# FREQUENCY CHANNEL ENCODING

## Angle Modulation (PM and FM)

Angle encoded signals have the general form

For phase modulation (PM), in general

so that, in particular, for

the signal becomes

For frequency modulation (FM), things are quite a bit more complicated and calculus rears its head.   To be quite specific

so that, in particular and most simply, for

TEMPORAL REPRESENTATIONS OF FM SIGNALS

FM Signal with b =5                                                   FM Signal with b =5
(includes an overlay of modulation)                                  (without overlay)

The spectral representation of angular modulation signals is a bit more complicated the representations of amplitude modulated signals discussed previously. The equation above may be rewritten as

So that in the limit of small modulation [1] -- i.e. for a narrowband signal [2]

PHASOR REPRESENTATIONS OF AM AND FM SIGNALS

A DSB-AM Signal                                                 A narrowband FM Signal

[1]    When the argument of a sinusoidal function, say A, is small

[2]    For the more mathematically inclined, it may be noted that going beyond the narrowband limitation, it can be shown that

where the is the so called nth order Bessel function of the first kind.  Thus,   is the value of the relative amplitude of the nth sideband. For example,