# Frequency Modulation, FM Sidebands & Bandwidth

### The bandwidth of a frequency modulated, FM signal depends on a variety of factors including the level and frequency of the modulation.

The bandwidth, sideband formation and spectrum of a frequency modulated signal are not as straightforward as they are for an amplitude modulated signal.

Nevertheless the sidebands and bandwidth of the FM signal are still very important.

Using a well know rule called Carson's Rule it is possible to provide a good estimate of the bandwidth of an FM signal.

## Frequency modulation sidebands

The modulation of any carrier in any way produces sidebands. For amplitude modulated signals, the way in which these sidebands are created and their bandwidth and amplitude are quite straightforward. The situation for frequency modulated signals is different.

The FM sidebands are dependent on both the level of deviation and the frequency of the modulation. In fact the total frequency modulation spectrum consists of the carrier plus an infinite number of sidebands spreading out on either side of the carrier at integral multiples of the modulating frequency. The values for the levels of the sidebands can be seen to rise and fall with varying values of deviation and modulating frequency as seen in the diagram below. The parameters for the FM sidebands are determined by a formula using Bessel functions of the first kind.

Fortunately outside the main signal area itself, the level of the sidebands falls away and for practical systems filtering all but removes them without any main detriment to the signal.

For small values of modulation index, when using narrow-band FM, NBFM, the signal consists of the carrier and the two sidebands spaced at the modulation frequency either side of the carrier. The sidebands further out are minimal and can be ignored. On a spectrum analyzer the signal looks very much like the spectrum of an AM signal. The difference is that the lower sideband is out of phase by 180°.

As the level of the modulation index is increased other sidebands at twice the modulation frequency start to appear. Further increases in modulation index result in the level of other sidebands increasing in level.

## Carson's Rule for FM bandwidth

The bandwidth of an FM signal is not as straightforward to calculate as that of an AM signal.

A very useful rule of thumb used by many engineers to determine the bandwidth of an FM signal is known as Carson's Rule. This rule states that 98% of the signal power is contained within a bandwidth equal to the deviation frequency, plus the modulation frequency doubled. Carson's Rule can be expressed simply as a formula:

$BT=2\left(\Delta f+fm\right)$

Where:
Δf = deviation
BT = total bandwidth (for 98% power)
fm = modulating frequency

To take the example of a typical broadcast FM signal that has a deviation of ±75kHz and a maximum modulation frequency of 15 kHz, the bandwidth of 98% of the power approximates to 2 (75 + 15) = 180kHz. To provide conveniently spaced channels 200 kHz is allowed for each station.

## Summary of frequency modulation bandwidth & sidebands

It is worth summarizing some of the highlight points about frequency modulation sidebands, FM spectrum & bandwidth.

• The bandwidth of a frequency modulated signal varies with both deviation and modulating frequency.
• Increasing modulating frequency increases the frequency separation between sidebands.

Frequency modulation bandwidth is a key issue as there is a growing level of importance associated with ensuring transmissions stay within their allocated channel. Accordingly FM signals need to be carefully tailored to ensure all the significant sidebands remain within the channel allocation.