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Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the
voltage Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...
or current
noise Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...
generated by the thermal agitation of the
charge carrier In solid state physics, a charge carrier is a particle or quasiparticle that is free to move, carrying an electric charge, especially the particles that carry electric charges in electrical conductors. Examples are electrons, ions and holes. ...
s (usually the
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s) inside an
electrical conductor In physics and electrical engineering, a conductor is an object or type of material that allows the flow of charge (electric current) in one or more directions. Materials made of metal are common electrical conductors. The flow of negatively c ...
at equilibrium, which happens regardless of any applied voltage. Thermal noise is present in all
electrical circuit An electrical network is an interconnection of electrical components (e.g., battery (electricity), batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e. ...
s, and in sensitive electronic equipment (such as
radio receiver In radio communications, a radio receiver, also known as a receiver, a wireless, or simply a radio, is an electronic device that receives radio waves and converts the information carried by them to a usable form. It is used with an antenna. ...
s) can drown out weak signals, and can be the limiting factor on sensitivity of electrical measuring instruments. Thermal noise is proportional to
absolute temperature Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion. Thermodynamic temperature is typically expres ...
, so some sensitive electronic equipment such as
radio telescope A radio telescope is a specialized antenna (radio), antenna and radio receiver used to detect radio waves from astronomical radio sources in the sky. Radio telescopes are the main observing instrument used in radio astronomy, which studies the r ...
receivers are cooled to
cryogenic In physics, cryogenics is the production and behaviour of materials at very low temperatures. The 13th International Institute of Refrigeration's (IIR) International Congress of Refrigeration (held in Washington, DC in 1971) endorsed a univers ...
temperatures to improve their
signal-to-noise ratio Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to noise power, often expressed in deci ...
. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium. Thermal noise in an ideal resistor is approximately
white White is the lightest color and is achromatic (having no chroma). It is the color of objects such as snow, chalk, and milk, and is the opposite of black. White objects fully (or almost fully) reflect and scatter all the visible wa ...
, meaning that its power
spectral density In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be decomposed into ...
is nearly constant throughout the
frequency spectrum In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be decomposed int ...
(Figure 2). When limited to a finite bandwidth and viewed in the
time domain In mathematics and signal processing, the time domain is a representation of how a signal, function, or data set varies with time. It is used for the analysis of mathematical functions, physical signals or time series of economic or environmental ...
(as sketched in Figure 1), thermal noise has a nearly Gaussian amplitude distribution. For the general case, this definition applies to charge carriers in any type of conducting medium (e.g. ions in an
electrolyte An electrolyte is a substance that conducts electricity through the movement of ions, but not through the movement of electrons. This includes most soluble Salt (chemistry), salts, acids, and Base (chemistry), bases, dissolved in a polar solven ...
), not just
resistor A resistor is a passive two-terminal electronic component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...
s. Thermal noise is distinct from
shot noise Shot noise or Poisson noise is a type of noise which can be modeled by a Poisson process. In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where s ...
, which consists of additional current fluctuations that occur when a voltage is applied and a macroscopic current starts to flow.


History of thermal noise

In 1905, in one of
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
's ''Annus mirabilis'' papers the theory of
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
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. Geertruida de Haas-Lorentz, daughter of
Hendrik Lorentz Hendrik Antoon Lorentz ( ; ; 18 July 1853 – 4 February 1928) was a Dutch theoretical physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for their discovery and theoretical explanation of the Zeeman effect. He derive ...
, 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. 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 Shot noise or Poisson noise is a type of noise which can be modeled by a Poisson process. In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where s ...
. 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 Nokia Bell Labs, commonly referred to as ''Bell Labs'', is an American industrial research and development company owned by Finnish technology company Nokia. With headquarters located in Murray Hill, New Jersey, Murray Hill, New Jersey, the compa ...
found the same kind of noise in communication systems, but described it in terms of frequencies. He described his findings to Harry Nyquist, also at Bell Labs, who used principles of
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
and
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
to explain the results, published in 1928.


Noise of ideal resistors for moderate frequencies

Johnson's experiment (Figure 1) found that the thermal noise from a resistance R at kelvin temperature T and bandlimited to a
frequency band Spectral bands are regions of a given spectrum, having a specific range of wavelengths or frequencies. Most often, it refers to electromagnetic bands, regions of the electromagnetic spectrum. More generally, spectral bands may also be means in ...
of bandwidth \Delta f (Figure 3) has a mean square voltage of: : \overline = 4 k_\text T R \, \Delta f where k_ is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
(
joules The joule ( , or ; symbol: J) is the unit of energy in the International System of Units (SI). In terms of SI base units, one joule corresponds to one kilogram- metre squared per second squared One joule is equal to the amount of work don ...
per
kelvin The kelvin (symbol: K) is the base unit for temperature in the International System of Units (SI). The Kelvin scale is an absolute temperature scale that starts at the lowest possible temperature (absolute zero), taken to be 0 K. By de ...
). 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.


Power spectral density

The mean square voltage per
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or Cycle per second, cycle) per second. The hertz is an SI derived unit whose formal expression in ter ...
of bandwidth is 4 k_\text T R and may be called the power spectral density (Figure 2). Its square root at room temperature (around 300 K) approximates to 0.13 \sqrt in units of . A 10 kΩ resistor, for example, would have approximately 13  at room temperature.


RMS noise voltage

The square root of the mean square voltage yields the
root mean square In mathematics, the root mean square (abbrev. RMS, or rms) of a set of values is the square root of the set's mean square. Given a set x_i, its RMS is denoted as either x_\mathrm or \mathrm_x. The RMS is also known as the quadratic mean (denote ...
(RMS) voltage observed over the bandwidth \Delta f : : V_\text = \sqrt = \sqrt \, . 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 A voltage source is a two-terminal (electronics), terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output Electric current, current. However, a r ...
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 \, \Delta f corresponds to almost one nanovolt of RMS noise.


RMS noise current

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 A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it. A current source is the dual of a voltage source. The term ''current sink'' is sometimes used for sources fed ...
with the following RMS current: : I_\text = = \sqrt .


Thermal noise on capacitors

Ideal
capacitors In electrical engineering, a capacitor is a device that stores electrical energy by accumulating electric charges on two closely spaced surfaces that are insulated from each other. The capacitor was originally known as the condenser, a term st ...
, 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 A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
) has what is called ''kTC'' noise. The equivalent noise bandwidth of an RC circuit is \Delta f \tfrac. 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 spectral density of the noise in the passband. The mean-square and RMS noise voltage generated in such a filter are: : \overline = = : V_\text = \sqrt = \sqrt. The noise charge Q_n is the
capacitance Capacitance is the ability of an object to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are two closely related ...
times the voltage: : Q_n = C \, V_n = C \sqrt = \sqrt : \overline = C^2 \, \overline = C^2 = k_\text 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:


Reset noise

An extreme case is the zero bandwidth limit called the reset noise left on a capacitor by opening an ideal
switch In electrical engineering, a switch is an electrical component that can disconnect or connect the conducting path in an electrical circuit, interrupting the electric current or diverting it from one conductor to another. The most common type o ...
. 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 In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
as given above. The reset noise of capacitive sensors is often a limiting noise source, for example in
image sensor An image sensor or imager is a sensor that detects and conveys information used to form an image. It does so by converting the variable attenuation of light waves (as they refraction, pass through or reflection (physics), reflect off objects) into s ...
s. Any system in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in t ...
has
state variable A state variable is one of the set of Variable (mathematics), variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behavi ...
s with a mean
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
of per degree of freedom. Using the formula for energy on a capacitor (''E'' = ''CV''2), mean noise energy on a capacitor can be seen to also be ''C'' = . Thermal noise on a capacitor can be derived from this relationship, without consideration of resistance.


Thermometry

The Johnson–Nyquist noise has applications in precision measurements, in which it is typically called "Johnson noise thermometry". For example, the
NIST The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical s ...
in 2017 used the Johnson noise thermometry to measure the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
with uncertainty less than 3 ppm. It accomplished this by using
Josephson voltage standard A Josephson voltage standard is a complex system that uses a superconducting integrated circuit chip operating at a temperature of 4 K to generate stable voltages that depend only on an applied frequency and fundamental constants. It is an int ...
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. 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.


Thermal noise on inductors

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: : \overline = \, .


Maximum transfer of noise power

The noise generated at a resistor ''R_\text'' can transfer to the remaining circuit. The maximum power transfer happens when the Thévenin equivalent resistance R_ of the remaining circuit matches ''R_\text''. 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_\text = k_\text \,T \Delta f \, . This maximum is independent of the resistance and is called the ''available noise power'' from a resistor.


Available noise power in decibel-milliwatts

Signal power is often measured in dBm (
decibels The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a ...
relative to 1 milliwatt). Available noise power would thus be 10\ \log_(\tfrac) in dBm. At room temperature (300 K), the available noise power can be easily approximated as 10\ \log_(\Delta f) - 173.8 in dBm for a bandwidth in hertz. Some example available noise power in dBm are tabulated below:


Nyquist's derivation of ideal resistor noise

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 Maxwell may refer to: People * Maxwell (surname), including a list of people and fictional characters with the name ** James Clerk Maxwell, mathematician and physicist * Justice Maxwell (disambiguation) * Maxwell baronets, in the Baronetage of N ...
to explain Johnson's experimental result. Nyquist's
thought experiment A thought experiment is an imaginary scenario that is meant to elucidate or test an argument or theory. It is often an experiment that would be hard, impossible, or unethical to actually perform. It can also be an abstract hypothetical that is ...
summed the energy contribution of each standing wave mode of oscillation on a long lossless
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
between two equal resistors (R_1 R_2). According to the conclusion of Figure 5, the total average power transferred over bandwidth \Delta f from R_1 and absorbed by R_2 was determined to be: : \overline = k_ T \, \Delta f \, . Simple application of
Ohm's law Ohm's law states that the electric current through a Electrical conductor, conductor between two Node (circuits), points is directly Proportionality (mathematics), proportional to the voltage across the two points. Introducing the constant of ...
says the current from V_1 (the thermal voltage noise of only R_1) through the combined resistance is I_1 \tfrac \tfrac, so the power transferred from R_1 to R_2 is the square of this current multiplied by R_2, which simplifies to: : P_\text = I_1^2 R_2 = I_1^2 R_1 = \left( \frac \right)^2 R_1 = \frac \, . Setting this P_\text equal to the earlier average power expression \overline allows solving for the average of V_1^2 over that bandwidth: : \overline = 4 k_\text T \, \Delta f \, . Nyquist used similar reasoning to provide a generalized expression that applies to non-equal and complex impedances too. And while Nyquist above used k_ T according to classical theory, Nyquist concluded his paper by attempting to use a more involved expression that incorporated the
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
h (from the new theory of
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
).


Generalized forms

The 4 k_\text 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 Passive may refer to: * Passive voice, a grammatical voice common in many languages, see also Pseudopassive * Passive language, a language from which an interpreter works * Passivity (behavior), the condition of submitting to the influence of ...
and linear.


Complex impedances

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 In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of Electrical_resistance, resistance and Electrical_reactance, reactance in a electrical circuit, circuit. Quantitatively, the impedan ...
Z(f). The formula for the power spectral density of the series noise voltage is : S_(f) = 4 k_\text T \eta(f) \operatorname (f) The function \eta(f) is approximately 1, except at very high frequencies or near absolute zero (see below). The real part of impedance, \operatorname (f)/math>, 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_\text = \sqrt. Alternatively, a parallel noise current can be used to describe Johnson noise, its power spectral density being : S_(f) = 4 k_\text T \eta(f) \operatorname (f) where Y(f) \tfrac is the electrical admittance; note that \operatorname (f) \tfrac \, .


Quantum effects at high frequencies or low temperatures

With proper consideration of quantum effects (which are relevant for very high frequencies or very low temperatures near
absolute zero Absolute zero is the lowest possible temperature, a state at which a system's internal energy, and in ideal cases entropy, reach their minimum values. The absolute zero is defined as 0 K on the Kelvin scale, equivalent to −273.15 ° ...
), the multiplying factor \eta(f) mentioned earlier is in general given by: :\eta(f) = \frac+\frac \frac \, . At very high frequencies (f \gtrsim \tfrac), the spectral density S_(f) now 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 \eta(f)=1 for conventional electronics work.


Relation to Planck's law

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. In other words, a hot resistor will create electromagnetic waves on a
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
just as a hot object will create electromagnetic waves in free space. In 1946, Robert H. Dicke elaborated on the relationship, and further connected it to properties of antennas, particularly the fact that the average antenna aperture over all different directions cannot be larger than \tfrac, where λ is wavelength. This comes from the different frequency dependence of 3D versus 1D Planck's law.


Multiport electrical networks

Richard Q. Twiss extended Nyquist's formulas to multi-
port A port is a maritime facility comprising one or more wharves or loading areas, where ships load and discharge cargo and passengers. Although usually situated on a sea coast or estuary, ports can also be found far inland, such as Hamburg, Manch ...
passive electrical networks, including non-reciprocal devices such as circulators and isolators. 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_(f) = 2 k_\text T \eta(f) (Z_(f) + Z_(f)^*) where the Z_ are the elements of the impedance matrix \mathbf. 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_(f) = 2 k_\text T \eta(f) (Y_(f) + Y_(f)^*) where \mathbf = \mathbf^ is the admittance matrix.


Notes


See also

* Fluctuation-dissipation theorem *
Shot noise Shot noise or Poisson noise is a type of noise which can be modeled by a Poisson process. In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where s ...
*
Pink noise Pink noise, noise, fractional noise or fractal noise is a signal (information theory), signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequenc ...
*
Langevin equation In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Lange ...
* Rise over thermal


References


External links


Amplifier noise in RF systems

Thermal noise (undergraduate) with detailed math
{{DEFAULTSORT:Johnson-Nyquist noise Noise (electronics) Electrical engineering Electronic engineering Electrical parameters Radar signal processing