Phase Modulation (PM)

The principle of phase modulation

Phase modulation (PM) is a form of angle modulation where the instantaneous of the carrier wave is varied in direct proportion to the instantaneous amplitude of the modulating signal. The information signal therefore sets the positive or negative phase deviation at each instant.

Throughout this process, the amplitude and nominal frequency of the remain constant, while the instantaneous frequency changes with the time derivative of phase, just as in Frequency Modulation (FM). For a sinusoidal carrier of peak voltage A across a resistive load R, the average carrier power is $P = A^2/(2R)$ ; PM does not change this envelope power.

When the modulating signal is at its maximum positive amplitude, the carrier wave experiences the maximum phase shift in one direction.
When the modulating signal is at its maximum negative amplitude, the carrier experiences the maximum phase shift in the opposite direction.
The instantaneous frequency of the PM wave reaches its largest deviation where the magnitude of the slope of the modulating signal is greatest, while the phase deviation is greatest at the signal's amplitude extremes.

The mathematics of PM

A phase-modulated signal can be described mathematically with the following formula:

y(t) = A \cdot \cos(2\pi f_c t + k_p \cdot x_m(t))

Reading the formula

$y(t)$ is the final modulated signal, and $t$ is time measured in seconds.
$A$ is the constant amplitude of the carrier wave.
$f_c$ is the nominal carrier frequency in Hertz.
$x_m(t)$ is the at time $t$ .
$k_p$ is the phase deviation constant (or modulator sensitivity). It determines the phase shift for a given modulating-signal amplitude. If $x_m(t)$ is measured in volts, its unit is rad/V. For a sinusoidal input with peak amplitude $A_m$ , the phase-modulation index is $\beta_p = k_p A_m$ radians; as a numerical phase ratio, it is dimensionless. If its frequency is $f_m$ , then $\Delta f = \beta_p f_m$ in hertz and Carson's estimate is $B \approx 2(\Delta f + f_m)$ hertz.

The mathematical relationship between PM and FM

Phase Modulation and Frequency Modulation are related angle-modulation forms. Instantaneous frequency is proportional to the time derivative of phase, so one form can be obtained from the other by integration or differentiation of the modulating signal.

Generating FM from PM: If you first pass the information signal through an and then use this integrated signal to phase-modulate a carrier, the resulting output is an FM signal. This principle is used in Armstrong's indirect FM method, often followed by frequency multiplication.
Generating PM from FM: Conversely, if you first pass the information signal through a and then use this differentiated signal to frequency-modulate a carrier, the resulting output is a PM signal.

Both forms use constant-envelope angle modulation, so oscillator, phase-locking, and demodulation techniques can often be adapted between them.

Challenges of analog phase modulation

Standalone analog phase modulation is less common than FM in many transmission systems. Two engineering challenges are relevant:

Implementation complexity: Modulating and accurately demodulating analog PM signals can require more involved phase-sensitive circuits than common AM or FM implementations.
Phase ambiguity: A coherent receiver may be unable to determine whether its recovered reference has phase $\theta$ or $\theta + \pi$ . A +180-degree shift and a -180-degree shift describe the same phase modulo 360 degrees; the ambiguity concerns the unknown absolute reference. The two references produce opposite BPSK decisions because $\cos(\theta + \pi) = -\cos(\theta)$ , which can invert all decoded symbols. Noise and channel phase error make carrier-phase recovery more difficult. This is the source of the inversion risk in coherent binary phase detection.

Differential encoding, as used in DPSK, avoids dependence on the absolute carrier phase by carrying information in phase changes between consecutive symbols.

PM and digital phase modulation

The same phase-control principle appears in digital Phase-Shift Keying (PSK), a widely used modulation family. PSK selects a finite set of phase angles, with each angle representing a symbol or binary pattern such as 00, 01, 10, or 11.

Wi-Fi, satellite links, 4G, and 5G use PSK and QAM, with QAM varying both phase and amplitude. PM therefore provides the continuous-time basis for understanding phase changes in these digital waveforms.