How do I calculate the total noise figure of a cascaded receiver chain with multiple stages?
The Friis Cascade Equation
When multiple RF components are connected in series, each one adds noise to the signal. The Friis equation, developed by Harald T. Friis in 1944, provides the method for calculating the total noise factor of the entire chain from the individual noise factors and gains of each stage.
The equation shows that the first stage has the greatest impact on overall system noise. This is why low-noise amplifiers (LNAs) are placed as close to the antenna as possible. If the first stage has high gain and low noise figure, the noise contributions of all subsequent stages become negligible because they are divided by the first stage gain.
In practice, this means a receiver with a 0.5 dB NF LNA providing 20 dB of gain will have a total system noise figure very close to 0.5 dB regardless of how noisy the downstream mixer and IF amplifier are. Conversely, placing a lossy cable or filter before the LNA directly adds to the system noise figure.
When cascading stages, always work in linear units (noise factor F, not NF in dB; power gain G, not dB). Convert each stage's noise figure to noise factor using F = 10^(NF/10), cascade using Friis, then convert the result back to dB.
Where:
F₁, F₂, F₃ = noise factor of each stage (linear)
G₁, G₂ = available power gain of each stage (linear)
Frequently Asked Questions
Does the Friis equation work for passive components?
Yes. A passive component has a noise factor equal to its loss factor (1/gain). For a 3 dB attenuator, F = 2 and G = 0.5. Insert these values into the Friis equation like any other stage.
What if the stages are not impedance matched?
The standard Friis equation assumes matched conditions. Impedance mismatch between stages changes the effective noise figure and gain, requiring modified versions of the equation or S-parameter based noise analysis.
How many stages do I need to include?
Include every component in the signal path: cables, connectors, filters, amplifiers, mixers, and attenuators. After the first high-gain stage, contributions become small, but lossy components anywhere in the chain should be accounted for.