Also known as: Johnson noise, Nyquist noise, Johnson–Nyquist noise, kTB noise, thermal agitation noise
Thermal noise — also called Johnson–Nyquist noise — is the electrical noise produced by the random thermal motion of charge carriers in any resistor or lossy element, with an available power of P = kTB that depends only on temperature and bandwidth, not on the material.1 It is the irreducible origin of a receiver’s noise floor: even a perfect radio delivers this much noise to its detector, so thermal noise is the ultimate limit on receiver sensitivity and on the signal-to-noise ratio you can achieve. Discovered experimentally by John B. Johnson and explained theoretically by Harry Nyquist at Bell Labs in 1928, it is a form of white noise — flat across the spectrum up to very high frequencies.
How it works
Charge carriers in a conductor never sit still: at any temperature above absolute zero they jostle with thermal energy, and that random motion appears at the terminals as a fluctuating voltage. Nyquist showed the available noise power a resistor can deliver to a matched load is
N = kTB
where k is Boltzmann’s constant (1.38 × 10⁻²³ J/K), T is the absolute temperature in kelvin, and B is the bandwidth in hertz. Two consequences matter:
- It is independent of resistance. The open-circuit noise voltage rises with R, but the power delivered to a matched load does not — so kTB is a property of temperature and bandwidth alone.
- It is white. Up to frequencies far above any radio band the spectrum is flat, meaning equal noise power in every hertz. Doubling the bandwidth doubles the noise.
At the standard reference temperature of 290 K (roughly room temperature), kT works out to a noise power spectral density of −174 dBm/Hz. This single number is the starting point of nearly every link and sensitivity budget: multiply by bandwidth (add 10·log₁₀ B in decibels) to get the thermal floor. A 1 Hz bandwidth sees −174 dBm; a 12.5 kHz land-mobile channel sees about −174 + 41 = −133 dBm of thermal noise.
In practice
Thermal noise is only the input floor. A real receiver adds its own noise, so the noise it presents at the detector is kTB raised by the noise figure of the front end. The sensitivity of a radio is therefore:
MDS = −174 dBm/Hz + 10·log₁₀(B) + NF + required SNR
Because the kTB term is fixed by physics, the only knobs an engineer controls are bandwidth (narrower channels see less noise), noise figure (a good low-noise amplifier first in the chain), and the required SNR of the modulation. Cooling the front end lowers T and hence the noise — the reason radio-astronomy and satellite ground stations use cryogenic LNAs, quantified through noise temperature rather than noise figure.
Relevance to SDR
Thermal noise sets the hard floor every software-defined radio works against. When you tune an RTL-SDR, Airspy, or SDRplay across an empty band, the grass on the waterfall is dominated by kTB plus the receiver’s noise figure and, in cheaper dongles, quantization and spur artifacts stacked on top. Understanding kTB is what tells you whether a marginal P25 or DMR control channel is limited by physics (too little signal above the thermal floor — fix it with antenna gain or an LNA) or by something you added (an under-driven ADC, a lossy feedline, or local interference).
GopherTrunk does not model thermal noise explicitly, but it lives with the consequences: its symbol-level SNR and EVM estimates are measuring how far a captured signal sits above this floor, and the decode thresholds it reports are, in the end, statements about the thermal-noise-limited SNR each vocoder and FEC scheme needs. No amount of DSP can recover a signal that has been allowed to fall to the kTB floor before it reaches the ADC — which is why front-end gain staging, not back-end filtering, is the first thing to check on a weak channel.
Sources
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Johnson–Nyquist noise — Wikipedia, derivation of the kTB relation and the −174 dBm/Hz reference density. ↩