# Third-order intercept point

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In telecommunications, a third-order intercept point (IP3 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 low-order polynomial, derived by means of Taylor series expansion. The third-order intercept point relates nonlinear products caused by the third-order nonlinear term to the linearly amplified signal, in contrast to the second-order 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 n-th 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 n-th order intermodulation products then appear at n times the frequency spacing of the input tones. This two-tone 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 log10 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 xn translates into a straight line with slope of n. Therefore, the linearly amplified signal will exhibit a slope of 1. A third-order 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 third-order 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 small-signal gain of the device.

### Practical considerations

The concept of intercept point is based on the assumption of a weakly nonlinear system, meaning that higher-order 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 device-under-test.