RF & Wireless Communications Engineering
Friis Cascaded RF Noise Figure (dB) and LNA Placement Formula
Calculate 3-stage cascaded RF receiver Noise Factor F_tot = F_1 + (F_2 - 1)/G_1 + (F_3 - 1)/(G_1*G_2) and Noise Figure NF_dB = 10*LOG10(F_tot) in Excel and Arte Sheets.
Never add dB noise figures directly — convert to linear F_k and G_k via Friis’s equation!
Friis (1944) proved that in a cascaded RF front-end, every downstream stage’s excess noise (F_k - 1) is divided by the cumulative linear power gain G_1 * ... * G_{k-1} of all preceding stages: F_tot = F_1 + (F_2 - 1) / G_1 + (F_3 - 1) / (G_1 * G_2), where F_k = 10^(NF_k_dB / 10) and G_k = 10^(Gain_k_dB / 10). Adding dB values directly is completely wrong — and placing a lossy coax cable ahead of the Low-Noise Amplifier (LNA) destroys receiver sensitivity!
Compare both layouts on the sheet: (1) LNA first (NF_1 = 3 dB, Gain_1 = 20 dB -> G_1 = 100 in B2:B3) followed by a 6 dB mixer (Gain_2 = 10 dB in B4:B5) and 10 dB IF amp (B6) yields linear F_tot = 1.9953 + 2.9811/100 + 9/1000 = 2.0341, or NF_tot = 3.0837 dB (downstream stages add just +0.0837 dB!); (2) put a 6 dB lossy feedline (NF_1 = 6 dB, Gain_1 = -6 dB) BEFORE the 3 dB / 20 dB LNA and cascaded NF_tot jumps to 9.1916 dB!
| RF Front-End Cascade Order | Linear Noise Factor F_tot | Cascaded Noise Figure NF_dB |
|---|---|---|
| Masthead LNA First (3dB/+20dB -> 6dB/+10dB -> 10dB) | 2.0341 | 3.0837 dB (Optimal Sensitivity) |
| Lossy Cable Before LNA (6dB/-6dB -> 3dB/+20dB -> 10dB) | 8.3016 | 9.1916 dB (+6.11 dB Penalty!) |
| Noiseless Downstream Stages (NF_2 = NF_3 = 0 dB) | 1.9953 | 3.0000 dB (= Stage 1 NF) |
Friis 3-Stage Cascaded Noise Figure NF_tot (dB) Formula
Require physical NF_k_dB >= 0 dB (since passive or active linear noise factor F_k >= 1.0000 at 290 K reference temperature; note passive attenuators of L dB loss have NF_dB = +L dB and Gain_dB = -L dB).
=LET(nf_1, B2, gn_1, B3, nf_2, B4, gn_2, B5, nf_3, B6, IF(OR(nf_1 < 0, nf_2 < 0, nf_3 < 0), "Invalid Noise Figure (Require NF_dB >= 0)", LET(f_1, POWER(10, nf_1 / 10), g_1, POWER(10, gn_1 / 10), f_2, POWER(10, nf_2 / 10), g_2, POWER(10, gn_2 / 10), f_3, POWER(10, nf_3 / 10), f_tot, f_1 + (f_2 - 1) / g_1 + (f_3 - 1) / (g_1 * g_2), ROUND(10 * LOG10(f_tot), 4))))Linear Cascaded Noise Factor F_tot Formula
Expose linear F_tot alongside NF_dB so link-budget sheets can also compute equivalent noise temperature T_eq = 290 * (F_tot - 1).
=LET(nf_1, B2, gn_1, B3, nf_2, B4, gn_2, B5, nf_3, B6, IF(OR(nf_1 < 0, nf_2 < 0, nf_3 < 0), "Invalid Noise Figure (Require NF_dB >= 0)", LET(f_1, POWER(10, nf_1 / 10), g_1, POWER(10, gn_1 / 10), f_2, POWER(10, nf_2 / 10), g_2, POWER(10, gn_2 / 10), f_3, POWER(10, nf_3 / 10), ROUND(f_1 + (f_2 - 1) / g_1 + (f_3 - 1) / (g_1 * g_2), 4))))Try the example
Use the original values or change them to test your understanding.