Phase Modulation (PM)
Phase Modulation (PM) is a form of angle modulation in which the phase of a carrier wave is varied proportionally to the instantaneous amplitude of a modulating signal. Although the nominal carrier frequency remains constant, phase variation causes instantaneous frequency changes, since frequency is the time derivative of phase.
Mathematical Representation
The PM signal is described by:
s(t) = A_c · cos(ω_c t + k_p · m(t))
Where:
s(t) = Modulated signal
A_c = Carrier amplitude
ω_c = Angular carrier frequency
k_p = Phase sensitivity (rad/V)
m(t) = Modulating signal
t = Time
The instantaneous frequency is:
ω(t) = ω_c + k_p · d/dt m(t)
This reflects that PM causes frequency variation based on the derivative of the modulating signal.
Modulation Index
The PM modulation index is defined as:
β = k_p × A_m
Where:
A_m = Peak amplitude of the modulating signal
In contrast to FM, the modulation index in PM is independent of the modulating frequency.
Spectral Characteristics
PM signals produce a carrier and multiple sidebands, with amplitudes determined by Bessel functions. The bandwidth increases with:
Modulation index β
Signal content and waveform
PM typically occupies less bandwidth than FM, but more than AM.
PM vs. FM – Technical Comparison
| Aspect | Phase Modulation (PM) | Frequency Modulation (FM) |
|---|---|---|
| Varies | Carrier phase | Carrier frequency |
| Instantaneous frequency | Proportional to d/dt m(t) | Proportional to m(t) |
| Modulation index | β = kp × Am | β = Δf / fm |
| Frequency deviation | Depends on signal slope | Depends on signal amplitude |
| Mathematical relation | PM = FM with integrated signal | FM = PM with differentiated signal |
| Analog use | Rare | Common (e.g. FM radio) |
| Digital relevance | Basis for PSK, QPSK, 8-PSK | Basis for FSK |
This comparison highlights the key technical differences between phase modulation and frequency modulation.
| Aspect | Details |
|---|---|
| Phase Modulation (PM) | |
| Varies | Carrier phase |
| Instantaneous frequency | Proportional to d/dt m(t) |
| Modulation index | β = kp × Am |
| Frequency deviation | Depends on signal slope |
| Mathematical relation | PM = FM with integrated signal |
| Analog use | Rare |
| Digital relevance | Basis for PSK, QPSK, 8-PSK |
| Frequency Modulation (FM) | |
| Varies | Carrier frequency |
| Instantaneous frequency | Proportional to m(t) |
| Modulation index | β = Δf / fm |
| Frequency deviation | Depends on signal amplitude |
| Mathematical relation | FM = PM with differentiated signal |
| Analog use | Common (e.g. FM radio) |
| Digital relevance | Basis for FSK |
This comparison highlights the key technical differences between phase modulation and frequency modulation.
Applications
PM is mainly used in:
Digital modulation schemes (e.g., BPSK, QPSK)
RFID and NFC
Satellite links and optical systems
Radar signal processing
High-speed modems
FM synthesis (via mathematical equivalence)
Example: BPSK
In Binary Phase Shift Keying, a binary signal m(t) with values ±1 produces:
m(t) = +1 → 0° phase
m(t) = –1 → 180° phase
The modulated signal becomes:
s(t) = A_c · cos(ω_c t + π · (1 – m(t))/2)
Advantages and Limitations
Advantages
Immunity to amplitude noise
High spectral efficiency (in digital systems)
Phase continuity can reduce spectral spreading
Limitations
Complex demodulation (requires phase reference)
Sensitive to phase noise
Not widely used for analog audio