The frequency modulation or MF or FM is a mode of modulation consisting in transmitting a Signal by the frequency modulation of a carrying signal (carrying).

One speaks about frequency modulation in opposition to the Amplitude modulation. In frequency modulation, information is carried by a modification of the frequency of carrying, and not by a variation of amplitude. The frequency modulation is more robust than the amplitude modulation to transmit a message under difficult conditions (Atténuation and important Bruit).

For numeric signals, one uses an alternative called Frequency-shift keying or FSK. The FSK uses discrete frequencies.

Examples

• the Modem S ( Mo dulateur- dem odulator) low flow use the frequency modulation;
• the analogical telephones use the frequency modulation to compose the number: each figure is coded by a combination of two frequencies to form a code DTMF. It is about a modulation FSK which uses more than two frequencies (MFSK, multiple frequency-shift keying);
• the radios of the “Bande FM” emit, as their name indicates it, in frequency modulation on the band VHF II.

Theory

It is supposed that the signal to be transmitted is:

$x\_m\; (T)\; \backslash ,$

with the following restriction on the amplitude:

$\backslash \; left|\; x\_m\; (T)\; \backslash \; right|\; \backslash \; the\; 1\; \backslash ,$

Carrying the sinusoidal is:

$x\_p\; (T)\; =\; has\; \backslash \; cos\; (2\; \backslash \; pi\; f\_p\; T)\; \backslash ,$

where f_p is the frequency of carrying in Hertz and has an arbitrary amplitude. The signal modulated out of FM is the following:

$x\_t\; (T)\; =\; has\; \backslash \; cos\; \backslash \; left\; (2\; \backslash \; pi\; \backslash \; int\_\; \{0\}\; ^\; \{T\}\; F\; (\backslash \; tau)\; \backslash ,\; D\; \backslash \; tau\; \backslash \; right)\; =\; has\; \backslash \; cos\; \backslash \; left\; (2\; \backslash \; pi\; \backslash \; int\_\; \{0\}\; ^\; \{T\}\; \backslash \; left\; f\_p\; +\; f\_\; \backslash \; Delta\; x\_m\; (\backslash \; tau)\; \backslash \; right\; \backslash ,\; D\; \backslash \; tau\; \backslash \; right)$

where $f\; (T)\; =\; f\_p\; +\; f\_\; \backslash \; Delta\; x\_m\; (T)$

In this equation, $f\; (T)$ is the instantaneous Fréquence of the oscillator and $f\_\; \backslash \; Delta$ the deviation in frequency , which corresponds to the maximum deviation compared to the frequency of carrying the $f\_p$, by supposing that $x\_m\; (T)$ is limited to interval +1.

Although at first sight one can imagine that the frequencies are limited to the interval $f\_p$ ± $f\_\; \backslash \; Delta$, this reasoning neglects the distinction between instantaneous frequency and spectral frequency . The harmonic Specter of a signal real FM has components which go until infinite frequencies, although they quickly become negligible.

In a simplified way, the spectrum of carrying sinusoidal modulated out of FM by a sinusoidal signal can be represented by a Fonction of Bessel, which makes it possible to formally model the spectral occupation of a modulation FM.

In an approached way, the Règle of Carson indicates that about all the power (~98%) of a signal modulated in frequency is included/understood in the waveband:

$2\; (f\_\; \backslash \; Delta\; +f\_m)\; \backslash ,$

where $f\_\; \backslash \; Delta$ is the maximum deviation of the instantaneous frequency $f\; (T)$ starting from the frequency of carrying the $f\_p$ (by supposing that $x\_m\; (T)$ is in interval +1), and $f\_m$ is the greatest frequency of the signal to transmit $x\_m\; (T)$.

Note: the frequency modulation can be seen as a particular case of the Phase modulation where the phase modulation of carrying is the temporal Intégrale signal to be transmitted.

In the everyday usage, the frequency of modulation is always lower than the carrier frequency, but not to follow this rule can give interesting results, in particular in sound Synthèse.

Case of a sinusoidal signal

The modulation of carrying a sinusoidal by a sinusoidal signal less frequency can be written as follows:

$x\_p\; (T)\; =\; has\; \backslash \; cos\; (2\; \backslash \; pi\; f\_pt\; +\; \backslash \; beta\; \backslash \; sin\; (2\; \backslash \; pi\; f\_m\; T))\; =\; \backslash \; sum\_\; \{n=-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; J\_n\; (\backslash \; beta)\; has\; \backslash \; cos\; (2\; \backslash \; pi\; (f\_p\; +\; N\; f\_m)\; T)\; \backslash ,\; \backslash !$

While varying $\backslash \; beta$, one varies the intensity of the modulation, therefore the variation between the smallest frequency largest and, which alternates at the frequency $f\_m$.

Case of the FSK

In FSK, the signal $x\_m$ a whole of discrete values $x\_i$ can take (for example two in the binary modulations), which gives during the transmission of a value $x\_i$:

$x\_p\; (T)\; =\; has\; \backslash \; cos\; \backslash \; left\; (2\; \backslash \; pi\; \backslash \; int\_\; \{0\}\; ^\; \{T\}\; \backslash \; left\; f\_p\; +\; f\_\; \backslash \; Delta\; x\_i\; \backslash \; right\; \backslash ,\; D\; \backslash \; tau\; \backslash \; right)\; =\; has\; \backslash \; cos\; \backslash \; left\; (2\; \backslash \; pi\; \backslash \; left\; f\_p\; +\; f\_\; \backslash \; Delta\; x\_i\; \backslash \; right\; T\; \backslash \; right)$

One sees thus that the instantaneous frequency can take only one discrete whole of values, a value for each value $x\_i$ of the signal to transmit.

See too

• Frequency-shift keying
• Amplitude modulation
• Edwin Howard Armstrong

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