Phase Modulation

For phase modulation, the signal of (1) encodes the message \(m(t)\) into the phase \(\phi(t)\) and sets \(A(t)\) and \(\omega(t)\) constant:

\[ s_{PM}(t) = A_C \cos(\omega_c t + \phi(t)) \]

where

\(A_C\) is the constant amplitude,
\(\omega_c\) is the constant carrier frequency, and
\(\phi(t)\) is the phase where \(m(t)\) is encoded.

The relationship between \(m(t)\) and \(\phi(t)\) can be set as

\[ \phi(t) = \beta_\phi \frac{m(t)}{\max | m(t) | } \]

where

\(\beta_\phi\) is the phase deviation chosen so that \(\phi(t) \in [-\pi, +\pi)\).