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Cascaded Noise Figure Calculator

Calculate cascaded noise figure and IP3 for multi-stage RF receiver chains using the Friis formula. Optimize LNA and filter ordering. Free, instant results.

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Formula

Ftotal=F1+F2−1G1+F3−1G1G2+⋯ ,1IIP3in=1IIP31+G1IIP32+⋯F_{total} = F_1 + \frac{F_2-1}{G_1} + \frac{F_3-1}{G_1 G_2} + \cdots, \quad \frac{1}{\mathrm{IIP3}_{in}} = \frac{1}{\mathrm{IIP3}_1} + \frac{G_1}{\mathrm{IIP3}_2} + \cdots

Reference: Friis, "Noise Figures of Radio Receivers" (1944); Pozar Chapter 10; Razavi "RF Microelectronics"

F_n— Noise factor of stage n (linear: 10^(NF_dB/10))
G_n— Power gain of stage n (linear: 10^(Gain_dB/10))
NF— Noise figure in dB: 10·log₁₀(F) (dB)
IIP3_n— Input IP3 of stage n (mW) (mW)
OIP3— IIP3_total + cascaded gain (dBm)

How It Works

Cascaded noise figure determines receiver sensitivity in RF systems — wireless engineers, radar designers, and satellite communication architects use the Friis formula to optimize signal chain performance. The cascade equation NF_total = NF_1 + (NF_2-1)/G_1 + (NF_3-1)/(G_1*G_2) + ... shows that the first stage dominates system noise performance because subsequent stages are divided by cumulative gain, per Pozar's 'Microwave Engineering' (4th ed.) and ITU-R P.372.

A typical receiver with 2 dB LNA (NF_1), 20 dB LNA gain (G_1), and 8 dB mixer (NF_2) achieves NF_total = 2 + (6.31-1)/100 = 2.05 dB — the 8 dB mixer adds only 0.05 dB because it's preceded by 20 dB gain. However, placing a 3 dB cable before the LNA degrades system NF to 3 + (1.58-1)/0.5 = 4.16 dB — every dB of loss before the LNA adds approximately 1 dB to system noise figure.

For cascaded linearity (IIP3), the formula inverts: IIP3_total^-1 = IIP3_1^-1 + G_1*IIP3_2^-1 + G_1*G_2*IIP3_3^-1, meaning the last stage (with highest preceding gain) dominates linearity. This creates the fundamental noise-linearity tradeoff in receiver design — high LNA gain improves noise figure but degrades IIP3 by boosting signals before the mixer.

Worked Example

Problem

Design a 2.4 GHz receiver front-end with NF < 2.5 dB and IIP3 > -15 dBm for WiFi application.

Component specifications:

  • Band filter: 1.5 dB insertion loss (NF = 1.5 dB, IIP3 = infinite)
  • LNA: NF = 1.2 dB, Gain = 18 dB, IIP3 = +5 dBm
  • Mixer: NF = 10 dB, Gain = -1 dB (conversion loss), IIP3 = +10 dBm
  • IF amplifier: NF = 4 dB, Gain = 20 dB, IIP3 = +15 dBm
Noise figure calculation (linear values, NF and gains):
  1. Filter contribution: NF_1 = 1.41 (1.5 dB), G_1 = 0.71 (-1.5 dB)
  2. LNA contribution: (NF_2 - 1)/G_1 = (1.32 - 1)/0.71 = 0.45
  3. Mixer contribution: (NF_3 - 1)/(G_1*G_2) = (10 - 1)/(0.71*63.1) = 0.20
  4. IF amp contribution: (NF_4 - 1)/(G_1*G_2*G_3) = (2.51 - 1)/(0.71*63.1*0.79) = 0.04
  5. NF_total = 1.41 + 0.45 + 0.20 + 0.04 = 2.10 linear = 3.22 dB
Result: NF = 3.22 dB exceeds 2.5 dB requirement. Solution: use lower-loss filter (0.8 dB) or higher-gain LNA (22 dB). With 0.8 dB filter: NF_total = 2.35 dB — meets spec.

IIP3 calculation confirms linearity: IIP3_total = -12 dBm (dominated by mixer after 16.5 dB LNA gain), meeting -15 dBm requirement.

Solution

use lower-loss filter (0.8 dB) or higher-gain LNA (22 dB). With 0.8 dB filter: NF_total = 2.35 dB — meets spec.

IIP3 calculation confirms linearity: IIP3_total = -12 dBm (dominated by mixer after 16.5 dB LNA gain), meeting -15 dBm requirement.

Practical Tips

  • ✓Place the lowest noise figure, highest gain amplifier first in the chain — a 0.5 dB NF LNA with 25 dB gain suppresses all following stage contributions by > 200:1
  • ✓Minimize loss between antenna and LNA — use short, low-loss cable (LMR-400 vs RG-58), mount LNA at antenna feedpoint for receive-critical applications like radio astronomy or GPS
  • ✓Budget NF degradation for manufacturing tolerance — if spec is 2.5 dB, design for 2.0 dB nominal; LNA NF varies +/- 0.3 dB unit-to-unit, cables add 0.1-0.2 dB connector variation

Common Mistakes

  • ✗Forgetting to convert dB to linear ratios — Friis formula requires linear noise factor and gain values; mixing dB and linear causes order-of-magnitude errors
  • ✗Neglecting loss before the LNA — every 1 dB of cable, filter, or switch loss before the first amplifier adds 1 dB to system NF; a 3 dB preselector filter degrades 1.5 dB LNA to 4.5 dB system NF
  • ✗Assuming high-NF stages don't matter — while their contribution is divided by preceding gain, insufficient gain still allows significant degradation; a 15 dB NF mixer after only 10 dB LNA gain adds 0.4 dB to system NF
  • ✗Ignoring the noise-linearity tradeoff — increasing LNA gain improves NF but degrades IIP3; receiver design requires balancing both specifications per Razavi's 'RF Microelectronics'

Frequently Asked Questions

The Friis formula divides each stage's noise contribution by the cumulative gain of all preceding stages. First stage noise (NF_1) is not divided by anything — it contributes fully. Second stage contribution is divided by G_1; third by G_1*G_2. With 20 dB (100x) LNA gain, a 10 dB (10x) mixer adds only (10-1)/100 = 0.09 to total noise factor (0.4 dB). This mathematical structure makes first-stage NF the dominant receiver parameter.
No — noise figure represents the ratio of actual output noise to ideal (thermal-limited) output noise, which is always >= 1 (0 dB). Negative NF would imply the device removes noise, violating thermodynamics. The theoretical minimum is 0 dB (noise factor = 1), achieved only by an ideal lossless passive device at the same temperature as the source. Practical LNAs achieve 0.3-0.5 dB NF using GaAs pHEMT or InP HEMT technology cooled or at room temperature with careful design.
Noise figure is defined at 290 K (17 C) standard temperature per IEEE. Actual noise power scales with physical temperature: P_noise = k*T*B. A device with 3 dB NF at 290 K has equivalent noise temperature T_e = 290*(NF-1) = 290 K. At 77 K (liquid nitrogen), the same device would show lower equivalent noise temperature. Cryogenic LNAs for radio astronomy achieve < 10 K equivalent temperature (< 0.15 dB NF) by cooling to 15-20 K physical temperature.
Noise factor (F) is the linear ratio: F = (SNR_in)/(SNR_out) = 1 + T_e/T_0 where T_e is equivalent noise temperature and T_0 = 290 K. Noise figure (NF) is noise factor expressed in decibels: NF = 10*log10(F). A device with F = 2 (noise factor) has NF = 3 dB (noise figure). The Friis formula uses linear noise factor; results are typically reported in dB as noise figure. Always clarify which is meant when NF values < 1 dB are discussed.
For sensitivity-limited applications (weak signals, long range), yes — each 1 dB NF improvement equals 1 dB better sensitivity. However, low-NF LNAs often have lower IIP3, risking intermodulation from strong interferers. In congested RF environments (urban cellular, WiFi), linearity may matter more than noise figure. Modern receiver architectures use digitally-controlled gain distribution to optimize NF when signals are weak and IIP3 when signals are strong.
Use Friis formula with linear values. Example: LNA NF = 2 dB (F1 = 1.58), Gain = 20 dB (G1 = 100); Mixer NF = 8 dB (F2 = 6.31). F_total = F1 + (F2 - 1)/G1 = 1.58 + (6.31 - 1)/100 = 1.58 + 0.053 = 1.633. NF_total = 10*log10(1.633) = 2.13 dB. The 8 dB mixer degrades system NF by only 0.13 dB because 20 dB LNA gain suppresses its contribution. This is why LNA NF and gain are the critical receiver parameters.
A 3 dB attenuator has NF = 3 dB (F = 2.0) and gain = -3 dB (G = 0.5). Friis formula: F_total = F_atten + (F_LNA - 1)/G_atten. For 1 dB LNA (F = 1.26): F_total = 2.0 + (1.26 - 1)/0.5 = 2.0 + 0.52 = 2.52 = 4.0 dB. The 3 dB attenuator degraded system NF by exactly 3 dB — loss before the LNA adds directly to system noise figure. This is why cable loss, preselector filters, and switches before the LNA are minimized in sensitive receivers.
Application-dependent targets per industry standards: GPS receiver: 1.5-2.5 dB (weak -130 dBm signals require low NF). LTE/5G base station: 2-3 dB (3GPP specifies reference sensitivity). WiFi: 4-6 dB (strong signals, less NF-critical). Amateur weak-signal: 0.5-1.5 dB (EME, satellite). Cellular handset: 5-7 dB (limited by handset antenna environment noise). Each 1 dB NF improvement increases receiver sensitivity 1 dB — for GPS, this extends coverage; for cellular, it reduces required base station density.

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