Editing Frequency modulation (section)

===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
|}