When two or more tones are present in a nonlinear device, intermodulation products are created as a result. A
power series describes all of the possible combinations of generated frequencies. 3rd-order products lie near in
frequency to the two input tones and are therefore very likely to fall inband at the output. As a device is driven
farther into its nonlinear region, the amplitudes of the third order products increase while the powers of the input
tones decrease. If the device was not limited in output power, then the powers of the intermodulation products would
increase in power until they were eventually equal in power with the input tones at the output.

Graph of P1dB, IP2, IP3, and Saturation

See cascade calculations for NF,
IP2, IP3, and
P1dB.

Click here to view an example of a cascaded system. |

Assuming a gain of 1 (0 dB) the slope of the fundamental gain line would be 1:1; the slope of the 3rd-order gain
line would be 3:1. Accordingly, the 3rd-order products increase in power at twice the rate of the input tones and
are always three times farther away from the IP3 than the input tones when not near saturation.

The power of the 3rd-order products can be predicted when the IP3 is known, or the IP3 can be predicted when
the relative amplitudes of the 3rd-order tones and the input tones are known.

Calculating the cascaded values for 3rd-order intercept point (IP3) for the system budget requires use of ratios
for gain and power levels for IP3 (do not use dB and dBm values, respectively). The standard format for indicating
*decibel* values is to use upper case letters; i.e., IP3 for units of dBm. The standard format for indicating
*power* values is to use lower case letters; i.e., ip3 for units of mW.

Conversions: ip3 = 10^{IP3/10} ↔ IP3 = 10 * log_{10}
(ip3)

where ip3 has units of mW and IP3 has units of dBm

A Typical Chain of Cascaded Components

Combining 2 Stages at a Time for Calculations

This equation gives the method for calculating cascaded **output** IP3 (oip3) values based
on the oip3 and gain of each stage. When using the formula in a software program or in a spreadsheet, it is more
convenient and efficient to calculate each successive cascaded stage with the one preceding it using the following
format, per the drawing (above-right).

These formulas are used to convert back and forth between input- and output-referenced IP3 values:

IP3_{Output} = (IP3_{Input} + Gain) {dBm}

IP3_{Input} = (IP3_{Output} - Gain) {dBm}

The following equation is a series expansion of the mixing (multiplying) of two pure tones:

P_{3rd-order products} = P_{input tones@output} - 2 · (IP3 - P_{input tones@output})
{dBm}

P_{3rd-order products} = 3 · P_{input tones@output} - 2 · IP3 {dBm}

IP3 =3/2 · P_{input tones@output} - 1/2 P_{3rd-order products} {dBm}

P_{L}
= P_{2} - 2*(IP3 - P_{1})

P_{U} = P_{1} - 2*(IP3 - P_{2})

where power units are kept constant in dBm or dBW.