# Heat noise

Thermal noise , thermal  noise , thermal noise , Nyquist noise , Johnson noise or Johnson-Nyquist noise called, is a largely white noise resulting from the thermal motion of the charge carriers in electrical circuits apparent. The frequency spectrum of the resistance noise was researched experimentally by John Bertrand Johnson and at the same time theoretically justified by Harry Theodor Nyquist .

## Manifestation

With unloaded ohmic resistors, heat noise expresses itself as thermal resistance noise , often simply called resistance noise. The thermal motion of the conduction electrons generated at the terminals of the dipole to the noise power and the noise voltage. The values ​​present in the event of a short circuit or open circuit can generally be specified as the spectral noise power density . They are proportional to the absolute temperature. In the case of an unloaded component, the noise power is independent of the electrically conductive medium; on the other hand, in the case of components through which direct current flows, current noise can occur, which can be far above the thermal noise with the carbon film resistor. In the case of current-carrying semiconductors , additional noise arises from modulation of the load current - with voltage impressions - due to thermally induced fluctuations in the number of carriers in the conduction band and valence band and thus the conductivity.

Johnson experimented in 1927/28 at temperatures between the boiling point of nitrogen and that of water with resistances of very different materials. Among other things, carbon film, copper and platinum resistors as well as capillaries filled with various electrolytes were used.

Johnson announced that in 1918 Schottky had recognized from theoretical considerations that heat noise from conduction electrons must be detectable with tube amplifiers, but with a resonance circuit at the amplifier input, the sought-after effect is masked by the shot noise . Nyquist cited Schottky's work because of the resulting stimulus to derive the electrodynamic noise power from thermodynamics and statistical mechanics.

## causes

The conduction electrons of electrically conductive materials (metals, semiconductors) take part in the largely disordered, thermally excited movement of the components of the atomic level and move randomly and undirected. At room temperature they make a small contribution to the specific heat, and their disordered movement provides the finite electrical noise power in question at the terminals of a two-pole terminal. The conduction electrons generate statistically independent voltage and current pulses of finite, short duration at a high rate, the superposition of which leads to the broad frequency distribution which in electrical engineering is mostly perceived as a noise source with a white spectrum. The noise power spectrum ranges from the frequency zero to a limit frequency, the value of which is determined by the quanta of the electromagnetic harmonic components that can still be excited thermally. Nyquist's first calculation of the noise spectrum makes use of the uniform distribution theorem of thermodynamics. - A finite DC voltage component is not observed; it could not be seen as a random component, cf. Thermoelectricity . To do this, a symmetry break would be necessary, for which there is no apparent reason, because thermodynamic equilibrium is assumed for drag noise .

The resistance noise is characterized here by the power spectrum, which is white within wide frequency limits. Another question is the description by the amplitude distribution of the instantaneous values ​​of voltage or current. Experience has shown that there is a normal distribution (Gaussian distribution) with a mean value of zero, the scattering parameter of which is given by the noise power. In particular, an arbitrarily large amplitude can accordingly be expected with an exponentially decreasing probability.

The stochastic amplitude statistics mean that noise voltages must be measured with true quadratic rectification. For this purpose Johnson used (after electronic amplification) a thermal converter in which the heat generated by the added noise power causes a temperature increase. This is measured with a thermocouple whose thermoelectric voltage, averaged linearly over time, is proportional to the mean value of the noise voltage square. This measurement specification is somewhat generalized through the definition of the autocorrelation function mathematically . The conversion factor of the thermal converter is measured with a power that can be easily defined using a direct voltage.

## Noise quantities

Analogous to the random fluctuations in Brownian movement , fluctuations in the open-circuit voltage are observed over the course of time on an ohmic resistor . The mean of these voltages is zero. After electronic amplification, the root mean square value of the voltage is measured as the noise quantity, which can be converted into the rms value . The mean voltage square is proportional to the absolute temperature , the size of the electrical resistance and the bandwidth of the measuring arrangement. ${\ displaystyle t}$${\ displaystyle u (t)}$${\ displaystyle {\ overline {u (t) ^ {2}}}}$${\ displaystyle T}$${\ displaystyle R}$${\ displaystyle \ Delta f}$

The influence of the bandwidth is not easily recognizable with a broadband structure, the amplitude statistics can be assessed quite well. Their variance is given by. The amplitude statistics can be determined well within a narrow band. In the narrow band, the influence of a centered bandwidth can clearly be seen proportionally from the attack and decay times , through which the components of the noise spectrum are modulated. ${\ displaystyle {\ overline {u (t) ^ {2}}}}$${\ displaystyle f}$${\ displaystyle {\ tfrac {1} {\ Delta f}}}$${\ displaystyle f}$

• Resistance noise is an expression of the coupling of thermal and electrodynamic fluctuations. It can be clarified by considering the range of services on the path chosen by Schottky and Nyquist.

The Nyquist formula establishes the following relationship for the noise voltage in no-load operation:

${\ displaystyle {\ overline {u ^ {2}}} = 4k _ {\ mathrm {B}} TR \ Delta f \,}$

with the effective no-load noise voltage

${\ displaystyle U_ {R, \, \ mathrm {eff}} = {\ sqrt {\ overline {u ^ {2}}}},}$

consequently

${\ displaystyle U_ {R, \, \ mathrm {eff}} = {\ sqrt {4k _ {\ mathrm {B}} TR \ Delta f}}.}$

Here are the Boltzmann constant , the absolute temperature and the ohmic resistance of the noisy two-pole. is the permitted bandwidth . ${\ displaystyle k _ {\ mathrm {B}}}$${\ displaystyle T}$${\ displaystyle R}$${\ displaystyle \ Delta f}$

In addition to this, the time-averaged noise current square is calculated in the event of a short circuit ${\ displaystyle {\ overline {i ^ {2}}}}$

${\ displaystyle {\ overline {i ^ {2}}} = {\ frac {4k _ {\ mathrm {B}} T \ Delta f} {R}} \,}$

with the effective short-circuit noise current

${\ displaystyle I_ {R, \, \ mathrm {eff}} = {\ sqrt {\ overline {i ^ {2}}}} = {\ sqrt {\ frac {4k _ {\ mathrm {B}} T \ Delta f} {R}}}.}$

Ginsburg provides extensive information on the general validity of Nyquist's formula and its importance for profound questions in physics.

## Noise level

The noise power can also be specified logarithmically as the noise level:

${\ displaystyle P _ {\ mathrm {dBm}} = 10 \ log _ {10} {(k _ {\ mathrm {B}} T \ Delta f \ cdot 1000)} = 10 \ log _ {10} ({k_ { \ mathrm {B}} T \ cdot 1000}) + 10 \ log _ {10} {(\ Delta f)}}$

At room temperature ( ) the following applies: ${\ textstyle T = 300 \, {\ text {K}}}$

${\ displaystyle P _ {\ mathrm {dBm}} = - 174 + 10 \ log _ {10} {(\ Delta f)}}$, with in Hz${\ displaystyle \ Delta f}$

The following table shows thermal noise levels for various bandwidths at room temperature:

Bandwidth ${\ displaystyle (\ Delta f)}$ Thermal noise level Hints
1 Hz −174 dBm
10 Hz −164 dBm
100 Hz −154 dBm
1 kHz −144 dBm
10 kHz −134 dBm FM channel of a radio
22 kHz −130.58 dBm AUDIO ITU-R 468-4 unrated, 22Hz-22kHz
100 kHz −124 dBm
180 kHz −121.45 dBm An LTE resource block
200 kHz −121 dBm GSM channel
1 MHz −114 dBm Bluetooth channel
2 MHz −111 dBm Public GPS channel
3.84 MHz −108 dBm UMTS channel
6 MHz −106 dBm Analog television
20 MHz −101 dBm WLAN 802.11
40 MHz −98 dBm WLAN 802.11n 40 MHz channel
80 MHz −95 dBm WLAN 802.11ac 80 MHz channel
160 MHz −92 dBm WLAN 802.11ac 160 MHz channel
1 GHz −84 dBm UWB

## Equivalent circuit and power balance

The equivalent circuit diagram of a noisy resistor as a concentrated component is the series connection of the resistor R , which is imagined to be noise-free, as a source resistor with the voltage source representing its noise , which emits the open-circuit voltage square. For representation with a noise current source , a source current generator of the short-circuit current square is connected in parallel with the ideal internal resistance . ${\ displaystyle {\ overline {u ^ {2}}}}$${\ displaystyle {\ overline {i ^ {2}}}}$${\ displaystyle R}$

In the event of a short circuit, the noisy ohmic resistance itself dissipates the generated power

${\ displaystyle P _ {\ Delta f, \; \ mathrm {short circuit}} = {\ overline {u ^ {2}}} / R = 4k _ {\ mathrm {B}} T \ Delta f,}$

because the full source voltage drops across it.

When adjusting the power, each of the two noisy ohmic resistors dissipates the power in the other and in itself

${\ displaystyle P _ {\ Delta f, \; \ mathrm {verf {\ ddot {u}} gbar}} = {\ overline {(u / 2) ^ {2}}} / R = k _ {\ mathrm {B }} T \ Delta f,}$

because half the source voltage drops across them. This is the maximum power that can be delivered by a source and is called the available power . This term makes you independent of the randomness of a circuit and is suitable for a general discussion, as the thermally activated but electrodynamically mediated energy exchange between the two noisy resistors, which are coupled to a heat bath of temperature, takes place symmetrically. ${\ displaystyle R}$${\ displaystyle T}$

These four dissipated noise powers together again result in the short-circuit power, which is consequently also generated overall in this arrangement. The two resistors connected together for power adjustment work, understood as a unit of resistance , in the short circuit and their dissipated power is of the size and thus also as for each resistor individually. ${\ displaystyle 2R}$${\ displaystyle {\ tfrac {\ overline {u ^ {2}}} {2R}}}$${\ displaystyle 4k _ {\ mathrm {B}} T \ Delta f,}$

• In a purely ohmic circuit, the dissipated power is independent of the size when the power is matched and is solely determined thermodynamically by the available power${\ displaystyle R}$${\ displaystyle k _ {\ mathrm {B}} T \ Delta f.}$
• With this formulation in square sizes as a power balance, the claim already recognized by Schottky is manifestly met, it is a matter of the above coupling of thermal fluctuations to electrodynamic phenomena. Fluctuation energy of the order of the mean thermodynamic quantum exchanges every electromagnetic mode with the heat bath.${\ displaystyle {\ tfrac {1} {2}} k _ {\ mathrm {B}} T}$${\ displaystyle f}$

The formulation as power balance eliminates the need to use the quantity of electrical resistance and, because of this generality, clarifies the proposed use of the lemma heat noise . Because of the necessary quadratic rectification, power is the actual measured variable anyway.

## Quantum Theory Extension

The integration of the above equations over the entire frequency range leads to the ultraviolet catastrophe . A strictly white spectrum also requires the unrealistic participation of impulses of any short duration to stimulate the harmonic components. Therefore the quantum-theoretical extension is necessary for high frequencies . Nyquist was already doing this. The quantum mechanical zero point energy recognized later is occasionally cited as a possible non-thermal source of noise.

### Nyquist formula

For sufficiently high frequencies or correspondingly low temperatures, the formula already given by Nyquist (*)

{\ displaystyle {\ begin {aligned} {\ overline {u ^ {2}}} & = 4k _ {\ mathrm {B}} TR \, {\ frac {hf / k _ {\ mathrm {B}} T} { \ mathrm {e} ^ {hf / k _ {\ mathrm {B}} T} -1}} \, \ Delta f, & 0 \ leqq f <\ infty \\ & = 4k _ {\ mathrm {B}} TR \ , {\ frac {f / f _ {\ mathrm {Q}}} {\ mathrm {e} ^ {f / f _ {\ mathrm {Q}}} - 1}} \, \ Delta f \ end {aligned}} }

be used. The quantum theoretical limit frequency was already used in the second expression , defined by

${\ displaystyle f _ {\ mathrm {Q}} = {\ frac {k _ {\ mathrm {B}} T} {h}}}$.

At room temperature (300 K) it is . ${\ displaystyle f _ {\ mathrm {Q}} = 6 {,} 25 \ cdot 10 ^ {12} \, {\ text {Hz}}}$

• Above the thermal resistance noise is no longer spectrally white (**) , but decreases exponentially with increasing frequency according to the Boltzmann factor.${\ displaystyle f _ {\ mathrm {Q}}}$
• For low frequencies or sufficiently high temperatures, the quantum-theoretically extended formula goes over to the low-frequency value as expected .${\ displaystyle {\ overline {u ^ {2}}} = 4k _ {\ mathrm {B}} TR \ Delta f}$
(*)Note: In Nyquist's original work, a factor is missing from his formula (8). Callen and Welton gave a derivation on the basis of quantum mechanics . The Nyquist formula applies to electrical or mechanical linear dissipative systems.${\ displaystyle \ nu}$
(**)The interval must be selected to be sufficiently small so that the changes caused by the frequency-dependent factor can be neglected with the desired accuracy in this measuring interval. With Nyquist his formula (4) is written differentially with the voltage spectrum (with Nyquist's designation) because a frequency-dependent resistance is realistically permitted; . A frequency-independent effective resistance is always assumed here.${\ displaystyle \ Delta f}$${\ displaystyle E _ {\ nu} ^ {2} \ mathrm {d} \ nu = 4R _ {\ nu} k _ {\ mathrm {B}} T \ mathrm {d} \ nu}$${\ displaystyle E _ {\ nu} ^ {2}}$${\ displaystyle R _ {\ nu}}$${\ displaystyle E _ {\ nu} ^ {2} \ mathrm {d} \ nu = \ mathrm {d} {\ overline {u ^ {2}}}}$${\ displaystyle R}$

### Zero point energy

A contribution of zero point energy to heat noise is occasionally put up for discussion. The zero point energy is required by Heisenberg's uncertainty and is in the harmonic oscillator . As a fully corrected quantum mechanical formula, ${\ displaystyle {\ tfrac {1} {2}} hf}$

{\ displaystyle {\ begin {aligned} {\ overline {u ^ {2}}} & = 4k _ {\ mathrm {B}} TR \ left ({\ frac {hf / k _ {\ mathrm {B}} T} {\ mathrm {e} ^ {hf / k _ {\ mathrm {B}} T} -1}} + {\ frac {1} {2}} {\ frac {hf} {k _ {\ mathrm {B}} T}} \ right) \ Delta f = 4k _ {\ mathrm {B}} TR \, {\ frac {{\ tfrac {1} {2}} hf / k _ {\ mathrm {B}} T} {\ tanh \ left ({\ frac {1} {2}} {hf / k _ {\ mathrm {B}} T} \ right)}} \, \ Delta f, & \ mathrm {} 0 \ leqq f <\ infty \ \ & = 4R \ left ({\ frac {hf} {\ mathrm {e} ^ {hf / k _ {\ mathrm {B}} T} -1}} + {\ frac {1} {2}} hf \ right) \ Delta f \ end {aligned}}}

often suggested. With this formula, the ultraviolet catastrophe would increasingly reintroduced.

The zero point energy is not available for thermal processes such as heat noise to exchange energy with a load resistor . The latter formulation, which expresses the quantum mechanical approach very directly , obviously requires that the available spectral power density , which at sufficiently high frequencies or sufficiently low temperatures, is solely attributable to the zero point oscillation and to be exchanged between source and load resistance when the power is matched . ${\ displaystyle R}$ ${\ displaystyle {\ overline {u ^ {2}}} (4R \ Delta f) = {\ tfrac {1} {2}} hf}$

• This required changes of state of half a quant.

For the maser it was shown that the zero point energy is not amplified.

## Range of services

The performance spectrum emphasizes the fact that each electromagnetic frequency component has to be granted its own thermal degree of freedom , independently of the oscillations of other frequencies , the equipartition theorem . Nyquist shows this conceptually for the electromagnetic case by connecting a (non-dissipative) reactance filter between the resistors which are in power adjustment. If the harmonic oscillations of different frequencies were not equally strongly coupled to the heat bath, then, contrary to the 2nd law of thermodynamics, the colder resistance could on average increase the temperature of the warmer one.

• Each electromagnetic spectral component is independently in detailed equilibrium with the heat bath via the noisy two-pole and has two degrees of thermal freedom due to its electromagnetic nature.${\ displaystyle f}$${\ displaystyle R}$
• The necessary quantum-theoretical supplement shows that these independent frequency components require the minimum energy of a photon , which becomes clear with large quanta, as their thermal excitation is hindered by “freezing” because the temperature is too low.${\ displaystyle hf}$

The power spectrum for the available power of any ohmic resistor is defined by (*)

${\ displaystyle W (f) = {\ frac {hf} {\ mathrm {e} ^ {hf / k _ {\ mathrm {B}} T} -1}} \ ,, \ qquad \ qquad \ quad 0 \ leqq f <\ infty}$

with the low frequency value

${\ displaystyle W (f) = k _ {\ mathrm {B}} T \ qquad \ qquad \ qquad \ qquad \ \ left (f \ ll f _ {\ mathrm {Q}} \ right).}$

Note: The spectral power density is of the dimension energy .

The following applies to performance adjustment

${\ displaystyle P (f) _ {\ Delta f, \; \ mathrm {verf {\ ddot {u}} gbar}} = {\ frac {\ overline {u ^ {2}}} {4R}} = \ int _ {f} ^ {f + \ Delta f} \! \! W (f) \, \ mathrm {d} f}$.

The total available power is

${\ displaystyle P = \ int _ {0} ^ {\ infty} \! \! W (f) \, \ mathrm {d} f = {\ tfrac {\ pi ^ {2}} {6}} {\ frac {(k _ {\ mathrm {B}} T) ^ {2}} {h}} = {\ tfrac {\ pi ^ {2}} {6}} k _ {\ mathrm {B}} Tf _ {\ mathrm {Q}}}$.

The effective bandwidth limited by quantum theory is assumed to be white, assuming a continuously constant spectral power${\ displaystyle k _ {\ mathrm {B}} T}$

${\ displaystyle \ Delta f _ {\ mathrm {Q, \, eff}} = {\ tfrac {\ pi ^ {2}} {6}} f _ {\ mathrm {Q}}}$

The total available power at room temperature (300 K) is . ${\ displaystyle P = 4 {,} 26 \ cdot 10 ^ {- 8} \ {\ text {Watt}}}$

(*)It should be mentioned again that for reasons of symmetry the thermal noise cannot excite a constant component, which would be determined; it would result in an additive component to the spectrum proportional to the Dirac collision function .${\ displaystyle \ delta (f)}$

### Black waveguide and black cavity radiation

Two ohmic bipoles with the same frequency-independent resistance in the heat bath of the absolute temperature are connected by a lossless line from the wave resistance , s. real wave impedance . Because of this adaptation according to the wave resistance, there are only advancing waves in both directions on the line. Influences from standing waves due to reflection are not present, as a result there is no frequency selectivity. With this wiring there is anyway power adjustment. ${\ displaystyle R}$${\ displaystyle T}$ ${\ displaystyle Z = R}$

• The ideal line - of any length and defined wave resistance - is interposed so that the coupling of thermal fluctuations to electrodynamic ones is supported by the thought of spatially extended electromagnetic waves .

The electromagnetic waves on the line are emitted by the noisy resistors and completely absorbed in the other.

• The simultaneously rushing and dissipating resistances mediate the setting and maintenance of the thermodynamic equilibrium between the energy content of the electromagnetic waves and the heat bath, cf. Fluctuation-dissipation theorem .

The power transferred to the other resistor does not disturb the thermodynamic equilibrium, on average there is no directed energy transport.

• This one-dimensional arrangement with regard to the propagation of the electromagnetic processes along the black waveguide , as the arrangement (*) is called here, is an electrical equivalent to three-dimensional black cavity radiation . (**)
• Nyquist obtained the low-frequency noise spectrum by considering the arrangement described above, by applying the equal distribution theorem to the spectral components of the electromagnetic waves , represented by the capacitive and inductive occupancy of the line with energy stores or per line length. He imagined an ideal coaxial cable with wave impedance as the line.${\ displaystyle C ^ {\ prime}}$${\ displaystyle L ^ {\ prime}}$${\ displaystyle Z = {\ sqrt {L ^ {\ prime} / C ^ {\ prime}}} = R}$
• At high frequencies he looked at quanta and corrected the formula of the white spectrum according to the results of Planck's formula .${\ displaystyle hf}$

In the low frequency range, the excitation of the electromagnetic waves is not reduced by quantum theory. The white spectrum says: by means of the line, the available fluctuation energy is transmitted from one resistor to the other through every spectral component of the frequency . It corresponds to two degrees of freedom, which is consistent with the electromagnetic nature of the transmission mechanism. Electric and magnetic fields each contribute a degree of freedom and therefore, according to the uniform distribution law, the mean fluctuation energy . ${\ displaystyle f}$${\ displaystyle k _ {\ mathrm {B}} T}$${\ displaystyle {\ tfrac {1} {2k _ {\ mathrm {B}} T}}}$

The low frequency approximation in the figure gives the number of excited photons with the factor . Almost 10 10 quanta are condensed in the electromagnetic wave of the frequency at room temperature , the potentially quantum character of the wave is not obvious. - The spectral component of an electromagnetic wave can accommodate any number of quanta , cf. Photons and bosons . ${\ displaystyle f \, \ ll \, f _ {\ mathrm {Q}}}$${\ displaystyle W (f) \ approx hf {\ tfrac {k _ {\ mathrm {B}} T} {hf}}}$${\ displaystyle {\ tfrac {k _ {\ mathrm {B}} T} {hf}}}$${\ displaystyle hf}$${\ displaystyle f = 1 \, {\ text {kHz}}}$${\ displaystyle f}$${\ displaystyle hf}$

The Hochfrequenznäherung with leads to the Boltzmann factor corresponding to the reduced availability of correspondingly large amounts of energy in the heat bath. The quanta can be excited thermodynamically with high yield only efficiently up to the order of magnitude , larger quanta are frozen at comparatively small thermally available energies in the sense of freezing, for example, the degrees of freedom of rotation of the specific heat at low temperatures. ${\ displaystyle f \, \ gg \, f _ {\ mathrm {Q}}}$${\ displaystyle W (f) \ approx hf \, \ mathrm {e} ^ {- hf \! / \! k _ {\ mathrm {B}} T}}$${\ displaystyle hf}$${\ displaystyle k _ {\ mathrm {B}} T}$${\ displaystyle hf}$${\ displaystyle k _ {\ mathrm {B}} T}$

With is and with the quantum-theoretical frequency limit would be clearly noticeable, the electromagnetic mode would only be occupied by a photon in around half the time. For frequencies up to 1 GHz, however, the ideal black waveguide can hardly be realized with sufficient accuracy using current electrical engineering means. ${\ displaystyle T = 0 {,} 05 \, {\ text {K}}}$${\ displaystyle f _ {\ mathrm {Q}} = 1 \, {\ text {GHz}}}$${\ displaystyle W (f _ {\ mathrm {Q}}) _ {T = 0.05 \; \ mathrm {K}} = {\ tfrac {1} {1.718}} = 0 {,} 58}$

A comparison: Above, the total power P  = 4.26 · 10 −8  watts for room temperature was calculated. Also at T  = 300 K, the black body emits  approximately the same power 4.6 · 10 −8 watts into the half-space from an area of ​​10 −10 m² according to Stefan-Boltzmann's law .

(*)'This “black body” thus allows the investigation of radiation itself, unaffected by the material properties of the radiating body, an almost ideal case of the experimental verification of a perfect abstraction, a theoretical concept.' The author Walther Gerlach (1936) has emphasized this quote. He continues to describe the way to research Planck's formula: the development of the connection between radiation energy and wavelength did not lead to numerous facts that had to be sorted out, but directly to physical law.
(**)A clear difference to cavity radiation is particularly emphasized, which simplifies the Nyquist formula accordingly. The finite spectral power of the resistance noise extends as a white spectrum down to any small frequencies, while that of the black cavity radiation disappears proportionally to for , because the frequency is included in the calculation of the density of states (number of oscillators in the frequency interval) due to the radiation in a finite solid angle . With the black waveguide, however , the radiation of the available power to the adapted load resistance is one-dimensional, so the number of oscillators per frequency interval is 1, see Fig. Nyquist. The densely lying states with energy can each be occupied with many photons according to the mean occupation density${\ displaystyle f ^ {2}}$${\ displaystyle f \, \ rightarrow 0}$${\ displaystyle f}$${\ displaystyle hf}$${\ displaystyle {\ tfrac {W (f)} {hf}} = {\ tfrac {1} {\ mathrm {e} ^ {hf / k _ {\ mathrm {B}} T} -1}}.}$

The noisy resistance works on the ideal capacitor of capacitance . ${\ displaystyle R}$${\ displaystyle C}$

The no-load voltage spectrum of the heat noise is reduced at the capacitive load by the square of the voltage divider factor. ${\ displaystyle 4RW (f)}$${\ displaystyle {\ tfrac {1} {1+ (f / f _ {\ mathrm {E}}) ^ {2}}}}$

${\ displaystyle f _ {\ mathrm {E}} = {\ tfrac {1} {2 \ pi RC}}}$   is the electrotechnical cutoff frequency of the RC arrangement to the time constant${\ displaystyle \ tau = RC.}$

Every ohmic resistor as a component has a small stray capacitance in parallel , the spectrum of its terminal voltage is in practice (*)

${\ displaystyle W _ {\ mathrm {Terminals}} (f) = {\ frac {4 \, Rk _ {\ mathrm {B}} T} {1+ (f / f _ {\ mathrm {E}}) ^ {2 }}}.}$

In thermal equilibrium, the formula for the energy on a capacitor at a capacitor voltage is the mean energy ${\ displaystyle {\ tfrac {1} {2}} CU ^ {2}}$${\ displaystyle U}$

{\ displaystyle {\ begin {aligned} {\ tfrac {1} {2}} \, C \, {\ overline {u ^ {2}}} & = {\ tfrac {1} {2}} C {\ int _ {0} ^ {\ infty} \! 4 \, R \, W (f) \, {\ frac {1} {1+ (f / f _ {\ mathrm {E}}) ^ {2}} } \; \ mathrm {d} f} \ approx {\ tfrac {1} {2}} C \; \ cdot \; 4 \, Rk _ {\ mathrm {B}} T {\ int _ {0} ^ { \ infty} \! \! {\ frac {1} {1+ (f / f _ {\ mathrm {E}}) ^ {2}}} \; \ mathrm {d} f} \\\\ & = { \ tfrac {1} {2}} C \; \ cdot \; 4 \, Rk _ {\ mathrm {B}} T \; \ Delta f _ {\ mathrm {eff}} = {\ tfrac {1} {2} } k _ {\ mathrm {B}} T \ end {aligned}}}

with the last replaced by the low frequency value. The capacitor is constantly supplied and withdrawn for around the duration , the correlation time , for example . ${\ displaystyle W (f)}$${\ displaystyle k _ {\ mathrm {B}} T}$${\ displaystyle \ tau}$${\ displaystyle {\ tfrac {1} {2k _ {\ mathrm {B}} T}}}$

The effective bandwidth of the RC element is defined by

${\ displaystyle \ Delta f _ {\ mathrm {eff}} = {\ int _ {0} ^ {\ infty} \! \! {\ frac {1} {1+ (f / f _ {\ mathrm {E}} ) ^ {2}}} \; \ mathrm {d} f} = {\ tfrac {\ pi} {2}} f _ {\ mathrm {E}} = {\ tfrac {1} {4}} (RC) ^ {- 1}.}$
• The capacitor is coupled to its heat bath via the resistor and stores the energy on average${\ displaystyle R}$${\ displaystyle {\ tfrac {1} {2}} k _ {\ mathrm {B}} T.}$
• In terms of thermodynamics, the condenser has a degree of freedom, like an energy store. Both statements apply accordingly to the inductance.

The energy that is complementary to the stored energy of the total energy thermally generated by in the effective frequency interval is dissipated in itself. ${\ displaystyle {\ tfrac {1} {2}} \, C \, {\ overline {u ^ {2}}}}$${\ displaystyle {\ tfrac {1} {2}} k _ {\ mathrm {B}} T}$${\ displaystyle R}$${\ displaystyle \ Delta f _ {\ mathrm {eff}}}$${\ displaystyle k _ {\ mathrm {B}} T}$${\ displaystyle R}$

This balance is known from the charging of a capacitor with a constant voltage and can be derived from the principle of minimum entropy production . Of course, the power generated outside of the effective bandwidth is dissipated in itself; because as the resistance increases, it works increasingly in short circuit. ${\ displaystyle R}$${\ displaystyle f \, \ gg \, f _ {\ mathrm {E}}}$

The time constant and thus the effective frequency band are just such that the one thermal degree of freedom of the capacitor is sufficient. ${\ displaystyle RC}$${\ displaystyle \ Delta f _ {\ mathrm {eff}}}$

Conclusion 1: In the equivalent circuit diagram, every real capacitor consists of an ideal capacitor with a finite insulation resistance connected in parallel, which means that it is coupled to a heat bath. The real capacitor therefore stores the supplied mean energy, which is only dependent on the temperature. According to the capacitor, the effective noise voltage is present on the capacitor, for which purpose electron charges are stored on average in terms of amount . On a capacitor of 1 pF, the effective noise voltage at room temperature is 64 µV, which requires 402 elementary charges, which are transported on average for the random voltage fluctuations. The fact is remembered and . ${\ displaystyle {\ tfrac {1} {2}} k _ {\ mathrm {B}} T.}$${\ displaystyle {\ tfrac {1} {2}} \, C \, {\ overline {u ^ {2}}} \, = \, {\ tfrac {1} {2}} \, {\ tfrac { \ overline {q ^ {2}}} {C}}}$${\ displaystyle {\ sqrt {\ overline {u ^ {2}}}} \, = \, {\ sqrt {\ tfrac {k _ {\ mathrm {B}} T} {C}}},}$${\ displaystyle {\ tfrac {\ sqrt {\ overline {q ^ {2}}}} {| e |}} \, = \, {\ tfrac {\ sqrt {Ck _ {\ mathrm {B}} T}} {| e |}}}$${\ displaystyle e}$${\ displaystyle {\ overline {u}} = 0}$${\ displaystyle {\ overline {q}} = 0}$

Conclusion 2: The fundamental proportionality of the noise power to the absolute temperature becomes immediately recognizable when the noise voltage square is measured with high resistance over a capacitor. A wire resistor is useful as a noisy resistor because it allows very large temperature changes; According to the formula, its unavoidable temperature dependence does not affect the measurement result with this circuit . ${\ displaystyle T}$${\ displaystyle {\ overline {u ^ {2}}} = k _ {\ mathrm {B}} T / C}$${\ displaystyle R}$

This arrangement is suitable for an impressive demonstration experiment . must always be large enough so that the amplifier's inherent noise does not interfere. ${\ displaystyle RT}$

• The result makes it particularly clear that the resistor component only serves as a mediator between the heat accumulator and the electrical accumulator. The same applies to magnetic storage.
(*)The stray capacitance of a resistance component practically limits the spectrum before an influence from the quantum-theoretical limit frequency becomes noticeable. At high frequencies, an inductive component must also be taken into account. However, this has reached the frequency limit from which the component can no longer be viewed as concentrated ; the rushing resistance would now have to be treated under the conditions of line theory . Ultimately, device terminals are no longer well defined and an antenna concept is more appropriate.${\ displaystyle f _ {\ mathrm {Q}}}$

### Dissipation and storage

In fact, the voltage spectrum would have to be integrated as a quantum-theoretical formula, but the frequency band of a real capacitor reaching up to the electrotechnical cut -off frequency limits the effective spectrum at 300 K, far below the quantum-theoretical cut-off frequency${\ displaystyle W (f)}$${\ displaystyle f _ {\ mathrm {E}}}$${\ displaystyle f _ {\ mathrm {Q}}}$

This fact is used in the following to calculate the power dissipated in the noisy resistor even under a capacitive load. In contrast to the above, the square of the voltage is to be considered here over the resistance itself, which is to be evaluated with the square of the magnitude of the complex voltage divider factor. The power dissipated in is ${\ displaystyle {\ tfrac {(f / f _ {\ mathrm {E}}) ^ {2}} {1+ (f / f _ {\ mathrm {E}}) ^ {2}}}}$${\ displaystyle R}$

${\ displaystyle P = {\ frac {\ overline {u ^ {2}}} {R}} = 4 {\ int _ {0} ^ {\ infty} \! W (f) \, {\ frac {( f / f _ {\ mathrm {E}}) ^ {2}} {1+ (f / f _ {\ mathrm {E}}) ^ {2}}} \; \ mathrm {d} f}}$.

By adding and subtracting 1 to the electrotechnical division factor in the integral and including −1 in this division factor, the quantum theoretical cutoff frequency is initially obtained

${\ displaystyle P = 4 \, k _ {\ mathrm {B}} T {\ int _ {0} ^ {\ infty} \! \! {\ frac {f / f _ {\ mathrm {Q}}} {\ mathrm {e} ^ {f / f _ {\ mathrm {Q}}} - 1}} \ left (1 - {\ frac {1} {1+ (f / f _ {\ mathrm {E}}) ^ {2 }}} \ right) \ mathrm {d} f}.}$

The integral over the first summand, the short-circuit power in R itself, has already been evaluated above, the integral over the second is calculated - mostly in an excellent approximation - by setting the factor equal to 1 for the sake of simplicity , because the frequency band is generally much wider than the electrotechnical up . The result obtained immediately is expressed with the bandwidths or the effective bandwidths ${\ displaystyle {\ tfrac {f \! / \! f _ {\ mathrm {Q}}} {\ mathrm {e} ^ {f \! / \! f _ {\ mathrm {Q}}} - 1}}}$${\ displaystyle f _ {\ mathrm {Q}}}$${\ displaystyle f _ {\ mathrm {E}}}$

{\ displaystyle {\ begin {aligned} P & \ approx 4 \, k _ {\ mathrm {B}} T \ left ({\ tfrac {\ pi ^ {2}} {6}} f _ {\ mathrm {Q}} - {\ tfrac {\ pi} {2}} f _ {\ mathrm {E}} \ right) = 4 \, k _ {\ mathrm {B}} T \ left (\ Delta f _ {\ mathrm {Q, \, eff}} - \ Delta f _ {\ mathrm {eff}} \ right) \\ & \ approx k _ {\ mathrm {B}} T \ left ({\ tfrac {2} {3}} \, \ pi ^ { 2} {\ frac {k _ {\ mathrm {B}} T} {h}} - {\ frac {1} {RC}} \ right) \!. \ End {aligned}}}

The second term is small compared to the first, which represents the mean total power dissipated in R in the event of a short circuit. This is reduced by the capacity due to the capacitive load   , as the capacitor voltage reduces the voltage drop across R and the current in the circuit. Capacitor voltage and current are out of phase, characterizing the storage of energy and the transport of reactive power over time . ${\ displaystyle {\ tfrac {\ overline {u ^ {2}}} {R}} = {\ tfrac {k _ {\ mathrm {B}} T} {RC}},}$${\ displaystyle k _ {\ mathrm {B}} T}$${\ displaystyle {\ tfrac {1} {2}} RC}$

## Autocorrelation function

The impact processes and the emission and absorption processes in the resistance material are on average evenly distributed over time, as long as the resistance does not age. In this respect, the resistance noise is stationary. The marking of a time stamp like t  = 0 has no meaning for the general characterization of the noise. This makes it unnecessary to distinguish between an odd and an even component of the original voltage , so that the tangent of a phase angle as the usual measure of their ratio is not an important characteristic for the stationary noise itself. Consequently, for a mathematically invariant description, instead of the Fourier transform of the amplitude spectrum, square sizes can be chosen, as above the performance spectrum. They already contain sufficient information about the time structure . ${\ displaystyle u (t)}$${\ displaystyle u (t)}$

As information about amplitudes, it facilitates the usual comparison with a direct voltage with the same heat generation. In addition to the temporal structure which can above mentioned amplitude distribution are evaluated. The two distributions are independent of each other, but a restriction in the frequency band influences the spread of the amplitude statistics. The white spectrum does not necessarily include a normal distribution of the instantaneous values, as is the case with resistance noise. (*)${\ displaystyle u _ {\ mathrm {eff}} \, = \, {\ sqrt {\ overline {u ^ {2}}}}}$${\ displaystyle {\ overline {u ^ {2}}}}$

To characterize the stationary noise over time, not only the mean voltage square remains . ${\ displaystyle {\ overline {u ^ {2}}}}$

• Rather, there is invariant to be described inner temporal relationship of represented by the autocorrelation function is measured:${\ displaystyle u (t)}$
${\ displaystyle \ rho (\ Delta t) = \ lim _ {T \ to \ infty} {\ frac {1} {2T}} {\ int _ {- T} ^ {+ T}} \! u (t ) u (t + \! \ Delta t) \, \ mathrm {d} t}$

The autocorrelation function, hereinafter referred to as AKF, is independent of the time direction: and have the same AKF. The definition formula shows immediately that marking any time as the new reference time has no influence. ${\ displaystyle u (+ t)}$${\ displaystyle u (-t)}$${\ displaystyle t = t_ {0}}$${\ displaystyle t \, '\, = \, t \, - \, t_ {0}}$

The AKF has its maximum ${\ displaystyle \ Delta t = 0}$

${\ displaystyle \ rho (0) = {\ overline {u ^ {2}}}}$.

${\ displaystyle {\ tfrac {\ rho (0)} {R}}}$is the power dissipated in the resistor by the terminal voltage . ${\ displaystyle R}$${\ displaystyle u (t)}$

The AKF is always an even function of . This means that no causal sequence is indexed by time . Nevertheless, and are not independent, cannot change at will. The power spectrum determines the effective, fastest possible change, for example through its upper limit frequency. ${\ displaystyle \ Delta t}$${\ displaystyle t}$${\ displaystyle u (t)}$${\ displaystyle u (t + \ Delta t)}$${\ displaystyle u (t)}$

With the AKF, the point-in-time (or local ) description level ( time domain ) corresponds to the frequency spectrum. The latter describes the internal relationship for the description level with harmonic oscillations ( frequency range ).

• One or the other of the equivalent representations is selected depending on the intention or the technical measurement requirements .
In order to verify the resistance noise experimentally, the possibility of frequency representation was important. At the time of Johnson's discovery, it was even necessary because the short-term and correlation technology were not as developed as the frequency-oriented filter technology due to the advances in radio technology with its knowledge of resonant circuits.

In fact, a mathematical transformation justifies the equivalent representation of the stationary process by the AKF or by the frequency spectrum. Wiener and Chintchin provided the proof by stating that the Fourier transform delivers the desired result:

{\ displaystyle {\ begin {aligned} & S (f) = \! \ int _ {- \ infty} ^ {+ \ infty} \! \! \ rho (\ Delta t) \, \ mathrm {e} ^ { -j2 \ pi f \ Delta t} \, \ mathrm {d} \ Delta t \ quad \ qquad \ qquad \ qquad \ qquad \; \, S (f) = 2 \ int _ {0} ^ {\ infty} \! \! \ rho (\ Delta t) \ cos (2 \ pi f \ Delta t) \, \ mathrm {d} \ Delta t \\ & \ rho (\ Delta t) = \! \ int _ {- \ infty} ^ {+ \ infty} \! \! S (f) \, \ mathrm {e} ^ {+ j2 \ pi f \ Delta t} \, \ mathrm {d} f \ quad \ qquad \ qquad \ qquad \ qquad \ quad \ rho (\ Delta t) = 2 \! \ int _ {0} ^ {\ infty} \! \! S (f) \ cos (2 \ pi f \ Delta t) \, \ mathrm {d} f \ end {aligned}}}

${\ displaystyle S (f)}$is defined for reasons of symmetry of the transformation formulas for negative frequencies. It should therefore be noted that it was only defined for above based on the measurement process . ${\ displaystyle S (f) \, = \, {\ tfrac {1} {2}} \, W (f)}$${\ displaystyle W (f)}$${\ displaystyle f \, \ geqq \, 0}$

As auto spectra, resistance noise spectra are real, straight functions of frequency. The position of the signs in the exponent is in this respect convention, it is chosen as indicated with regard to cross-correlation functions , in which the causal linkage is an aim of the analysis.

In the transformation pair on the right, the complex exponential function is replaced by in the integrand and the integration limits are 0 and because functions are transformed. This is the classic Viennese chintchin formulation, although it is often replaced by something closer to measurement technology . ${\ displaystyle 2 \ cos {(2 \ pi f \ Delta t)}}$${\ displaystyle \ infty,}$${\ displaystyle 2S (f)}$${\ displaystyle W (f)}$

(*) Note: As the pre-factor 2 is omitted in the formulation with the (measured) spectrum instead of in the formula below right, a factor 4 arises in the formula above right.${\ displaystyle W (f)}$${\ displaystyle 2S (f)}$

### Resistance with parallel capacitance

The AKF for the spectrum of the terminal voltage of the resistor with stray capacitance lying in parallel is ${\ displaystyle W _ {\ mathrm {Terminals}} (f)}$

${\ displaystyle \ rho _ {\ mathrm {Terminals}} (\ Delta t) = {\ tfrac {1} {2}} \! \ int _ {- \ infty} ^ {+ \ infty} \! \! { \ frac {4 \, Rk _ {\ mathrm {B}} T} {1+ (f / f _ {\ mathrm {E}}) ^ {2}}} \, \ mathrm {e} ^ {j2 \ pi f \ Delta t} \, \ mathrm {d} f = {\ tfrac {2} {\ pi}} {\ frac {Rk _ {\ mathrm {B}} T} {\ tau}} \! \ Int _ {0 } ^ {\ infty} \! \! {\ frac {\ mathrm {d} x} {1 + x ^ {2}}} \, \ cos \ left (x {\ tfrac {\ Delta t} {\ tau }} \ right) = {\ frac {Rk _ {\ mathrm {B}} T} {\ tau}} \, \ mathrm {e} ^ {- {\ frac {| \ Delta t |} {\ tau}} }.}$

The power that is dissipated in the noisy resistor itself when the capacitor is parallel to the capacitance${\ displaystyle C}$

${\ displaystyle \ rho _ {\ mathrm {Terminals}} (0) \! / \! R = k _ {\ mathrm {B}} T \! / \! \ tau.}$

The standardized ACF is determined solely by the statistical relationship

${\ displaystyle {\ frac {\ rho _ {\ mathrm {Terminals}} (\ Delta t)} {\ rho _ {\ mathrm {Terminals}} (0)}} = \ mathrm {e} ^ {- {\ frac {| \ Delta t |} {\ tau}}}.}$

The mean correlation duration is defined by

${\ displaystyle {\ overline {\ Delta t}} = {\ frac {\ int _ {- \ infty} ^ {+ \ infty} \! | \ Delta t | \, {\ frac {\ rho _ {\ mathrm {Terminals}} (\ Delta t)} {\ rho _ {\ mathrm {Terminals}} (0)}} \, \ mathrm {d} \ Delta t} {\ int _ {- \ infty} ^ {+ \ infty} \! {\ tfrac {\ rho _ {\ mathrm {Terminals}} (\ Delta t)} {\ rho _ {\ mathrm {Terminals}} (0)}} \, \ mathrm {d} \ Delta t }} = \ tau \; {\ frac {\ int _ {0} ^ {+ \ infty} \! {\ frac {\ Delta t} {\ tau}} \, \ mathrm {e} ^ {- {\ frac {\ Delta t} {\ tau}}} \, \ mathrm {d} {\ frac {\ Delta t} {\ tau}}} {\ int _ {0} ^ {+ \ infty} \ mathrm {e } ^ {- {\ frac {\ Delta t} {\ tau}}} \, \ mathrm {d} {\ tfrac {\ Delta t} {\ tau}}}} = \ tau = RC = {\ frac { 1} {2 \ pi f _ {\ mathrm {E}}}}.}$
• This wiring of the noisy resistor forces the noise to have an average correlation period; it is equal to its time constant , cf. above .${\ displaystyle \ tau = RC}$
• Long correlation periods are represented with exponentially decreasing weight.

Excursus on the metrological significance of the correlation time. Measuring the noisy deflection of a measuring instrument requires many independent readings for sufficient statistics to calculate the mean value and its error with the desired accuracy. (Gaussian noise is beneficial for this.)

• The minimum required measurement duration is calculated from the number of individual measurements required for the desired accuracy multiplied by a small multiple of the correlation time of the disturbance.

### Quantum-theoretically limited AKF of the resistance noise

The ACF for the quantum-theoretically limited spectrum of the available power is calculated below. ${\ displaystyle W (f)}$

Note 1: defined on goes into this formula. ${\ displaystyle S (f) \, = \, {\ tfrac {1} {2}} \, W (f)}$${\ displaystyle - \ infty \, <\, f \, <\, + \ infty}$

Note 2: The correlation function of the terminal voltage has been dealt with above , now it is of the dimension power . ${\ displaystyle \ rho (\ Delta t)}$

{\ displaystyle {\ begin {aligned} \ rho (\ Delta t) & = {\ tfrac {1} {2}} k _ {\ mathrm {B}} T \! \ int _ {- \ infty} ^ {+ \ infty} \! \! {\ tfrac {| f / f _ {\ mathrm {Q}} |} {\ mathrm {e} ^ {| f / f _ {\ mathrm {Q}} |} \; - \; 1}} \, \ mathrm {e} ^ {j2 \ pi f \ Delta t} \, \ mathrm {d} f \\ & = k _ {\ mathrm {B}} Tf _ {\ mathrm {Q}} \! \ int _ {0} ^ {\ infty} \! \! {\ tfrac {f / f _ {\ mathrm {Q}}} {\ mathrm {e} ^ {f / f _ {\ mathrm {Q}}} \ ; - \; 1}} \, \ cos {\ left [2 \ pi {\ tfrac {f} {f _ {\ mathrm {Q}}}} (f _ {\ mathrm {Q}} \ Delta t) \ right ]} \, \ mathrm {d} {\ tfrac {f} {f _ {\ mathrm {Q}}}} = k _ {\ mathrm {B}} Tf _ {\ mathrm {Q}} \! \ int _ {0 } ^ {\ infty} \! \! {\ frac {x} {\ mathrm {e} ^ {x} \; - \; 1}} \, \ cos {\ left (2 \ pi xf _ {\ mathrm { Q}} \ Delta t \ right)} \, \ mathrm {d} x \ end {aligned}}}

The total available power already calculated above follows from this

${\ displaystyle P = \ rho (0) = {\ tfrac {\ pi ^ {2}} {6}} \, k _ {\ mathrm {B}} Tf _ {\ mathrm {Q}}.}$

The normalized ACF of the quantum-mechanically limited noise spectrum again describes the internal temporal structure alone

{\ displaystyle {\ begin {aligned} {\ frac {\ rho (\ Delta t)} {\ rho (0)}} & = 3 \ left [(\ omega _ {\ mathrm {Q}} \ Delta t) ^ {- 2} - \ sinh ^ {- 2} (\ omega _ {\ mathrm {Q}} \ Delta t) \ right] \\ & = 3 \ sinh ^ {- 2} (\ omega _ {\ mathrm {Q}} \ Delta t) \ left [\ left ({\ frac {\ sinh (\ omega _ {\ mathrm {Q}} \ Delta t)} {\ omega _ {\ mathrm {Q}} \ Delta t }} \ right) ^ {2} -1 \ right] \\ {\ text {with}} \ omega _ {\ mathrm {Q}} & = 2 \ pi f _ {\ mathrm {Q}} = {\ frac {k _ {\ mathrm {B}} T} {\ hbar}}. \ end {aligned}}}

shows, that

${\ displaystyle {\ frac {\ rho (\ Delta t)} {\ rho (0)}} = {\ begin {cases} 1 \ ,, & {\ text {if}} \ quad \ Delta t \ to 0 \ \\ 0 {,} 522 \ ,, & {\ text {if}} \ quad \ omega _ {\ mathrm {Q}} | \ Delta t | = 2 \ \\ 3 \, (\ omega _ {\ mathrm {Q}} \ Delta t) ^ {- 2} \ ,, & {\ text {if}} \ quad \ omega _ {\ mathrm {Q}} | \ Delta t | \ gg 1 {\ text {. }} \ end {cases}}}$
• The noise, limited by quantum theory, has a correlation duration of approx ${\ displaystyle | \ Delta t | = 2 / \ omega _ {\ mathrm {Q}} = 2 \, {\ frac {\ hbar} {k _ {\ mathrm {B}} T}}.}$
• The long correlation periods are weighted proportionally .${\ displaystyle \ Delta t ^ {- 2}}$

This makes it clear, for example, that a weak drop in the spectrum results in a steep correlation function and vice versa. The spectrum, which is limited in a capacitive manner , is linked to an exponential decrease in the statistical weight of increasing correlation times. - The spectrum, limited by quantum theory, drops practically exponentially with increasing frequency, its correlation function ultimately only approximately correspondingly${\ displaystyle f ^ {- 2}}$${\ displaystyle | \ Delta t | ^ {- 2}.}$

### White noise

The extreme case is cited for the question of the broad spectrum in the case of an internal context of short duration and vice versa. Any short-term processes correspond to the white spectrum. An impulse that disappears as it arises can serve this purpose and is mathematically well defined with the Dirac distribution . For this arbitrarily short-term object, only the values for finite can be specified. Nevertheless, this delta distribution is suitable because of the mean value property ${\ displaystyle \ delta (t)}$${\ displaystyle \ delta (t) = 0}$${\ displaystyle t \, \ neq \, 0}$

${\ displaystyle \ int _ {- \ infty} ^ {+ \ infty} \! \ delta (t) \, \ mathrm {d} t = 1}$

for the representation of physical facts. (*)

${\ displaystyle \ delta (t)}$necessarily leads to correlation functions: Because no square of the distribution can be formed, the power must be calculated on the AKF, cf. Convolution integral , can be used:

${\ displaystyle \ delta (t) = \ int _ {- \ infty} ^ {+ \ infty} \! \ delta (t) \ delta (t + \ theta) \, \ mathrm {d} \ theta}$

The voltage pulse at the time ${\ displaystyle t_ {0}}$

${\ displaystyle u (t) = p \ delta (t-t_ {0}) \, \!}$

generates the voltage surge

${\ displaystyle \ int _ {- \ infty} ^ {+ \ infty} \! \! u (t) \, \ mathrm {d} \ Delta t \ = \ int _ {- \ infty} ^ {+ \ infty } \! \! p \, \ delta (t-t_ {0}) \, \ mathrm {d} \ Delta t = p}$

the unit 1 Vs and the AKF has any short correlation time

${\ displaystyle \ rho (\ Delta t) = p ^ {2} \! \! \ int _ {- \ infty} ^ {+ \ infty} \! \ delta (t-t_ {0}) \ delta (t ) \, \ mathrm {d} \ Delta t = p ^ {2} \ delta (\ Delta t)}$

as well as the white frequency spectrum

{\ displaystyle {\ begin {alignedat} {2} S (f) & = p ^ {2} \! \! \ int _ {- \ infty} ^ {+ \ infty} \! \ delta (\ Delta t) \, \ mathrm {e} ^ {- j \, 2 \ pi \! f \ Delta t} \, \ mathrm {d} \ Delta t = p ^ {2}, & \ quad & \ - \ infty \ leq f \ leq + \ infty \,. \\\ end {alignedat}}}

Conversely, the arbitrarily narrow frequency band leads to ${\ displaystyle f_ {0}}$

{\ displaystyle {\ begin {alignedat} {2} S (f) & = {\ tfrac {1} {2}} {\ hat {u}} ^ {2} \ left [\ delta (f-f_ {0 }) + \ delta (f + f_ {0}) \ right], \ quad \ - \ infty

periodic correlation of any length on the AKF

${\ displaystyle \ rho (\ Delta t) = {\ hat {u}} ^ {2} \ cos (2 \ pi f_ {0} t) \ ,.}$

With the correlation time becomes arbitrarily long. The following applies to direct voltage${\ displaystyle f_ {0} \, \ rightarrow 0}$${\ displaystyle u (t) = {\ hat {u}}}$

{\ displaystyle {\ begin {alignedat} {2} S (f) & = {\ hat {u}} ^ {2} \ delta (f), \ \ quad - \ infty

Here one can simply speak of an infinitely long correlation duration with a spectrum that is also strictly localized.

(*) Remark: The impact always has the reciprocal value of the dimension of its argument to the physical dimension: it is therefore of dimension 1. Here the time with the unit 1 s and expressions with dimensional argument such as the object that each application ultimately amounts to, always mean the expression ; because the distribution is defined purely mathematically. The formula is in the argument.${\ displaystyle \ delta (t)}$${\ displaystyle \ delta (t) \ mathrm {d} t}$${\ displaystyle t}$${\ displaystyle \ delta (t) \ mathrm {d} t,}$${\ displaystyle \ delta ({\ tfrac {t} {\ mathrm {s}}}) \, \ mathrm {d} ({\ tfrac {t} {\ mathrm {s}}})}$${\ displaystyle \ delta (\ alpha t) \, = \, | \ alpha | ^ {- 1} \, \ delta (t), \ \ alpha \, \ neq \, 0.}$ ${\ displaystyle \ delta (t) = \ delta (-t)}$

#### Stationary sequence of collision functions

Voltage pulses defined above should be generated independently of one another at any times with the same probability with the average number density per time interval ; they form a stationary sequence. Let the voltage surges p be given the same number of positive or negative signs so that the linear mean value, the constant component, disappears. The pulses are statistically independent. Such a construction could serve as a first approach for a description of the heat noise. However, the instantaneous values ​​obviously do not meet a normal distribution (bell curve). ${\ displaystyle \ lambda}$

The statistical independence allows the simple specification of the ACF of this sequence with the help of Campbell's theorem:

${\ displaystyle \ rho (\ Delta t) = \ lambda \, p ^ {2} \ delta (\ Delta t) \ ,.}$

The AKF (dimension of power of the SI unit 1 W after division by a resistor R ) does not change its course, the correlation time remains vanishingly small. The frequency spectrum ( energy dimension of the unit 1 Ws after division by the resistance R , as power per frequency bandwidth) does not change either, except for the factor${\ displaystyle \ lambda}$

${\ displaystyle S (f) = \ lambda \, p ^ {2}.}$

#### Exponential momentum

• Under the assumptions of Campbell's theorem, the quadratic quantities power and energy add up without changing the mean internal temporal relationship of the pulse train - measured by the AKF - statistical overlapping of pulses of finite duration (incoherent superposition) is permitted, although the resulting amplitude spectrum is changed .

To illustrate, in the pulse sequence described above - under appropriate conditions - the impact functions are represented by exponential pulses

${\ displaystyle h (t) = {\ begin {cases} {\ hat {u}} \, \ mathrm {e} ^ {- t / \ tau}, & \ t \ geq 0 \\ 0, & \ t <0 \ end {cases}}}$

replaced. The AKF and the frequency spectrum, a Lorentz profile , of the modified voltage are:

{\ displaystyle {\ begin {aligned} \ rho (\ Delta t) & = \ lambda {\ frac {{\ hat {u}} ^ {2} \ tau} {2}} \, \ mathrm {e} ^ {- | \ Delta t | / \ tau} + \ left [(\ lambda {\ hat {u}} \ tau) ^ {2} \ right] \\ S (f) & = \ lambda ({\ hat { u}} \ tau) ^ {2} {\ frac {1} {1+ (2 \ pi f \ tau) ^ {2}}} + \ left [(\ lambda {\ hat {u}} \ tau) ^ {2} \, \ delta (f) \ right] \ end {aligned}}}

For the terms in square brackets, see Comment. (*)

${\ displaystyle {\ tfrac {\ rho (0)} {R}} = {\ tfrac {{\ hat {u}} ^ {2}} {2R}} \ lambda \ tau}$is the power dissipated at the resistor . The degree of overlap can be adjusted by the product . ${\ displaystyle R}$${\ displaystyle \ lambda \ tau}$

AKF and spectrum have the same dependency on or as with the noise of the resistor with a parallel capacitor, s. above , although the individual pulses are certainly significantly different. Correspondingly , with the time constant, a capacitor discharges through a resistor. ${\ displaystyle \ Delta t}$${\ displaystyle f}$${\ displaystyle h (t)}$${\ displaystyle \ tau = RC}$

• While the RC -filtered resistance noise of the invariant inner connection as is impressed, he is here from single process ago determined before.${\ displaystyle {\ tfrac {\ rho (\ Delta t)} {\ rho (0)}} = \ mathrm {e} ^ {- | \ Delta t | / \ tau}}$
• From or from the spectrum, it is not possible to infer determined individual processes or random ones.${\ displaystyle \ rho (\ Delta t)}$
(*) Note: A finite constant component arises when all exponential pulses with a fixed (positive) sign are included in the sequence . DC components of the individual pulses as here are coherent and must therefore be added as amplitude: . As a result, the DC power (after division by ) is added to the AKF and the corresponding term is added to the spectrum.${\ displaystyle {\ overline {u (t)}} \, = \, \ lambda {\ hat {u}} \ tau}$${\ displaystyle u (t)}$${\ displaystyle {\ overline {h (t)}} \, = \, {\ hat {u}} \ tau}$${\ displaystyle {\ overline {u (t)}} \, = \, \ lambda \, {\ overline {h (t)}}}$${\ displaystyle R}$${\ displaystyle {\ overline {u}} ^ {2} \, = \, (\ lambda {\ hat {u}} \ tau) ^ {2}}$${\ displaystyle {\ overline {u}} ^ {2} \, \ delta (f) \, = \, (\ lambda {\ hat {u}} \ tau) ^ {2} \, \ delta (f) }$

## literature

• Heinz Beneking : Practice of Electronic Noise (= BI university scripts 734 / 734a-d, ). Bibliographisches Institut, Mannheim u. a. 1971.
• Heinz Bittel , Leo Storm: Noise. An introduction to understanding electrical fluctuations. Springer, Berlin a. a. 1971, ISBN 3-540-05055-8 .
• Rudolf Müller: Rauschen (= semiconductor electronics. Vol. 15). 2nd, revised and expanded edition. Springer, Berlin a. a. 1990, ISBN 3-540-51145-8 .

## Individual evidence

1. JB Johnson: Thermal Agitation of Electricity in Conductors. In: Physical Review . Vol. 32, No. 1, 1928, pp. 97-109, doi: 10.1103 / PhysRev.32.97 .
2. a b c d H. Nyquist: Thermal Agitation of Electric Charge in Conductors. In: Physical Review. Vol. 32, No. 1, 1928, pp. 110-113, doi: 10.1103 / PhysRev.32.110 .
3. W. Schottky: About spontaneous current fluctuations in different electricity conductors. In: Annals of Physics . Vol. 362, No. 23, 1918, pp. 541-567, doi: 10.1002 / andp.19183622304 .
4. a b В. Л. Гинзбург: Некоторые вопросы теории электрических флуктуации. In: Успехи физических наук. Vol. 46, No. 3, 1952, , pp. 348-387; in German: WL Ginsburg: Some problems from the theory of electrical fluctuation phenomena. In: Advances in Physics. Vol. 1, No. 1953, , pp. 51-87, here p. 67, doi: 10.1002 / prop.19530010202 .
5. ^ A b Herbert B. Callen , Theodore A. Welton: Irreversibility and generalized noise. In: Physical Review. Vol. 83, No. 1, 1951, pp. 34-40, doi: 10.1103 / PhysRev . 83.34 .
6. ^ A. van der Ziel: The effect of zero point energy noise in Maser amplifiers. In: Physica B C. Vol. 96, No. 3, 1979, , pp. 325-326, doi: 10.1016 / 0378-4363 (79) 90015-9 .
7. Walther Gerlach : Theory and Experiment in Exact Science. In: Natural Sciences. Vol. 24, No. 46/47, 1936, , pp. 721-741, here p. 732, doi: 10.1007 / BF01504074 .
8. ^ H. Nyquist: Thermal Agitation of Electric Charge in Conductors. In: Physical Review. Vol. 32, No. 1, 1928, pp. 110–113, here p. 112. Please note: At that time the author applied the term degree of freedom to the modes of vibration .${\ displaystyle f}$