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Noise temperature
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In electronics, noise temperature is one way of expressing the level of available noise power introduced by a component or source. The power spectral density of the noise is expressed in terms of the temperature (in kelvins) that would produce that level of Johnson–Nyquist noise, thus:
where:
- is the noise power (in W, watts)
- is the total bandwidth (Hz, hertz) over which that noise power is measured
- is the Boltzmann constant (1.381×10−23 J/K, joules per kelvin)
- is the noise temperature (K, kelvin)
Thus the noise temperature is proportional to the power spectral density of the noise, . That is the power that would be absorbed from the component or source by a matched load. Noise temperature is generally a function of frequency, unlike that of an ideal resistor which is simply equal to the actual temperature of the resistor at all frequencies.
Noise voltage and current
[edit]A noisy component may be modelled as a noiseless component in series with a noisy voltage source producing a voltage of vn, or as a noiseless component in parallel with a noisy current source producing a current of in. This equivalent voltage or current corresponds to the above power spectral density , and would have a mean squared amplitude over a bandwidth B of:
where R is the resistive part of the component's impedance or G is the conductance (real part) of the component's admittance. Speaking of noise temperature therefore offers a fair comparison between components having different impedances rather than specifying the noise voltage and qualifying that number by mentioning the component's resistance. It is also more accessible than speaking of the noise's power spectral density (in watts per hertz) since it is expressed as an ordinary temperature which can be compared to the noise level of an ideal resistor at room temperature (290 K).
Note that one can only speak of the noise temperature of a component or source whose impedance has a substantial (and measurable) resistive component. Thus it does not make sense to talk about the noise temperature of a capacitor or of a voltage source. The noise temperature of an amplifier refers to the noise that would be added at the amplifier's input (relative to the input impedance of the amplifier) in order to account for the added noise observed following amplification.
System noise temperature
[edit]An RF receiver system is typically made up of an antenna and a receiver, and the transmission line(s) that connect the two together. Each of these is a source of additive noise. The additive noise in a receiving system can be of thermal origin (thermal noise) or can be from other external or internal noise-generating processes. The contributions of all noise sources are typically lumped together and regarded as a level of thermal noise. The noise power spectral density generated by any source () can be described by assigning to the noise a temperature as defined above:[1]
In an RF receiver, the overall system noise temperature equals the sum of the effective noise temperature of the receiver and transmission lines and that of the antenna.[2]
The antenna noise temperature gives the noise power seen at the output of the antenna. The composite noise temperature of the receiver and transmission line losses represents the noise contribution of the rest of the receiver system. It is calculated as the effective noise that would be present at the antenna input terminals if the receiver system were perfect and created no noise. In other words, it is a cascaded system of amplifiers and losses where the internal noise temperatures are referred to the antenna input terminals. Thus, the summation of these two noise temperatures represents the noise input to a "perfect" receiver system.
Noise factor and noise figure
[edit]One use of noise temperature is in the definition of a system's noise factor or noise figure. The noise factor specifies the increase in noise power (referred to the input of an amplifier) due to a component or system when its input noise temperature is .
is customarily taken to be room temperature, 290 K.
The noise factor (a linear term) is more often expressed as the noise figure (in decibels) using the conversion:
The noise figure can also be seen as the decrease in signal-to-noise ratio (SNR) caused by passing a signal through a system if the original signal had a noise temperature of 290 K. This is a common way of expressing the noise contributed by a radio frequency amplifier regardless of the amplifier's gain. For instance, assume an amplifier has a noise temperature 870 K and thus a noise figure of 6 dB. If that amplifier is used to amplify a source having a noise temperature of about room temperature (290 K), as many sources do, then the insertion of that amplifier would reduce the SNR of a signal by 6 dB. This simple relationship is frequently applicable where the source's noise is of thermal origin since a passive transducer will often have a noise temperature similar to 290 K.
However, in many cases the input source's noise temperature is much higher, such as an antenna at lower frequencies where atmospheric noise dominates. Then there will be little degradation of the SNR. On the other hand, a good satellite dish looking through the atmosphere into space (so that it sees a much lower noise temperature) would have the SNR of a signal degraded by more than 6 dB. In those cases a reference to the amplifier's noise temperature itself, rather than the noise figure defined according to room temperature, is more appropriate.
Effective noise temperature
[edit]The noise temperature of an amplifier is commonly measured using the Y-factor method. If there are multiple amplifiers in cascade, the noise temperature of the cascade can be calculated using the Friis equation:[3]
where
- = resulting noise temperature referred to the input
- = noise temperature of the first component in the cascade
- = noise temperature of the second component in the cascade
- = noise temperature of the third component in the cascade
- = power gain of the first component in the cascade
- = power gain of the second component in the cascade
Therefore, the amplifier chain can be modelled as a black box having a gain of and a noise figure given by . In the usual case where the gains of the amplifier's stages are much greater than one, then it can be seen that the noise temperatures of the earlier stages have a much greater influence on the resulting noise temperature than those later in the chain. One can appreciate that the noise introduced by the first stage, for instance, is amplified by all of the stages whereas the noise introduced by later stages undergoes lesser amplification. Another way of looking at it is that the signal applied to a later stage already has a high noise level, due to amplification of noise by the previous stages, so that the noise contribution of that stage to that already amplified signal is of less significance.
This explains why the quality of a preamplifier or RF amplifier is of particular importance in an amplifier chain. In most cases only the noise figure of the first stage need be considered. However one must check that the noise figure of the second stage is not so high (or that the gain of the first stage is so low) that there is SNR degradation due to the second stage anyway. That will be a concern if the noise figure of the first stage plus that stage's gain (in decibels) is not much greater than the noise figure of the second stage.
One corollary of the Friis equation is that an attenuator prior to the first amplifier will degrade the noise figure due to the amplifier. For instance, if stage 1 represents a 6 dB attenuator so that , then . Effectively the noise temperature of the amplifier has been quadrupled, in addition to the (smaller) contribution due to the attenuator itself (usually room temperature if the attenuator is composed of resistors). An antenna with poor efficiency is an example of this principle, where would represent the antenna's efficiency.
See also
[edit]References
[edit]- ^ Skolnik, Merrill I., Radar Handbook (2nd Edition). McGraw-Hill, 1990. ISBN 978-0-07-057913-2
- ^ Skolnik, Merrill I. (2001). Introduction to Radar Systems (Third ed.). Boston: McGraw-Hill. pp. 731–732. ISBN 0-07-290980-3.
- ^ McClaning, Kevin, and Tom Vito. Radio Receiver Design. Atlanta, GA: Noble Publishing Corporation, 2000. ISBN 1-884932-07-X.
Noise temperature
View on GrokipediaBasic Concepts
Definition
Noise temperature is defined as the temperature at which a resistor would produce the same available noise power as the actual noise source.[6] The available noise power is given by where is Boltzmann's constant, is the noise temperature, and is the bandwidth.[6] It is expressed in Kelvin (K) and is independent of the actual physical temperature of the device or system.[7] For a resistor at room temperature (290 K), the noise temperature equals the physical temperature under thermal equilibrium.[6] Noise temperature serves as a complementary metric to noise figure in characterizing noise performance.[7]Physical basis
Noise temperature originates from the thermodynamic principle that the thermal fluctuations in an electrical conductor produce noise power equivalent to that emitted by a blackbody radiator in the Rayleigh-Jeans limit of Planck's law, valid in the microwave frequency regime where the photon energy . In this classical approximation, the spectral radiance of blackbody radiation simplifies to , leading to an available noise power independent of frequency for a matched load. This equivalence allows noise in electronic systems to be characterized by an effective temperature, linking electrical engineering to fundamental thermodynamics.[8] The foundational mechanism is Johnson-Nyquist noise, resulting from the random thermal agitation of electrons in a resistor, akin to Brownian motion in a viscous medium. Harry Nyquist derived this in 1928 by applying the equipartition theorem to the degrees of freedom in an equivalent transmission line model of the resistor, ensuring each mode has average energy . The resulting open-circuit mean-square noise voltage is , where is the resistance, is Boltzmann's constant, is the physical temperature, and is the bandwidth; the corresponding available power delivered to a matched load is . This power spectral density remains flat (white noise) across frequencies in the classical regime, providing the basis for defining noise temperature as the temperature a resistor would need to match the observed noise power from any noisy source.[9] Quantum mechanically, the Nyquist formula receives corrections from the full Planck distribution via the Callen-Welton fluctuation-dissipation theorem, yielding the voltage noise spectral density . The term reduces to in the Rayleigh-Jeans limit, recovering the classical expression, while the term arises from zero-point energy—the ground-state fluctuations of quantum harmonic oscillators—and persists even at absolute zero, introducing a frequency-dependent component . These quantum effects become pronounced at high frequencies () or low temperatures, deviating from classical predictions by up to 50% or more.[10] In engineering contexts, the classical limit dominates due to the low frequencies and room temperatures typically encountered, but quantum contributions, particularly zero-point noise, are relevant as of 2025 in cryogenic systems for quantum computing, where millikelvin operation highlights these irreducible fluctuations, setting fundamental limits on amplifier noise and qubit readout fidelity in devices like Josephson parametric amplifiers.[11]Noise in Components
Noise voltage and current
In passive components such as resistors, noise manifests as random fluctuations in voltage and current, primarily due to thermal agitation of charge carriers. The root-mean-square (RMS) noise voltage across an open-circuited resistor of resistance at physical temperature , within bandwidth , is given by where is Boltzmann's constant. This expression arises from Nyquist's thermodynamic derivation, which equates the available noise power from the resistor to the blackbody radiation power in a one-dimensional transmission line mode. Specifically, the mean-square voltage fluctuation must satisfy the condition that the maximum power deliverable to a matched load is , independent of . For an open-circuit condition, this available power is , yielding .[12][13] Dually, the RMS noise current through a short-circuited resistor is This follows from the available power equivalence under short-circuit conditions, where , so . The invariance of the available noise power ensures that the noise temperature remains constant at regardless of termination (open or short), as the expressions for and are related by . This duality highlights the resistor's role as a noise source with equivalent voltage or current representations.[12][14] For a practical example, consider a 50 resistor at 290 K in a 1 GHz bandwidth. The RMS noise voltage is , while the RMS noise current is . These values demonstrate the scale of thermal fluctuations in typical RF components.[13] In real resistors, particularly those made from carbon or semiconductors, additional noise mechanisms contribute to voltage and current fluctuations beyond pure thermal noise. Shot noise arises from the discrete nature of charge carriers crossing potential barriers, such as in biased junctions within the resistor material, with RMS current , where is the elementary charge and is the average DC current (zero in unbiased passive resistors, but present if minute leakage occurs).[15][14] Flicker noise, or 1/f noise, introduces low-frequency excess fluctuations due to material defects or trapping-detrapping processes, with voltage spectral density and integrated mean-square voltage , often parameterized as for bias voltage and corner frequency . In unbiased resistors, this manifests as resistance fluctuations, elevating the total noise above thermal levels at frequencies below ~1 kHz. The contribution is equivalently modeled via an excess noise temperature where the additional power is , allowing the total effective noise temperature to represent the combined effects.[16][17]Thermal noise equivalence
The equivalence principle in noise temperature modeling states that any linear noisy two-port network, such as an amplifier or mixer, can be represented equivalently by an ideal noiseless two-port network augmented with a thermal noise source at an effective temperature , which accounts for all internal noise contributions referred to the input.[18] This representation simplifies analysis by transforming diverse noise mechanisms—thermal, shot, flicker, or others—into a single equivalent thermal source, facilitating cascade calculations and performance predictions in RF systems. The principle, foundational to modern noise theory, originates from the chain noise matrix formulation that decomposes the network's noise into input-referred uncorrelated voltage and current sources.[18] For passive attenuators, which introduce loss without gain, the output noise temperature is modeled as , where is the power loss factor, is the input noise temperature, and is the physical (ambient) temperature of the attenuator.[19] This equation arises from the attenuator's partial transmission of input noise power (scaled by ) combined with the thermal noise generated internally by its dissipative elements, which approaches as loss increases. At thermal equilibrium (), the output noise temperature equals , preserving detailed balance and preventing unphysical noise amplification.[19] In active devices like low-noise amplifiers (LNAs), excess noise beyond the physical temperature's thermal contribution is quantified by defining as the additional input-referred temperature required to match the observed output noise when the device is noiseless otherwise. For an amplifier at physical temperature , the total effective input noise temperature is , where captures non-thermal sources such as shot noise in transistors or generation-recombination processes; typically exceeds (often by factors of 10 or more in practical RF designs) due to these excess mechanisms. This approach enables direct comparison of device performance across technologies, independent of specific noise origins. In modern 5G and mm-wave systems operating above 20 GHz, non-thermal noise often dominates the effective temperature in active components, driven by high-frequency limitations like carrier transit times and avalanche effects in semiconductors. This dominance underscores the utility of the equivalence principle for optimizing noise budgets in bandwidth-constrained, high-data-rate links.System-Level Noise
System noise temperature
The system noise temperature, denoted , quantifies the total noise contribution in a radio receiving system as an equivalent temperature referred to the input, encompassing both external and internal sources. It is defined as , where is the antenna noise temperature from environmental sources and is the receiver's internal noise temperature. This parameter is crucial for assessing overall system performance, as it directly influences the signal-to-noise ratio at the receiver input.[20] In multi-stage receiver chains, such as those in communication links or radio telescopes, is aggregated using the Friis formula for cascaded noise temperatures: where is the noise temperature of the -th stage and is its available power gain (with ).[21] This formulation highlights the dominance of the first stage's noise, emphasizing the need for low-noise amplifiers early in the chain to minimize . The antenna noise temperature integrates contributions from sky noise (e.g., cosmic microwave background, typically 2.7 K at low frequencies but higher due to galactic emissions), ground noise (from thermal radiation of the Earth, often 200–300 K for low-elevation antennas), and atmospheric effects, weighted by the antenna's radiation pattern.[4] The system noise temperature sets fundamental sensitivity limits for radio receivers, where the minimum detectable signal power is approximately (with as Boltzmann's constant and as bandwidth), determining the threshold for weak signal detection in applications like deep-space communication or radio astronomy.[20] In modern phased array systems for satellite communications, correlated noise—arising from mutual coupling between elements and shared local oscillator phases—complicates calculations, potentially increasing effective noise beyond uncorrelated Friis predictions and requiring array-specific models for accurate assessment.[22][23]Effective noise temperature
The effective noise temperature of a device, such as an amplifier, represents the equivalent temperature of a noise-free source that would produce the same additional noise power at the output as the internal noise contributions of the device itself.[24] It is defined by the relation , where is the noise factor of the device and K is the standard reference temperature.[24] This metric isolates the device's intrinsic noise, independent of the source temperature, allowing for standardized comparisons across components.[25] In amplifiers, quantifies the noise added by the device when referred to its input, enabling the assessment of performance without reliance on external source conditions.[26] For instance, low-noise amplifiers (LNAs) used in radio astronomy or satellite communications often achieve K, such as 2 K at 4.5 GHz or 8 K at higher frequencies, minimizing degradation of weak input signals.[27] Unlike system noise temperature, which aggregates contributions across an entire receiver chain (e.g., antenna, amplifiers, and mixers), focuses solely on the isolated noise of a single device, facilitating targeted design optimizations.[28] Recent advancements in cryogenic cooling have further reduced in specialized applications, particularly quantum sensors where thermal noise must be suppressed to near quantum limits.[29] For example, cryogenic LNAs operating at temperatures around 4 K can attain as low as 3 K across 4–16 GHz, enhancing sensitivity in quantum readout circuits by minimizing added noise from active components.[30] In microwave quantum sensors, such cooling techniques have enabled effective noise temperatures below 6 K over wide bandwidths, supporting high-fidelity measurements in superconducting qubit systems since the early 2020s.[31]Performance Metrics
Noise factor
The noise factor , a dimensionless linear ratio greater than or equal to unity, quantifies the degradation in signal-to-noise ratio (SNR) caused by a device or system. It is formally defined according to IEEE standards as the ratio of the total available noise power per unit bandwidth at the output to the portion of that noise power attributable to thermal noise from the input termination at the standard temperature K, when the input is driven by a matched source. This can be expressed as , where is the total output noise power, is the available power gain, and is the input noise power with Boltzmann's constant and bandwidth . Equivalently, the noise factor relates to the effective input noise temperature by , highlighting its direct connection to noise temperature concepts. The choice of K as the reference temperature stems from its close approximation to typical room temperature conditions in laboratory environments, ensuring consistent and reproducible measurements across devices. This value also yields a convenient thermal noise power spectral density of dBm/Hz at room temperature, serving as a standard benchmark for noise floor calculations in RF systems. As a linear metric, the noise factor facilitates straightforward computations in multi-stage systems, particularly through the Friis cascade formula, which determines the overall noise factor of a series of stages as , where is the noise factor and is the available gain of the -th stage. This form underscores the importance of minimizing noise in early stages with high gain to optimize system performance. For instance, an ideal noiseless device exhibits , amplifying only the input signal and noise without addition; a practical low-noise amplifier with effectively doubles the output noise relative to the amplified input thermal noise alone. While broadband noise factor specifications integrate performance over a wide frequency range, the spot noise factor measures degradation at a specific frequency using a narrow bandwidth, making it essential for frequency-selective applications. In emerging 6G systems, which leverage narrowband channels at millimeter-wave and terahertz frequencies for high-capacity communications, spot noise factor evaluations are vital to ensure low degradation in targeted spectral bands where selectivity is paramount.Noise figure
The noise figure (NF) is defined as the logarithmic expression of the noise factor (F), quantifying the degradation of the signal-to-noise ratio in decibels:This metric standardizes noise performance relative to a reference temperature of 290 K, as per IEEE definitions.[32] A noise figure of 3 dB corresponds to a noise factor of 2, indicating that the device effectively doubles the input noise power. Similarly, NF values of 0 dB, 6 dB, and 10 dB equate to F = 1, 4, and 10, respectively, illustrating how the logarithmic scale compresses wide ranges of noise degradation for practical engineering use. The following table provides common conversions:
| Noise Figure (dB) | Noise Factor (F) |
|---|---|
| 0 | 1 |
| 1 | 1.26 |
| 3 | 2 |
| 6 | 4 |
| 10 | 10 |
| 20 | 100 |
where and are the measured output noise powers for the hot () and cold () states, respectively. The effective input noise temperature is then derived as
from which the noise figure follows as . This method achieves high accuracy for amplifiers and receivers across RF and microwave frequencies.[33] By 2025, vector network analyzers (VNAs) have become a standard tool for noise figure measurement, employing vector correction techniques to determine the full noise parameter matrix (four noise parameters) without requiring a separate noise source. These VNA-based methods, such as those using source pull corrections, enable precise noise figure extraction from S-parameter and noise figure sweeps, particularly for high-frequency devices up to 125 GHz, offering advantages in integration and reduced setup complexity over traditional Y-factor setups. For frequency-converting devices like mixers, noise figure is specified as either single-sideband (SSB) or double-sideband (DSB), reflecting whether noise from one or both sidebands contributes to the output. In DSB measurements, noise from both upper and lower sidebands folds into the intermediate frequency, typically yielding a 3 dB lower noise figure than the SSB case, where only one sideband is considered; this distinction is critical for receiver design, as SSB figures better represent image-rejecting applications.[34] The decibel scale of noise figure facilitates cascading analysis in multi-stage systems, where the total noise figure approximates the first stage's NF plus the second stage's NF minus the first stage's gain (all in dB), simplifying optimization of receiver chains without linear conversions.[3]