 Thirdorder intercept point

In telecommunications, a thirdorder intercept point (IP_{3} or TOI) is a measure for weakly nonlinear systems and devices, for example receivers, linear amplifiers and mixers. It is based on the idea that the device nonlinearity can be modeled using a loworder polynomial, derived by means of Taylor series expansion. The thirdorder intercept point relates nonlinear products caused by the thirdorder nonlinear term to the linearly amplified signal, in contrast to the secondorder intercept point that uses second order terms.
The intercept point is a purely mathematical concept, and does not correspond to a practically occurring physical power level. In many cases, it lies beyond the damage threshold of the device.
Contents
Definitions
Two different definitions for intercept points are in use:
 Based on harmonics: The device is tested using a single input tone. The nonlinear products caused by nth order nonlinearity appear at n times the frequency of the input tone.
 Based on intermodulation products: The device is fed with two sine tones with a small frequency difference. The nth order intermodulation products then appear at n times the frequency spacing of the input tones. This twotone approach has the advantage that it is not restricted to broadband devices and is commonly used for radio receivers.
It is worth noticing that these definitions differ by 4.8 dB (10 log_{10} 3), so care should be taken when using existing equations, models or measurement data.
The intercept point is obtained graphically by plotting the output power versus the input power both on logarithmic scales (e.g., decibels). Two curves are drawn; one for the linearly amplified signal at an input tone frequency, one for a nonlinear product. On a logarithmic scale, the function x^{n} translates into a straight line with slope of n. Therefore, the linearly amplified signal will exhibit a slope of 1. A thirdorder nonlinear product will increase by 3 dB in power when the input power is raised by 1 dB.
Both curves are extended with straight lines of slope 1 and n (3 for a thirdorder intercept point). The point where the curves intersect is the intercept point. It can be read off from the input or output power axis, leading to input or output intercept point, respectively (IIP3/OIP3).
Input and output intercept point differ by the smallsignal gain of the device.
Practical considerations
The concept of intercept point is based on the assumption of a weakly nonlinear system, meaning that higherorder nonlinear terms are small enough to be negligible. In practice, the weakly nonlinear assumption may not hold for the upper end of the input power range, be it during measurement or during use of the amplifier. As a consequence, measured or simulated data will deviate from the ideal slope of n. The intercept point according to its basic definition should be determined by drawing the straight lines with slope 1 and n through the measured data at the smallest possible power level (possibly limited towards lower power levels by instrument or device noise). It is a frequent mistake to derive intercept points by either changing the slope of the straight lines, or fitting them to points measured at too high a power level. In certain situations such a measure can be useful, but it is not an intercept point according to definition. Its value depends on the measurement conditions that need to be documented, whereas the IP according to definition is mostly unambiguous; although there is some dependency on frequency and tone spacing, depending on the physics of the deviceundertest.
One of the useful applications of third order intercept point is as a ruleofthumb measure to estimate nonlinear products. It can be seen that the spacing between two straight lines with slopes of 3 and 1 closes with slope 2.
For example, assume a device with an inputreferred thirdorder 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 2x15 dB below the test signal power at the device output (in other words, 3×15 dB below the outputreferred thirdorder intercept point).
A ruleofthumb that holds for many linear radio frequency amplifiers is that the 1 dB compression point falls approximately 10 dB below the thirdorder intercept point.
Theory
The thirdorder intercept point (TOI) is a property of the device transfer function O (see diagram). This transfer function relates the output signal voltage level to the input signal voltage level. We assume a “linear” device having a transfer function whose small signal form may be expressed in terms of a power series containing only odd terms, making the transfer function an odd function of input signal voltage, i.e., O[−s(t)] = −O[s(t)]. Where the signals passing through the actual device are modulated sinusoidal voltage waveforms (e.g., RF amplifier), device nonlinearities can be expressed in terms of how they affect individual sinusoidal signal components. For example, say the input voltage signal is the sine wave
and the device transfer function produces an output of the form
where G is the amplifier gain and D_{3} is cubic distortion. We may substitute the first equation into the second and, using the trigonometric identity
we obtain the device output voltage waveform as
The output waveform contains the original waveform, cos(ωt), plus a new harmonic term, cos(3ωt), the thirdorder. The coefficient of the cos(ωt) harmonic has two terms, one that varies linearly with V and one that varies with the cube of V. In fact, the coefficient of cos(ωt) has nearly the same form as the transfer function, except for the factor ¾ on the cubic term. In other words, as signal level V is increased, the level of the cos(ωt) term in the output eventually levels off, similar to how the transfer function levels off. Of course, the coefficients of the higherorder harmonics will increase (with increasing V) as the coefficient of the cos(ωt) term levels off (the power has to go somewhere).
If we now restrict our attention to that portion of the cos(ωt) coefficient which varies linearly with V, and then ask ourselves, at what input voltage level, V, will the coefficients of the first and third order terms have equal magnitudes (i.e., where the magnitudes intersect), we find that this happens when
which is the ThirdOrder Intercept Point (TOI). So, we see that the TOI input power level is simply 4/3 times the ratio of the gain and the cubic distortion term in the device transfer function. The smaller the cubic term is in relation to the gain, the more linear the device is and the higher the TOI is, which clearly makes sense. The TOI, being related to the magnitude squared of the input voltage waveform, is a power quantity, typically measured in milliwatts (mW). The TOI is always beyond operational power levels because the output power saturates before reaching this level.
The TOI is closely related to the amplifier's "1 dB compression point," which is defined as that point at which the total coefficient of the cos(ωt) term is 1 dB below the linear portion of that coefficient. We can relate the 1 dB compression point to the TOI as follows. Since 1 dB = 20 log_{10} 1.122, we may say, in a voltage sense, that the 1dB compression point occurs when
or
or
In a power sense (V^{2} is a power quantity), a factor of 0.10875 corresponds to −9.636 dB, so by this approximate analysis, the 1 dB compression point occurs roughly 9.6 dB below the TOI.
Recall: decibel figure = 10 dB × log_{10}(power ratio) = 20 dB × log_{10}(voltage ratio).
Notes
 The thirdorder intercept point is an extrapolated convergence – not directly measurable – of intermodulation distortion products in the desired output.
 It indicates how well a receiver performs in the presence of strong nearby signals.
 It is sometimes used (interchangeably with the 1 dB compression point) to define the upper limit of the dynamic range of an amplifier.
 Determination of a thirdorder intercept point of a superheterodyne receiver is accomplished by using two test frequencies that fall within the first intermediate frequency mixer passband. Usually, the test frequencies are about 20 to 30 kHz apart.
 The concept of intercept point has no meaning for strongly nonlinear systems, such as when an output signal is clipped due to limited supply voltage.
See also
 Intermodulation Intercept Point
 Secondorder intercept point
References
 This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C" (in support of MILSTD188).
 Understanding RF Power Amplifiers
 The Relationship of Intercept Points and Composite Distortions
Categories: Amplifiers
Wikimedia Foundation. 2010.
Look at other dictionaries:
Secondorder intercept point — The Second Order Intercept Point, also known as the SOI, IP2, or IIP2 (Input Intercept Point), is a measure of linearity that quantifies the second order distortion generated by nonlinear systems and devices. Examples of frequently used devices… … Wikipedia
Intercept Point — Der Intercept Point, kurz IP (dt. Schnittpunkt) stammt aus der elektronischen Messtechnik und stellt eine nicht direkt messbare Größe zur Charakterisierung der nichtlinearen Eigenschaften von Übertragungsgliedern (z. B. Verstärkern) der… … Deutsch Wikipedia
Third AngloDutch War — Infobox Military Conflict conflict=Third Anglo Dutch War partof=Franco Dutch War caption= The Battle of Texel, 11 ndash;21 August 1673 by Willem van de Velde, the younger, painted 1683, depicts a battle of the Third Anglo Dutch War date=1672 1674 … Wikipedia
Point of No Return (Babylon 5) — Infobox Television episode Title = Point of No Return Series = Babylon 5 Caption = Season = 3 Episode = 9 Airdate = 26 February 1996 Production = 309 Writer = J. Michael Straczynski Director = Jim Johnston Guests = Majel Barrett (Lady Morella)… … Wikipedia
Order of battle in the Atlantic campaign of 1806 — The Battle of San Domingo, 6th February 1806, with H.M.S. Canopus Joining the Action, Thomas Lyde Hornbrook The Atlantic campaign of 1806 was one of the most important and complex naval campaigns of the post Tra … Wikipedia
New World Order (professional wrestling) — New World Order Stable Members See below Name(s) New World Order nWo nWo Hollywood nWo Wolfpac The Band … Wikipedia
Intermodulation — A frequency spectrum plot showing intermodulation between two injected signals at 270 and 275 MHz (the large spikes). Visible intermodulation products are seen as small spurs at 280 MHz and 265 MHz. Intermodulation or intermodulation distortion… … Wikipedia
IIP3 — Der Intercept Point, kurz IP (dt. Schnittpunkt) stammt aus der elektronischen Messtechnik und stellt eine nicht direkt messbare Größe zur Charakterisierung der nichtlinearen Eigenschaften von Übertragungsgliedern (z. B. Verstärkern) der… … Deutsch Wikipedia
IP2 — Der Intercept Point, kurz IP (dt. Schnittpunkt) stammt aus der elektronischen Messtechnik und stellt eine nicht direkt messbare Größe zur Charakterisierung der nichtlinearen Eigenschaften von Übertragungsgliedern (z. B. Verstärkern) der… … Deutsch Wikipedia
IP5 — Der Intercept Point, kurz IP (dt. Schnittpunkt) stammt aus der elektronischen Messtechnik und stellt eine nicht direkt messbare Größe zur Charakterisierung der nichtlinearen Eigenschaften von Übertragungsgliedern (z. B. Verstärkern) der… … Deutsch Wikipedia