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{{Modulation techniques}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
[[File:Amfm3-en-de.gif|thumb|right|250px|A signal may be carried by an [[Amplitude modulation|AM]] or FM radio wave.|alt=Animation of audio, AM and FM signals]]


In [[telecommunications]] and [[signal processing]], '''frequency modulation''' ('''FM''') is the encoding of [[information]] in a [[carrier wave]] by varying the [[instantaneous frequency]] of the wave. (Compare with [[amplitude modulation]], in which the [[amplitude]] of the carrier wave varies, while the frequency remains constant.)
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
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In [[analog signal]] applications, the difference between the instantaneous and the base frequency of the carrier is directly proportional to the instantaneous value of the input-signal amplitude.
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[[Digital data]] can be encoded and transmitted via a carrier wave by shifting the carrier's frequency among a predefined set of frequencies—a technique known as [[frequency-shift keying]] (FSK). FSK is widely used in [[modem]]s and [[fax modem]]s, and can also be used to send [[Morse code]].<ref>{{Cite book
'''MathML'''
|title = Teach yourself electricity and electronics
:<math forcemathmode="mathml">E=mc^2</math>
|author = Stan Gibilisco
|publisher = McGraw-Hill Professional
|year = 2002
|isbn = 978-0-07-137730-0
|page = 477
|url = http://books.google.com/?id=-Q6SBAKsmXkC&pg=PA477&dq=morse-code+frequency-shift-keying+sent-using-fsk
}}</ref> [[Radioteletype]] also uses FSK.<ref>{{Cite book
|title = The Electronics of Radio
|author = David B. Rutledge
|publisher = Cambridge University Press
|year = 1999
|isbn = 978-0-521-64645-1
|page = 310
|url = http://books.google.com/?id=ZvJYLhk4N64C&pg=RA2-PA310&dq=radio-teletype+fsk
}}</ref>


Frequency modulation is used in [[radio]], [[telemetry]], [[radar]], seismic prospecting, and monitoring [[newborn]]s for seizures via [[EEG]].<ref>B. Boashash, editor, “Time-Frequency Signal Analysis and Processing – A Comprehensive Reference”, Elsevier Science, Oxford, 2003; ISBN 0-08-044335-4</ref> FM is widely used for [[FM broadcasting|broadcasting]] music and speech, [[two-way radio]] systems, magnetic tape-recording systems and some video-transmission systems. In radio systems, frequency modulation with sufficient [[Bandwidth (signal processing)|bandwidth]] provides an advantage in cancelling naturally-occurring noise.
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


Frequency modulation is known as [[phase modulation]] when the carrier phase modulation is the time [[integral]] of the FM signal.{{clarify|date=July 2013}}
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


==Theory==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
If the information to be transmitted (i.e., the [[baseband signal]]) is <math>x_m(t)</math> and the [[sinusoidal]] carrier is <math>x_c(t) = A_c \cos (2 \pi f_c t)\,</math>, where ''f<sub>c</sub>'' is the carrier's base frequency, and ''A<sub>c</sub>'' is the carrier's amplitude, the modulator combines the carrier with the baseband data signal to get the transmitted signal:


:<math>y(t) = A_c \cos \left( 2 \pi \int_{0}^{t} f(\tau) d \tau \right)</math>
==Demos==
::<math>= A_{c} \cos \left( 2 \pi \int_{0}^{t} \left[ f_{c} + f_{\Delta} x_{m}(\tau) \right] d \tau \right)</math>
::<math> = A_{c} \cos \left( 2 \pi f_{c} t + 2 \pi f_{\Delta} \int_{0}^{t}x_{m}(\tau) d \tau \right) </math>


In this equation, <math>f(\tau)\,</math> is the ''[[instantaneous phase#Instantaneous frequency|instantaneous frequency]]'' of the oscillator and <math>f_{\Delta}\,</math> is the ''[[frequency deviation]]'', which represents the maximum shift away from ''f<sub>c</sub>'' in one direction, assuming ''x''<sub>''m''</sub>(''t'') is limited to the range ±1.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


While most of the energy of the signal is contained within  ''f<sub>c</sub>'' ± ''f''<sub>Δ</sub>, it can be shown by [[Fourier analysis]] that a wider range of frequencies is required to precisely represent an FM signal. The [[frequency spectrum]] of an actual FM signal has components extending infinitely, although their amplitude decreases and higher-order components are often neglected in practical design problems.<ref name=TGTSCS05/>


===Sinusoidal baseband signal===
* accessibility:
Mathematically, a baseband modulated signal may be approximated by a [[Sine wave|sinusoid]]al [[continuous wave]] signal with a frequency ''f<sub>m</sub>''. The integral of such a signal is:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


:<math>\int_{0}^{t}x_m(\tau)d \tau = \frac{A_m \cos (2 \pi f_m t)}{2 \pi f_m}\,</math>
==Test pages ==


In this case, the expression for y(t) above simplifies to:
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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*[[Help:Formula]]


:<math> y(t) = A_{c} \cos \left( 2 \pi f_{c} t + \frac{f_{\Delta}}{f_{m}} \cos \left( 2 \pi f_{m} t \right) \right)\,</math>
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
where the amplitude <math>A_{m}\,</math> of the modulating [[Sine wave|sinusoid]] is represented by the peak deviation <math>f_{\Delta}\,</math> (see [[frequency deviation]]).
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
The [[harmonic]] distribution of a [[sine wave]] carrier modulated by such a [[sinusoidal]] signal can be represented with [[Bessel function]]s; this provides the basis for a mathematical understanding of frequency modulation in the frequency domain.
 
===Modulation index===
As in other modulation systems, the value of the modulation index indicates by how much the modulated variable varies around its unmodulated level. It relates to variations in the carrier frequency:
 
:<math>h = \frac{\Delta{}f}{f_m} = \frac{f_\Delta |x_m(t)|}{f_m} \ </math>
 
where <math>f_m\,</math> is the highest frequency component present in the modulating signal ''x''<sub>''m''</sub>(''t''), and <math>\Delta{}f\,</math> is the peak frequency-deviation—i.e. the maximum deviation of the ''[[instantaneous phase#Instantaneous frequency|instantaneous frequency]]'' from the carrier frequency. If <math>h \ll 1</math>, the modulation is called ''narrowband FM'', and its bandwidth is approximately <math>2 f_m\,</math>.
 
If <math>h \gg 1</math>, the modulation is called ''wideband FM'' and its bandwidth is approximately <math>2 f_\Delta\,</math>. While wideband FM uses more bandwidth, it can improve the [[signal-to-noise ratio]] significantly; for example, doubling the value of <math>\Delta{}f\,</math>, while keeping <math>f_m</math> constant, results in an eight-fold improvement in the signal-to-noise ratio.<ref>Der, Lawrence, Ph.D., ''Frequency Modulation (FM) Tutorial'', http://www.silabs.com/Marcom%20Documents/Resources/FMTutorial.pdf, Silicon Laboratories, Inc., accessed 2013 February 24, p. 5</ref> (Compare this with [[Chirp spread spectrum]], which uses extremely wide frequency deviations to achieve processing gains comparable to traditional, better-known spread-spectrum modes).
 
With a tone-modulated FM wave, if the modulation frequency is held constant and the modulation index is increased, the (non-negligible) bandwidth of the FM signal increases but the spacing between spectra remains the same; some spectral components decrease in strength as others increase. If the frequency deviation is held constant and the modulation frequency increased, the spacing between spectra increases.
 
Frequency modulation can be classified as narrowband if the change in the carrier frequency is about the same as the signal frequency, or as wideband if the change in the carrier frequency is much higher (modulation index >1) than the signal frequency.
<ref>B. P. Lathi, ''Communication Systems'', John Wiley and Sons, 1968 ISBN 0-471-51832-8, p, 214–217</ref> For example, narrowband FM is used for [[two way radio]] systems such as [[Family Radio Service]], in which the carrier is allowed to deviate only 2.5&nbsp;kHz above and below the center frequency with speech signals of no more than 3.5&nbsp;kHz bandwidth. Wideband FM is used for [[FM broadcasting]], in which music and speech are transmitted with up to 75&nbsp;kHz deviation from the center frequency and carry audio with up to a 20-kHz bandwidth.
 
===Bessel functions===
For the case of a carrier modulated by a single sine wave, the resulting frequency spectrum can be calculated using [[Bessel function]]s of the first kind, as a function of the sideband number and the modulation index. The carrier and sideband amplitudes are illustrated for different modulation indices of FM signals. For particular values of the modulation index, the carrier amplitude becomes zero and all the signal power is in the sidebands.<ref name=TGTSCS05>T.G. Thomas, S. C. Sekhar  ''Communication Theory'', Tata-McGraw Hill 2005, ISBN 0070590915  page 136</ref>
 
Since the sidebands are on both sides of the carrier, their count is doubled, and then multiplied by the modulating frequency to find the bandwidth. For example, 3 kHz deviation modulated by a 2.2 kHz audio tone produces a modulation index of 1.36. Examining the chart shows this modulation index will produce three sidebands. These three sidebands, when doubled, gives us (6 * 2.2 kHz) or a 13.2 kHz required bandwidth.
<!--   and   are spaces the width of a digit and punctuation, respectively; see [[Space (punctuation)#Table of spaces]]-->
{| class="wikitable" style="text-align:right;"
|-
!rowspan=2| Modulation<br />index
!colspan=17| Sideband
|-
! Carrier
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
! 16
|-
! 0.00
| 1.00
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 0.25
| 0.98
| 0.12
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 0.5 
| 0.94
| 0.24
| 0.03
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 1.0 
| 0.77
| 0.44
| 0.11
| 0.02
|
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 1.5 
| 0.51
| 0.56
| 0.23
| 0.06
| 0.01
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 2.0 
| 0.22
| 0.58
| 0.35
| 0.13
| 0.03
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 2.41
| 0   
| 0.52
| 0.43
| 0.20
| 0.06
| 0.02
|
|
|
|
|
|
|
|
|
|
|
|-
! 2.5 
|−0.05
| 0.50
| 0.45
| 0.22
| 0.07
| 0.02
| 0.01
|
|
|
|
|
|
|
|
|
|
|-
! 3.0 
|−0.26
| 0.34
| 0.49
| 0.31
| 0.13
| 0.04
| 0.01
|
|
|
|
|
|
|
|
|
|
|-
! 4.0 
|−0.40
|−0.07
| 0.36
| 0.43
| 0.28
| 0.13
| 0.05
| 0.02
|
|
|
|
|
|
|
|
|
|-
! 5.0 
|−0.18
|−0.33
| 0.05
| 0.36
| 0.39
| 0.26
| 0.13
| 0.05
| 0.02
|
|
|
|
|
|
|
|
|-
! 5.53
| 0   
| −0.34
| −0.13
| 0.25
| 0.40
| 0.32
| 0.19
| 0.09
| 0.03
| 0.01
|
|
|
|
|
|
|
|-
! 6.0 
| 0.15
|−0.28
|−0.24
| 0.11
| 0.36
| 0.36
| 0.25
| 0.13
| 0.06
| 0.02
|
|
|
|
|
|
|
|-
! 7.0 
| 0.30
| 0.00
|−0.30
|−0.17
| 0.16
| 0.35
| 0.34
| 0.23
| 0.13
| 0.06
| 0.02
|
|
|
|
|
|
|-
! 8.0 
| 0.17
| 0.23
|−0.11
|−0.29
|−0.10
| 0.19
| 0.34
| 0.32
| 0.22
| 0.13
| 0.06
| 0.03
|
|
|
|
|
|-
! 8.65
| 0   
| 0.27
| 0.06
|−0.24
|−0.23
| 0.03
| 0.26
| 0.34
| 0.28
| 0.18
| 0.10
| 0.05
| 0.02
|
|
|
|
|-
! 9.0 
|−0.09
| 0.25
| 0.14
|−0.18
|−0.27
|−0.06
| 0.20
| 0.33
| 0.31
| 0.21
| 0.12
| 0.06
| 0.03
| 0.01
|
|
|
|-
! 10.0 
|−0.25
| 0.04
| 0.25
| 0.06
|−0.22
|−0.23
|−0.01
| 0.22
| 0.32
| 0.29
| 0.21
| 0.12
| 0.06
| 0.03
| 0.01
|
|
|-
! 12.0 
| 0.05
|−0.22
|−0.08
| 0.20
| 0.18
|−0.07
|−0.24
|−0.17
| 0.05
| 0.23
| 0.30
| 0.27
| 0.20
| 0.12
| 0.07
| 0.03
| 0.01
|}
 
===Carson's rule===
{{Main|Carson bandwidth rule}}
 
A [[rule of thumb]], ''Carson's rule'' states that nearly all (~98 percent) of the power of a frequency-modulated signal lies within a [[bandwidth (signal processing)|bandwidth]] <math> B_T\, </math> of:
 
:<math>\ B_T = 2(\Delta f +f_m)\,</math>
 
where <math>\Delta f\,</math>, as defined above, is the peak deviation of the instantaneous frequency <math>f(t)\,</math> from the center carrier frequency <math>f_c\,</math>.
 
==Noise Reduction==
A major advantage of FM in a communications circuit, compared for example with [[Amplitude modulation|AM]], is the possibility of improved [[Signal-to-noise ratio]] (SNR). Compared with an optimum AM scheme, FM typically has poorer SNR below a certain signal level called the noise threshold, but above a higher level – the full improvement or full quieting threshold – the SNR is much improved over AM. The improvement depends on modulation level and deviation.  For typical voice communications channels, improvements are typically 5-15 dB.  FM broadcasting using wider deviation can achieve even greater improvements.  Additional techniques, such as pre-emphasis of higher audio frequencies with corresponding de-emphasis in the receiver, are generally used to improve overall SNR in FM circuits. Since FM signals have constant amplitude, FM receivers normally have limiters that remove AM noise, further improving SNR.<ref>{{cite book |title=Reference Data for Radio Engineers |edition=Fifth |page=21-11 |year=1970 |publisher=Howard W. Sams & Co. |editor=H. P. Westman}}</ref><ref>{{cite book |title=The ARRL Handbook for Radio Communications |publisher=American Radio Relay League |year=2010 |editor=H. Ward Silver and Mark J. Wilson (Eds) |author=Alan Bloom |chapter=Chapter 8. Modulation |page=8.7 |isbn=0-87259-144-8}}</ref>
 
=={{anchor|Practical Implementation}}Implementation==
 
===Modulation===
FM signals can be generated using either direct or indirect frequency modulation:
* Direct FM modulation can be achieved by directly feeding the message into the input of a [[Voltage-controlled oscillator|VCO]].
* For indirect FM modulation, the message signal is integrated to generate a [[phase modulation|phase-modulated signal]]. This is used to modulate a [[crystal oscillator|crystal-controlled oscillator]], and the result is passed through a [[frequency multiplier]] to give an FM signal.<ref>"Communication Systems" 4th Ed, Simon Haykin, 2001</ref>
 
===Demodulation===
{{See also|Detector_(radio)#Frequency_and_phase_modulation_detectors|l1=Detectors}}
Many FM detector circuits exist. A common method for recovering the information signal is through a [[Foster-Seeley discriminator]]. A [[phase-locked loop]] can be used as an FM demodulator. ''Slope detection'' demodulates an FM signal by using a tuned circuit which has its resonant frequency slightly offset from the carrier. As the frequency rises and falls the tuned circuit provides a changing amplitude of response, converting FM to AM. AM receivers may detect some FM transmissions by this means, although it does not provide an efficient means of [[Detector (radio)|detection]] for FM broadcasts.
 
==Applications==
 
===Magnetic tape storage===
FM is also used at [[Intermediate frequency|intermediate frequencies]] by analog [[Video cassette recorder|VCR]] systems (including [[VHS]]) to record the [[Luminance (video)|luminance]] (black and white) portions of the video signal. Commonly, the chrominance component is recorded as a conventional AM signal, using the higher-frequency FM signal as bias. FM is the only feasible method of recording the luminance ("black and white") component of video to (and retrieving video from) [[magnetic tape]] without distortion; video signals have a large range of frequency components – from a few [[hertz]] to several [[megahertz]], too wide for [[Equalization|equalizers]] to work with due to electronic noise below −60&nbsp;[[decibel|dB]]. FM also keeps the tape at saturation level, acting as a form of [[noise reduction]]; a [[audio level compression|limiter]] can mask variations in playback output, and the [[FM capture]] effect removes [[print-through]] and [[pre-echo]]. A continuous pilot-tone, if added to the signal – as was done on [[V2000]] and many Hi-band formats – can keep mechanical jitter under control and assist [[timebase correction]].
 
These FM systems are unusual, in that they have a ratio of carrier to maximum modulation frequency of less than two; contrast this with FM audio broadcasting, where the ratio is around 10,000. Consider, for example, a 6-MHz carrier modulated at a 3.5-MHz rate; by Bessel analysis, the first sidebands are on 9.5 and 2.5&nbsp;MHz and the second sidebands are on 13&nbsp;MHz and −1&nbsp;MHz. The result is a reversed-phase sideband on +1&nbsp;MHz; on demodulation, this results in unwanted output at 6−1 = 5&nbsp;MHz. The system must be designed so that this unwanted output is reduced to an acceptable level.<ref>: "FM Systems Of Exceptional Bandwidth" Proc. IEEE vol 112, no. 9, p. 1664, September 1965</ref>
 
===Sound===
FM is also used at [[audio frequency|audio frequencies]] to synthesize sound. This technique, known as [[frequency modulation synthesis|FM synthesis]], was popularized by early digital [[synthesizer]]s and became a standard feature in several generations of [[personal computer]] [[sound card]]s.
 
===Radio===
[[File:FM Broadcast Transmitter High Power.jpg|200px|thumb|An American FM radio transmitter in Buffalo, NY at WEDG]]
{{Main|FM broadcasting}}
[[Edwin Howard Armstrong]] (1890–1954) was an American electrical engineer who invented wideband frequency modulation (FM) radio.<ref>{{Cite book
|title = Principles of modern communications technology
|author = A. Michael Noll
|publisher = Artech House
|year = 2001
|isbn = 978-1-58053-284-6
|page = 104
|url = http://books.google.com/books?id=6tDEZlwiMK0C&pg=PA104
}}</ref>
He patented the regenerative circuit in 1914, the superheterodyne receiver in 1918 and the super-regenerative circuit in 1922.<ref>{{patent|US|1342885}}</ref> Armstrong presented his paper, "A Method of Reducing Disturbances in Radio Signaling by a System of Frequency Modulation", (which first described FM radio) before the New York section of the [[Institute of Radio Engineers]] on November 6, 1935. The paper was published in 1936.<ref>{{Cite journal
|first = E. H.
|last = Armstrong
|title = A Method of Reducing Disturbances in Radio Signaling by a System of Frequency Modulation
|journal = Proceedings of the IRE
|volume = 24
|issue = 5
|pages = 689–740
|publisher = IRE
|date= May 1936
|doi = 10.1109/JRPROC.1936.227383
}}</ref>
 
As the name implies, wideband FM (WFM) requires a wider [[signal bandwidth]] than [[amplitude modulation]] by an equivalent modulating signal; this also makes the signal more robust against [[Noise (radio)|noise]] and [[Interference (communication)|interference]]. Frequency modulation is also more robust against signal-amplitude-fading phenomena. As a result, FM was chosen as the modulation standard for high frequency, [[high fidelity]] [[radio]] transmission, hence the term "[[FM radio]]" (although for many years the [[BBC]] called it "VHF radio" because commercial FM broadcasting uses part of the [[VHF]] band—the [[FM broadcast band]]). FM [[receiver (radio)|receivers]] employ a special [[Detector (radio)|detector]] for FM signals and exhibit a phenomenon known as the ''[[capture effect]]'', in which the [[Tuner (radio)|tuner]] "captures" the stronger of two stations on the same frequency while rejecting the other (compare this with a similar situation on an AM receiver, where both stations can be heard simultaneously). However, [[frequency drift]] or a lack of [[Electronic selectivity|selectivity]] may cause one station to be overtaken by another on an [[adjacent channel]]. Frequency [[drift (telecommunication)|drift]] was a problem in early (or inexpensive) receivers; inadequate selectivity may affect any tuner.
 
An FM signal can also be used to carry a [[stereophonic sound|stereo]] signal; this is done with [[multiplexing]] and demultiplexing before and after the FM process. The FM modulation and demodulation process is identical in stereo and monaural processes. A high-efficiency radio-frequency [[switching amplifier]] can be used to transmit FM signals (and other [[constant envelope|constant-amplitude signals]]). For a given signal strength (measured at the receiver antenna), switching amplifiers use [[low-power electronics|less battery power]] and typically cost less than a [[linear amplifier]]. This gives FM another advantage over other modulation methods requiring linear amplifiers, such as AM and [[Quadrature amplitude modulation|QAM]].
 
FM is commonly used at [[VHF]] [[radio frequencies]] for [[high-fidelity]] [[radio broadcasting|broadcasts]] of music and [[Speech communication|speech]]. Analog TV sound is also broadcast using FM. Narrowband FM is used for voice communications in commercial and [[amateur radio]] settings. In broadcast services, where audio fidelity is important, wideband FM is generally used. In [[two-way radio]], narrowband FM (NBFM) is used to conserve bandwidth for land mobile, marine mobile and other radio services.
 
==See also==
* [[Amplitude modulation]]
* [[Continuous-wave frequency-modulated radar]]
* [[Chirp]]
* [[FM broadcasting]]
* [[FM stereo]]
* [[FM-UWB]] (FM and Ultra Wideband)
* [[History of radio]]
* [[Modulation]], for a list of other modulation techniques
 
==References==
{{Reflist|35em}}
 
==Further reading==
* A. Bruce Carlson. ''Communication Systems, 4th edition.'' McGraw-Hill Science/Engineering/Math. 2001. ISBN 0-07-011127-8, ISBN 978-0-07-011127-1.
* Gary L. Frost. ''Early FM Radio: Incremental Technology in Twentieth-Century America.'' Baltimore: Johns Hopkins University Press, 2010. ISBN 0-8018-9440-9, ISBN 978-0-8018-9440-4.
* Ken Seymour, AT&T Wireless (Mobility). ''Frequency Modulation, The Electronics Handbook, pp 1188-1200, 1st Edition, 1996. 2nd Edition, 2005'' CRC Press, Inc., ISBN 0-8493-8345-5 (1st Edition).
 
{{Analogue TV transmitter topics}}
{{Telecommunications}}
{{Audio broadcasting}}
 
[[Category:Radio modulation modes]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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