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{{Short description|non-linear effect in amplitude modulation}}
{{about|signal processing|the album by Bill Evans and Jim Hall|Intermodulation (album)}}
{{mergefrom|IMD3|discuss=Talk:IMD3#Merger proposal|date=January 2019}}
[[Image:RF Intermodulation at 280 MHz.jpg|thumb|288px|A [[frequency spectrum]] plot showing intermodulation between two injected signals at 270 and 275 MHz (the large spikes).  Visible intermodulation products are seen as small spurs at 280 MHz and 265 MHz.]]
[[File:3rd order intermod animation (thumbnail).png|frame|link=File:Two_carrier_3rd_order_intermod_measurement.png#/media/File:Two_carrier_3rd_order_intermod_measurement.png|3rd order active intermodulation animation showing the rapid growth of 3rd order intermodulation products as the power level of 2 CW tones is successively increased at the input of an amplifier operated into its nonlinear region. Fifth, seventh, and ninth order intermodulation products, normally below the noise floor, become visible when the amplifier is overdriven.]]

'''Intermodulation''' ('''IM''') or '''intermodulation distortion''' ('''IMD''') is the [[amplitude modulation]] of [[Signal (electrical engineering)|signals]] containing two or more different [[frequencies]], caused by [[non-linear|nonlinearities]] or time variance in a system.  The intermodulation between frequency components will form additional components at frequencies that are not just at [[harmonic]] frequencies ([[integer]] [[multiple (mathematics)|multiple]]s) of either, like [[harmonic distortion]], but also at the sum and difference frequencies of the original frequencies and at sums and differences of multiples of those frequencies.

Intermodulation is caused by non-linear behaviour of the [[signal processing]] (physical equipment or even algorithms) being used. The [[theoretical]] outcome of these non-linearities can be calculated by generating a [[Volterra series]] of the characteristic, or more approximately by a [[Taylor series]].<ref>{{cite book|last1=Rouphael|first1=Tony J.|title=Wireless Receiver Architectures and Design: Antennas, RF, Synthesizers, Mixed Signal, and Digital Signal Processing|date=2014|publisher=Academic Press|page=244|url=https://books.google.com/books?id=Hs8wZIg8z2gC&pg=PA244&dq=taylor-series-expansion+volterra-series+weakly-nonlinear&hl=en&sa=X&ved=0ahUKEwiMgrbQsZ7aAhWTFXwKHZNXDeEQ6AEINDAC#v=onepage&q=taylor-series-expansion%20volterra-series%20weakly-nonlinear&f=false}}</ref>

Practically all audio equipment has some non-linearity, so it will exhibit some amount of IMD, which however may be low enough to be imperceptible by humans. Due to the characteristics of the human [[auditory system]], the same percentage of IMD is perceived as more bothersome when compared to the same amount of harmonic distortion.<ref name="RumseyMccormick2012">{{cite book|author1=Francis Rumsey|author2=Tim Mccormick|title=Sound and Recording: An Introduction|year=2012|publisher=Focal Press|isbn=978-1-136-12509-6|edition=5th|page=538}}</ref><ref name="DavisDavis1989">{{cite book|author1=Gary Davis|author2=Ralph Jones|title=The Sound Reinforcement Handbook|year=1989|publisher=Yamaha / Hal Leonard Corporation|isbn=978-0-88188-900-0|page=85|edition=2nd}}</ref>{{Dubious|date=May 2015|reason = harmonic distortion with high-order components (like crossover distortion) can be just as objectionable at a given percent}}

Intermodulation is also usually undesirable in radio, as it creates unwanted [[spurious emission]]s, often in the form of [[sidebands]]. For radio transmissions this increases the occupied bandwidth, leading to adjacent channel [[interference (communication)|interference]], which can reduce audio clarity or increase spectrum usage.

IMD is only distinct from [[harmonic distortion]] in that the stimulus signal is different.  The same nonlinear system will produce both [[total harmonic distortion]] (with a solitary sine wave input) and IMD (with more complex tones).  In music, for instance, IMD is [[Distortion (music)|intentionally applied]] to electric guitars using overdriven [[Guitar amplifier|amplifiers]] or [[Effects unit|effects pedals]] to produce new tones at ''sub''harmonics of the tones being played on the instrument.  See [[Power chord#Analysis]].

IMD is also distinct from intentional modulation (such as a [[frequency mixer]] in [[superheterodyne receiver]]s) where signals to be modulated are presented to an intentional nonlinear element ([[Analog multiplier|multiplied]]).  See [[non-linear]] [[Electronic mixer#Product mixers|mixers]] such as mixer [[diode]]s and even single-[[transistor]] oscillator-mixer circuits. However, while the intermodulation products of the received signal with the local oscillator signal are intended, superheterodyne mixers can, at the same time, also produce unwanted intermodulation effects from strong signals near in frequency to the desired signal that fall within the passband of the receiver.

==Causes of intermodulation==
A linear system cannot produce intermodulation. If the input of a [[LTI system theory|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 of 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|400px|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_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.

==Passive intermodulation (PIM)==<!-- This section is linked from [[PIM]]. Hence, do not change the title w/o fixing the link from the disambiguation page -->
{{Main article|Rusty bolt effect}}
{{unreferenced section|date=December 2014}}
As explained in [[intermodulation#Causes of intermodulation|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.<ref>Passive Intermodulation Interference in Communication Systems, P. L. Lui,  Electronics & Communication Engineering Journal, Year: 1990, Volume: 2, Issue: 3,  Pages: 109 - 118.</ref> 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.

It is also possible for a single broadband carrier to generate PIM if it passes through a PIM generating surface or defect. These distortions would show up as side lobes in a telecommunication signal and interfere with adjacent channels and impede reception.

PIM can be severe problem in modern communication systems. Paths that share both high power transmission and the receive signal are most susceptible to this kind of interference. Once PIM interference finds its way to receive path, it can not be filtered or separated.<ref>"Passive Intermodulation Characteristics", M. Eron, Microwave Journal, March, 2014.</ref>

=== Sources of PIM ===

Ferromagnetic materials are the most common materials to avoid and include ferrites, nickel, (including nickel plating) and steels (including some stainless steels). These materials exhibit [[Magnetic hysteresis|hysteresis]] when exposed to reversing magnetic fields, resulting in PIM generation.

PIM can also be generated in components with manufacturing or workmanship defects, such as cold or cracked solder joints or poorly made mechanical contacts. If these defects are exposed to high RF currents, PIM can be generated. As a result, RF equipment manufacturers perform factory PIM tests on components, to eliminate PIM caused by these design and manufacturing defects.

PIM can also be inherent in the design of a high power RF component where RF current is forced to narrow channels or restricted.

In the field, PIM can be caused by components that were damaged in transit to the cell site, installation workmanship issues and by external PIM sources. Some of these include:
 
* Contaminated surfaces or contacts due to dirt, dust, moisture or oxidation.
* Loose mechanical junctions due to inadequate torque, poor alignment or poorly prepared contact surfaces.
* Loose mechanical junctions caused during transportation, shock or vibration. 
* Metal flakes or shavings inside RF connections.
* Inconsistent metal-to-metal contact between RF connector surfaces caused by any of the following:
** Trapped dielectric materials (adhesives, foam, etc.), cracks or distortions at the end of the outer conductor of coaxial cables, often caused by overtightening the back nut during installation, solid inner conductors distorted in the preparation process, hollow inner conductors excessively enlarged or made oval during the preparation process.
* PIM can also occur in connectors, or when conductors made of two [[Galvanization|galvanically]] unmatched metals come in contact with each other.   
* Nearby metallic objects in the direct beam and side lobes of the transmit antenna including rusty bolts, roof flashing, vent pipes, guy wires, etc.

=== PIM Testing ===
IEC 62037 is the international standard for PIM testing and gives specific details as to PIM measurement setups. The standard specifies the use of two +43 dBm (20W) tones for the test signals for PIM testing. This power level has been used by RF equipment manufacturers for more than a decade to establish PASS / FAIL specifications for RF components.

== Intermodulation in electronic circuits ==
[[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 Modulation|amplitude-modulating]] the second signal. If SID only occurs for a portion of the signal, it is called "transient" intermodulation distortion.<ref name="AES">[http://www.aes.org/par/i/#IM AES Pro Audio Reference for IM]</ref>

== Measurement ==

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 RMS voltage, although it may be specified in terms of individual component strengths, in [[decibel]]s, as is common with [[Radio frequency|RF]] work. Audio IMD [[Audio system measurements|standard tests]] include SMPTE standard RP120-1994<ref name="AES"/> where two signals (at 60&nbsp;Hz and 7&nbsp;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,<ref>http://www.leonaudio.com.au/3-4.ratio.distortion.measurement.pdf Graeme John Cohen: 3-4 Ratio; A method of measuring distortion products</ref> or almost — but not exactly — 3:1 for example<!-- where was that article?? -->).

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 [[Fast Fourier transform|Fourier Transformations]] in software or a dedicated [[spectrum analyser]], 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.

== See also ==
* [[Beat (acoustics)]]
* [[Audio system measurements]]
* [[Second-order intercept point]]
* [[Third-order intercept point]], a metric of an amplifier or system related to intermodulation
* [[Luxemburg–Gorky effect]]

==References==
<references/>
{{FS1037C MS188}}

== External links ==
*{{cite web
|url=http://users.tpg.com.au/ldbutler/Intermodulation.htm 
|title=Intermodulation Performance and Measurement of Intermodulation Components
|author= Lloyd Butler
|year=1997 
|work=VK5BR
|publisher="Amateur Radio," August 1997 
|accessdate=30 January 2012}}

[[Category:Audio amplifier specifications]]
[[Category:Waves]]
[[Category:Radio electronics]]