Johnson Noise (Thermal Noise): What Is It

— and Why Every Receiver Designer Must Care

TL;DR: Johnson noise is the random voltage or current generated by the thermal motion of charge carriers in any resistive element. It is white, Gaussian, and unavoidable. In receivers, it defines the kTB noise floor that ultimately limits sensitivity. Most of RX design is about not letting anything raise that floor.

The Physical Idea

Any resistor at absolute temperature T has electrons constantly jostling. That agitation appears as a zero-mean random voltage even with no applied signal. Because the effect arises from thermodynamics—not manufacturing—you can’t build it out. You can only lower the temperature, narrow the bandwidth, or minimize added noise in the first stage.

At everyday RF or baseband frequencies, the classical model holds: the noise spectral density is flat up to roughly f ≪ kT/h ≈ 6 THz at 290 K. Only far into the THz range do quantum corrections matter.

The Three Formulas You’ll Use Every Day

• Open-circuit noise voltage: vₙ(rms) = √(4 k T R B) [V]
  Example: 50 Ω, 290 K, 1 MHz → vₙ ≈ 0.895 µV rms.
  Noise density: √(4 k T R) → 50 Ω → 0.895 nV/√Hz, 1 kΩ → 4.00 nV/√Hz, 10 kΩ → 12.7 nV/√Hz.
• Available noise power: Pₙ = k T B [W]
  At 290 K: N₀ ≈ −174 dBm/Hz → Power in bandwidth B = N₀ + 10 log₁₀ B.
• Noise figure and equivalent noise temperature:
  F = (SNR_in / SNR_out), NF = 10 log₁₀ F [dB], Tₑ = T₀ (F − 1) with T₀ = 290 K.
  Conversions: 0.5 dB → 35 K, 1 dB → 75 K, 2 dB → 170 K, 3 dB → 289 K, 5 dB → 627 K.

Why Johnson Noise Matters in an RX

It sets the absolute sensitivity limit.
Pₘᵢₙ ≈ −174 dBm/Hz + 10 log₁₀ B + NF + SNR_req.
Example A (10 MHz, NF = 4 dB, SNR req = 10 dB): Pₘᵢₙ ≈ −90 dBm.
Example B (200 kHz, NF = 5 dB, SNR req = 12 dB): Pₘᵢₙ ≈ −104 dBm.
Doubling bandwidth adds 3 dB of noise power.

The first stage is king (Friis equation).
F_total = F₁ + (F₂ − 1)/G₁ + (F₃ − 1)/(G₁ G₂) + …
Loss before the LNA is especially costly: 1 dB of pre-LNA loss adds ≈ 1 dB system NF.

It ties RX design to impedance and noise matching.
Maximum available noise power from the source is k T B only if the source is matched. Minimum amplifier NF often occurs at a source impedance Z_opt ≠ 50 Ω, so front ends must balance power vs noise match.

It interacts with mixers and images.
A double-sideband mixer folds noise from both RF sidebands into IF, adding ≈ 3 dB noise versus an SSB or image-reject scheme.

It accumulates with the antenna and environment.
System noise temperature T_sys ≈ T_A + Tₑ₁/L_pre + (other terms). Cryogenic LNAs exist because most of T_sys sits in that first stage.

Bandwidth Really Means Noise Bandwidth

Real filters aren’t brick-walls. Noise power after a filter is Pₙ = k T ∫₀^∞ |H(f)|² df = k T Bₙ, where Bₙ is the equivalent noise bandwidth (ENBW). Example: a 1-pole RC low-pass with −3 dB cut-off f_c has Bₙ = (π/2) f_c ≈ 1.57 f_c.

Practical Design Levers

• Start with a low-NF, high-gain LNA and put lossy stuff after it.
• Shape bandwidth tightly at RF, IF and baseband — narrower Bₙ lowers the noise floor.
• Choose resistor values with noise in mind (Johnson noise ∝ R in voltage mode, 1/R in current mode).
• Prefer SSB mixers if sensitivity is tight; budget the 3 dB penalty for DSB.
• Control temperature — noise scales with absolute T.
• Don’t confuse Johnson noise with shot noise (iₙ² = 2 q I B) or 1/f noise at low frequencies.

How to Measure It in Practice

• Noise floor via spectrum analyzer: Use RMS detector + noise marker. Reading ≈ −174 dBm/Hz + NF + 10 log₁₀ (RBW). Correct for detector averaging (~2.5 dB).
• Noise figure (Y-factor) method: Measure hot/cold source ratio Y = P_hot/P_cold = (Tₑ + T_hot)/(Tₑ + T_cold). Then Tₑ = (T_hot − Y T_cold)/(Y − 1), F = 1 + Tₑ/T₀.
• A passive attenuator of loss L at temperature T has F = 1 + (L − 1) T/T₀ (= L if T = T₀).

Worked Micro-Examples

• 50 Ω thermal noise density at 290 K: 0.895 nV/√Hz. Over 1 MHz → 0.895 µV_rms → −114 dBm.
• RX sensitivity (20 MHz channel, NF = 3 dB, SNR req = 5 dB): −174 + 10 log₁₀ (20e6) + 3 + 5 ≈ −93 dBm.
• Friis intuition: ≥ 15–20 dB low-NF gain up front buries later stages’ noise.

Common Pitfalls

• Using −174 dBm/Hz without temperature correction (10 log₁₀ T/290).
• Using −3 dB bandwidth instead of ENBW.
• Ignoring image-band noise in DSB mixers (~3 dB hidden loss).
• Summing dBm powers directly instead of in linear units.
• Mixing voltage noise density (nV/√Hz) with power density (dBm/Hz).

Quick Reference

• k = 1.380649×10⁻²³ J/K
• Room-temp noise floor: −174 dBm/Hz
• vₙ = √(4 k T R B), Pₙ = k T B
• Tₑ = 290 (10^(NF/10) − 1)
• Friis: F_total = F₁ + (F₂ − 1)/G₁ + …
• RC low-pass ENBW: Bₙ = (π/2) f_c

Bottom Line for RX

Johnson noise is the inescapable baseline. The receiver that wins is the one that:

• Keeps pre-LNA losses near zero.
• Uses a low-NF, high-gain first stage.
• Filters early and tightly (true Bₙ).
• Budgets image and mixer noise correctly.

Do those four things, and you’ll be operating as close to the kTB limit as physics allows.

Interested in more technical content? Subscribe to our updates for deep-dive RF articles and lab notes.

Questions or experiences to share? Contact RF.Guru.