Talk:Third-order intercept point

I put this page in the category of amplifiers. It probably also belongs in a mixer category if there were one, but I couldn't find one and I can't justify creating the category for one article (two including the main Frequency mixer page). --Christaj 02:01, 6 September 2006 (UTC)

At the end of the text a small example is given:

" For example, assume a device with an input-referred 3rd order intercept point of 10 dBm is driven with a test signal of -5 dBm. This power is 15 dB below the intercept point, therefore nonlinear products will appear at approximately another 15 dB below the test signal power at the device output (in other words, 2*15 dB below the output-referred 3rd order intercept point). "

I think it's wrong: When the assumed device is driven with a signal of -5 dBm, the test signal at the output will appear 15 dB below the output-referred 3rd order intercept point (OIP3) and nonlinear (3rd order) products will appear at another 30 dB below the test signal (and 45 dB below the OIP3). Unless 2nd order nonlinear products are meant in the example, which would appear outside the frequency band of most narrow-band systems and would have no effect.

I have a question about this subject: can the 3rd order intercept point be used to specify the capability of the system to deal witch strong out of band signals?

IP3 comment
I also feel that the example given for tones 15dB below the intercept point is wrong for 3rd order products. In this case, the 3rd order distortion products will be 30dB down on the signal and 45dB below the intercept point.

Series expansion
According to the article, the nonlinearity is modeled using polynomial series expansion (Taylor series). I don't think this is a good model for a non-linearity. It works well when the signal gets smaller, but for growing input values, a polynomial grows without bounds. A real nonlinearity doesn't, but is rather bounded by the supply voltage or available current. There must be a better series expansion than a polynomial, that still fits with the IP3 concept.

A simple model for the nonlinearity of a bipolar differential stage is the tanh function. A Taylor series approximates tanh well only in a small range close to the origin, for a practical polynomial order. --HelgeStenstrom (talk) 10:52, 21 April 2008 (UTC)


 * The polynomial model allows one to easily work out the power of intermodulation products, and hence IP3. Oli Filth(talk) 11:10, 21 April 2008 (UTC)


 * Yes, it's easy to work out the IM products of the polynomial, but it's only a model, and perhaps not a very good one, in particular at high amplitude levels, such as close to compression. Real amplifiers and mixers don't have a measured IP3 that is independent on the power level, like a polynomial does. --HelgeStenstrom (talk) 08:18, 30 April 2008 (UTC)

TOI Equation
I'm no expert in RF stuff, but should the equation "V^2 = (4G)/(3D)" really be "V^2 = (4G)/(D)"? The article also points out that the TOI is roughly 4 times the ratio of the gain over the third distortion term. Secondly if you take into account the "(3/4)D(V^3)" term in the linear coefficient, my math seems to show the TOI being approximately "V^2 = 2G/D". Any comments? Guerberj (talk) 03:59, 25 March 2009 (UTC)

Calculation Error
I have copied/pasted the following from the article page. Talk pages exist for exactly this kind of discussion - please argue it out here, and then correct the main article if that is required. GyroMagician (talk) 10:38, 13 July 2012 (UTC)

Although the result of this derivation is correct, the mathematical derivation is WRONG. TOI has to be calculated from the third-order INTERMODULATION between two sinusoidal inputs. Hence, a correct drivation is as follows:
 * $$\ s(t) = V \cos(\omega1 t)+V \cos(\omega2 t)$$
 * $$\ O[s(t)] = G s(t) - D_3 s^3(t) + \ldots$$

Calculate now O(s) and you find that there are responses at frequencies omega1, omega2, 3*omega1, 3*omega2, 2*omega1+omega2, 2*omega2+omega1, 2*omega1-omega2 and 2*omega2-omega1. The response at frequency 2*omega1-omega2 (and also a 2*omega2-omega1) is proportional to V^3, while the fundamental responses at frequencies omega1 and omega2 are mainly proportional to V^1. The (extrapolated) value of V where these two coincide, is defined as the TOI. The calculations reveal that this value is given by:
 * $$\ V^2 = \frac{4 G}{3 D_3}$$

Conclusion: the presented result is correct, but the derivation is completely wrong.


 * REBUTTAL TO THIS STATEMENT:
 * It is not a theoretical or mathematical requirement to use a two tone intermodulation test to determine the intercept point of a non-linear device. The single tone derivation as presented is perfectly correct - perhaps it should be extended to the multiple tone case for clarity.  The use of the two-tone test is more one of measurement practicality, particularly where the third harmonic would be at a frequency outside the range of the measurement equipment used.  For example if the IP3 of a 10GHz amplifier would require a spectrum analyser capable of 30GHz if a single tone were to be used.  The use of the two tone test places a third (and 5th, 7th ..) order product distanced only by the chosen tone separation.  The intrinsic non-linearity of a device is not influenced by the number of tones presented to it, so it comes as no surprise that the same result is obtained.
 * It is not a theoretical or mathematical requirement to use a two tone intermodulation test to determine the intercept point of a non-linear device. The single tone derivation as presented is perfectly correct - perhaps it should be extended to the multiple tone case for clarity.  The use of the two-tone test is more one of measurement practicality, particularly where the third harmonic would be at a frequency outside the range of the measurement equipment used.  For example if the IP3 of a 10GHz amplifier would require a spectrum analyser capable of 30GHz if a single tone were to be used.  The use of the two tone test places a third (and 5th, 7th ..) order product distanced only by the chosen tone separation.  The intrinsic non-linearity of a device is not influenced by the number of tones presented to it, so it comes as no surprise that the same result is obtained.

It's not just the measurement equipment! Often we are talking about narrow-band amplifiers, which will not have any useful response at the third harmonic. Given 2 tones within their bandwidth, 3rd order intermods are generated at 2f1 +/- f2 and 2f2 +/- f1. Two of these are close to the carriers f1 and f2. Or for old-fashioned analogue TV we had 3 tone tests, one for the video carrier, one colour subcarrier, and the sound carrier. — Preceding unsigned comment added by 81.143.19.9 (talk) 15:28, 13 December 2012 (UTC)

While the effect is similar, the 2f1-f2 type products grow 3 times faster than the 3f1 or 3f2 terms in the expansion of (cos(w1*t) + cos(w2*t))^3. The coefficients follow the familiar binomial expansion of 1 3 3 1. All third order cosine results have the 1/4 multiplier shown in the article, but the 2f1-f2 type terms also get multiplied by 3. Since these are the fastest growing products, they are of most interest and hence are used for the TOI definition.Quantumken (talk) 01:01, 27 February 2021 (UTC)

Error in TOI Calculation
The article makes a error in calculating the TOI by mistakenly equating the two terms multiplying $$ \cos( \omega t )$$.

Instead of equating $$ G V = \frac{3}{4}D_3 V^3$$ where both sides come from the term at the original frequency, ω, it should equate be $$ G V = D_3 \frac{V^3}{4}$$. Here the left side comes from the term at the original frequency, ω and the right side comes from the term at the harmonic frequency, 3ω.

Then $$V^2 = \frac{4G}{D_3}$$ rather than $$V^2 = \frac{4 G}{3 D_3}$$.

Intuitively, this determines the point where the small signal gain transfer function $$ \ O_1 (V) = G V $$ intercepts the transfer function of the third order term $$ \ O_3(V) = D_3 \frac{V^3}{4} $$.