Editing Intermodulation (section)

==Causes of intermodulation==
A [[LTI system theory|linear time-invariant]] system cannot produce intermodulation. If the input of a linear time-invariant system is a signal of a single frequency, then the output is a signal of the same frequency; only the [[amplitude]] and [[Phase (waves)|phase]] can differ from the input signal.

Non-linear systems generate [[harmonic]]s in response to sinusoidal input, meaning that if the input of a non-linear system is a signal of a single frequency, <math>~f_a,</math> then the output is a signal which includes a number of integer multiples of the input frequency signal; (i.e. some of <math>~ f_a, 2f_a, 3f_a, 4f_a, \ldots</math>).

Intermodulation occurs when the input to a non-linear system is composed of two or more frequencies. Consider an input signal that contains three frequency components at<math>~f_a</math>, <math>~ f_b</math>, and <math>~f_c</math>; which may be expressed as

:<math>\ x(t) = M_a \sin(2 \pi f_a t + \phi_a) + M_b \sin(2 \pi f_b t + \phi_b) + M_c \sin(2 \pi f_c t + \phi_c)</math>

where the <math>\ M</math> and <math>\ \phi</math> are the amplitudes and phases of the three components, respectively.

We obtain our output signal, <math>\ y(t)</math>, by passing our input through a non-linear function <math>G</math>:
:<math>\ y(t) = G\left(x(t)\right)\,</math>

<math>\ y(t)</math> will contain the three frequencies of the input signal, <math>~f_a</math>, <math>~ f_b</math>, and <math>~f_c</math> (which are known as the ''fundamental'' frequencies), as well as a number of [[linear combination]]s of the fundamental frequencies, each in the form

:<math>\ k_af_a + k_bf_b + k_cf_c</math>

where <math>~k_a</math>, <math>~ k_b</math>, and <math>~k_c</math> are arbitrary integers which can assume positive or negative values.  These are the '''intermodulation products''' (or '''IMPs''').

In general, each of these frequency components will have a different amplitude and phase, which depends on the specific non-linear function being used, and also on the amplitudes and phases of the original input components.

More generally, given an input signal containing an arbitrary number <math>N</math> of frequency components <math>f_a, f_b, \ldots, f_N</math>, the output signal will contain a number of frequency components, each of which may be described by

:<math>k_a f_a + k_b f_b + \cdots + k_N f_N,\,</math>

where the coefficients <math>k_a, k_b, \ldots, k_N</math> are arbitrary integer values.

=== Intermodulation order ===
[[Image:Imps thirdorder.png|thumb|upright=1.8|Distribution of third-order intermodulations: in blue the position of the fundamental carriers, in red the position of dominant IMPs, in green the position of specific IMPs.]]

The ''order'' <math>\ O</math> of a given intermodulation product is the sum of the absolute values of the coefficients,

:<math>\ O = \left|k_a\right| + \left|k_b\right| + \cdots + \left|k_N\right|,</math>

For example, in our original example above, third-order intermodulation products (IMPs) occur where <math>\ |k_a|+|k_b|+|k_c| = 3</math>:

* <math>f_a + f_b + f_c</math>
* <math>f_a + f_b - f_c</math>
* <math>f_a + f_c - f_b</math>
* <math>f_b + f_c - f_a</math>
* <math>2f_a - f_b</math>
* <math>2f_a - f_c</math>
* <math>2f_b - f_a</math>
* <math>2f_b - f_c</math>
* <math>2f_c - f_a</math>
* <math>2f_c - f_b</math>

In many radio and audio applications, odd-order IMPs are of most interest, as they fall within the vicinity of the original frequency components, and may therefore interfere with the desired behaviour. For example, intermodulation distortion from the third order ('''IMD3''') of a circuit can be seen by looking at a signal that is made up of two [[sine wave]]s, one at <math>f_1</math> and one at <math>f_2</math>.  When you cube the sum of these sine waves you will get sine waves at various [[frequency|frequencies]] including <math>2\times f_2-f_1</math> and <math>2\times f_1-f_2</math>.  If <math>f_1</math> and <math>f_2</math> are large but very close together then <math>2\times f_2-f_1</math> and <math>2\times f_1-f_2</math> will be very close to <math>f_1</math> and <math>f_2</math>.