Editing Frequency modulation (section)

==Theory==
{{more citations needed|section|date=November 2017}}
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:<ref>{{Cite book |last=Faruque |first=Saleh |url=https://nvhrbiblio.nl/biblio/boek/Faruque%20-%20Radio%20Frequency%20Modulation%20made%20easy.pdf |title=Radio Frequency Modulation Made Easy |publisher=Springer Cham |year=2017 |isbn=978-3-319-41200-9 |pages=33–37 |language=en}}</ref> {{citation needed|date=April 2017}}

:<math>\begin{align} 
  y(t) &= A_c \cos\left(2\pi \int_0^t f(\tau) d\tau\right) \\
       &= A_c \cos\left(2\pi \int_0^t \left[f_c + f_\Delta x_m(\tau)\right] d\tau\right) \\ 
       &= A_c \cos\left(2\pi f_c t + 2\pi f_\Delta \int_0^t x_m(\tau) d\tau\right) \\ 
\end{align}</math>

where <math>f_\Delta = K_f A_m</math>, <math>K_f</math> being the sensitivity of the frequency modulator and <math>A_m</math> being the amplitude of the modulating signal or baseband signal.

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.

This process of integrating the instantaneous frequency to create an instantaneous phase is different from adding the modulating signal to the carrier frequency
:<math>\begin{align} 
  y(t) &= A_c \cos\left(2\pi \left[f_c + f_\Delta x_m(t)\right] t \right) 
\end{align}</math>
which would result in a modulated signal that has spurious local minima and maxima that do not correspond to those of the carrier.

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===
Mathematically, a baseband modulating signal may be approximated by a [[Sine wave|sinusoid]]al [[continuous wave]] signal with a frequency ''f<sub>m</sub>''. This method is also named as single-tone modulation. The integral of such a signal <math>x_m(t) = cos(2\pi f_m t)</math> is:

:<math>\int_0^t x_m(\tau)d\tau = \frac{\sin\left(2\pi f_m t\right)}{2\pi f_m}\,</math>

In this case, the expression for y(t) above simplifies to:

:<math>y(t) = A_c \cos\left(2\pi f_c t + \frac{f_\Delta}{f_m} \sin\left(2\pi f_m t\right)\right)\,</math>

where the amplitude <math>A_m\,</math> of the modulating [[sine wave|sinusoid]] is represented in the peak deviation <math>f_\Delta = K_f A_m</math> (see [[frequency deviation]]).

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 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 \left|x_m(t)\right|}{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{{snd}}i.e. the maximum deviation of the ''[[instantaneous phase#Instantaneous frequency|instantaneous frequency]]'' from the carrier frequency. For a sine wave modulation, the modulation index is seen to be the ratio of the peak frequency deviation of the carrier wave to the frequency of the modulating sine wave.

{{anchor|narrowband_FM_anchor}}If <math>h \ll 1</math>, the modulation is called '''narrowband FM''' (NFM), and its bandwidth is approximately <math>2f_m\,</math>. Sometimes modulation index <math>h < 0.3</math>&nbsp;is considered NFM and other modulation indices are considered wideband FM (WFM or FM).

For digital modulation systems, for example, binary frequency shift keying (BFSK), where a binary signal modulates the carrier, the modulation index is given by:

:<math>h = \frac{\Delta{}f}{f_m} = \frac{\Delta{}f}{\frac{1}{2T_s}} = 2\Delta{}fT_s \ </math>

where <math>T_s\,</math> is the symbol period, and <math>f_m = \frac{1}{2T_s}\,</math> is used as the highest frequency of the modulating binary waveform by convention, even though it would be more accurate to say it is the highest ''fundamental'' of the modulating binary waveform. In the case of digital modulation, the carrier <math>f_c\,</math> is never transmitted. Rather, one of two frequencies is transmitted, either <math>f_c + \Delta f</math> or <math>f_c - \Delta f</math>, depending on the binary state 0 or 1 of the modulation signal.

If <math>h \gg 1</math>, the modulation is called ''wideband FM'' and its bandwidth is approximately <math>2f_\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>{{cite web |last=Der |first=Lawrence |title=Frequency Modulation (FM) Tutorial |url=http://www.silabs.com/Marcom%20Documents/Resources/FMTutorial.pdf |archive-url=https://web.archive.org/web/20141021093250/http://www.silabs.com/Marcom%20Documents/Resources/FMTutorial.pdf |archive-date=2014-10-21 |website=Silicon Laboratories |s2cid=48672999 |access-date=17 October 2019}}</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&nbsp;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.

{{anchor|narrowband FM}}
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&nbsp;>&nbsp;1) than the signal frequency.<ref>Lathi, B. P. (1968). ''Communication Systems'', pp.&nbsp;214–17. New York: John Wiley and Sons, {{ISBN|0-471-51832-8}}.</ref> For example, narrowband FM (NFM) 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&nbsp;kHz bandwidth and subcarriers up to 92&nbsp;kHz.

===Bessel functions===
[[File:Waterfall FM.jpg|thumb|Frequency spectrum and [[waterfall plot]] of a 146.52{{nbsp}}MHz carrier, frequency modulated by a 1,000{{nbsp}}Hz sinusoid.  The modulation index has been adjusted to around 2.4, so the carrier frequency has small amplitude. Several strong sidebands are apparent; in principle an infinite number are produced in FM but the higher-order sidebands are of negligible magnitude.]]

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|0-07-059091-5}}  p. 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&nbsp;kHz deviation modulated by a 2.2&nbsp;kHz audio tone produces a modulation index of 1.36. Suppose that we limit ourselves to only those sidebands that have a relative amplitude of at least 0.01. Then, examining the chart shows this modulation index will produce three sidebands. These three sidebands, when doubled, gives us (6 × 2.2&nbsp;kHz) or a 13.2&nbsp;kHz required bandwidth.
{| class="wikitable" style="text-align:right;"
|-
! rowspan=2  | Modulation<br />index
! colspan=17 | Sideband amplitude
|-
! 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
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 2.40483
| 0.00
| 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
|
|
|
|
|
|
|
|
|-
!  5.52008
|  0.00   
| −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
|
|
|
|
|
|-
!  8.65373
|  0.00   
|  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\left(\Delta f + f_m\right) = 2f_m(h + 1)</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>, <math>h</math> is the modulation index which is the ratio of frequency deviation to highest frequency in the modulating signal, and <math>f_m\,</math>is the highest frequency in the modulating signal.
Carson's rule can only be applied to sinusoidal signals. For non-sinusoidal signals:

:<math>B_T = 2(\Delta f + W) = 2W(D + 1)</math>

where W  is the highest frequency in the modulating signal but non-sinusoidal in nature and D is the Deviation ratio which is the ratio of frequency deviation to highest frequency of modulating non-sinusoidal signal.