# 2.9 IP2 and IP3: Second- and Third-Order Nonlinearities

## 2.9.1 Harmonics (Single-Tone Test)

If a monochromatic signal is fed into a nonlinear system, the output is composed of the input frequency (fundamental) and harmonics that are integer multiples of the fundamental frequency, as depicted in Figure 2.48. By only considering second- and third-order nonlinearities of the system, the output of a nonlinear block can be modeled as a simple polynomial function:

2.152

where α_{1} is the small-signal gain and α_{2} and α_{3} are positive factors related to the second- and third-order nonlinearities, respectively.

The negative term in front of α_{3} is due to the fact that the third-order term compresses the gain of the fundamental and to avoid any confusion arising from the assumption α_{3} < 0.

If *x*(*t*) = *A* cos(ω_{1}*t*), we can develop Equation 2.152:

2.153

in which we find a DC component, the input frequency called the fundamental with amplitude H1, and the second- and third-order terms called the “harmonics” with amplitudes H2 and H3, respectively.

As the input signal amplitude increases, the output level of the fundamental ...