Short note: Phase modulation

Phase modulation (PM) is angle modulation in which the instantaneous phase of the carrier varies linearly with the message signal, while the amplitude stays constant.

$$s_{PM}(t) = A_c\cos\left[2\pi f_c t + k_p\, m(t)\right]$$

where $k_p$ is the phase sensitivity (rad/V).

Instantaneous frequency:

$$f_i(t) = f_c + \frac{k_p}{2\pi}\frac{dm(t)}{dt}$$

so in PM the frequency deviation depends on the rate of change of the message.

Single tone: for $m(t) = A_m\cos 2\pi f_m t$,

$$s(t) = A_c\cos\left[2\pi f_c t + \beta_p\cos 2\pi f_m t\right], \qquad \beta_p = k_p A_m$$

The PM index $\beta_p$ is independent of $f_m$, while the frequency deviation $\Delta f = k_p A_m f_m$ increases with $f_m$.

Relation with FM:

• PM of $m(t)$ = FM of $\frac{dm}{dt}$ (differentiator + FM modulator gives PM).
• FM of $m(t)$ = PM of $\int m\,dt$ (integrator + phase modulator gives FM, used in Armstrong's method).

Features: constant envelope and power $A_c^2/2$; spectrum given by Bessel functions as in FM; bandwidth by Carson's rule $2(\Delta f + f_m)$. PM can be demodulated by a phase detector (e.g. a PLL) or by an FM discriminator followed by an integrator. Digital forms of PM (BPSK, QPSK) are widely used in modern data communication.