Thermal Noise
Every resistor in every circuit — in a microphone preamplifier, in a sensor bridge, in an op-amp's feedback network, in an antenna and its cable — is constantly generating a tiny random voltage. You cannot hear it as a separate signal. You cannot switch it off. It is an unavoidable consequence of the fact that electrons in any conductor above absolute zero temperature are in constant, random thermal motion. This random motion produces random voltage fluctuations: thermal noise, also called Johnson noise or Johnson-Nyquist noise. It sets the absolute lower limit on how small a signal any amplifier, receiver or measuring instrument can ever detect.
Understanding thermal noise from first principles is the foundation of everything else in this module. Once you understand where noise comes from and how to calculate how much of it any circuit element generates, you can understand noise figure, noise floor, dynamic range, and the many ways that an amplifier, receiver or measurement system can be designed to work closer to that fundamental physical limit.
Thermal noise has a flat power spectral density from DC to extremely high frequencies. The total noise power available in any bandwidth B is simply P = kTB. The narrower the bandwidth, the less total noise power — which is why narrower filters improve reception.
View LargerThe Physical Origin of Thermal Noise
Thermal noise originates in the thermal (heat) energy stored in a material. All matter above absolute zero (0 K or −273.15 °C) contains thermal energy — the kinetic energy of atoms and electrons in random motion. In a conductor, the free electrons that carry current are not stationary between uses. They are constantly moving at high velocities in random directions, colliding with the crystal lattice of the conductor billions of times per second. This is true even with no external voltage applied and no current flowing through the circuit.
Each random collision and each segment of random electron motion involves a small displacement of charge. A displacement of charge is the definition of a current, and a current through a resistance produces a voltage. Because the motion is random, the resulting voltage is also random — it has no specific frequency, no specific phase, and averages to zero over any long period of time. But it is not zero at any given instant. It fluctuates continuously around zero, and that fluctuation is what we call thermal noise.
The key insight is that this process is driven entirely by temperature. Hotter conductors have more energetic electrons, more violent random motion, and therefore more noise. Colder conductors have less. At absolute zero, the random thermal motion would stop and so would the thermal noise — but absolute zero is physically unreachable, so some thermal noise is always present in any practical circuit.
This phenomenon was first measured experimentally by John B. Johnson at Bell Labs in 1928, and the theoretical explanation was provided almost simultaneously by Harry Nyquist. Their names are attached to the effect: Johnson noise or Johnson-Nyquist noise. The terms thermal noise, resistor noise, and thermal agitation noise all refer to the same thing.
Noise Power: P = kTB
Nyquist derived the fundamental formula for the available noise power from any resistor at temperature T, measured over a bandwidth B:
Where:
P = available noise power (watts)
k = Boltzmann's constant = 1.38 × 10−23 joules per kelvin (J/K)
T = absolute temperature of the resistor in kelvins (K)
B = noise bandwidth in hertz (Hz)
This formula has several remarkable properties that are worth understanding fully.
It does not depend on the resistance value. A 1 Ω resistor and a 1 MΩ resistor at the same temperature generate the same available noise power in the same bandwidth. The resistance affects the noise voltage (more resistance, more voltage) but the available power is the same. This is a consequence of thermodynamics: a resistor at temperature T in thermal equilibrium with a matched load delivers exactly P = kTB to that load, regardless of the resistance value. If the power depended on resistance, you could extract net energy from a thermal equilibrium system, violating the second law of thermodynamics.
It depends linearly on temperature. Double the temperature in kelvins, and you double the noise power. Room temperature (around 17 °C) is approximately 290 K. A cryogenically cooled low-noise amplifier at 77 K (liquid nitrogen temperature) generates only 77/290 = 0.27 times as much thermal noise as one at room temperature — which is why astronomers use cryogenic cooling to achieve very low noise in radio telescope receivers.
It depends linearly on bandwidth. Double the bandwidth, and you double the noise power. This is the most useful practical consequence: narrow bandwidths mean less noise. A circuit limited to 1 kHz passes 20 times less noise than one passing the full 20 kHz audio band — 13 dB less. That is why sensor amplifiers are filtered to the narrowest bandwidth the signal needs, why a spectrum analyzer shows a lower noise floor when you narrow its resolution bandwidth, and why narrow-band communication modes can work with far less signal power than wide-band ones.
Worked Example: Noise Power at Room Temperature
Solution:
P = kTB
P = (1.38 × 10−23) × 290 × 2400
P = (1.38 × 10−23) × 696,000
P = 9.60 × 10−18 watts
P = 9.60 × 10−15 milliwatts
Converting to dBm: P(dBm) = 10 × log10(9.60 × 10−15)
P = 10 × (−14.02)
P = −140.2 dBm
Result: The noise floor in a 2.4 kHz bandwidth at room temperature (with a perfectly noiseless amplifier) is approximately −140 dBm. Any signal weaker than this cannot be detected by simple measurement in that bandwidth, even with perfect electronics.
Noise Voltage and Noise Current
The formula P = kTB gives the available noise power. For circuit design purposes, it is often more useful to think in terms of the equivalent noise voltage or noise current produced by a resistor. The Thevenin equivalent circuit of a noisy resistor is a noiseless resistor of the same value in series with a noise voltage source. The Norton equivalent is a noiseless resistor in parallel with a noise current source.
Noise Current (RMS): In = √(4kTB/R)
Where R is the resistance in ohms.
Notice that unlike the noise power (which is independent of R), the noise voltage increases with the square root of resistance, and the noise current decreases with the square root of resistance. A high-resistance source generates more noise voltage but less noise current than a low-resistance source of the same temperature. The available noise power remains P = kTB regardless.
Worked Example: Noise Voltage of a Resistor
Solution:
Vn = √(4kTBR)
Vn = √(4 × 1.38 × 10−23 × 290 × 20,000 × 10,000)
Vn = √(4 × 8.00 × 10−17 × 10,000)
Vn = √(3.20 × 10−12)
Vn = 1.79 × 10−6 V
Vn = 1.79 microvolts (µV) RMS
Result: About 1.8 µV RMS of random noise appears across the resistor. A moving-coil phono cartridge produces only 200 to 500 µV, so its signal would be just 41 to 49 dB above this noise. With a 200 Ω source instead, the noise in the same bandwidth falls to 0.25 µV — which is why low-level audio inputs use low source and input resistances.
Noise Temperature
Noise temperature is an alternative way to express the noise contribution of a component or system. Instead of specifying noise in dB or in noise power, you express it as the temperature (in kelvins) of a resistor that would produce the same amount of noise power.
The concept is particularly useful in two contexts. First, real antennas do not simply present a resistive source impedance with a pure thermal noise of 290 K. An antenna pointed at the sky picks up background radiation from space, galactic noise, and any local noise sources. The effective noise temperature of an antenna might be anywhere from a few kelvins (a highly directive dish pointing at cold sky) to tens of thousands of kelvins (an antenna near power lines). Second, noise temperature allows the noise contribution of an amplifier to be expressed independently of any reference temperature, making it easier to apply the Friis formula for cascaded stages.
The relationship between noise factor F (a dimensionless ratio, covered in the next lesson) and noise temperature Te is:
where T0 = 290 K (the standard reference temperature)
and F = SNRin / SNRout (the noise factor of the amplifier)
A perfectly noiseless amplifier has F = 1 and Te = 0 K. A preamplifier with a noise figure of 1 dB has F = 1.259, giving Te = (1.259 − 1) × 290 = 75 K. This means it adds noise equivalent to a 75 K resistor at its input — far less than room temperature, which is why 1 dB NF preamplifiers are considered very good for weak-signal work.
The Noise Spectrum: White Noise
One of the most useful properties of thermal noise is that its power spectral density — the noise power per unit bandwidth — is essentially constant across all frequencies from DC to several hundred gigahertz. This flat spectrum is the defining characteristic of what engineers call white noise, by analogy with white light that contains all visible wavelengths equally.
Why is thermal noise white? Because the random collisions between electrons and the crystal lattice are extremely short events — much shorter than the period of any radio frequency. A very short, sharp impulse in the time domain corresponds to a very flat, broad spectrum in the frequency domain. The collision events are so brief that the noise they generate is flat across all frequencies that matter for radio communications, from VLF to millimeter-wave.
The practical consequence is that the P = kTB formula can be applied to any bandwidth, anywhere in the frequency spectrum, and gives the correct result. Whether a 2.4 kHz bandwidth is centered at 1 kHz, 3.5 MHz or 144 MHz, the noise power in it at 290 K is the same: approximately −140 dBm.
This flatness does eventually break down at very high frequencies (above about 10 THz) where quantum mechanical effects become important. But for all practical electronics, thermal noise is perfectly white.
The −174 dBm/Hz Reference
The most important single number in receiver noise analysis is −174 dBm/Hz, the thermal noise power density at room temperature (290 K). This comes directly from the P = kTB formula with T = 290 K and B = 1 Hz:
P0(dBm) = 10 × log10(4.00 × 10−18) = −173.98 ≈ −174 dBm/Hz
This is the starting point for all noise floor calculations. To find the noise floor in any bandwidth B (in Hz), simply add 10 × log10(B) to −174 dBm:
For some common bandwidths:
| Bandwidth | Typical use | 10·log₁₀(BW) | Noise Floor (noiseless system) |
|---|---|---|---|
| 10 Hz | Slow sensor or lock-in measurement | +10 dB | −164 dBm |
| 1 kHz | Narrow analyzer resolution bandwidth | +30 dB | −144 dBm |
| 2400 Hz | Voice channel | +33.8 dB | −140 dBm |
| 20 kHz | Full audio band | +43 dB | −131 dBm |
| 200 kHz | FM broadcast channel | +53 dB | −121 dBm |
| 20 MHz | Wi-Fi channel | +73 dB | −101 dBm |
These are the noise floors of a perfectly noiseless amplifier or receiver. Every real one adds noise on top of this, characterized by its noise figure (covered in the next lesson). The actual noise floor is always worse than these values by an amount equal to the noise figure in dB.
Thermal Noise in Practice
Thermal noise has direct consequences for almost every sensitive circuit you will design or troubleshoot.
Source Resistance in Audio and Sensor Amplifiers
Noise voltage rises with the square root of resistance, so the resistance a signal source presents to an amplifier sets a floor that no amplifier can get below. A 200 Ω dynamic microphone generates only 0.25 µV of thermal noise across the 20 kHz audio band, while a 10 kΩ source generates 1.8 µV — 17 dB more. This is why professional microphones use low impedances, why low-noise preamplifiers use low-value input and feedback resistors where they can, and why a high-value resistor in the input of a sensitive op-amp stage can quietly become the largest noise source on the board.
Bandwidth Is a Design Choice
Because noise power is proportional to bandwidth, every circuit should pass only the bandwidth its signal needs. A temperature sensor that changes over seconds needs a few hertz of bandwidth, not the full 100 kHz an op-amp can deliver; a simple RC low-pass filter after the amplifier can cut its noise by 30 to 40 dB. The same principle explains why a spectrum analyzer's noise floor drops as you narrow its resolution bandwidth, and why averaging many ADC samples gives a quieter reading.
Cables, Attenuators and Lossy Filters
Any lossy component generates thermal noise. A cable or attenuator with 3 dB of loss at room temperature halves the signal while its own resistance replaces the noise it absorbs, so the signal-to-noise ratio falls by 3 dB. Loss in front of the first amplifier — a long cable from a sensor or an antenna, a lossy input filter, a pad on a spectrum analyzer — directly reduces sensitivity. The next lesson shows how to calculate this with the Friis formula, and why an amplifier placed right at the source, before the cable, recovers almost all of it.
Cooling for Extreme Sensitivity
Because noise power is proportional to absolute temperature, cooling a sensor or amplifier reduces its noise. Astronomical and scientific cameras cool their image sensors to −20 °C to −100 °C to reduce dark noise, and the receivers in radio telescopes and deep-space tracking stations are cooled to around 15 K or below. Cooling an amplifier from 290 K to 4 K (liquid helium) reduces its thermal noise by a factor of 290/4 = 72.5, about 18.6 dB. Few projects justify that effort, but it shows that thermal noise is a physical phenomenon set by temperature.
⚖ Experiment: Observe Thermal Noise with a Spectrum Analyzer
This experiment makes thermal noise visible on a spectrum display, demonstrating the flat noise floor and how bandwidth changes the measured noise power.
- A low-cost spectrum analyzer such as a TinySA
- The analyzer display (no extra software needed)
- A 50 Ω SMA terminator (available for under $2)
- A short SMA-to-SMA cable or direct connection
- Connect the 50 Ω terminator directly to the analyzer input. This replaces the antenna with a resistor at room temperature — pure thermal noise with no external signals.
- Set the analyzer to a centre frequency of 100 MHz. Set the gain to maximum. Set the bandwidth to 2 MHz.
- Observe the spectrum. You will see a flat, noisy baseline — this is the combination of the analyzer's own internal noise and the thermal noise from the 50 Ω terminator.
- Note the approximate noise level in dBm. Write it down.
- Now change the resolution bandwidth from 300 kHz to 30 kHz. What happens to the displayed noise level?
- The noise level should drop by approximately 10 × log10(300,000 / 30,000) = 10 dB. This directly demonstrates that noise power is proportional to bandwidth.
- Now connect a real antenna instead of the terminator and compare. You will see the noise floor rise as the antenna picks up thermal and man-made noise from the environment.
The 50 Ω terminator produces a flat noise floor with no signal peaks. Reducing bandwidth reduces the displayed noise level by exactly the amount the formula predicts. This confirms that noise power is proportional to bandwidth (P = kTB) and that the noise is spectrally flat (white noise). The difference between the terminator noise floor and the antenna noise floor shows you the external noise arriving from the environment.
Frequently Asked Questions
Why can't we filter out thermal noise the way we filter out interference?
Interference has a specific frequency, so a filter can remove it without affecting the desired signal. Thermal noise is spectrally white — it has equal power at every frequency. A filter does reduce thermal noise, but it reduces the signal by exactly the same fraction. The only way to improve the signal-to-noise ratio against thermal noise is to either increase the signal (a stronger source, better coupling, less cable loss) or reduce the bandwidth — which you can only do if the signal itself is narrowband enough to fit in the narrower bandwidth.
Does the noise power really not depend on the resistance value?
Correct — the available noise power P = kTB is independent of resistance. This can seem counterintuitive because a higher resistance produces a higher noise voltage (Vn = √(4kTBR)). However, when you connect that higher-resistance source to a load, the load must be matched to the source (same resistance) to extract maximum power. A high-resistance source needs a high-resistance load, and the current through that load is correspondingly smaller. The extra voltage and the smaller current cancel out, leaving the same available power regardless of resistance value. This is required by thermodynamics.
Is the noise floor of −174 dBm/Hz an absolute limit that cannot be beaten?
−174 dBm/Hz is the thermal noise floor at 290 K (approximately room temperature). It can be beaten by cooling the receiving system to a lower temperature. At 29 K (one-tenth of 290 K), the thermal noise floor is −184 dBm/Hz. At liquid helium temperature (4 K), it is approximately −197 dBm/Hz. These very low noise temperatures are used in specialized weak-signal work such as deep-space communication and radio astronomy. For practical electronics, treating −174 dBm/Hz as a floor is appropriate.
Why does my receiver still pick up signals weaker than the noise floor?
The noise floor is the level below which a signal cannot be detected using simple amplitude measurement in a single bandwidth. Signals below the noise floor can still be recovered using techniques that average or correlate over time. GPS receivers, lock-in amplifiers and digital weak-signal communication modes collect many milliseconds to minutes of data and use signal processing to detect signals 20 to 30 dB below the instantaneous noise floor. These modes exploit the fact that the signal is coherent (consistent, predictable) while the noise is random, so averaging over time pulls the signal up relative to the noise.
Test Your Knowledge
Answer the questions below to check your understanding of this lesson. Every answer can be found in the lesson above.