Engineering & Scientific
Friis Free-Space Path Loss (FSPL), Nyquist Noise Floor & RF Link SNR Margin
Build an end-to-end RF wireless link budget in dB/dBm combining exact speed-of-light Friis FSPL, Johnson-Nyquist k_B*T*B thermal noise floor plus receiver noise figure, and carrier-to-noise ratio (SNR).
Combining Friis inverse-square aperture spreading with Johnson-Nyquist thermal noise
Harald Friis’s (1946) transmission equation expresses line-of-sight isotropic Free-Space Path Loss over distance d (km) and carrier frequency f (MHz) using exact vacuum light speed c = 299,792,458 m/s as FSPL_dB = 20 * LOG10((4 * PI() * (d_km * 1000) * (f_MHz * 10^6)) / 299792458). Subtracting FSPL_dB from Effective Isotropic Radiated Power plus receive antenna gain (P_tx_dBm + G_net_dBi) yields received carrier power P_rx_dBm.
At the receiver input, thermal agitation across temperature T (Kelvin) and channel bandwidth B (Hz) generates Johnson-Nyquist noise power P_n = k_B * T * B Watts (where Boltzmann’s constant k_B = 1.380649E-23 J/K). Converting Watts to milliwatts (* 1000), taking 10 * LOG10(...), and adding receiver Noise Figure NF_dB establishes the effective noise floor N_floor_dBm (~ -174 dBm/Hz + 10*LOG10(B_Hz) + NF_dB) and resulting SNR = P_rx_dBm - N_floor_dBm.
| RF Link Stage (10 km, 2,400 MHz, 20 MHz BW) | Formula | Decibel Level |
|---|---|---|
| Free-Space Path Loss FSPL | 20 * LOG10(4*PI()*d_m*f_Hz / 299792458) | 120.05 dB |
| Received Carrier Power P_rx | P_tx (20 dBm) + G_net (16 dBi) - FSPL | -84.05 dBm |
| Receiver Noise Floor (T=290 K, NF=5 dB) | 10*LOG10(k_B*T*B*1000) + NF_dB | -95.96 dBm |
| Post-Receiver Carrier SNR | P_rx_dBm - N_floor_dBm | 11.91 dB |
Exact Speed-of-Light Friis FSPL Formula (dB)
Convert B2 (km) to meters and B3 (MHz) to Hz and evaluate 20 * LOG10(4 * PI() * d * f / c).
=ROUND(20 * LOG10((4 * PI() * (B2 * 1000) * (B3 * 1000000)) / 299792458), 2)Johnson-Nyquist Receiver Noise Floor Formula (dBm)
Multiply exact SI Boltzmann constant 1.380649E-23 by Kelvin temperature B6, Hz bandwidth B7 * 10^6, and 1,000 mW/W, convert via 10 * LOG10, and add receiver Noise Figure B8.
=ROUND(10 * LOG10(1.380649E-23 * B6 * (B7 * 1000000) * 1000) + B8, 2)Try the example
Use the original values or change them to test your understanding.