Johnson–Nyquist Noise: The Receiver's Thermal Baseline
Johnson–Nyquist Noise: The Receiver's Thermal Baseline
A resistor needs no applied voltage to produce a random terminal voltage. Thermal agitation is enough. That unavoidable noise does not explain every raised spectrum trace, but it gives RF engineers the reference against which loss, bandwidth and receiver noise are judged.
Johnson noise is a property of dissipative impedance at finite temperature. In the classical RF range it is well described by the Nyquist equation. The design objective is not to “remove” that noise; it is to avoid adding unnecessary loss and noise while keeping the front end linear.
What the Random Voltage Represents
Microscopic thermal motion produces fluctuating current and voltage in a conductor. The mean voltage is zero, but the mean-square voltage is not. NIST's Johnson-noise thermometry work uses this relationship as a primary temperature measurement: the noise power is predictable from resistance, thermodynamic temperature and bandwidth.
At ordinary HF and VHF frequencies and room temperature, the classical approximation is extremely accurate. NIST reports the 4kTR relationship accurate to better than 5 × 10−9 below 100 MHz for temperatures above 25 K. Quantum corrections matter only when frequency is no longer small compared with kT/h.
The Voltage and Power Forms Describe Different Terminations
For a real resistance R at thermodynamic temperature T over equivalent noise bandwidth B, the open-circuit RMS noise voltage is:
vn = √(4kTRB)
The corresponding one-sided open-circuit voltage-noise density is √(4kTR) V/√Hz. Its Norton equivalent is a parallel current-noise density √(4kT/R) A/√Hz.
When that resistor is conjugately matched to a load, half the open-circuit voltage appears across the load and the available noise power becomes:
Pn,av = kTB
At the standard noise reference temperature T0 = 290 K, kT is 4.004 × 10−21 W/Hz, or approximately −173.98 dBm/Hz. The familiar “−174 dBm/Hz” is therefore a rounded matched available-power density—not the open-circuit voltage of every circuit node.
| Example at 290 K | Result | Interpretation |
|---|---|---|
| 50 Ω voltage-noise density | 0.895 nV/√Hz | Open-circuit voltage density |
| 50 Ω over 1 MHz | 0.895 µV RMS | Open-circuit integrated voltage |
| Matched available power over 1 MHz | −113.98 dBm | kTB delivered under conjugate match |
| 1 kΩ voltage-noise density | 4.00 nV/√Hz | Higher voltage density, not higher matched kTB power |
Bandwidth Means Equivalent Noise Bandwidth
Noise passes through the complete magnitude-squared response of a filter. The relevant bandwidth is therefore:
Bn = ∫ |H(f)|² df / |H|max²
A −3 dB bandwidth equals the noise bandwidth only for an ideal rectangular response. For a first-order low-pass response integrated from DC to infinity, Bn = (π/2)fc. Doubling equivalent noise bandwidth adds 3.01 dB of integrated white-noise power.
Noise Figure Refers Added Noise to the Input
Noise factor F is the ratio of input SNR to output SNR under the stated source temperature, source impedance and bandwidth. Noise figure is 10log10F. The equivalent input noise temperature is:
Te = T0(F−1)
At T0 = 290 K, a 1 dB noise figure corresponds to about 75 K and a 3 dB noise figure to about 289 K. That conversion is a reference-plane tool; it does not claim that the amplifier is physically at Te.
The First Stage Matters, but Gain Cannot Excuse Overload
For cascaded stages expressed as linear noise factors and power gains, Friis gives:
Ftotal = F1 + (F2−1)/G1 + (F3−1)/(G1G2) + …
Useful low-noise gain in the first stage suppresses the noise contribution of later stages. Excessive gain can instead reduce blocking margin, compress the front end or generate intermodulation. “Maximum gain first” is not a receiver design rule. Choose enough gain to meet the noise budget while preserving the strongest expected signals.
A passive loss L before the LNA is especially costly in a receiver-noise-limited system. At physical temperature Tp, it contributes (L−1)Tp and multiplies the following LNA noise temperature by L when both are referred to the loss input. On noisy HF bands, external antenna noise may still dominate; establish that margin instead of assuming it.
Sensitivity Is a Budget, Not One Magic Number
For a matched source at 290 K, a useful input-noise-floor estimate is:
Nin ≈ −173.98 dBm/Hz + 10log10(Bn) + NF
A detection or demodulation threshold then adds the required SNR for the stated probability, modulation, coding and measurement criterion. A 10 MHz noise bandwidth, 4 dB NF and 10 dB required SNR gives about −90 dBm. A 200 kHz noise bandwidth, 5 dB NF and 12 dB required SNR gives about −104 dBm.
Those are examples, not receiver ratings. Real antennas deliver external noise; mismatches change transducer gain and amplifier noise parameters; filters have finite ENBW; AGC and detectors use particular statistics; and interference is not Johnson noise.
Do Not Put Every Noise Mechanism in the kTB Bucket
- Shot noise follows charge crossing a potential barrier and depends on DC current.
- Flicker noise rises toward low frequency and depends strongly on device technology and bias.
- Oscillator phase noise can convert a strong nearby signal into reciprocal-mixing noise.
- Atmospheric, galactic and man-made noise enters through the antenna and environment; ITU-R P.372 treats these external sources.
- Intermodulation and clipping are nonlinear products, not a thermal floor.
Measure at a Declared Reference Plane
A calibrated noise source and Y-factor measurement can determine receiver noise temperature when the source temperatures, ENR, mismatch, loss, detector linearity and bandwidth are controlled. Spectrum-analyzer noise markers can be useful, but detector type, RBW filter ENBW, preamplifier state and instrument noise floor must be included. A universal detector “correction” is not a substitute for the instrument's measurement method.
For a hot/cold source ratio Y = Phot/Pcold, the equivalent input noise temperature under the stated conditions is:
Te = (Thot − YTcold)/(Y−1)
Bottom line: Johnson–Nyquist noise is the thermal baseline, not the whole noise story. Keep the voltage and matched-power forms straight, use equivalent noise bandwidth, refer every contribution to one plane, and preserve enough linearity to hear the noise floor you worked to achieve.
Primary sources checked
Mini-FAQ
- What is Johnson–Nyquist noise? It is the random terminal voltage or current associated with a dissipative impedance at finite thermodynamic temperature.
- Why is −174 dBm/Hz used? It is the rounded matched available noise-power density kT at the 290 K reference temperature.
- Why is the open-circuit voltage based on 4kTRB? The resistor's Thévenin source has that mean-square open-circuit voltage; a matched load receives half the voltage and kTB power.
- Does a higher resistance produce more available thermal-noise power? No. It produces more open-circuit voltage noise and less Norton current noise, while the conjugately matched available power remains kTB.
- Can shielding reduce Johnson noise? It can reduce external coupling, but it does not remove the resistor's own thermal noise.
- What is the most effective design lever? Use the smallest necessary equivalent noise bandwidth, control pre-LNA loss and choose first-stage noise, gain and linearity together.