Intermodulation

A 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 can differ from the input signal.

Non-linear systems generate harmonics in response to sinusoidal input, meaning that if the input of a non-linear system is a signal of a single frequency, ${\displaystyle ~f_{a},}$ then the output is a signal which includes a number of integer multiples of the input frequency signal; (i.e. some of ${\displaystyle ~f_{a},2f_{a},3f_{a},4f_{a},\ldots }$).

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${\displaystyle ~f_{a}}$, ${\displaystyle ~f_{b}}$, and ${\displaystyle ~f_{c}}$; which may be expressed as

${\displaystyle \ x(t)=M_{a}\sin(2\pi f_{a}t+\phi _{a})+M_{b}\sin(2\pi f_{b}t+\phi _{b})+M_{c}\sin(2\pi f_{c}t+\phi _{c})}$

where the ${\displaystyle \ M}$ and ${\displaystyle \ \phi }$ are the amplitudes and phases of the three components, respectively.

We obtain our output signal, ${\displaystyle \ y(t)}$, by passing our input through a non-linear function ${\displaystyle G}$:

${\displaystyle \ y(t)=G\left(x(t)\right)\,}$

${\displaystyle \ y(t)}$ will contain the three frequencies of the input signal, ${\displaystyle ~f_{a}}$, ${\displaystyle ~f_{b}}$, and ${\displaystyle ~f_{c}}$ (which are known as the fundamental frequencies), as well as a number of linear combinations of the fundamental frequencies, each in the form

${\displaystyle \ k_{a}f_{a}+k_{b}f_{b}+k_{c}f_{c}}$

where ${\displaystyle ~k_{a}}$, ${\displaystyle ~k_{b}}$, and ${\displaystyle ~k_{c}}$ 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 ${\displaystyle N}$ of frequency components ${\displaystyle f_{a},f_{b},\ldots ,f_{N}}$, the output signal will contain a number of frequency components, each of which may be described by

${\displaystyle k_{a}f_{a}+k_{b}f_{b}+\cdots +k_{N}f_{N},\,}$

where the coefficients ${\displaystyle k_{a},k_{b},\ldots ,k_{N}}$ are arbitrary integer values.

Intermodulation order

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 ${\displaystyle \ O}$ of a given intermodulation product is the sum of the absolute values of the coefficients,

${\displaystyle \ O=\left|k_{a}\right|+\left|k_{b}\right|+\cdots +\left|k_{N}\right|,}$

For example, in our original example above, third-order intermodulation products (IMPs) occur where ${\displaystyle \ |k_{a}|+|k_{b}|+|k_{c}|=3}$:

• ${\displaystyle f_{a}+f_{b}+f_{c}}$
• ${\displaystyle f_{a}+f_{b}-f_{c}}$
• ${\displaystyle f_{a}+f_{c}-f_{b}}$
• ${\displaystyle f_{b}+f_{c}-f_{a}}$
• ${\displaystyle 2f_{a}-f_{b}}$
• ${\displaystyle 2f_{a}-f_{c}}$
• ${\displaystyle 2f_{b}-f_{a}}$
• ${\displaystyle 2f_{b}-f_{c}}$
• ${\displaystyle 2f_{c}-f_{a}}$
• ${\displaystyle 2f_{c}-f_{b}}$

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 waves, one at ${\displaystyle f_{1}}$ and one at ${\displaystyle f_{2}}$. When you cube the sum of these sine waves you will get sine waves at various frequencies including ${\displaystyle 2\times f_{2}-f_{1}}$ and ${\displaystyle 2\times f_{1}-f_{2}}$. If ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ are large but very close together then ${\displaystyle 2\times f_{2}-f_{1}}$ and ${\displaystyle 2\times f_{1}-f_{2}}$ will be very close to ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$.

As explained in a previous section, intermodulation can only occur in non-linear systems. Non-linear systems are generally composed of active components, meaning that the components must be biased with an external power source which is not the input signal (i.e. the active components must be "turned on").

Passive intermodulation (PIM), however, occurs in passive devices (which may include cables, antennas etc.) that are subjected to two or more high power tones.[4] The PIM product is the result of the two (or more) high power tones mixing at device nonlinearities such as junctions of dissimilar metals or metal-oxide junctions, such as loose corroded connectors. The higher the signal amplitudes, the more pronounced the effect of the nonlinearities, and the more prominent the intermodulation that occurs — even though upon initial inspection, the system would appear to be linear and unable to generate intermodulation.

The requirement for "two or more high power tones" need not be discrete tones. Passive intermodulation can also occur between different frequencies (i.e. different "tones") within a single broadband carrier. These PIMs would show up as sidebands in a telecommunication signal, which interfere with adjacent channels and impede reception.

Passive intermodulations are a major concern in modern communication systems in cases when a single antenna is used for both high power transmission signals as well as low power receive signals (or when a transmit antenna is in close proximity to a receive antenna). Although the power in the passive intermodulation signal is typically many orders of magnitude lower than the power of the transmit signal, the power in the passive intermodulation signal is often times on the same order of magnitude (and possibly higher) than the power of the receive signal. Therefore, if a passive intermodulation finds its way to receive path, it cannot be filtered or separated from the receive signal. The receive signal would therefore be clobbered by the passive intermodulation signal.[5]

Slew-induced distortion (SID) can produce intermodulation distortion (IMD) when the first signal is slewing (changing voltage) at the limit of the amplifier's power bandwidth product. This induces an effective reduction in gain, partially amplitude-modulating the second signal. If SID only occurs for a portion of the signal, it is called "transient" intermodulation distortion.[6]

Intermodulation distortion in audio is usually specified as the root mean square (RMS) value of the various sum-and-difference signals as a percentage of the original signal's root mean square voltage, although it may be specified in terms of individual component strengths, in decibels, as is common with radio frequency work. Audio system measurements (Audio IMD) include SMPTE standard RP120-1994[6] where two signals (at 60 Hz and 7 kHz, with 4:1 amplitude ratios) are used for the test; many other standards (such as DIN, CCIF) use other frequencies and amplitude ratios. Opinion varies over the ideal ratio of test frequencies (e.g. 3:4,[7] or almost — but not exactly — 3:1 for example).

After feeding the equipment under test with low distortion input sinewaves, the output distortion can be measured by using an electronic filter to remove the original frequencies, or spectral analysis may be made using Fourier transformations in software or a dedicated spectrum analyzer, or when determining intermodulation effects in communications equipment, may be made using the receiver under test itself.

In radio applications, intermodulation may be measured as adjacent channel power ratio. Hard to test are intermodulation signals in the GHz-range generated from passive devices (PIM: passive intermodulation). Manufacturers of these scalar PIM-instruments are Summitek and Rosenberger. The newest developments are PIM-instruments to measure also the distance to the PIM-source. Anritsu offers a radar-based solution with low accuracy and Heuermann offers a frequency converting vector network analyzer solution with high accuracy.

• Beat (acoustics)
• Audio system measurements
• Second-order intercept point (SOI)
• Third-order intercept point (TOI), a metric of an amplifier or system related to intermodulation
• Luxemburg–Gorky effect

•  This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22. (in support of MIL-STD-188).

• Butler, Lloyd (August 1997). "Intermodulation Performance and Measurement of Intermodulation Components". Amateur Radio. VK5BR. Retrieved 2012-01-30.