A heterodyne is a
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
that is created by combining or mixing two other frequencies using a
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
technique called ''heterodyning'', which was invented by Canadian inventor-engineer
Reginald Fessenden
Reginald Aubrey Fessenden (October 6, 1866 – July 22, 1932) was a Canadian-born inventor, who did a majority of his work in the United States and also claimed U.S. citizenship through his American-born father. During his life he received hundre ...
.
Heterodyning is used to shift signals from one
frequency range
A frequency band is an interval in the frequency domain, delimited by a lower frequency and an upper frequency. The term may refer to a radio band or an interval of some other spectrum.
The frequency range of a system is the range over which ...
into another, and is also involved in the processes of
modulation and
demodulation.
The two input frequencies are combined in a
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
signal-processing device such as a
vacuum tube
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric potential difference has been applied.
The type known as ...
,
transistor
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
, or
diode, usually called a ''
mixer''.
In the most common application, two signals at frequencies and are mixed, creating two new signals, one at the sum of the two frequencies , and the other at the difference between the two frequencies .
The new signal frequencies are called ''heterodynes''. Typically, only one of the heterodynes is required and the other signal is
filtered
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter m ...
out of the output of the mixer. Heterodyne frequencies are related to the phenomenon of "
beat
Beat, beats or beating may refer to:
Common uses
* Patrol, or beat, a group of personnel assigned to monitor a specific area
** Beat (police), the territory that a police officer patrols
** Gay beat, an area frequented by gay men
* Battery (c ...
s" in acoustics.
A major application of the heterodyne process is in the
superheterodyne radio receiver circuit, which is used in virtually all modern radio receivers.
History
![Heterodyne radio receiver circuit 1920](https://upload.wikimedia.org/wikipedia/commons/b/b8/Heterodyne_radio_receiver_circuit_1920.png)
In 1901,
Reginald Fessenden
Reginald Aubrey Fessenden (October 6, 1866 – July 22, 1932) was a Canadian-born inventor, who did a majority of his work in the United States and also claimed U.S. citizenship through his American-born father. During his life he received hundre ...
demonstrated a
direct-conversion heterodyne receiver or
beat receiver
Beat, beats or beating may refer to:
Common uses
* Patrol, or beat, a group of personnel assigned to monitor a specific area
** Beat (police), the territory that a police officer patrols
** Gay beat, an area frequented by gay men
* Battery ( ...
as a method of making
continuous wave radiotelegraphy signals audible. Fessenden's receiver did not see much application because of its local oscillator's stability problem. A stable yet inexpensive local oscillator was not available until
Lee de Forest invented the
triode vacuum tube oscillator.
[, stating "Fessenden's circuit was ahead of its time, however, as there simply was no technology available then with which to build the required local oscillator with the necessary frequency stability." Figure 7.10 shows a simplified 1907 heterodyne detector.] In a 1905 patent, Fessenden stated that the frequency stability of his local oscillator was one part per thousand.
In radio telegraphy, the characters of text messages are translated into the short duration dots and long duration dashes of
Morse code that are broadcast as radio signals.
Radio telegraphy
Wireless telegraphy or radiotelegraphy is transmission of text messages by radio waves, analogous to electrical telegraphy using cables. Before about 1910, the term ''wireless telegraphy'' was also used for other experimental technologies for t ...
was much like ordinary
telegraphy
Telegraphy is the long-distance transmission of messages where the sender uses symbolic codes, known to the recipient, rather than a physical exchange of an object bearing the message. Thus flag semaphore is a method of telegraphy, whereas ...
. One of the problems was building high power transmitters with the technology of the day. Early transmitters were
spark gap transmitter
A spark-gap transmitter is an obsolete type of radio transmitter which generates radio waves by means of an electric spark."Radio Transmitters, Early" in Spark-gap transmitters were the first type of radio transmitter, and were the main type use ...
s. A mechanical device would make sparks at a fixed but audible rate; the sparks would put energy into a resonant circuit that would then ring at the desired transmission frequency (which might be 100 kHz). This ringing would quickly decay, so the output of the transmitter would be a succession of
damped wave
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples i ...
s. When these damped waves were received by a simple detector, the operator would hear an audible buzzing sound that could be transcribed back into alpha-numeric characters.
With the development of the
arc converter
The arc converter, sometimes called the arc transmitter, or Poulsen arc after Danish engineer Valdemar Poulsen who invented it in 1903, was a variety of spark transmitter used in early wireless telegraphy. The arc converter used an electric arc ...
radio transmitter in 1904,
continuous wave (CW) modulation began to be used for radiotelegraphy. CW Morse code signals are not amplitude modulated, but rather consist of bursts of sinusoidal carrier frequency. When CW signals are received by an AM receiver, the operator does not hear a sound. The direct-conversion (heterodyne) detector was invented to make continuous wave radio-frequency signals audible.
The "heterodyne" or "beat" receiver has a
local oscillator
In electronics, a local oscillator (LO) is an electronic oscillator used with a mixer to change the frequency of a signal. This frequency conversion process, also called heterodyning, produces the sum and difference frequencies from the frequenc ...
that produces a radio signal adjusted to be close in frequency to the incoming signal being received. When the two signals are mixed, a "beat" frequency equal to the difference between the two frequencies is created. Adjusting the local oscillator frequency correctly puts the beat frequency in the
audio
Audio most commonly refers to sound, as it is transmitted in signal form. It may also refer to:
Sound
* Audio signal, an electrical representation of sound
*Audio frequency, a frequency in the audio spectrum
* Digital audio, representation of sou ...
range, where it can be heard as a tone in the receiver's
earphone
Headphones are a pair of small loudspeaker drivers worn on or around the head over a user's ears. They are electroacoustic transducers, which convert an electrical signal to a corresponding sound. Headphones let a single user listen to an au ...
s whenever the transmitter signal is present. Thus the Morse code "dots" and "dashes" are audible as beeping sounds. This technique is still used in radio telegraphy, the local oscillator now being called the
beat frequency oscillator
In a radio receiver, a beat frequency oscillator or BFO is a dedicated oscillator used to create an audio frequency signal from Morse code radiotelegraphy ( CW) transmissions to make them audible. The signal from the BFO is mixed with the receive ...
or BFO. Fessenden coined the word ''heterodyne'' from the Greek roots ''hetero-'' "different", and ''dyn-'' "power" (cf.
δύναμις or dunamis).
Superheterodyne receiver
An important and widely used application of the heterodyne technique is in the
superheterodyne receiver (superhet), which was invented by U.S. engineer
Edwin Howard Armstrong in 1918. In the typical superhet, the incoming
radio frequency
Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the ...
signal from the antenna is mixed (heterodyned) with a signal from a local oscillator (LO) to produce a lower fixed frequency signal called the
intermediate frequency (IF) signal. The IF signal is amplified and filtered and then applied to a
detector
A sensor is a device that produces an output signal for the purpose of sensing a physical phenomenon.
In the broadest definition, a sensor is a device, module, machine, or subsystem that detects events or changes in its environment and sends ...
that extracts the audio signal; the audio is ultimately sent to the receiver's loudspeaker.
The superheterodyne receiver has several advantages over previous receiver designs. One advantage is easier tuning; only the RF filter and the LO are tuned by the operator; the fixed-frequency IF is tuned ("aligned") at the factory and is not adjusted. In older designs such as the
tuned radio frequency receiver (TRF), all of the receiver stages had to be simultaneously tuned. In addition, since the IF filters are fixed-tuned, the receiver's selectivity is the same across the receiver's entire frequency band. Another advantage is that the IF signal can be at a much lower frequency than the incoming radio signal, and that allows each stage of the IF amplifier to provide more gain. To first order, an amplifying device has a fixed
gain-bandwidth product. If the device has a gain-bandwidth product of 60 MHz, then it can provide a voltage gain of 3 at an RF of 20 MHz or a voltage gain of 30 at an IF of 2 MHz. At a lower IF, it would take fewer gain devices to achieve the same gain. The
regenerative radio receiver
A regenerative circuit is an amplifier circuit that employs positive feedback (also known as regeneration or reaction). Some of the output of the amplifying device is applied back to its input so as to add to the input signal, increasing the am ...
obtained more gain out of one gain device by using positive feedback, but it required careful adjustment by the operator; that adjustment also changed the selectivity of the regenerative receiver. The superheterodyne provides a large, stable gain and constant selectivity without troublesome adjustment.
The superior superheterodyne system replaced the earlier TRF and regenerative receiver designs, and since the 1930s most commercial radio receivers have been superheterodynes.
Applications
Heterodyning, also called ''frequency conversion'', is used very widely in
communications engineering
Telecommunications Engineering is a subfield of electrical engineering which seeks to design and devise systems of communication at a distance. The work ranges from basic circuit design to strategic mass developments. A telecommunication enginee ...
to generate new frequencies and move information from one frequency channel to another. Besides its use in the superheterodyne circuit found in almost all radio and television receivers, it is used in
radio transmitter
In electronics and telecommunications, a radio transmitter or just transmitter is an electronic device which produces radio waves with an antenna. The transmitter itself generates a radio frequency alternating current, which is applied to the ...
s,
modem
A modulator-demodulator or modem is a computer hardware device that converts data from a digital format into a format suitable for an analog transmission medium such as telephone or radio. A modem transmits data by Modulation#Digital modulati ...
s,
satellite
A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioi ...
communications and set-top boxes,
radar
Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, we ...
,
radio telescopes,
telemetry
Telemetry is the in situ collection of measurements or other data at remote points and their automatic transmission to receiving equipment (telecommunication) for monitoring. The word is derived from the Greek roots ''tele'', "remote", an ...
systems, cell phones, cable television converter boxes and
headends,
microwave relay
Microwave transmission is the transmission of information by electromagnetic waves with wavelengths in the microwave frequency range of 300MHz to 300GHz(1 m - 1 mm wavelength) of the electromagnetic spectrum. Microwave signals are normally limi ...
s,
metal detector
A metal detector is an instrument that detects the nearby presence of metal. Metal detectors are useful for finding metal objects on the surface, underground, and under water. The unit itself, consist of a control box, and an adjustable shaft, ...
s,
atomic clock
An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwe ...
s, and military
electronic countermeasure (jamming) systems.
Up and down converters
In large scale
telecommunication network
A telecommunications network is a group of nodes interconnected by telecommunications links that are used to exchange messages between the nodes. The links may use a variety of technologies based on the methodologies of circuit switching, mess ...
s such as
telephone network
A telephone network is a telecommunications network that connects telephones, which allows telephone calls between two or more parties, as well as newer features such as fax and internet. The idea was revolutionized in the 1920s, as more and mor ...
trunks,
microwave relay
Microwave transmission is the transmission of information by electromagnetic waves with wavelengths in the microwave frequency range of 300MHz to 300GHz(1 m - 1 mm wavelength) of the electromagnetic spectrum. Microwave signals are normally limi ...
networks, cable television systems, and
communication satellite
A communications satellite is an artificial satellite that relays and amplifies radio telecommunication signals via a transponder; it creates a communication channel between a source transmitter and a receiver at different locations on Earth. C ...
links, large
bandwidth
Bandwidth commonly refers to:
* Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range
* Bandwidth (computing), the rate of data transfer, bit rate or thr ...
capacity links are shared by many individual communication channels by using heterodyning to move the frequency of the individual signals up to different frequencies, which share the channel. This is called
frequency division multiplexing
In telecommunications, frequency-division multiplexing (FDM) is a technique by which the total bandwidth available in a communication medium is divided into a series of non-overlapping frequency bands, each of which is used to carry a separate ...
(FDM).
For example, a
coaxial cable used by a cable television system can carry 500 television channels at the same time because each one is given a different frequency, so they do not interfere with one another. At the cable source or
headend, electronic upconverters convert each incoming television channel to a new, higher frequency. They do this by mixing the television signal frequency, ''f
CH'' with a
local oscillator
In electronics, a local oscillator (LO) is an electronic oscillator used with a mixer to change the frequency of a signal. This frequency conversion process, also called heterodyning, produces the sum and difference frequencies from the frequenc ...
at a much higher frequency , creating a heterodyne at the sum , which is added to the cable. At the consumer's home, the cable
set top box
A set-top box (STB), also colloquially known as a cable box and historically television decoder, is an information appliance device that generally contains a TV-tuner input and displays output to a television set and an external source of si ...
has a downconverter that mixes the incoming signal at frequency with the same local oscillator frequency creating the difference heterodyne frequency, converting the television channel back to its original frequency: . Each channel is moved to a different higher frequency. The original lower basic frequency of the signal is called the
baseband
In telecommunications and signal processing, baseband is the range of frequencies occupied by a signal that has not been modulated to higher frequencies. Baseband signals typically originate from transducers, converting some other variable int ...
, while the higher channel it is moved to is called the
passband
A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
.
Analog videotape recording
Many analog
videotape
Videotape is magnetic tape used for storing video and usually sound in addition. Information stored can be in the form of either an analog or digital signal. Videotape is used in both video tape recorders (VTRs) and, more commonly, videocasse ...
systems rely on a downconverted color subcarrier to record color information in their limited bandwidth. These systems are referred to as "heterodyne systems" or "color-under systems". For instance, for
NTSC
The first American standard for analog television broadcast was developed by National Television System Committee (NTSC)National Television System Committee (1951–1953), Report and Reports of Panel No. 11, 11-A, 12–19, with Some supplement ...
video systems, the
VHS (and
S-VHS
, the common initialism for Super VHS, is an improved version of the VHS standard for consumer-level video recording. Victor Company of Japan introduced S-VHS in Japan in April 1987, with their JVC-branded HR-S7000 VCR, and in certain overse ...
) recording system converts the color subcarrier from the NTSC standard 3.58 MHz to ~629 kHz.
[ Videotape formats using tape]
; Retrieved 2007-01-01 PAL
Phase Alternating Line (PAL) is a colour encoding system for analogue television. It was one of three major analogue colour television standards, the others being NTSC and SECAM. In most countries it was broadcast at 625 lines, 50 fields (25 ...
VHS color subcarrier is similarly downconverted (but from 4.43 MHz). The now-obsolete 3/4"
U-matic
U-matic is an analogue recording Videocassette#Cassette formats, videocassette format first shown by Sony in prototype in October 1969, and introduced to the market in September 1971. It was among the first video formats to contain the vide ...
systems use a heterodyned ~688 kHz subcarrier for NTSC recordings (as does
Sony
, commonly stylized as SONY, is a Japanese multinational conglomerate corporation headquartered in Minato, Tokyo, Japan. As a major technology company, it operates as one of the world's largest manufacturers of consumer and professiona ...
's
Betamax, which is at its basis a 1/2″ consumer version of U-matic), while PAL U-matic decks came in two mutually incompatible varieties, with different subcarrier frequencies, known as Hi-Band and Low-Band. Other videotape formats with heterodyne color systems include
Video-8 and
Hi8
The 8mm video format refers informally to three related videocassette formats. These are the original Video8 (analog recording) format and its improved successor Hi8 (analog video and analog audio but with provision for digital audio), as well as ...
.
The heterodyne system in these cases is used to convert quadrature phase-encoded and amplitude modulated sine waves from the broadcast frequencies to frequencies recordable in less than 1 MHz bandwidth. On playback, the recorded color information is heterodyned back to the standard subcarrier frequencies for display on televisions and for interchange with other standard video equipment.
Some U-matic (3/4″) decks feature 7-pin mini-
DIN connector
The DIN connector is an electrical connector that was standardized by the ' (DIN), the German Institute for Standards, in the early 1970s. The male DIN connectors (plugs) feature a 13.2 mm diameter metal shield with a notch that limits the ...
s to allow dubbing of tapes without conversion, as do some industrial VHS, S-VHS, and Hi8 recorders.
Music synthesis
The
theremin
The theremin (; originally known as the ætherphone/etherphone, thereminophone or termenvox/thereminvox) is an electronic musical instrument controlled without physical contact by the performer (who is known as a thereminist). It is named afte ...
, an
electronic musical instrument, traditionally uses the heterodyne principle to produce a variable
audio frequency
An audio frequency or audible frequency (AF) is a periodic vibration whose frequency is audible to the average human. The SI unit of frequency is the hertz (Hz). It is the property of sound that most determines pitch.
The generally accepted ...
in response to the movement of the musician's hands in the vicinity of one or more antennae, which act as capacitor plates. The output of a fixed radio frequency oscillator is mixed with that of an oscillator whose frequency is affected by the
variable capacitance between the antenna and the musician's hand as it is moved near the pitch control antenna. The difference between the two oscillator frequencies produces a tone in the audio range.
The
ring modulator
In electronics, ring modulation is a signal processing function, an implementation of frequency mixing, in which two signals are combined to yield an output signal. One signal, called the carrier, is typically a sine wave or another simple ...
is a type of
frequency mixer
In electronics, a mixer, or frequency mixer, is an electrical circuit that creates new frequencies from two signals applied to it. In its most common application, two signals are applied to a mixer, and it produces new signals at the sum and di ...
incorporated into some synthesizers or used as a stand-alone audio effect.
Optical heterodyning
Optical heterodyne detection Optical heterodyne detection is a method of extracting information encoded as modulation of the phase, frequency or both of electromagnetic radiation in the wavelength band of visible or infrared light. The light signal is compared with standard o ...
(an area of active research) is an extension of the heterodyning technique to higher (visible) frequencies. Guerra (1995) first published the results of what he called a "form of optical heterodyning" in which light patterned by a 50 nm pitch grating illuminated a second grating of pitch 50 nm, with the gratings rotated with respect to each other by the angular amount needed to achieve magnification. Although the illuminating wavelength was 650 nm, the 50 nm grating was easily resolved. This showed a nearly 5-fold improvement over the Abbe resolution limit of 232 nm that should have been the smallest obtained for the numerical aperture and wavelength used. This super-resolution microscopic imaging through optical heterodyning later came to be know by many as "structured illumination microscopy".
In addition to super-resolution optical microscopy, optical heterodyning could greatly improve
optical modulator
An optical modulator is a device which is used to modulate a beam of light. The beam may be carried over free space, or propagated through an optical waveguide (optical fibre). Depending on the parameter of a light beam which is manipulated, modul ...
s, increasing the density of information carried by
optical fiber
An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass ( silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a mea ...
s. It is also being applied in the creation of more accurate
atomic clock
An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwe ...
s based on directly measuring the frequency of a laser beam. See NIST subtopic 9.07.9-4.R for a description of research on one system to do this.
Since optical frequencies are far beyond the manipulation capacity of any feasible electronic circuit, all visible frequency photon detectors are inherently energy detectors not oscillating electric field detectors. However, since energy detection is inherently "
square-law" detection, it intrinsically mixes any optical frequencies present on the detector. Thus, sensitive detection of specific optical frequencies necessitates optical heterodyne detection, in which two different (close-by) wavelengths of light illuminate the detector so that the oscillating electrical output corresponds to the difference between their frequencies. This allows extremely narrow band detection (much narrower than any possible color filter can achieve) as well as precision measurements of phase and frequency of a light signal relative to a reference light source, as in a
laser Doppler vibrometer
A laser Doppler vibrometer (LDV) is a scientific instrument that is used to make non-contact vibration measurements of a surface. The laser beam from the LDV is directed at the surface of interest, and the vibration amplitude and frequency are extr ...
.
This phase sensitive detection has been applied for Doppler measurements of wind speed, and imaging through dense media. The high sensitivity against background light is especially useful for
lidar.
In
optical Kerr effect
The Kerr effect, also called the quadratic electro-optic (QEO) effect, is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index chan ...
(OKE) spectroscopy, optical heterodyning of the OKE signal and a small part of the probe signal produces a mixed signal consisting of probe, heterodyne OKE-probe and homodyne OKE signal. The probe and homodyne OKE signals can be filtered out, leaving the heterodyne frequency signal for detection.
Heterodyne detection is often used in
interferometry but usually confined to single point detection rather than widefield interferometry, however, widefield heterodyne interferometry is possible using a special camera. Using this technique which a reference signal extracted from a single pixel it is possible to build a highly stable widefield heterodyne interferometer by removing the piston phase component
caused by
microphonics
Microphonics, microphony, or microphonism describes the phenomenon wherein certain components in electronic devices transform mechanical vibrations into an undesired electrical signal (noise). The term comes from analogy with a microphone, which ...
or vibrations of the optical components or object.
Mathematical principle
Heterodyning is based on the
trigonometric identity
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
:
:
The product on the left hand side represents the multiplication ("mixing") of a
sine wave with another sine wave. The right hand side shows that the resulting signal is the difference of two
sinusoidal terms, one at the sum of the two original frequencies, and one at the difference, which can be considered to be separate signals.
Using this trigonometric identity, the result of multiplying two sine wave signals
and
at different frequencies
and
can be calculated:
:
The result is the sum of two sinusoidal signals, one at the sum and one at the difference of the original frequencies.
Mixer
The two signals are combined in a device called a ''
mixer''. As seen in the previous section, an ideal mixer would be a device that multiplies the two signals. Some widely used mixer circuits, such as the
Gilbert cell, operate in this way, but they are limited to lower frequencies. However, any ''
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
'' electronic component also multiplies signals applied to it, producing heterodyne frequencies in its output—so a variety of nonlinear components serve as mixers. A nonlinear component is one in which the output current or voltage is a
nonlinear function
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
of its input. Most circuit elements in communications circuits are designed to be
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
. This means they obey the
superposition principle; if
is the output of a linear element with an input of
:
:
So if two sine wave signals at frequencies and are applied to a linear device, the output is simply the sum of the outputs when the two signals are applied separately with no product terms. Thus, the function
must be nonlinear to create mixer products. A perfect multiplier only produces mixer products at the sum and difference frequencies , but more general nonlinear functions produce higher order mixer products: for integers and . Some mixer designs, such as double-balanced mixers, suppress some high order undesired products, while other designs, such as
harmonic mixer
The harmonic mixer and subharmonic mixer are a type of frequency mixer, which is a circuit that changes one signal frequency to another. The ordinary mixer has two input signals and one output signal. If the two input signals are sinewaves at freq ...
s exploit high order differences.
Examples of nonlinear components that are used as mixers are
vacuum tube
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric potential difference has been applied.
The type known as ...
s and
transistor
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
s biased near cutoff (
class C), and
diodes.
Ferromagnetic core inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s driven into
saturation
Saturation, saturated, unsaturation or unsaturated may refer to:
Chemistry
* Saturation, a property of organic compounds referring to carbon-carbon bonds
**Saturated and unsaturated compounds
** Degree of unsaturation
**Saturated fat or fatty aci ...
can also be used at lower frequencies. In
nonlinear optics
Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in ''nonlinear media'', that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typic ...
, crystals that have nonlinear characteristics are used to mix
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fi ...
light beams to create
optical heterodyne frequencies.
Output of a mixer
To demonstrate mathematically how a nonlinear component can multiply signals and generate heterodyne frequencies, the nonlinear function
can be expanded in a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
(
MacLaurin series
Maclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin ...
):
:
To simplify the math, the higher order terms above are indicated by an ellipsis (". . .") and only the first terms are shown. Applying the two sine waves at frequencies and to this device:
:
:
:
It can be seen that the second term above contains a product of the two sine waves. Simplifying with
trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
:
:
:
So the output contains sinusoidal terms with frequencies at the sum and difference of the two original frequencies. It also contains terms at the original frequencies and at multiples of the original frequencies , , , , etc.; the latter are called
harmonics
A harmonic is a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', ...
, as well as more complicated terms at frequencies of , called
intermodulation products. These unwanted frequencies, along with the unwanted heterodyne frequency, must be filtered out of the mixer output by an
electronic filter to leave the desired frequency.
See also
*
Electroencephalography
*
Homodyne
In electrical engineering, homodyne detection is a method of extracting information encoded as modulation of the phase and/or frequency of an oscillating signal, by comparing that signal with a standard oscillation that would be identical to the ...
*
Transverter
*
Intermodulation
Intermodulation (IM) or intermodulation distortion (IMD) is the amplitude modulation of signals containing two or more different frequencies, caused by nonlinearities or time variance in a system. The intermodulation between frequency comp ...
– a problem with strong higher-order terms produced in some non-linear mixers
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{{Authority control
Frequency mixers
Signal processing