# intermodulation

 Dictionary: in·ter·mod·u·la·tion   (ĭn'tər-mŏj'ə-lā'shən)
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n.
Modulation of the frequencies of electromagnetic waves occurring when the waves interact as they are transmitted through a nonlinear electronic system.

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## Intermodulation

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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.

'Intermodulation or intermodulation distortion (IMD), or intermod for short, is the unwanted amplitude modulation of signals containing two or more different frequencies, each component modulating other components, in a system with nonlinearities. This will form additional signals at frequencies that are not, in general, at harmonic frequencies (integer multiples) of either, but instead often at sum and difference frequencies of the original frequencies.

Intermodulation is caused by non-linear behaviour of the signal processing being used. The theoretical outcome of these non-linearities can be calculated by conducting a Volterra series of the characteristic, while the usual approximation of those non-linearities is obtained by conducting a Taylor series.

Intermodulation is rarely desirable in radio or audio processing, as it essentially creates spurious emissions (sidebands; frequency components at the sum and difference of the input frequencies, and possibly more), which can create minor to severe interference to other operations on the signal (cross modulation). It should not be confused with general harmonic distortion (which does have widespread use in audio effects processing), nor confused with intentional modulation (such as a frequency mixer in superheterodyne receivers) where signals to be modulated (multiplied) are presented to different inputs of the mixing elements, although the distinction is difficult in non-linear mixers such as mixer diodes and even single-transistor oscillator-mixer circuits. Intermodulation specifically creates non-harmonic tones ("off-key" notes, in the audio case) due to unwanted mixing of closely spaced frequencies.

## Causes of intermodulation

A linear 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. However, non-linear systems generate harmonics, meaning that if the input of a non-linear system is a signal of a single frequency, $~f_a,$ then the output is a signal which includes a number of integer multiples of the input frequency; (i.e some of $~ f_a, 2f_a, 3f_a, 4f_a, \ldots$).

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$~f_a$, $~ f_b$, and $~f_c$; which may be expressed as

$\ x(t) = M_a \sin(2 \pi f_a t + \phi_a) + M_b \sin(2 \pi f_b t + \phi_b) + M_c \sin(2 \pi f_c t + \phi_c)$

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

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

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

$\ y(t)$ will contain the three frequencies of the input signal, $~f_a$, $~ f_b$, and $~f_c$ (which are known as the fundamental frequencies), as well as a number of linear combinations of the fundamental frequencies, each of the form

$\ k_af_a + k_bf_b + k_cf_c$

where $~k_a$, $~ k_b$, and $~k_c$ 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 N of frequency components $f_a, f_b, \ldots, f_N$, the output signal will contain a number of frequency components, each of which may be described by

$k_a f_a + k_b f_b + \cdots + k_N f_N,\,$

where the coefficients $k_a, k_b, \ldots, k_N$ 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 $\ O$ of a given intermodulation product is the sum of the absolute values of the coefficients,

$\ O = \left|k_a\right| + \left|k_b\right| + \cdots + \left|k_N\right|,$

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

$\ (f_a + f_b - f_c), (f_a + f_c - f_b), (f_b + f_c - f_a)$
$\ (2f_a - f_b), (2f_a - f_c), (2f_b - f_a), (2f_b - f_c), (2f_c - f_a), (2f_c - f_b)$

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.

## Intermodulation noise

In a transmission path or device, intermodulation noise is noise, generated during modulation and demodulation, that results from nonlinear characteristics in the path or device. Intermodulation noise occurs when the frequency sum or difference of a particular signal, S1, interferes with the component frequency sum or difference of another signal, S2.

Someone listening to a car radio while driving close by an AM or FM radio transmission tower may hear two types of 'interference' / distortion:

• 'front-end overload', where the transmission from the near station overwhelms the car radio; and
• intermodulation, where the distortion products are heard.

## Passive intermodulation

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"). However, even passive components can perform in a non-linear manner and cause intermodulation. Diodes are widely known for their passive nonlinear effects, but parasitic nonlinearity can arise in other components as well. For example, audio transformers exhibit non-linear behavior near their saturation point, electrolytic capacitors can start to behave as rectifiers under large-signal conditions, and RF connectors and antennas can exhibit non-linear characteristics. Even the air itself can behave in a non-linear fashion, which can be exploited to produce audible sound from intermodulation of ultrasonic frequencies.

Passive intermodulation (PIM) occurs in passive systems (i.e. the input signal is the only source of energy to the system) when the input signal is very high power, and the system consists of junctions of dis-similar metals or junctions of metals and oxides. These junctions effectively form diodes, which are non-linear. The higher the signal amplitude, the more pronounced the effect of the non-linearities, and the more prominent the intermodulation may occur, even though upon initial inspection, the system would appear to be linear and unable to generate intermodulations.

PIM can also occur in connectors, or when conductors made of two galvanically unmatched metals come in contact with each other. However, the most common source of passive intermodulation in connectors comes from the conduction of signal current through ferromagnetic metals such as nickel, which has a nonlinear magnetization-inductance hysteresis. This effect has been exploited to make reliable sources of PIM[1], which can be used to cancel unwanted PIM from a system[2].

## Intermodulation in electronic circuits

Intermodulation can be caused by many common amplifier limitations, such as slew rate, crossover distortion and reduced transistor hFE or saturation of collector-emitter junctions near clipping. Slew-induced distortion (SID; also named slewing induced distortion or slew-rate induced distortion) can produce intermodulation distortion (IMD) because the gain reduction, during times when the first signal is slewing (changing voltage) as fast as the amplifier's power bandwidth can drive it, partially amplitude-modulates the second signal. Similarly, transient intermodulation distortion (TIM or TID, and sometimes referred to as simply transient distortion or dynamic intermodulation distortion[3]) is the name given to intermodulation resulting from compression that may come at the peaks of the signal, but often the transients are accompanied by high slew rate so it is sometimes said that transient intermodulation distortion is normally nothing more than SID[4].

## Intermodulation in audio applications

Audio engineers usually strive to avoid intermodulation, as for anything other than extremely simple input waveforms, it introduces frequency components that are not harmonically related, which tends to sound unmusical and unpleasant. However, certain audio effects rely on amplitude modulation; these include tremolo and ring modulation, and one way to generate such effects is through deliberate intermodulation in a non-linear device (although more often it may be achieved by an analog multiplier without intermodulation). Harmonic distortion occurs through the same mechanism (non-linearity in an amplifier or loudspeaker for instance) as intermodulation distortion, but involves an input waveform containing only a single frequency component, or several frequency components that are already harmonically related. All audio devices give rise to distortion to some extent; harmonic distortion and intermodulaton distortion tests highlight different aspects of imperfections, and one type of distortion may be inaudibly low while the other is significantly high for some equipment under certain conditions.

## 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 decibels, as is common with RF work. Audio IMD Audio system measurements standard tests include SMPTE standard RP120-1994 [5] where two signals (at 70Hz and 6kHz, 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[6], 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 a electronic filter to remove the original frequencies, or spectral analysis may be made using 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.

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 IMD (abbreviation) product modulator (electronics) electromagnetic interference

 How does intermodulation interference between two repeater transmitters usually occur?

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