The point the device is never at

The distortion this field has measured so far has been harmonic. One tone in, and the question is how much of it comes back out at twice and three times the frequency. That is the right question for an amplifier whose signal occupies less than an octave, because everything it makes lands somewhere a filter can reach.

It is the wrong question for nearly everything else. Feed a device two tones a little way apart and the third-order terms produce components at $2f_1 - f_2$ and $2f_2 - f_1$, which are beside the tones rather than above them — inside whatever band the signal was in, at a spacing set by the spacing of the tones. No filter reaches them. They are what limits a receiver’s dynamic range, and they are invisible to every measurement in this field until now.

The number the world quotes for them is the third-order intercept, and it is a strange kind of number. It is not a measurement. It is where two straight lines would cross if both went on being straight, and the whole content of this essay is that neither does, that the crossing is therefore a place the device is never in, and that for a bounded device the distance from it is exactly computable.

Three for one, and where it stops being three #

The mechanism is one line of algebra. Any odd nonlinearity has a cubic term; a cubic term fed $\cos\omega_1 t + \cos\omega_2 t$ produces components at $2\omega_1 - \omega_2$ among others; and because it is cubic, the amplitude of that component is proportional to the cube of the drive. So on logarithmic axes the product rises three decibels for every one the fundamental rises, and the two lines have slopes 3 and 1.

Measured on a differential pair over a decade of drive from 0.01 to 0.1 millivolts, the fitted slopes are 1.0000 and 3.0000. That is not a coincidence and it is not luck about the fit: it is what the cubic term does when it is the only thing happening.

Where it stops being three is a boundary worth putting a number on, because it is the boundary that makes the intercept an extrapolation rather than a point. The local slope of the product against drive is 2.999 at one millivolt, 2.988 at three, 2.870 at ten, 2.160 at thirty and 1.487 at fifty. By a millivolt or two the assumption behind the intercept has already started to fail, and the intercept itself is at a hundred and three.

A product a harmonic measurement understates by 9.5 decibels #

Before the extrapolation, one comparison that costs nothing and settles a common confusion. The same drive that makes a third-order intermodulation product also makes a third harmonic, and the two are not the same size.

Measured on all three curves in this essay, at every drive inside the cubic region, the intermodulation product is three times the third harmonic — 3.000, 2.999, 3.000 — which is 9.54 decibels. The factor comes from counting: the cube of a sum of two cosines has three ways of producing $2\omega_1 - \omega_2$ where the cube of a single cosine has one way of producing $3\omega$.

So a bench measurement of third-harmonic distortion, taken honestly and reported honestly, understates the thing that will actually limit the circuit by nearly ten decibels. That is the practical reason two-tone testing exists, and it is a fact about arithmetic rather than about any device.

Where the products land, which is the whole problem #

It is worth being concrete about why these particular components matter, because the arithmetic that places them is short and is the entire practical argument.

Two tones at 100.0 and 100.1 megahertz produce second-order products at 0.1 and 200.1 megahertz and third-order products at 99.9 and 100.2. The second-order pair is a long way from the signal and a filter that passes the band at all will reject both. The third-order pair is a hundred kilohertz from the tones — inside the band by construction, because the spacing of the products is the spacing of the tones and nothing else.

That is why the odd order is the one that gets a specification. It is not that odd-order terms are larger; on a bare exponential the second harmonic is much larger than the third at every drive. It is that a filter can be built for one and cannot be built for the other, so the third-order coefficient of a device sitting in front of a receiver is a hard limit on what the receiver can do, and the second is a design detail.

The same counting says what happens with more than two tones, and it is worse in a way that is easy to underestimate. Every triple of tones contributes products at $f_i + f_j - f_k$, so the number of in-band products grows as the cube of the number of tones while each one stays the same size. A device carrying a hundred channels is carrying a noise-like floor made entirely of its own third-order term, and the intercept is still the number that predicts it.

Why the measurement needs no window #

The transform underneath every number here is taken on exactly one period of a record whose two tones sit at integer bin numbers — the sixtieth and the sixty-fifth of two thousand and forty-eight samples. That is a choice with consequences worth stating, because it is what makes the products readable at all.

If the tones are coherent with the record, every product is coherent with it too: $2m - n$ and $2n - m$ are integers when $m$ and $n$ are, so each product lands wholly in one bin with nothing leaking into its neighbours. No window is needed, and none is used. A window would spread each tone over several bins with a skirt that falls at some finite rate, and a product a hundred and twenty decibels below the tones would be sitting underneath that skirt rather than on the noise floor.

The alternative — incoherent tones plus a window — is what a spectrum analyser does, because it cannot choose the transmitter’s frequency. It costs about forty decibels of usable range and a good deal of argument about which window. Here the tones are chosen along with the record, so the coherent arrangement is free, and the products this essay measures reach $10^{-16}$ of the fundamental before the arithmetic gives out rather than the window.

The crossing, and the ceiling #

Extrapolate the two fitted lines and they meet. For a differential pair the crossing is at a drive of exactly four thermal voltages — 103.4 millivolts — and an output of exactly 2, in units where the pair’s tail current is 1 and its small-signal output is the drive over two thermal voltages.

Now ask what the device can actually do. A pair’s output is $\tanh$ of its drive, so at large drive it is a hard limiter, and two equal tones through a hard limiter come out with each tone at $8/\pi^2$ of full scale — 0.8106, the classical two-tone suppression result, and it is the largest fundamental this device produces at any drive whatever. Measured by scanning two thousand thermal voltages of drive: 0.8108.

So the intercept sits above the ceiling by

$$\frac{2}{8/\pi^2} = \frac{\pi^2}{4} = 2.4674$$

which is 7.845 decibels, exactly, with no tail current in it, no temperature in it and no fabrication in it. Both quantities scale with the tail current and the ratio does not. A specification that gives a third-order intercept for a differential pair is quoting a point 7.8 decibels outside the range of the thing it is specifying, and the amount by which it is outside is a property of the hyperbolic tangent.

The rule of thumb, and what it is worth #

Everybody who uses intercepts also uses a rule connecting the intercept to the one-decibel compression point: the intercept is about 9.6 decibels above it. The rule comes from assuming the device is exactly a cubic — that the same coefficient produces both the compression and the product — and it is worth checking against a device rather than against the assumption.

For the differential pair, the drive at which the fundamental has fallen one decibel below its small-signal value is 1.425 thermal voltages, and the intercept is at 4.000. The ratio is 8.963 decibels, against the rule’s 9.636. So the rule is 0.67 decibels optimistic on a $\tanh$, which is about as good as a rule of thumb gets and is a number rather than a hope.

For a bare exponential the rule has no meaning at all, and that is the more interesting failure. An exponential’s incremental gain rises with amplitude, because the curve is convex, so the fundamental never falls one decibel below its small-signal value at any drive; there is no compression point to be 9.6 decibels below. The rule silently assumes the compression is compression.

What degeneration does to the intercept #

A resistor in the emitter divides the drive that reaches the junction and linearises what the junction does with it, and the essay that measured those two effects found the distortion edge moving as the square of the degeneration factor. The intercept moves too, and by a different expression, which comes out of the same series reversion:

$$\hat v_{\text{IP3}} = 2V_T\sqrt{\frac{2D^4}{|3 - 2D|}}$$

Measured against a coherent two-tone transform of a Newton-solved curve, which shares no arithmetic with the reversion: 11.314 thermal voltages at $D = 2$ against a predicted 11.314; 33.940 at $D = 6$ against 33.941; 78.505 at $D = 11$ against 78.515.

The intercept that does not exist #

The $|3 - 2D|$ in the denominator is the interesting part, and it says something no measurement of a single device would suggest: at $D = 3/2$ the expression divides by zero.

That is not an artefact. The third-order coefficient of a degenerated exponential vanishes at exactly three halves, because the exponential’s own convexity and the local feedback’s compression are both second order in the drive and cancel there. With no cubic term there is no three-for-one line, so there is nothing for the one-for-one line to cross.

Measured, the fitted slope of the product against drive at $D = 3/2$ comes out 4.96 rather than 3 — the fifth-order term, which is what is left — and the extrapolation returns infinity. A specification sheet cannot print that, and a device sitting near that degeneration has an intercept that is enormous, fragile, and a strong function of a resistor ratio nobody thought was critical.

Why an extrapolated number is still worth having #

Nothing above is an argument for abandoning the intercept, and it is worth saying why not.

An intercept is a single number that summarises a whole curve, and it does so correctly wherever the curve is cubic — which is where a receiver’s small signals live, several tens of decibels below anything that compresses. Two devices with the same intercept really do produce the same product at the same small drive, whatever their ceilings are. The extrapolation is a way of naming the coefficient of the cubic term in units a systems engineer can add up, and cascading intercepts down a chain is the same arithmetic that cascades noise figures.

What it is not is a description of the device near its limits, and the two facts that make that concrete are on the plot above: the intercept is 7.8 decibels above anything the device can produce, and by a hundredth of the intercept drive the slope it assumes has already left.

The four amplitudes a device has #

An intercept is an extrapolation, and this field measures four amplitudes that are not. The distortion a linear model cannot have is where the harmonics come from and how fast they grow. What a pair cancels, and what it only halves removes the even ones exactly and leaves the odd ones, which is what an intercept is made of. What a resistor in the emitter buys moves both amplitude boundaries and the distance between them. The exponent that is a square is the same accounting on a device whose law is not an exponential, and A bias point is a solution, not a choice is where every one of these amplitudes is measured from. The intercept is above all four, which is why it is a figure of merit and not a specification.

What is checked #

Four assertions, and the third is the one that turns a rule of thumb into a number.

That the product rises three for one and the fundamental one for one over the decade of drive the fit is taken on, with the straightness of both fits reported rather than assumed — the fit window is chosen by the device rather than by the author, bisected to where the product reaches a stated fraction of the fundamental, because a heavily degenerated stage makes a product below the transform’s own arithmetic floor at the drives that suit a bare exponential.

That an intermodulation product is three times the harmonic the same drive makes, on every curve tested, which is the 9.54 decibels a one-tone measurement leaves out.

That a pair’s largest output at any drive at all is $8/\pi^2$ of its tail current, and that the extrapolated intercept sits $\pi^2/4$ above it — asserted as the ratio, so that a device model with a different saturation would fail it.

And that the degenerated stage’s intercept is $2V_T\sqrt{2D^4/|3-2D|}$, which the reversion gives and the transform confirms to four figures.

A specification at a place on no curve, twice more #

An intercept sitting 7.84 decibels above the largest output a device can ever produce is the sharpest instance in this collection of a specification defined by extrapolation, and it is not the only one.

The constant that is a window is a specification defined by a derivative rather than by an extrapolation, and it fails the same way: an ideality factor has a value at every current and no value anywhere, with eight one-decade fits to one curve returning factors from 1.23 to 1.98 and two of them straight to a few parts in a thousand, so the residual gives no warning at all.

A floor, or a line is a specification defined by a total, and two clocks with the same total put their error twenty-eight decibels apart.

The three between them describe how a number stops being a measurement. It can be taken at a point that does not exist, which is this essay; taken at a point that exists and is not stated, which is the ideality factor; or taken as a summary that several different objects share, which is the jitter figure. In all three the number is arithmetically impeccable and answers a question nobody asked.

What makes the third-order intercept the most defensible of the three is that its extrapolation is declared. Everybody who quotes one knows the lines do not go on being straight, which is more than can be said for a derivative quoted as a constant — and the reason it survives is that the comparison between two devices is meaningful even where the number itself is not, which is a legitimate use of a quantity that has no operating point.

That defence has a condition, and it is the one this essay’s second result supplies. A comparison between two extrapolated numbers is meaningful only if both were extrapolated from the same shape of curve. The degenerated stage’s intercept — $2V_T\sqrt{2D^4/|3-2D|}$ — has a different functional form from the bare pair’s, so comparing a degenerated device’s intercept with an undegenerated one’s is comparing two extrapolations of two different families, and the ratio between them is not the ratio of anything either device does.

Which is the practical caution to end on. Two intercepts quoted for two similar devices are comparable; an intercept quoted for a device and an intercept quoted for a stage built around one are not, and the difference between those two cases is exactly the difference between the two expressions where the two exponents come from derives.

Which is a small result and it is the one a specification actually rests on. An intercept is used to compare, and a comparison is valid when both quantities are the same function of the same variable. On this page the two are not, and the number that says so — $\pi^2/4$, containing no tail current and no temperature — is what makes the incompatibility exact rather than a matter of degree. A number that contains none of the parameters a designer varies is either a universal constant or an artefact of an extrapolation, and this one is the second.