Friis Formula

The Friis formula provides the method to calculate the overall noise figure of a cascade of amplifier or network stages. In any multi-stage receiver or signal chain, each stage adds its own noise, but their contributions to the total noise figure are not equal. The Friis formula precisely quantifies how each stage contributes, and shows why the first stage noise figure dominates the cascade.

This formula is directly tested in GATE and is essential for RF system design, satellite receiver analysis, and understanding why low-noise amplifiers are always placed first in a receive chain.

Core Concept Explanation

Consider a chain of N amplifier stages, each with noise factor Fi and available power gain Gi. When the signal passes through the first stage, it is amplified by G1 but the stage also adds noise. The second stage sees an amplified signal and amplified noise from stage 1, plus it adds its own noise. The key insight is that the noise added by stage 2 appears at the output of a chain with gain G1 already applied, so when referred back to the input, stage 2's noise is divided by G1.

This means if the first stage has a high gain G1, all subsequent stages' noise contributions become negligible at the input-referred level. This is why in satellite receivers, the LNA (low-noise amplifier) with the lowest noise figure is always the first component after the antenna, and why increasing its gain suppresses all downstream noise.

If stage 1 has low gain (or is a passive attenuating device like a cable or filter with loss L > 1), the noise figure of stage 1 becomes L itself, and stage 2's noise contribution is divided by only 1/L, so it has a significant effect. High-loss front-end elements are therefore very damaging to receiver sensitivity.

Mathematical Expression

The Friis formula for total noise factor of N cascaded stages is: Ftotal = F1 + (F2-1)/G1 + (F3-1)/(G1G2) + (F4-1)/(G1G2G3) + .... All Fi and Gi are linear (not dB) quantities. The total noise figure in dB is then NF_total = 10 log10(Ftotal).

For a passive attenuator (lossy element) with insertion loss L (power loss ratio, L > 1) at physical temperature T0, the noise factor is F = L. If the attenuator is at temperature T0, this is exact. A cable with 3 dB loss has L = 2, so its noise factor is 2 (NF = 3 dB).

In terms of noise temperatures, the Friis formula becomes: Tsys = T1 + T2/G1 + T3/(G1G2) + ..., where Ti = (Fi - 1) x T0. This form is more convenient in satellite link budget calculations where noise temperatures are the natural unit.

Practical Understanding

Consider a typical satellite receiver front end: antenna, low-noise amplifier (LNA), bandpass filter (BPF), and downconverter mixer. Even though the BPF has a noise figure of several dB, if the LNA gain is 20 dB, the filter's contribution to overall system noise is divided by 100, making it almost negligible. This is the practical power of the Friis formula.

The formula also explains why inserting a lossy element (attenuator, long cable) before the LNA is very harmful. The attenuator has F = L and G = 1/L. After the attenuator, when the Friis formula is applied, the LNA's noise factor is divided by only 1/L, meaning the LNA contributes L times more noise than it would if it were directly at the input.

Given:
Stage 1: F1 = 2 (NF1 = 3 dB), G1 = 10 (10 dB)
Stage 2: F2 = 4 (NF2 = 6 dB), G2 = 20 (13 dB)
Stage 3: F3 = 8 (NF3 = 9 dB)

Why this formula applies:
Friis formula calculates total noise factor of cascaded stages with input-referred contributions

Formula:
Ftotal = F1 + (F2-1)/G1 + (F3-1)/(G1 x G2)

Substitution:
Ftotal = 2 + (4-1)/10 + (8-1)/(10 x 20)
       = 2 + 3/10 + 7/200

Calculation:
Ftotal = 2 + 0.3 + 0.035 = 2.335

Final Answer with units:
Ftotal = 2.335 (linear)
NF_total = 10 log10(2.335) = 3.68 dB

Exam Tip: In Friis formula GATE problems, always convert NF (dB) to linear F before applying the formula. Compute gain G in linear as well. After finding Ftotal, convert back to dB at the end. Mixing dB and linear in the formula is the most common exam mistake.

Key Implications of Friis Formula

• The first stage noise figure dominates the cascade noise figure when G1 is large.
• A high-gain, low-noise first stage (LNA) minimizes overall system noise figure effectively.
• A passive lossy element before the LNA multiplies all subsequent noise contributions and should be avoided or minimized.
• All Fi and Gi in the Friis formula must be in linear form, not dB.
• In noise temperature form: Tsys = T1 + T2/G1 + T3/(G1G2) + ... — useful for satellite link budget analysis.

Quick Revision

• Friis formula: Ftotal = F1 + (F2-1)/G1 + (F3-1)/(G1G2) + ...
• All F and G values must be linear (not dB) when applying the formula.
• Stage 1 dominates: high G1 suppresses contributions from all later stages.
• Attenuator with loss L at front end: F = L, G = 1/L — very damaging to system NF.
• Noise temperature version: Tsys = T1 + T2/G1 + T3/(G1G2) + ...
• Exam trap: Convert dB to linear first, apply formula, then convert result back to dB.
• LNA must be the first active stage in any low-noise receiver — this is a direct consequence of the Friis formula.