | −144.0 dBm | −141.0 dBm | −138.0 dBm | −134.0 dBm |
| 12.5 kHz | −133.0 dBm | −130.0 dBm | −127.0 dBm | −123.0 dBm |
| 125 kHz | −123.0 dBm | −120.0 dBm | −117.0 dBm | −113.0 dBm |
| 200 kHz | −121.0 dBm | −118.0 dBm | −115.0 dBm | −111.0 dBm |
| 1 MHz | −114.0 dBm | −111.0 dBm | −108.0 dBm | −104.0 dBm |
| 10 MHz | −104.0 dBm | −101.0 dBm | −98.0 dBm | −94.0 dBm |
| 20 MHz | −101.0 dBm | −98.0 dBm | −95.0 dBm | −91.0 dBm |
| 100 MHz | −94.0 dBm | −91.0 dBm | −88.0 dBm | −84.0 dBm |
Where the model stops
The Friis formula assumes each stage is matched to the next, uses available power gain and takes noise figures specified with a 290 K source. A mismatched interface changes both the real gain and the real noise figure, so use values measured in your impedance environment. Enter a passive loss as a negative gain with the same value as its noise figure, and remember that a hot cable or filter is noisier than its loss suggests.
An ADC can be treated as a last stage with 0 dB gain and its own noise figure, as TI does. Quote a mixer’s noise figure exactly as its data sheet specifies it, since single-sideband and double-sideband figures differ. And keep an eye on compression: the lowest noise figure is not useful if a strong nearby signal overloads the first stage, which is why many receivers accept a filter ahead of the LNA anyway.
Glossary
- Noise factor (F)
- Input SNR divided by output SNR with the source at 290 K, as a linear ratio. A noiseless stage has F = 1.
- Noise figure (NF)
- The noise factor in decibels: 10 log10(F).
- Equivalent noise temperature (Te)
- The temperature of a source at the input that would produce the same added noise: 290 × (F − 1) kelvin.
- T0
- The 290 K reference temperature used when a noise figure is specified.
- Noise floor
- Total noise power referred to the input over the system bandwidth.
- Sensitivity
- The weakest input signal that reaches the required SNR.
- LNA
- Low-noise amplifier, usually the first active stage of a receiver.
Related guides
- dBm to watts calculator: convert a noise floor or signal level.
- Free space path loss calculator: how much margin the sensitivity buys.
Frequently asked questions
What is the noise factor formula?
F = input SNR ÷ output SNR with the source at 290 K, as a linear ratio, and NF = 10 log10(F) puts it in dB. A 3 dB noise figure is a noise factor of 1.995, and a noiseless stage has F = 1.
Is this also a noise factor calculator?
Yes. Alongside the cascaded noise figure it shows the total noise factor F as a linear ratio. Enter each stage’s gain and noise figure in dB.
Why does the first stage dominate the noise figure?
Each later stage’s noise is divided by all the gain ahead of it. In TI’s example chain, a 3 dB second LNA behind 12 dB of gain raises the noise figure by only 0.2 dB.
Can I add noise figures in dB?
No. Convert each noise figure and gain to a linear ratio first, apply the Friis formula, and convert the total back to dB. Gains can be added in dB, noise figures cannot.
What is the noise figure of a cable, filter or attenuator?
For a passive part at 290 K the noise figure equals its loss. A cable with 3 dB of loss has a 3 dB noise figure, and if it sits ahead of the first amplifier that 3 dB adds directly to the system noise figure.
How do you convert noise figure to noise temperature?
Te = 290 × (F − 1), with F the linear noise factor. A 3 dB noise figure is 288.6 K and a 1 dB noise figure is 75.09 K.
How do you calculate receiver sensitivity from noise figure?
Sensitivity = −174 dBm + 10 log10(BW) + NF + the required SNR. With a 1.95 dB noise figure, 1 MHz of bandwidth and 10 dB of SNR that gives −102.0 dBm.
Why is thermal noise −174 dBm/Hz?
It is kT at the 290 K reference temperature, about 4 × 10⁻²¹ W/Hz. Calculated with the exact Boltzmann constant it is −173.98 dBm/Hz, which this calculator uses and which rounds to −174.
Should a filter go before or after the LNA?
For noise figure, after: a filter ahead of the LNA adds its insertion loss directly. In the example below, moving a 2 dB filter ahead of the LNA changes the noise figure from 1.95 dB to 3.61 dB. Strong out-of-band signals can force the filter forward anyway, to keep the LNA out of compression.
More free PCB tools
See all free electronics tools →
- dBm to watts converterConvert dBm, dBW, watts and milliwatts, plus the RMS and peak-to-peak voltage in 50 Ω or any impedance.
- Frequency to wavelength calculatorWavelength from frequency and back, in free space or in a cable or PCB with a velocity factor or dielectric constant.
- VSWR calculatorConvert between VSWR, return loss, reflection coefficient and mismatch loss, or compute them from a load impedance.
- Antenna length calculatorWavelength, half-wave dipole and quarter-wave monopole lengths for any frequency, with an end-effect factor.
- Free-space path loss calculatorFree-space path loss between two antennas from distance and frequency, in dB, with the distance for a given loss.
- Pi attenuator calculator (and T pad)Resistor values for Pi and T attenuator pads for any attenuation and impedance, with nearest standard values and their real loss.
Sources
- Rohde & Schwarz 1EE33, Understanding Noise in EMI Receivers: definitions, Friis and sensitivity
- Texas Instruments SLYT839, How the ADC noise figure impacts RF receiver designs: Friis equation and worked chains
- ELEX 7860 Wireless System Design, RF Design: Noise (course notes): T0 = 290 K, noise temperature, cascades
- U.S. Navy NAWCWD TP 8347, Electronic Warfare and Radar Systems Engineering Handbook, 4th ed. (2013), section 5-2: sensitivity with the required S/N term (PDF copy hosted by RF Cafe)
Design the whole board, not one number
ProtoFlow is free AI PCB design software: describe a circuit and get a schematic with real parts, PCB layout with an autorouter, SPICE simulation and Gerber files. Native KiCad, Altium and EAGLE import and export. Free on macOS, Windows and Linux.
Get for Mac
Free to start. No card required.
