The loss in front, counted twice

The floor a resistor sets established the one boundary in this collection that bounds a model from below. A kilohm at room temperature produces four nanovolts per root hertz, gain does not help because gain amplifies it too, and the quantity has no manufacturing in it — only a temperature and a resistance.

That essay is about a resistor. This one is about the sentence hidden inside it: only the real part of an impedance is warm. A reactance stores energy and does not dissipate it, so it has no fluctuation to go with a dissipation and it contributes no noise at all. Everything below follows from that one line, and one of the things that follows is a theorem with no fitting in it.

The theorem, and why it is not a fit #

Take any passive network at absolute temperature T, with an available power gain of G — which for a lossy thing is less than one, and L = 1/G is what a data sheet calls its insertion loss.

Its available output noise power is kTB. Not approximately: exactly, for any passive network at one temperature, whatever its topology. That is the fluctuation–dissipation statement, and the network’s own arrangement cannot change it, because whatever the arrangement dissipates it also generates.

Its available signal power at the output is G times what went in.

Noise figure is the ratio of input signal-to-noise to output signal-to-noise, and the source is already at kTB. So the numerator falls by G and the denominator does not move:

F = 1/G = L.

The noise figure of a passive loss is its loss, exactly, with no dependence on what kind of loss it is or how the network is built. A three-decibel attenuator has a three-decibel noise figure. So does three decibels of cable, three decibels of connector, three decibels of a switch’s on-resistance and three decibels of a transformer’s copper.

Sweep the slider and the arithmetic is exact at every point: nought point five decibels of cable gives one point eight one, one gives two point three one, four gives five point three one, eight gives nine point three one. Every decibel of loss in front is a decibel of noise figure, and it never interacts with anything else in the chain.

Twice, and both times in full #

The reason this matters more than the arithmetic suggests is that the two decibels are separate decibels.

The signal loss is obvious and everybody budgets for it: two decibels of cable means two decibels less signal at the amplifier, and a link budget has a line for it. What the same two decibels also do is raise the noise figure of everything behind them by two decibels — and Friis’s formula gives the first stage full weight, so there is no later stage that can recover it.

So the end-to-end penalty is four decibels of signal-to-noise, from a component whose specification says two.

That is the whole argument for putting the amplifier at the antenna rather than at the receiver, and the mismatch that the cable hides is the other half of the same observation: a lossy cable flatters a reflection measurement for exactly the reason it destroys a noise measurement, because attenuation is a two-way road that the signal only travels once.

The reactance that costs nothing #

Now put a lossless network in the same place.

A capacitive divider, an LC filter made of good components, an ideal transformer — each of them attenuates. A ten-to-one capacitive divider is twenty decibels of attenuation. And each of them adds no noise whatever, because there is no real part to be warm.

The catch, and it is not a small one: the source resistance the amplifier sees is transformed too. A lossless network cannot add noise and it can certainly move the operating point on the noise-versus-source-resistance curve that the floor a circuit has measures — which has a minimum at 6.67 kilohms for that amplifier and rises on both sides of it.

So the honest comparison is not “resistive attenuator bad, reactive attenuator free”. It is: a resistive attenuator costs its loss in noise figure and moves the source impedance; a reactive one costs only the second. That is usually a large win and it is never nothing.

The condition the theorem depends on #

F = L holds at the temperature the reference is defined at, and that condition is doing real work.

The available noise power out is kTB where T is the network’s own temperature. Noise figure is referred to 290 kelvin by convention. So a passive loss at 290 kelvin has a noise figure equal to its loss; a passive loss at 77 kelvin has a noise figure below its loss, and one in an oven has a noise figure above it.

That is not a curiosity. It is why the first components of a radio-astronomy front end are cooled, and it is why the same three-decibel attenuator can have a noise temperature of 290 kelvin on a bench and 77 in a dewar while its insertion loss has not changed by a thousandth of a decibel.

The general form is Tₑ = (L − 1)·Tₚₕᵧₛ, and setting Tₚₕᵧₛ = 290 recovers F = L. Everything on the slider above is that expression at room temperature.

Two routes to the same decibel #

The identity above is derived, and this collection’s habit is not to trust a derivation that has not been measured against something that shares no arithmetic with it. There are two independent routes to the same number and both are worth stating, because they fail differently.

The available-power route is the one above: the output noise is kTB by fluctuation–dissipation, the signal is attenuated by G, and the ratio of ratios is 1/G. It never mentions a resistor. It is a statement about a two-port and it holds for a network of a hundred elements as readily as for one.

The element route is to build the attenuator out of actual resistors, give each one its own 4kTR generator, solve the network at each generator in turn, and sum the powers at the output. That is what the floor a resistor sets does for a single network, and it produces the same answer with a great deal more work and one extra thing: it says which resistor contributed what.

The two agreeing is the check. Where they would disagree is instructive: the element route knows nothing about matching, so if the attenuator is not terminated in its design impedance the available-power argument’s G is no longer the insertion loss, and the identity F = L stops being about the number on the label. That is the same distinction the mismatch that the cable hides turns on, arriving from the other direction.

Where the loss usually is #

Three places, and none of them looks like an attenuator.

A connector and its cable is the obvious one, and it is the one people budget for. A filter in front of the amplifier is not: a band-pass filter placed ahead of the first stage to keep strong out-of-band signals out has an insertion loss in its pass band, and that loss is added to the noise figure of everything after it. The trade is real and it goes both ways — what the filter in front costs prices the other side of it — but the noise cost is exact and computable rather than a matter of judgement.

And a switch or a relay is the one that is forgotten entirely. Half an ohm of contact resistance in a fifty-ohm system is nought point nought four decibels and nobody cares; the same half ohm in series with a high-impedance source is not a loss at all in the same sense, because the network is no longer matched and the available-power argument has to be redone from the impedances.

Why the second decibel is the one that gets lost #

There is a documentation reason this mistake survives, and it is worth naming because it is not about electronics.

Insertion loss and noise figure live in different tables. Insertion loss is on the passive component’s data sheet, in decibels, beside its return loss and its power rating. Noise figure is on the amplifier’s data sheet, in decibels, beside its gain and its input intercept point. A link budget is assembled by adding up the first column and taking the noise figure from the row that has one — and every row without a noise figure column is silently treated as having a noise figure of zero.

The correct entry for a passive row is not zero and it is not blank. It is the same number that is already in the loss column, written again.

That is the whole of the fix, and stating it as a habit is more use than stating it as a theorem: a passive component’s noise figure equals its insertion loss, so copy the number across. The identity is exact, it needs no measurement, and it takes a column.

What this does not say #

It does not say a lossy element anywhere costs its loss in noise figure. Friis weights each stage by the gain in front of it, so a two-decibel loss after forty decibels of gain contributes essentially nothing — that is what the floor a circuit has measures when it reorders three stages and finds 8.70 decibels sitting in the ordering alone. The figure at the top of this page holds that gap fixed at 8.70 while the cable slides from nought to eight, which is the two effects being independent, demonstrated rather than asserted.

It also does not say anything about bandwidth. Noise figure is a ratio at a frequency, and the noise a receiver actually collects is that figure integrated over the bandwidth it happens to have — which the bandwidth noise sees shows is not the corner frequency and is π/2 times it for a single pole.

The transformer, which is the interesting case #

A transformer is the one passive element where the two halves of this argument meet, and it is worth a section because the answer is not what the turns ratio suggests.

An ideal transformer is lossless and it changes the source impedance by the square of its turns ratio. So it is the pure reactive case: no noise added, and the amplifier moved along its noise-versus-source-resistance curve — which is exactly why input transformers exist in front of microphone preamplifiers and moving-coil cartridges. A two-hundred-ohm source into an amplifier whose optimum is six and a half kilohms is transformed there by a ratio of about six, and the signal-to-noise improves without anything being amplified.

A real transformer has copper resistance in both windings and core loss in between, and both are real parts and therefore warm. The resistance that grows with frequency measures the first and shows it is not the direct-current value at any frequency worth using; the second is what the area a curve cannot have turned from an omission into a measured loop area.

So the design trade is: a step-up ratio buys a better position on the amplifier’s noise curve and costs the winding’s own loss in noise figure, and the optimum ratio is not the one that lands on the amplifier’s optimum source resistance. It is somewhat below it, because the loss needed to build a large ratio grows faster than the improvement it buys.

The number worth carrying #

One decibel per decibel, exactly, with no coefficient.

It is a rare shape of result in this collection: almost everything here is a measurement with a range attached, and this is an identity that holds for every topology, every impedance and every kind of dissipation, subject to one condition about temperature that is usually satisfied and occasionally is the whole design.

The practical form of it is a habit rather than a number. When a passive component goes in front of an amplifier, its insertion loss appears twice in the link budget, and the second appearance is in a different table from the first — which is exactly why it is the one that gets left out.

Which components in this collection are in front of something #

The habit is only useful if it is clear what counts as being in front, and several arrangements elsewhere on this site are exactly this one without being described as it.

What the filter in front costs is the clearest and the one where the arithmetic has consequences: an anti-alias filter is a passive network between a source and a converter, so whatever it dissipates in its passband is charged twice — once as signal and once as noise figure — before the converter’s own floor is reached. That essay prices the filter in clock rate and this one adds the term it does not carry.

The two resistors a ladder was designed between is where the loss is deliberate and large. A doubly terminated ladder’s passband loss is six decibels by construction, and half of that is a real resistor at the source end fitted to make the termination correct. Six decibels of insertion loss in front of an amplifier is six decibels of noise figure by this essay’s identity — so the arrangement that buys the best tolerance sensitivity in the filters field is the worst one available in the noise budget, and neither essay can settle the trade alone.

And the reactive exception matters most where it is easiest to forget. The q the components allow is what decides whether a matching or filtering network in front of an amplifier is lossless enough to be free: the reciprocals of the component quality factors add and the total sits below the smallest of them, so a network built around one mediocre inductor is a lossy network wherever that inductor’s current flows, and the decibels it loses are charged twice like any others.

The temperature clause, which is occasionally the whole design #

The identity carries one condition — that the lossy element is at the same temperature as the reference the noise figure is defined against — and it is worth saying what happens when it is not, because the exception is not exotic.

A lossy element that is cold contributes less noise than its loss implies, which is the whole basis of cryogenic front ends: the signal loss is unchanged and the noise contribution falls in proportion to the absolute temperature. A lossy element that is hot contributes more, and the ordinary case of that is an element heating itself. The loss that depends on what it causes is where this collection measures a component whose losses set its own temperature, finding a stable fixed point at 89 degrees for the case drawn — which is a hundred and twenty kelvin above ambient, and is a noise contribution 1.4 times what the same loss at room temperature would give.

Which is the shape the clause takes in practice. The identity is exact and its condition is a measurement, so a link budget written with insertion losses and no temperatures is right for everything at ambient and understates the contribution of anything that is not. The reactive exception is the clean case at the other end: a lossless network contributes nothing whatever its temperature, because only the real part of an impedance is warm.

Which is why a transformer in front of an amplifier is the one component that can improve a noise figure rather than spend it. It is very nearly lossless, so it costs almost nothing by this essay’s identity, and it transforms the source impedance — which the floor a circuit has shows is the quantity with an optimum, 6.67 kΩ for an ordinary part, being the ratio of the device’s two noise generators. A ratio a transformer can supply, for the price of a component that is not warm.