Johnson–Nyquist noise#general form

In 1905, in one of Albert Einstein's Annus mirabilis papers the theory of Brownian motion was first solved in terms of thermal fluctuations. The following year, in a second paper about Brownian motion, Einstein suggested that the same phenomena could be applied to derive thermally-agitated currents, but did not carry out the calculation as he considered it to be untestable.{{Cite journal |last=Dörfel |first=G. |date=2012-08-15 |title=The early history of thermal noise: The long way to paradigm change |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.201200736 |journal=Annalen der Physik |language=en |volume=524 |issue=8 |pages=117–121 |doi=10.1002/andp.201200736 |issn=0003-3804}}

Geertruida de Haas-Lorentz, daughter of Hendrik Lorentz, in her doctoral thesis of 1912, expanded on Einstein stochastic theory and first applied it to the study of electrons, deriving a formula for the mean-squared value of the thermal current.{{Citation |last=Van Der Ziel |first=A. |title=Advances in Electronics and Electron Physics Volume 50 |chapter=History of Noise Research |date=1980-01-01 |volume=50 |pages=351–409 |editor-last=Marton |editor-first=L. |chapter-url=https://www.sciencedirect.com/science/article/pii/S0065253908610665 |access-date=2024-03-16 |publisher=Academic Press |doi=10.1016/s0065-2539(08)61066-5 |isbn=978-0-12-014650-5 |editor2-last=Marton |editor2-first=C.}}

Walter H. Schottky studied the problem in 1918, while studying thermal noise using Einstein's theories, experimentally discovered another kind of noise, the shot noise.

Frits Zernike working in electrical metrology, found unusual random deflections while working with high-sensitive galvanometers. He rejected the idea that the noise was mechanical, and concluded that it was of thermal nature. In 1927, he introduced the idea of autocorrelations to electrical measurements and calculated the time detection limit. His work coincided with de Haas-Lorentz' prediction.

The same year, working independently without any knowledge of Zernike's work, John B. Johnson working in Bell Labs found the same kind of noise in communication systems, but described it in terms of frequencies.{{Cite journal |doi = 10.1103/PhysRev.29.350|title = Minutes of the Philadelphia Meeting December 28, 29, 30, 1926|journal = Physical Review|volume = 29|issue = 2|pages = 350–373|year = 1927|last1 = Anonymous|bibcode = 1927PhRv...29..350.}}{{cite journal|first=J.|last=Johnson|title=Thermal Agitation of Electricity in Conductors|journal= Physical Review|volume=32|pages=97–109|number=97|date=1928|doi=10.1103/physrev.32.97|bibcode=1928PhRv...32...97J}} He described his findings to Harry Nyquist, also at Bell Labs, who used principles of thermodynamics and statistical mechanics to explain the results, published in 1928.{{cite journal|first=H.|last=Nyquist|title=Thermal Agitation of Electric Charge in Conductors|journal= Physical Review|volume=32|pages=110–113|number=110|date=1928|doi=10.1103/physrev.32.110|bibcode=1928PhRv...32..110N}}

File:Effect-of-bandwidth-on-selecting-noise.svg with bandwidth $\backslash Delta\; f\; \{=\}\; f\_\backslash text\{upper\}\; \{-\}\; f\_\backslash text\{lower\}$ passes only the shaded area of height $4\; k\_\backslash text\{B\}\; T\; R$ and width $\backslash Delta\; f$. Note: practical filters don't have brickwall cutoffs, so the left and right edges of this area are not perfectly vertical.]]

Johnson's experiment (Figure 1) found that the thermal noise from a resistance $R$ at kelvin temperature $T$ and bandlimited to a frequency band of bandwidth $\backslash Delta\; f$ (Figure 3) has a mean square voltage of:

: $\backslash overline\; \{V\_n^2\}\; =\; 4\; k\_\backslash text\{B\}\; T\; R\; \backslash ,\; \backslash Delta\; f$

where $k\_\{\backslash rm\; B\}$ is the Boltzmann constant ({{val|1.380649|e=-23}} joules per kelvin). While this equation applies to ideal resistors (i.e. pure resistances without any frequency-dependence) at non-extreme frequency and temperatures, a more accurate general form accounts for complex impedances and quantum effects. Conventional electronics generally operate over a more limited bandwidth, so Johnson's equation is often satisfactory.

The mean square voltage per hertz of bandwidth is $4\; k\_\backslash text\{B\}\; T\; R$ and may be called the power spectral density (Figure 2).{{NoteTag|This article is using "one-sided" (positive-only frequency) not "two-sided" frequency.}} Its square root at room temperature (around 300 K) approximates to 0.13 $\backslash sqrt\{R\}$ in units of {{Sfrac|nanovolts|{{sqrt|hertz}}}}. A 10 kΩ resistor, for example, would have approximately 13 {{Sfrac|nanovolts|{{sqrt|hertz}}}} at room temperature.

File:JohnsonNoiseEquivalentCircuits.svg circuit: a noiseless resistor in series with a noise voltage source;

(C) Its Norton equivalent circuit: a noiseless resistance in parallel with a noise current source.|303x303px]]The square root of the mean square voltage yields the root mean square (RMS) voltage observed over the bandwidth $\backslash Delta\; f$:

: $V\_\backslash text\{rms\}\; =\; \backslash sqrt\{\backslash overline\; \{V\_n^2\}\}\; =\; \backslash sqrt\{\; 4\; k\_\backslash text\{B\}\; T\; R\; \backslash ,\; \backslash Delta\; f\; \}\; \backslash ,\; .$

A resistor with thermal noise can be represented by its Thévenin equivalent circuit (Figure 4B) consisting of a noiseless resistor in series with a gaussian noise voltage source with the above RMS voltage.

Around room temperature, 3 kΩ provides almost one microvolt of RMS noise over 20 kHz (the human hearing range) and 60 Ω·Hz for $R\; \backslash ,\; \backslash Delta\; f$ corresponds to almost one nanovolt of RMS noise.

A resistor with thermal noise can also be converted into its Norton equivalent circuit (Figure 4C) consisting of a noise-free resistor in parallel with a gaussian noise current source with the following RMS current:

: $I\_\backslash text\{rms\}\; =\; \{V\_\backslash text\{rms\}\; \backslash over\; R\}\; =\; \backslash sqrt\; \{\{4\; k\_\backslash text\{B\}\; T\; \backslash Delta\; f\; \}\; \backslash over\; R\}.$

Ideal capacitors, as lossless devices, do not have thermal noise. However, the combination of a resistor and a capacitor (an RC circuit, a common low-pass filter) has what is called kTC noise. The noise bandwidth of an RC circuit is $\backslash Delta\; f\; \{=\}\; \backslash tfrac\{1\}\{4RC\}.${{cite web |last=Lundberg |first=Kent H. |title=Noise Sources in Bulk CMOS |url=http://web.mit.edu/klund/www/papers/UNP_noise.pdf |page=10}} When this is substituted into the thermal noise equation, the result has an unusually simple form as the value of the resistance (R) drops out of the equation. This is because higher R decreases the bandwidth as much as it increases the noise.

The mean-square and RMS noise voltage generated in such a filter are:

{{cite journal |last1=Sarpeshkar |first1=R. |last2=Delbruck |first2=T. |last3=Mead |first3=C. A. |date=November 1993 |title=White noise in MOS transistors and resistors |url=http://users.ece.gatech.edu/~phasler/Courses/ECE6414/Unit1/Rahul_noise01.pdf |journal=IEEE Circuits and Devices Magazine |volume=9 |issue=6 |pages=23–29 |doi=10.1109/101.261888 |s2cid=11974773}}

: $$

\overline {V_n^2} = {4 k_\text{B} T R \over 4 R C} = {k_\text{B} T \over C}

: $$

V_\text{rms} = \sqrt{4 k_\text{B} T R \over 4 R C} = \sqrt{ k_\text{B} T \over C }.

The noise charge $Q\_n$ is the capacitance times the voltage:

: $Q\_n\; =\; C\; \backslash ,\; V\_n\; =\; C\; \backslash sqrt\{\; k\_\backslash text\{B\}\; T\; \backslash over\; C\; \}\; =\; \backslash sqrt\{\; k\_\backslash text\{B\}\; T\; C\; \}$

: $$

\overline{Q_n^2} = C^2 \, \overline{V_n^2} = C^2 {k_\text{B} T \over C} = k_\text{B} T C

This charge noise is the origin of the term "kTC noise". Although independent of the resistor's value, 100% of the kTC noise arises in the resistor. Therefore, it would incorrect to double-count both a resistor's thermal noise and its associated kTC noise, and the temperature of the resistor alone should be used, even if the resistor and the capacitor are at different temperatures. Some values are tabulated below:

An extreme case is the zero bandwidth limit called the reset noise left on a capacitor by opening an ideal switch. Though an ideal switch's open resistance is infinite, the formula still applies. However, now the RMS voltage must be interpreted not as a time average, but as an average over many such reset events, since the voltage is constant when the bandwidth is zero. In this sense, the Johnson noise of an RC circuit can be seen to be inherent, an effect of the thermodynamic distribution of the number of electrons on the capacitor, even without the involvement of a resistor.

The noise is not caused by the capacitor itself, but by the thermodynamic fluctuations of the amount of charge on the capacitor. Once the capacitor is disconnected from a conducting circuit, the thermodynamic fluctuation is frozen at a random value with standard deviation as given above. The reset noise of capacitive sensors is often a limiting noise source, for example in image sensors.

Any system in thermal equilibrium has state variables with a mean energy of {{Sfrac|kT|2}} per degree of freedom. Using the formula for energy on a capacitor (E = {{sfrac|1|2}}CV²), mean noise energy on a capacitor can be seen to also be {{sfrac|1|2}}C{{Sfrac|kT|C}} = {{Sfrac|kT|2}}. Thermal noise on a capacitor can be derived from this relationship, without consideration of resistance.

The Johnson–Nyquist noise has applications in precision measurements, in which it is typically called "Johnson noise thermometry".{{Cite journal |last1=White |first1=D R |last2=Galleano |first2=R |last3=Actis |first3=A |last4=Brixy |first4=H |last5=Groot |first5=M De |last6=Dubbeldam |first6=J |last7=Reesink |first7=A L |last8=Edler |first8=F |last9=Sakurai |first9=H |last10=Shepard |first10=R L |last11=Gallop |first11=J C |date=August 1996 |title=The status of Johnson noise thermometry |url=https://iopscience.iop.org/article/10.1088/0026-1394/33/4/6 |journal=Metrologia |volume=33 |issue=4 |pages=325–335 |doi=10.1088/0026-1394/33/4/6 |issn=0026-1394}}

For example, the NIST in 2017 used the Johnson noise thermometry to measure the Boltzmann constant with uncertainty less than 3 ppm. It accomplished this by using Josephson voltage standard and a quantum Hall resistor, held at the triple-point temperature of water. The voltage is measured over a period of 100 days and integrated.{{Cite journal |last1=Qu |first1=Jifeng |last2=Benz |first2=Samuel P |last3=Coakley |first3=Kevin |last4=Rogalla |first4=Horst |last5=Tew |first5=Weston L |last6=White |first6=Rod |last7=Zhou |first7=Kunli |last8=Zhou |first8=Zhenyu |date=2017-08-01 |title=An improved electronic determination of the Boltzmann constant by Johnson noise thermometry |journal=Metrologia |volume=54 |issue=4 |pages=549–558 |doi=10.1088/1681-7575/aa781e |issn=0026-1394 |pmc=5621608 |pmid=28970638}}

This was done in 2017, when the triple point of water's temperature was 273.16 K by definition, and the Boltzmann constant was experimentally measurable. Because the acoustic gas thermometry reached 0.2 ppm in uncertainty, and Johnson noise 2.8 ppm, this fulfilled the preconditions for a redefinition. After the 2019 redefinition, the kelvin was defined so that the Boltzmann constant is 1.380649×10⁻²³ J⋅K⁻¹, and the triple point of water became experimentally measurable.{{Cite press release |date=2016-11-15 |title=Noise, Temperature, and the New SI |url=https://www.nist.gov/news-events/news/2016/11/noise-temperature-and-new-si |website=NIST |language=en}}{{Cite press release |date=2017-06-29 |title=NIST 'Noise Thermometry' Yields Accurate New Measurements of Boltzmann Constant |url=https://www.nist.gov/news-events/news/2017/06/nist-noise-thermometry-yields-accurate-new-measurements-boltzmann-constant |website=NIST |language=en}}{{Cite journal |last1=Fischer |first1=J |last2=Fellmuth |first2=B |last3=Gaiser |first3=C |last4=Zandt |first4=T |last5=Pitre |first5=L |last6=Sparasci |first6=F |last7=Plimmer |first7=M D |last8=de Podesta |first8=M |last9=Underwood |first9=R |last10=Sutton |first10=G |last11=Machin |first11=G |last12=Gavioso |first12=R M |last13=Ripa |first13=D Madonna |last14=Steur |first14=P P M |last15=Qu |first15=J |date=2018 |title=The Boltzmann project |journal=Metrologia |volume=55 |issue=2 |pages=10.1088/1681–7575/aaa790 |doi=10.1088/1681-7575/aaa790 |issn=0026-1394 |pmc=6508687 |pmid=31080297}}

Inductors are the dual of capacitors. Analogous to kTC noise, a resistor with an inductor $L$ results in a noise current that is independent of resistance:{{cite journal |last=Pierce |first=J. R. |year=1956 |title=Physical Sources of Noise |url=https://ieeexplore.ieee.org/document/4052064 |journal=Proceedings of the IRE |volume=44 |issue=5 |pages=601–608 |doi=10.1109/JRPROC.1956.275123 |s2cid=51667159}}

: $$

\overline {I_n^2} = {k_\text{B} T \over L} \, .

The noise generated at a resistor $R\_\backslash text\{S\}$ can transfer to the remaining circuit. The maximum power transfer happens when the Thévenin equivalent resistance $R\_\{\backslash rm\; L\}$ of the remaining circuit matches $R\_\backslash text\{S\}$. In this case, each of the two resistors dissipates noise in both itself and in the other resistor. Since only half of the source voltage drops across any one of these resistors, this maximum noise power transfer is:

: $P\_\backslash text\{max\}\; =\; k\_\backslash text\{B\}\; \backslash ,T\; \backslash Delta\; f\; \backslash ,\; .$

This maximum is independent of the resistance and is called the available noise power from a resistor.

Signal power is often measured in dBm (decibels relative to 1 milliwatt). Available noise power would thus be $10\backslash \; \backslash log\_\{10\}(\backslash tfrac\{k\_\backslash text\{B\}\; T\; \backslash Delta\; f\}\{\backslash text\{1\; mW\}\})$ in dBm. At room temperature (300 K), the available noise power can be easily approximated as $10\backslash \; \backslash log\_\{10\}(\backslash Delta\; f)\; -\; 173.8$ in dBm for a bandwidth in hertz.{{Citation |last=Vizmuller |first=Peter |title=RF Design Guide |year=1995 |publisher=Artech House |isbn=0-89006-754-6}}{{rp|260}} Some example available noise power in dBm are tabulated below:

File:Nyquist-transmission-line-derivation-of-johnson-noise-with-voltage-sources.svg 1928 thought experiment using two noisy resistors (each represented here by a noise-free resistor in series with a noise voltage source) connected via a long lossless transmission line of length $l$. Each resistor's noise signal propagates across the line at velocity $\backslash text\{v\}$. All impedances are identical, so both signals are absorbed by the opposite resistor instead of being reflected.

Nyquist then imagined shorting both ends of the line, thereby trapping in-flight energy on the line. Because all in-flight energy is now completely reflected (due to the now-mismatched impedance), the in-flight energy can be represented as a summation of sinusoidal standing waves. For a band of frequencies $\backslash Delta\; f$, there are $2l\; \backslash ,\; \backslash Delta\; f\; /\; \backslash text\{v\}$ modes of oscillation.{{NoteTag|A standing wave occurs with frequency equal to every integer multiple of $\backslash tfrac\{\backslash text\{v\}\}\{2l\}$. The line is sufficiently long to make the number of modes within the bandwidth very large, such that the modes will be close enough in frequency to approximate a continuous frequency spectrum.}} Each mode provides $k\_\{\backslash rm\; B\}\; T$ of energy on average, of which $k\_\{\backslash rm\; B\}\; T\; /\; 2$ is electric and $k\_\{\backslash rm\; B\}\; T\; /\; 2$ is magnetic, so the total energy in that bandwidth on average is $k\_\{\backslash rm\; B\}\; T\; \backslash cdot\; 2l\; \backslash ,\; \backslash Delta\; f\; /\; \backslash text\{v\}\; .$ Each resistor contributed $k\_\{\backslash rm\; B\}\; T\; \backslash cdot\; l\; \backslash ,\; \backslash Delta\; f\; /\; \backslash text\{v\}$ (half of that total energy).

But since before the shorting there were originally no reflections, the value of that total in-flight energy also equals the combined energy that was transferred from both resistors to the line during the transit time interval of $l\; /\; \backslash text\{v\}$. Dividing the average energy transferred from each resistor to the line by the transit time interval results in a total power of $k\_\{\backslash rm\; B\}\; T\; \backslash ,\; \backslash Delta\; f$ transferred over bandwidth $\backslash Delta\; f$ on average from each resistor.]]

Nyquist's 1928 paper "Thermal Agitation of Electric Charge in Conductors" used concepts about potential energy and harmonic oscillators from the equipartition law of Boltzmann and Maxwell{{Cite book |last=Tomasi |first=Wayne |url=https://books.google.com/books?id=LXPWxmakFVgC |title=Electronic Communication |date=1994 |publisher=Prentice Hall PTR |isbn=9780132200622 |language=en}} to explain Johnson's experimental result. Nyquist's thought experiment summed the energy contribution of each standing wave mode of oscillation on a long lossless transmission line between two equal resistors ($R\_1\; \{=\}\; R\_2$). According to the conclusion of Figure 5, the total average power transferred over bandwidth $\backslash Delta\; f$ from $R\_1$ and absorbed by $R\_2$ was determined to be:

: $\backslash overline\; \{P\_1\}\; =\; k\_\{\backslash rm\; B\}\; T\; \backslash ,\; \backslash Delta\; f\; \backslash ,\; .$

Simple application of Ohm's law says the current from $V\_1$ (the thermal voltage noise of only $R\_1$) through the combined resistance is $I\_1\; \{=\}\; \backslash tfrac\{V\_1\}\{R\_1\; +\; R\_2\}\; \{=\}\; \backslash tfrac\{V\_1\}\{2R\_1\}$, so the power transferred from $R\_1$ to $R\_2$ is the square of this current multiplied by $R\_2$, which simplifies to:

: $P\_\backslash text\{1\}\; =\; I\_1^2\; R\_2\; =\; I\_1^2\; R\_1\; =\; \backslash left(\; \backslash frac\{V\_1\}\{2R\_1\}\; \backslash right)^2\; R\_1\; =\; \backslash frac\{V\_1^2\}\{4R\_1\}\; \backslash ,\; .$

Setting this $P\_\backslash text\{1\}$ equal to the earlier average power expression $\backslash overline\; \{P\_1\}$ allows solving for the average of $V\_1^2$ over that bandwidth:

: $\backslash overline\{V\_1^2\}\; =\; 4\; k\_\backslash text\{B\}\; T\; \{R\_1\}\; \backslash ,\; \backslash Delta\; f\; \backslash ,\; .$

Nyquist used similar reasoning to provide a generalized expression that applies to non-equal and complex impedances too. And while Nyquist above used $k\_\{\backslash rm\; B\}\; T$ according to classical theory, Nyquist concluded his paper by attempting to use a more involved expression that incorporated the Planck constant $h$ (from the new theory of quantum mechanics).

The $4\; k\_\backslash text\{B\}\; T\; R$ voltage noise described above is a special case for a purely resistive component for low to moderate frequencies. In general, the thermal electrical noise continues to be related to resistive response in many more generalized electrical cases, as a consequence of the fluctuation-dissipation theorem. Below a variety of generalizations are noted. All of these generalizations share a common limitation, that they only apply in cases where the electrical component under consideration is purely passive and linear.

Nyquist's original paper also provided the generalized noise for components having partly reactive response, e.g., sources that contain capacitors or inductors. Such a component can be described by a frequency-dependent complex electrical impedance $Z(f)$. The formula for the power spectral density of the series noise voltage is

:$$

S_{v_n v_n}(f) = 4 k_\text{B} T \eta(f) \operatorname{Re}[Z(f)].

The function $\backslash eta(f)$ is approximately 1, except at very high frequencies or near absolute zero (see below).

The real part of impedance, $\backslash operatorname\{Re\}[Z(f)]$, is in general frequency dependent and so the Johnson–Nyquist noise is not white noise. The RMS noise voltage over a span of frequencies $f\_1$ to $f\_2$ can be found by taking the square root of integration of the power spectral density:

:$V\_\backslash text\{rms\}\; =\; \backslash sqrt\{\backslash int\_\{f\_1\}^\{f\_2\}\; S\_\{v\_n\; v\_n\}(f)\; df\}$.

Alternatively, a parallel noise current can be used to describe Johnson noise, its power spectral density being

:$$

S_{i_n i_n}(f) = 4 k_\text{B} T \eta(f) \operatorname{Re}[Y(f)].

where $Y(f)\; \{=\}\; \backslash tfrac\{1\}\{Z(f)\}$ is the electrical admittance; note that $\backslash operatorname\{Re\}[Y(f)]\; \{=\}\; \backslash tfrac\{\backslash operatorname\{Re\}[Z(f)]\}\{|Z(f)|^2\}\; \backslash ,\; .$

With proper consideration of quantum effects (which are relevant for very high frequencies or very low temperatures near absolute zero), the multiplying factor $\backslash eta(f)$ mentioned earlier is in general given by:{{Cite journal |last1=Callen |first1=Herbert B. |last2=Welton |first2=Theodore A. |date=1951-07-01 |title=Irreversibility and Generalized Noise |url=https://link.aps.org/doi/10.1103/PhysRev.83.34 |journal=Physical Review |volume=83 |issue=1 |pages=34–40 |doi=10.1103/PhysRev.83.34}}

:$\backslash eta(f)\; =\; \backslash frac\{hf/k\_\backslash text\{B\}\; T\}\{e^\{hf/k\_\backslash text\{B\}\; T\}\; -\; 1\}+\backslash frac\{1\}\{2\}$

\frac{h f}{k_\text{B} T} \, .

At very high frequencies ($f\; \backslash gtrsim\; \backslash tfrac\{k\_\backslash text\{B\}\; T\}\{h\}$), the function $\backslash eta(f)$ starts to exponentially decrease to zero. At room temperature this transition occurs in the terahertz, far beyond the capabilities of conventional electronics, and so it is valid to set $\backslash eta(f)=1$ for conventional electronics work.

Nyquist's formula is essentially the same as that derived by Planck in 1901 for electromagnetic radiation of a blackbody in one dimension—i.e., it is the one-dimensional version of Planck's law of blackbody radiation.{{cite book |title=Fundamentals of Microwave Photonics |page=63 |url=https://books.google.com/books?id=mg91BgAAQBAJ&pg=PA63 |first1=V. J.|last1=Urick|first2=Keith J.|last2=Williams|first3=Jason D.|last3=McKinney|isbn=9781119029786 |date=2015-01-30 |publisher=John Wiley & Sons }} In other words, a hot resistor will create electromagnetic waves on a transmission line just as a hot object will create electromagnetic waves in free space.

In 1946, Robert H. Dicke elaborated on the relationship,{{Cite journal| doi = 10.1063/1.1770483| volume = 17| issue = 7| pages = 268–275| last = Dicke| first = R. H.| title = The Measurement of Thermal Radiation at Microwave Frequencies| journal = Review of Scientific Instruments| date = 1946-07-01| pmid=20991753| bibcode = 1946RScI...17..268D| s2cid = 26658623| doi-access = free}} and further connected it to properties of antennas, particularly the fact that the average antenna aperture over all different directions cannot be larger than $\backslash tfrac\{\backslash lambda^2\}\{4\backslash pi\}$, where λ is wavelength. This comes from the different frequency dependence of 3D versus 1D Planck's law.

Richard Q. Twiss extended Nyquist's formulas to multi-port passive electrical networks, including non-reciprocal devices such as circulators and isolators.{{Cite journal | doi = 10.1063/1.1722048| title = Nyquist's and Thevenin's Theorems Generalized for Nonreciprocal Linear Networks| journal = Journal of Applied Physics| volume = 26| issue = 5| pages = 599–602| year = 1955| last1 = Twiss | first1 = R. Q.| bibcode = 1955JAP....26..599T}}

Thermal noise appears at every port, and can be described as random series voltage sources in series with each port. The random voltages at different ports may be correlated, and their amplitudes and correlations are fully described by a set of cross-spectral density functions relating the different noise voltages,

: $S\_\{v\_m\; v\_n\}(f)\; =\; 2\; k\_\backslash text\{B\}\; T\; \backslash eta(f)\; (Z\_\{mn\}(f)\; +\; Z\_\{nm\}(f)^*)$

where the $Z\_\{mn\}$ are the elements of the impedance matrix $\backslash mathbf\{Z\}$.

Again, an alternative description of the noise is instead in terms of parallel current sources applied at each port. Their cross-spectral density is given by

: $S\_\{i\_m\; i\_n\}(f)\; =\; 2\; k\_\backslash text\{B\}\; T\; \backslash eta(f)\; (Y\_\{mn\}(f)\; +\; Y\_\{nm\}(f)^*)$

where $\backslash mathbf\{Y\}\; =\; \backslash mathbf\{Z\}^\{-1\}$ is the admittance matrix.

{{NoteFoot}}

• Fluctuation-dissipation theorem
• Shot noise
• Pink noise
• Langevin equation
• Rise over thermal

{{reflist}}

{{FS1037C MS188}}

• [http://www4.tpgi.com.au/users/ldbutler/AmpNoise.htm Amplifier noise in RF systems] {{Webarchive|url=https://web.archive.org/web/20080704180832/http://www4.tpgi.com.au/users/ldbutler/AmpNoise.htm |date=2008-07-04 }}
• [http://www.physics.utoronto.ca/~phy225h/experiments/thermal-noise/Thermal-Noise.pdf Thermal noise (undergraduate) with detailed math]

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