Noise Figure One and Two, Friis and IEEE

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New for February 2019: This page was written by John W McCorkle, who has also written extensively elsewhere, including in peer-reviewed IEEE journals, and who holds many patents. Thanks, John, we really appreciate your expertise and contribution!

There are two entirely different definitions for noise-figure in common use, the original from Friis [ [1] , [2] , [3] ], and the other from the IEEE [ [4] ]. By Friis's definition, noise figure (NF) and noise factor (F) are measures of degradation of the signal-to-noise ratio ( SNR), between the input and output of a component or an entire signal chain. F is the ratio of input to output SNR. NF is the number of dB that the SNR has dropped by. By the IEEE's definition, F and NF are not SNR measures, but are measures that are nonlinearly proportional to the noise temperature of a port on a device. They are numbers by which the noise made by an antenna, noise source, amplifier, or a radio receiver can be specified, which can equally be specified with an effective noise temperature. Each definition must be used consistently with its definition. The values cannot be interchanged―the IEEE NF cannot be substituted into Friis's NF = SNR_in-dB−SNR _out-dB formula.

Contents

Common Nomenclature. 1

First Definition, From Friis. 2

Second Definition, From IEEE.. 3

Mathematical Language Rigor 4

In Practice, Does The Difference In Definitions Matter?. 5

Key Formulas. 7

T_e For A Signal Chain. 7

T_e For An Attenuator Or Cable. 8

Example Usage. 8

Rules For Using The Two Definitions. 9

Antenna Noise Figure. 10

Summary. 11

Appendix 1 - A Walk Through Of The IEEE-Standard Source Document 11

Appendix 2 - Definition of underlying terms. 12

What is equivalent noise bandwidth B.. 12

SNR Terms. 12

What Is T_s 13

What Is Effective Input Noise Temperature T_e 13

Common Nomenclature

In both definitions, the nomenclature used is F for "noise factor", which is related to power, and NF for "noise figure", which expresses F in dB, so,

Noise Figure = NF = 10 log₁₀(F) and

Noise Factor = F = 10^NF/10

A few terms are required to clearly state and understand the noise-figure definitions. The terms are: (1) an equivalent noise bandwidth (B) of a block, (2) terms for signals, noise and signal-to-noise ratio (SNR ), (3) an effective noise temperature of a source (T_s) ("s" for source) that drives a block's input (which is defined by the system within which a block is used), and (4) an effective input noise temperature (T_e) ("e" for effective) of a block. Appendix 2 defines these. If they are familiar, great; if not, jump to Appendix 2 when you need clarification.

First Definition, From Friis

In 1942 [1,2] Friis, working at Bell Labs in Holmdel NJ, invented the concept of, and terms, "noise-figure" and "noise-factor". Reference [1] is the original and authoritative source the definition of noise-figure. The Friis definition of Noise Figure (NF) will be referred to asNF_gen ("gen" for general). Fundamentally, Friis's NF_gen "noise-figure" is a function of two variables (two temperatures, T_s and T_e) that is a dB metric that answers the question, "If I connect a particular block (like a single transistor amplifier, or an entire complex receiver chain) to a source (like an antenna) that is putting out noise power in addition to the signal, how much worse (lower in dB) will the block's output SNR be, relative to the SNR going into the block? Or in other words, how much worse (lower in dB) will the output SNR be from this particular block relative to an identical but perfectly noiseless block? Friis's NF_gen "noise-figure" is a system specific metric (i.e. it depends on the application and the SNR coming into that system) allowing different amplifiers and different receiver configurations to be compared based on a metric that captures exactly how many dB the SNR drops by in that system with that incoming SNR.

The function of two variables Friis developed is,

f (T_e,T_s) = 1 +  T_e/T_s

such that,

Friis's Noise Factor = F_gen =f(T_e,T_s)  = 1 + T_e/T_s = SNR_in/SNR_out = (s_in/(kT_sB)) /  SNR_out

Noise Factor F_gen is a ratio of power-ratios, and represents the factor by which the SNR is reduced across a device. Since F_gen is a power ratio (not a voltage ratio), the conversion to dB is 10 log (not 20 log), and therefore Friis's noise-figure NF_gen can be stated these ways,

NF_gen  = 10 log₁₀(F_gen)                                    _{_}

=  10 log₁₀ (SNR_in/SNR_out)

=  SNR_in-dB −  SNR_{out-dB    _}

Solving for T_e, when using Friis's F_gen definition, we get,

T_e  = (F_gen − 1) T_s

Below is a pictorial illustrating the derivation of Friis's NF and SNR formula. It is important to grasp that these formulas do not allow one to set T_s to a particular temperature, like 290°K on one side of the equation (i.e. on the F=1+T_e/T_s side), and at the same time let SNR_in or SNR_out on the other side of the equation be variables that are allowed to take on an arbitrary values- i.e. values based on a different T_s. T_s is on both sides of the equation and is buried within both the SNR _in term and the SNR_out term. All the terms must stay consistent.

Friis's f(T_e,T_s) function allows engineers to calculate the impact (or degradation) that an arbitrary block has on the signal-to-noise ratio (SNR) of a signal passing through it. NF_gen cannot be applied to a receiver alone or an amplifier alone because the SNR degradation across it depends on what the amplifier or receiver is connected to―i.e. how noisy the source is. In order to compute "SNR reduction", the "noise-figure" function must be a function of the two variables T_s, and T_e. From these inputs, Friis's noise-figureNF_gen characterizes how much a block, as used in the system it is connected to, degrades the signal-to-noise ratio (SNR). The same cannot be said of the second definition.

Second Definition, From IEEE

On June 11, 1959, the IRE published a standard [4] for measuring and marking devices like amplifiers, with a "noise figure" (which we will call NF_std), even though the amplifier was not connected to, or being used in, a system where T_s is known and could be factored in. On January 1, 1963, the AIEE and the IRE merged to form the Institute of Electrical and Electronics Engineers (IEEE), and this second definition for "noise-figure" remains an IEEE standard. Reference [4] remains the authoritative source document for the IEEE's noise-figure definition or standard. This IEEE "noise-figure", NF_std, is the number you will find listed on a receiver's or amplifier's specification sheet.

Fundamentally, rather than a function of two variables, F_std and NF_std are functions of only one variable, a single temperature in degrees K, which may be either an effective input noise temperature, or an effective output noise temperature. The function is,

h (T_e) = 1 + T_e/ 290°K = F_std.

To get back to an effective noise temperature with the above IEEE definition,

T_e  = (F_std + 1) 290°K.

Appendix 1 walks through the source document [4] to show how the function h(T_e) comes out of its language.

A common use for this IEEE "noise-figure" number, is for it to be listed on a data sheet for, or stamped on, a block like an amplifier, or a receiver, where its effective input noise temperature is simply converted to an IEEE "noise-figure" NF_std number. Another common use is for it to be stamped on, or listed on a specification sheet for, a 1-port device, like an antenna or a noise source that only has an output. In this case, the effective output noise temperature is simply converted to an IEEE "noise-figure" NF_std number.

The Friis definition of noise-figure NF_gen cannot apply to 1-port devices, because it captures the difference, in dB, between an input SNR and an output SNR, which requires the device being characterized to have an input and an output―as is shown in the above pictorial derivation. But the IEEE definition of "noise-figure" NF_std can be applied to a 1-port networks because the IEEE's noise-figure does not represent an SNR reduction. Instead, the IEEE definition simply defines a forward and backward transform of a temperature. It can be applied to blocks with one port (like a noise source or antenna) or blocks with any number of ports, where each port can have its own NF_std "noise-figure".

The IEEE's noise-figure NF_std carries no more information than simply a temperature―as is evident in the defining function, where T_e is its lone argument. Laboratory measurements of an amplifier's effective input noise temperature T_e and the amplifier's NF_std use the exact same steps, apart from the final step of using the h(T_e) function to convert T_e to F_std and NF_std.

Clearly the functions f and h are not the same―i.e. the two "noise-figures" are not the same. Functionally, and in general,

h (T_e) ≠ f(T_e, T_s)

Unlike the first definition from Friis, the IEEE "noise-figure" NF_std definition has nothing to do with SNR or a ratio of SNR's. NF_std is simply a number that is monotonically representative of a noise temperature. It is simply a non-linear mapping function or transform that changes or warps an effective input noise temperature of a device, (T_e, with units of temperature in degrees K), into other numbers that have no fixed meaning relative to an arbitrary system's SNR. The values are not a dB value representing a ratio of SNR power ratios―which is what NF_gen is. A 1 dB change in NF_std carries no quantitative meaning regarding how much an amplifier would change the SNR coming out of an arbitrary system―i.e. it does not mean there is a 1 dB change in SNR like NF_gen does. This fact will be illustrated by way of example later.

On systems that happen to have a source with a temperature ofT_s = 290°K, the IEEE's non-linearh(T_e) mapping makes the values of F_std and F_gen equal at that singular source temperature T_s = 290°K point. But the formula, or fundamental equation, h(T_e ) = 1 + T_e/290°K  = F_std has nothing to do with SNR reduction. In general,

F_std  ≠ F_gen

and as for formulas,

F_gen  = SNR_in/SNR_out,

but

F_std  ≠ SNR_in/SNR_out.

Mathematical Language Rigor

The fact that F_std ≠  SNR_in/SNR_out may surprise many readers given the existence of voluminous literature that begin by saying, correctly, that F_gen = SNR_in/SNR_out and then go on to erroneously sayF_std =  SNR_in/SNR_out.  Let me try to illustrate the problem of starting IEEE noise-figure descriptions with the erroneousF_std =  SNR_in/SNR_out formula by way of an example. Suppose T_F and T_C are temperatures in Fahrenheit and Celsius respectively. Mathematical formulas like T_F = 1.8T_C  − 32 or F_gen =  SNR_in/SNR_out do not just imply, but are always and inherently understood to mean that the terms can take on arbitrary values and moreover, that slopes and curvatures and partial derivatives associated with the various terms are accurate. Operations must be applied to both sides of the equations to keep them valid. These concepts are fundamental to mathematical language.

To start a description of IEEE standard noise figure by saying

F_std = SNR_in/SNR_out (which is wrong)

and then qualifying that equation by saying that n_in is equal to kT_refB where T_ref  = 290°K, is like starting an explanation of the Fahrenheit and Centigrade temperature scales by saying

T_F = T_C (which is wrong)

and then adding qualifying words that T_C  = −40 degrees―where the qualifying words, in both cases, is supposed to make the erroneous equation be ok. Regardless of the qualifying words, the formula T_F =  T_C is simply not a valid mathematical formula to convert these temperatures and will most definitely lead users of the formula to make erroneous derivations and calculations.

Similarly, by fundamental mathematical language, just like the formulaT_F = T_C is wrong, the formulaF_std =  SNR_in/SNR_out is also wrong. Neither of these formulas work when the formula's terms take on arbitrary values. Both are only true at a single point. They express a single unique condition or solution and are not a general formula. Application notes, book chapters, and literature that imply the erroneous F_std =  SNR_in/SNR_out formula in a general formula context has lead many trained engineers and physicists as well as laymen to create and use derivative formulas that are erroneous―simply because they assumed that fundamental mathematical language was operative.F_std ≠ SNR_in/SNR_out because the two definitions,f(T_e, T_s) and h( T_e), are fundamentally different and cannot be interchanged.

In Practice, Does The Difference In Definitions Matter?

Yes―the difference matters in two ways. First, it matters when the wrong formula is used to derive erroneous derivative formulas. Second, the values themselves can be significantly different in practice.

The text in [4], immediately following its definition of NF (quoted above), understates the discrepancy between the original Friis NF_gen and the new NF_std definition when it states,

"The standard noise temperature 290°K approximates the actual noise temperature of most input terminations. An alternative but related measure of performance useful for very low-noise transducers designed to operate from input terminations with noise temperatures substantially below 290°K is the "effective input noise temperature.""

Let's look at the claim that implies NF_std (which definesT_s to be 290°K regardless of reality)approximates NF_gen (where T_s is the actual source temperature). In practice, in applications where T_s is far from the IEEE's 290° K the IEEE standard NF stamped on a receiver or an amplifier, or calculated for a component like a lossy cable, is not even close to being a metric of SNR reduction. In low noise applications like satellite communications, radio astronomy, and applications using tiny "active antennas" where the tiny antenna puts out very little energy, a 0.1 dB change in the IEEE noise figure (NF_std) can result in over 10 dB of change in SNR. That is far from being approximately equal. It is not just low temperatures that are a problem, as suggested in this text in [4], it is a high temperature problem as well. In high temperature applications, like short wave antennas operating at low single digit MHz frequencies, antenna noise temperatures, like from a ¼ wave monopole, exceed 1 million degrees. In this case a 10 dB change in the IEEE noise figure (NF_std) can result in less than 0.1dB change in the actual system's SNR. That too is far from being approximately equal.

In the above practical examples, in one case F_std<<SNR_in/SNR_out i.e. F_std<<F_gen, while in another case, just the reverse, F_std>>SNR_in/SNR_out i.e. F_std>>F_gen. Based on these real-world examples, the first sentence in the above statement from [4] is shown to be far from true for many systems. One cannot simply assume or assert that NF_std "approximates" NF_gen. Often, NF_std and NF_gen are in fact, dramatically different.

Moreover, the text does not highlight or communicate the importance of the fact that while the Friis definition is a function that is fundamentally and exactly, an SNR degradation, contrarily, the new definition is based on a function that is fundamentally related to noise power alone and not SNR.

Continuing reading in the text from [4] above, we have,

"An alternative but related measure of performance useful for very low-noise transducers designed to operate from input terminations with noise temperatures substantially below 290°K is the "effective input noise temperature.""

which acknowledges that there is a problem, suggests that the problem matters, and offers a way to deal with the problem. It gives the solution when it says, "An alternative but related measure … is the effective input noise temperature." Indeed, this assertion takes the reader back to the original Friis definition for noise-figure that had been in use for 15 years, and back to the "Friis formula for noise temperature" (described below). This sentence from [4] tells a careful reader that to get accurate SNR-reduction values, they should make their noise calculations based on the actual source temperature T_s and the T_e's and G's of the components in the signal chain. The text is telling readers that to guarantee accuracy, either use T_e directly, or convert NF_std back to an effective noise temperature T_e, and then make all SNR calculations using the "Friis formula for noise temperature" and use Friis's noise-figure NF_gen formula, which uses T_s and T_e and calculates the SNR reduction.

Key Formulas

T_e For A Signal Chain

The total effective input noise T_e from a series of cascaded devices with gains G_n and effective noise input temperatures of T_n is easily derived as it is simply the solution to an equation for the noise coming out of a string of amplifiers, where the left side of the equation has a single effective noise temperature for the chain, and the right side has the individual temperatures of each stage in the chain, and the gains on the two sides match. Pictorially, it looks like this,

The equation to be solved is:

T_eG ₁ G ₂ G ₃ ∙∙∙ = [(T₁G₁+T ₂)G₂+T₂]G₃ ∙∙∙

The solution is called the "Friis formula  for noise temperature", and is,

T_e = T ₁ + T₂/G₁ + T₃/(G₁G₂) + T₄/( G₁G₂G₃) + ∙∙∙ + T_n/(G₁G₂G₃∙∙∙G _n-1)

This formula is completely general and applies to all blocks and all systems. The terms are simple and well defined. Importantly, there are no hidden assumptions―i.e. it is risk free, it simply always works. The SNR_dB loss metric (i.e. Friis's NF_gen defined noise-figure) can be computed by using the result of this formula ( T_e for the system) together with the actual source temperature T_s.

Easily derived from the above Friis formula for noise temperature, there is also the "Friis formula for noise factor" , which is,

F_total = F₁ + (F₂−1)/G₁ + (F₃−1)/(G₁G₂) + (F₄−1)/(G ₁G₂G₃) + ∙∙∙ + (F_n-1−1)/(G₁G₂G₃∙∙∙ G_n-1)

Just be aware that this formula has a buried assumption that all the F_n have been computed using the same source noise temperature T_s. Because of its buried and potentially illegitimate assumption, the author recommends simply always using the risk free and completely general "Friis formula for noise temperature".

T_e For An Attenuator Or Cable

An attenuator stage, like a cable, with a physical temperature T_a (a for ambient) and a gain of G  = P_out/P_in (ratio of output power to input power), has an effective input noise temperature of,

T _e = T_a (1/G - 1).

Example Usage

We will evaluate a system with a series connected antenna, lossy cable, and receiver and look at the SNR impact of the components and from the system. Suppose we have a room temperature (290°K) cable with 0.4dB of loss. In this case, G = 10^{− 0.4/10} = 0.912, so (1/G  − 1) = 0.096 and the cable would have an effective temperature T_e = 290°K (0.096) = 28°K. As such, it would have and IEEE standard NF_std = 10 log₁₀(1+28/290)  = 0.4 dB. Note that NF_std equals the loss if and only if the cable (or attenuator) temperature equals the IEEE standard's 290°K.

Next we will use the "Friis formula for noise temperature" to look at combinations of components. Since the above 0.4 dB loss cable is the first component in the chain, T₁ = 28°K and all the other noise contributions are amplified by the cable loss 1/G ₁, or ~1.1―i.e. relatively speaking, the noise from other components is effectively raised up 10% due to the loss in the cable.

Suppose this cable is in front of a receiver, and the receiver has a T_e = 10°K, then the overall T_e is

T_e = T ₁ + T₂/G₁ = 28°K + 10° K/0.912 = 39°K

Assuming T_s is the noise temperature of the source for the system, the SNR degradation caused by the system is,

F_gen  = 1 + T_e/T_s

Continuing the example, suppose the small antenna in our system has an output noise temperature of T_s = 2° K, and it connects to the previous 0.4 dB loss room temperature cable, and then to the previous T_e = 10° K receiver. Since the system comprised of the cable and receiver has a combined T_e = 39°K, the total SNR degradation from the system would be computed as,

F_gen = 1 + T_e/T_s = 1 + 39/2 = 20.5, and

NF_gen = 10 log₁₀(F_gen) = 10 log₁₀(20.5)= 13.1 dB

Without the 0.4 dB loss cable, the SNR degradation from the T_e = 10°K receiver alone would be reduced to,

F_gen = 1 + T_e/T_s = 1 + 10/2 = 5, and

NF_gen = 10 log₁₀(F_gen) = 10 log₁₀(5) = 7 dB

which is 13.1 − 7 = 6.1 dB less degradation. Note that this 6.1 dB is the SNR reduction caused by adding the cable to the above system, and that this 6.1 dB loss is different from the loss in SNR across the cable alone, which, from the T_e  = 28°K cable, is,

F_gen = 1 + T_e/T_s = 1 + 28/2 = 15, and

NF_gen = 10 log₁₀(F_gen) = 10 log₁₀(15) = 11.8 dB

The effective noise temperature, T_co, at the cable output, of the T_e = 28°K cable is

T_co = G(T_s+T_e) = 0.912(2°K + 28° K) = 27.3°K

The additional SNR reduction caused by the T_e  = 10°K receiver is

F_gen = 1 + T_e/T_co = 1 + 10/27.3 = 1.37, and

NF_gen = 10 log₁₀(F_gen) = 10 log₁₀(1.37) = 1.3 dB

So the total SNR reduction from the cable (11.8 dB) and the receiver (1.3 dB) is

11.8 dB + 1.3 dB = 13.1 dB,

which matches the 13.1 dB we initially calculated for the entire system.

Takeaways from this example/illustration are:

a)      The 0.4 dB loss cable, with anNF_std = 0.4 dB, actually loses NF_gen = 11.8 dB of SNR between its input and its output in this example.

b)      From (a), NF_std is grossly erroneous as an SNR loss metric! It is off by 11.4dB! NF_std is tremendously misleading and of no value. Again, yes, the difference between these definitions matter.

c)      In this particular system, the cable causes an extra 6.1 dB of SNR loss relative to having no cable at all. If I worked hard to reduce the 0.4 dB cable loss in this system (i.e. same receiver), the best possible improvement I could make in the SNR is 6.1 dB, even though the initial SNR loss across the cable is 11.8 dB. In other words, the more I reduce the SNR loss in the cable, the more SNR loss there is in the receiver. Only by improving both the receiver and the cable can the SNR be improved by more than 6.1 dB.

d)      If I worked hard to improve the receiver in this system (i.e. with no cable change), the best possible improvement I could make in the SNR is only 1.3 dB. Only if the cable loss were reduced would the way be opened for a receiver improvement to make more than 1.3 dB of SNR improvement.

e)      Only NF_gen provided reliable and accurate information regarding each component's impact to the system's SNR. It exactly captured the SNR reductions caused by the components, and as a result, gave clear insight into which components matter most and how to improve the system.

Rules For Using The Two Definitions

Rule Summary: (1) convert all NF_std values a T_e and (2) then use Friis's formulas everywhere else for SNR calculations . Given the existence and common use of the two entirely different definitions for noise-figure, it is vital that the reader (a) keep this fact in mind when reading, and (b) keep values and equation derivations coordinated and consistent. This section gives two simple rules that if followed, will always result in sound calculations. You will ace every test. You will notice these two rules were followed in the above example.

Rule-1 , Only use the IEEE standard to do the only thing it can do, to go back and forth between a temperature and an F_std or NF_std value―and nothing else

Anywhere you have a component with an NF_std , convert it back to a T_e. And otherwise do not use NF_std for anything else.

Never use NF_std directly in any SNR calculation or to represent an SNR reduction.

Rule-2 , Base all SNR degradation calculations on (a) the actual input source noise (T_s) for the system being analyzed, (b) the component noise temperatures (T_e's) and gains (G 's), and (c) Friis's function to compute NF_gen. In other words, Rule-2 is, always start with temperatures (which is what you have if you have followed Rule-1), if necessary use the "Friis formula for noise temperature" to find the combined T_e for a block to be assessed, then use the system T_e, plus theT_s driving the block, with Friis's NF_gen formula, to compute SNR reductions.

Corollary - Always follow the rules. Do not be tempted to use special cases or non-general formulas. To reduce risk of errors, it is much safer to never depend on special cases but instead, simply always use the same reliable and generic steps captured in the 2 rules―(1) convert all NF_std values a T_e and (2) then use Friis's formulas everywhere.

Antenna Noise Figure

Antennas sometimes come with a noise figure specification. It is easy to see how an antenna can have an effective output noise temperature T_out that would be the source temperature to a receiver connected to the antenna. Whatever noise an antenna picks up appears as some power coming out of its terminals. We can measure the noise coming out of the antenna through a system's equivalent noise bandwidth B and compute an effective noise temperature T_out such that k T_out B produces the measured noise power. But how can an antenna have a classical "Friis" noise figure, i.e. an SNR_in-dB − SNR_out-dB, or NF_gen, since:

(1)   it only has an output;

(2)   it does not have a unitless power-ratio gain G that is required by the "Friis formula for noise temperature" (It is not like an amplifier where the gain G is a unitless power ratio―such as 10 watts out per 1 watt in making G=10. An antenna has watts out, per watts/m ² power density impinging upon it, which means the gain number G needed, has units of m²―which is not a simple unitless number); and

(3)   It does not have a single gain number. (It does not have the same gain all directions. The gain number varies depending on where the signal is coming from, so there cannot be a single SNR reduction metric that applies to the antenna).

Excellent question. The answer is that it cannot have a classical (Friis)NF_gen noise figure, but it can have an IEEE NF_std "noise-figure" because the IEEE standard has nothing to do with an SNR and it also applies to 1-port networks. The antenna's output noise temperature T_out is simply converted to an IEEE noise factor and noise figure with the IEEE standard's equations:

F_std = 1 + T_out/290°K  and

NF_std = 10 log₁₀(F_std)

Given an antenna's NF_std, a system SNR analysis process starts in the standard way, by converting the antenna's NF_std back to an effective noise temperature, and then doing all the SNR calculations based on based on Friis's formulas―i.e. the antenna's effective output temperature T_out and the T_e's and gains of the signal chain components. In this case, the antenna's T_out simply becomes the source noise to the signal chain―i.e. T_s = T_out.

Summary

In 1959 the IRE standards group created an opposing definition for the term noise-figure, a term which was invented by Friis and universally understood to be his SNR metric since its publication in 1944. Errors continue to be made due to having two different definitions in common use at the same time. Given that we have and must live with two different definitions for Noise Figure, the best way to deal with this situation is to clearly illuminate and teach the differences so that usage is kept consistent with each definition. This article is intended to serve this purpose.

The two noise-figure metrics were precisely defined, and rules were given to make sure each metric was used consistently with its definition. An example case was given to illustrate the proper use of both definitions and how different their resulting numbers are. Hopefully this article will help electronics enthusiasts, engineers, and physicists to properly use both definitions and keep their use consistent with their definition.

An effective input noise temperature can be stamped on an amplifier or receiver, or listed on its data sheet to give the exact same information as stamping or listing the IEEE's NF_std on them. As such, nothing is gained by using the IEEE's noise-figure. On the other hand, much is lost by the IEEE's standard―an unambiguous, unconfused, unique, definition―the historical Friis definition that defines noise figure to be the SNR reduction across a block as used in an arbitrary system, which is not duplicated by any other metric or standard.

Appendix 1 ― A Walk Through Of The IEEE-Standard Source Document

Reference [4] says,

"The noise factor, at a specified input frequency, is defined as the ratio of 1) the total noise power per unit bandwidth at a corresponding output frequency available at the output port when the noise temperature of the input termination is standard (290°K) to 2) that portion of 1) engendered at the input frequency by the input termination."

The use of the terms "output frequency" and "input frequency" are there to allow blocks to use frequency conversion stages like mixers.

The "noise temperature of the input termination" is the source noise T_s applied to (or going into) the input, and is defined to be T_s = 290°K. Literature sometimes refers to this as the reference temperature, or T_ref = 290°K.

Assuming the block has a gain of G, and an effective input noise temperature of T_e, the "total noise power per unit bandwidth…available at the output port" [i.e. part "1)…"] is G(kT_s + kT_e ) = G(k 290°K + k  T_e).

The ratio of "1)…" (i.e. the above) to "2)…" is,

[G(k 290°K + k  T_e)] / [Gk 290°K ] = 1+T_e/290°K

So the IEEE "noise-figure" definition is

F_std  = h(T_e)= 1+T_e /290°K

And the invers to find T_e is,

T_e  = (F_std − 1) 290° K

Appendix 2 ― Definition of underlying terms

Definitions of equivalent noise bandwidth (B) , SNR, effective noise temperature of a source (T_s), and effective input noise temperature (T_e) are provided here.

What is equivalent noise bandwidth B

The equivalent noise bandwidth B in Hz, of a block is defined as follows. First, the block has a transfer function of H(f), where the maximum value of H(f) is H(f) _max. This function might have a very steep skirts on a passband, or might be a slow rolloff 2-pole filter. Either way, there is a maximum value of linear power gain P_out(f) /P_in(f) which is G_max  = |H(f)|²_max. A block so defined has an equivalent noise bandwidth B, in Hz, of

B = (1/|H(f)|²_max)∫|H( f)|²df,

where the integration limits cover frequencies from 0 to ∞.

When wideband noise with a power of n Watts/Hz is applied to the input of this block, using this definition for the noise-bandwidth, the measured output noise power will follow the simple formula P_out = nBG_max.

Another way B is defined is as follows. Suppose we substitute for the above block, a "brick wall" vertical skirt filter having a frequency range f covering (f₀ −  B/2) < f  < (f₀ + B/2)―i.e. a bandwidth that is the same as the noise-bandwidth of the above block. And suppose within this frequency range, the gain of this substituted filter is a flat G_max (i.e. the same as the above block), and the gain is zero outside this range. When wideband noise with a power of n Watts/Hz is applied to the input of this substituted "brick-wall filter", the output noise power will be nBG_max, same as the block above.

SNR Terms

Standard terms representing signals and noise that will be used are

s_in , n_in, s_out, and n_out

which represent signal and noise powers going into an input, or coming out of an output, respectively, as limited by the equivalent noise bandwidth ( B) of the block.

Using these terms, the input and output signal to noise power ratios (SNR) will be

SNR_in = s_in/n_in, and SNR_out = s_out/n_out.

These power ratios are expressed in dB as,

SNR_in-dB  = 10 log₁₀(SNR_in )  and  SNR_out-dB  = 10 log₁₀(SNR_out).

What Is T_s

T_s is an effective source temperature. The concept of an effective noise temperature is simply that the power of the noise going into or coming out of a circuit node, say P watts, with a noise-bandwidth of B, can be characterized as coming from heat quantified as an effective noise temperature T. In this case, the effective noise temperature T is simply

T  = P / kB

where k is Boltzmann's constant.

When the effective noise temperature T is known, the noise power, P watts, can be computed as

P  = kTB.

In any particular system, the energy from a source (like an antenna) driving a block will be some signal power s_in accompanied by some noise power n_in.  n_in is the noise power in watts coming out of the source, going into the input of the block. It is exactly like the P term above. The power of interest is limited by the noise-bandwidth B of the system, as that is the power that will pass through the system. As such, the source noise (e.g. noise coming out of the antenna) can be characterized by an effective noise temperature, just like the T above, but we call it T_s, where the subscript s is for "source". The noise power coming out of an antenna over a noise-bandwidth of B, might be n_in watts. If n_in and B are known, the effective source temperature T_s= n_in/(kB). To allow an analysis over arbitrary bandwidths, generally sources are characterized as having a source temperature T_s such that n_in can be calculated as

n_in  = kT_sB.

What Is Effective Input Noise Temperature T_e

The "effective input noise temperature" T_e of a block or device under test (DUT) is defined this way. Assume a noise free signal s_in going into the block and that we measure the output SNR of the block. It is the T_e that generates a noise power of kT_eB that when added to the input signals_in would make the input SNR, s_in/ kT_eB, equal to the measured output SNR. Effectively, all the noise sources that might be within the block are lumped together to form an equivalent "effective input noise" equal to kT_eB, where T_e is the "effective input noise temperature" of the block.

Another way of expressing this concept is to assume the energy driving the block's input is known and comprised of s_in and n_in. With these known quantities, the output SNR is measured. In this case, it is the temperature T_e such that the block's measured output SNR equals s_in/( n_in + kT_eB).  The effective input noise from the block, n_block =  kT_eB, is added to n_in so that the output SNR can be computed directly from the input signal level relative to the sum of the noise of the source plus the equivalent input noise of the block.