# Bessel Functions & Graphs

Bessel functions of the first kind are shown in the graph below. In frequency modulation (FM), the carrier and sideband frequencies disappear when the modulation index (β) is equal to a zero crossing of the function for the nth sideband. For example, the carrier (0th sideband) disappears when the Jn(0,β) plot equals zero. It is this feature that broadcasters exploit to suppress the carrier rather than simply inserting a bandstop filter between the transmitter and the antenna.

Using a filter greatly reduces the efficiency of the system since the power amplifier is outputting the carrier signal only to have it shorted to ground via the filter. Adjusting the modulation index to the proper value causes all of the output power to be concentrated in the usable signal, thus increasing efficiency. See FM. The 1st sideband disappears when the Jn(1,β) plot equals zero, the 2nd sideband disappears when the Jn(2,β) equals zero, etc., etc.

Bessel filter pole values can be found here. Bessel filter prototype values can be found here.

 Sample of Bessel Function Zero Crossings J0(β) J1(β) J2(β) J3(β) J4(β) J5(β) J6(β) β = 2.40β = 5.49β = 8.65β = 11.8 β = 3.83β = 7.05β = 10.2 β = 5.14β = 8.42β = 11.6 β = 6.38β = 8.42β = 11.6 β = 7.59β = 11.1β = 14.4 β = 8.77β = 12.3β = 15.7 β = 9.94β = 13.6β = 17.0

Note: Graph generated using Mathcad 4.0.

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