This Demonstration illustrates frequency modulation (FM) and phase modulation (PM) using one sinusoidal tone as the modulating signal.

For FM and PM, the modulating signals

are defined by

and

, respectively, where

is the signal frequency in Hz, and

is its amplitude. This definition for

is used to simplify the spectra of the modulated carrier

by using Bessel functions of the first kind (

BesselJ in

*Mathematica*).

With

defined as above, the modulated carrier

can now be defined as

, where

is the modulation index,

is the carrier frequency in Hz, and

is the carrier amplitude.

For FM modulation,

, where

is the deviation constant in Hz per volt, and for PM modulation,

, where

is in radians per volt. (The above units for

assumes that the unit of

is volts.)

The bandwidth of the modulated carrier

is defined as approximately

. This bandwidth contains 98% of the power. For very small

the modulated carrier spectra become a narrow band and for a large

the spectra becomes wideband.

The parameters

can be adjusted and the effect on the spectra of the modulated carrier can be observed.

This Demonstration also calculates and plots the (normalized) power content of the modulated carrier as a function of the bandwidth. This is also called the power ratio, and is defined as

, where

is a Bessel function of the first kind and

is the number of sidebands on each side of the carrier frequency

. This plot is useful in the design of FM and PM modulators as it allows one to determine the size of the bandwidth needed for a given power ratio.