Question

Write a single equation to express amplitude, frequency, phase modulation

Answer 1

PM changes the phase angle of the complex envelope in direct proportion to the message signal.

If m(t) is the message signal to be transmitted and the carrier onto which the signal is modulated is

{\displaystyle c(t)=A_{c}\sin \left(\omega _{\mathrm {c} }t+\phi _{\mathrm {c} }\right).},

then the modulated signal is

{\displaystyle y(t)=A_{c}\sin \left(\omega _{\mathrm {c} }t+m(t)+\phi _{\mathrm {c} }\right).}

This shows how {\displaystyle m(t)} modulates the phase - the greater m(t) is at a point in time, the greater the phase shift of the modulated signal at that point. It can also be viewed as a change of the frequency of the carrier signal, and phase modulation can thus be considered a special case of FM in which the carrier frequency modulation is given by the time derivative of the phase modulation.

The modulation signal could here be

{\displaystyle m(t)=\cos \left(\omega _{\mathrm {c} }t+h\omega _{\mathrm {m} }(t)\right)\ }

The mathematics of the spectral behavior reveals that there are two regions of particular interest:

For small amplitude signals, PM is similar to amplitude modulation (AM) and exhibits its unfortunate doubling of basebandbandwidth and poor efficiency.For a single large sinusoidal signal, PM is similar to FM, and its bandwidth is approximately{\displaystyle 2\left(h+1\right)f_{\mathrm {M} }},where {\displaystyle f_{\mathrm {M} }=\omega _{\mathrm {m} }/2\pi } and {\displaystyle h} is the modulation index defined below. This is also known as Carson's Rule for PM.