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# Intermodulation

[in-ter-moj-uh-ley-shuhn, -mod-yuh-]

Intermodulation or intermodulation distortion (IMD), or intermod for short, is the result of two or more signals of different frequencies being mixed together, forming additional signals at frequencies that are not, in general, at harmonic frequencies (integer multiples) of either.

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, which can create minor to severe interference to other operations on the signal. Intermodulation should not be confused with general harmonic distortion (which does have widespread use in audio effects processing). 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

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

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

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

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

$y\left(t\right)$ 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

The order $O$ of a given intermodulation product is the sum of the absolute values of the coefficients,

$O = left|k_aright| + left|k_bright| + cdots + left|k_Nright|,$

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

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

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:

• 'break-through', where the transmission from the near station overwhelms the car radio; and
• intermodulation, where another station entirely is heard.

On musical instruments, it is the beat frequency produced when two other notes are produced.

## 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. For example, audio transformers exhibit non-linear behavior near their saturation point and electrolytic capacitors can start to behave as rectifiers under large-signal conditions.

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.

PIMs can also occur in connectors, or when conductors made of two galvanically unmatched metals come in contact with each other.

## Intermodulation in audio applications

Intermodulation is a very specific type of distortion, and should not be confused with harmonic distortion in general.

Audio engineers and producers will often intentionally add harmonic distortion to a recorded track to create a desired sound. This may be intended to replicate the spectral character of a particular amplifier (such as a vacuum tube amplifier). However, these techniques almost exclusively rely on harmonic distortion and gain compression. It is extremely unlikely that any audio application uses intermodulation distortion to improve the acoustic effects of recorded music, because intermodulating the high spectral content of a complex audio signal would result in a very garbled output. The product frequencies that occur specifically in intermodulation are sums or differences of desired frequencies (and not harmonic multiples). The result would mix atonal (off-key) notes into the music. Furthermore, unlike a fuzztone synthesizer or pedal, the spectrum is not flat white noise; instead, the resulting sound would be unpleasant.