PM Basics

Phase modulation (PM) is the second form of angle modulation, alongside frequency modulation. In PM, the instantaneous phase of the carrier signal is varied in proportion to the message signal amplitude, while the carrier amplitude remains constant. PM and FM are closely related, sharing the same general angle modulation framework, but they differ in how the modulation index depends on message frequency. Understanding PM is essential both independently and as a foundation for understanding FM, since FM can be generated indirectly through PM.

Core Concept Explanation

In phase modulation, the instantaneous phase of the carrier is varied directly in proportion to the message signal: phi(t) = 2pi fc t + kp m(t), where kp is the phase sensitivity in radians per volt. The resulting PM signal is s(t) = Ac cos[2pi fc t + kp m(t)]. The amplitude Ac remains constant, just as in FM. The term kp m(t) is the phase deviation at time t.

The PM modulation index is defined as mp = kp * Am, where Am is the peak message amplitude. This is the maximum phase deviation in radians. Unlike FM where the modulation index mf = delta_f / fm depends on the message frequency fm, the PM modulation index mp = kp Am is independent of the message frequency. This is the most critical distinction between FM and PM.

The instantaneous frequency in PM is fi(t) = fc + (kp / 2pi) * d/dt[m(t)]. This means the PM signal's instantaneous frequency depends on the derivative of the message, not the message itself. Conversely, in FM, the instantaneous frequency is fc + kf m(t), proportional to the message directly. This reciprocal relationship means that an FM modulator preceded by a differentiator (pre-emphasis circuit) produces PM, and a PM modulator preceded by an integrator produces FM.

For a sinusoidal message m(t) = Am cos(2pi fm t), the PM signal becomes s(t) = Ac cos[2pi fc t + mp cos(2pi fm t)]. The peak frequency deviation in PM is delta_f = kp Am fm / (2pi) = mp fm / (2pi) (using angular form) or delta_f = kp Am fm (using conventional Hz form with appropriate kp units). This shows that in PM, the frequency deviation increases with message frequency, whereas in FM it does not.

Mathematical Expression

The general angle modulation expression is s(t) = Ac cos[theta_i(t)], where theta_i(t) is the instantaneous phase. For FM: theta_i(t) = 2pi fc t + 2pi kf integral(m(tau) d tau). For PM: theta_i(t) = 2pi fc t + kp m(t). These two expressions show the mathematical connection. Differentiating the FM instantaneous phase gives fi(t) = fc + kf m(t). Differentiating the PM instantaneous phase gives fi(t) = fc + (kp / 2pi) m'(t).

Because PM and FM share the same form of s(t) for a sinusoidal message, their spectra look identical: a carrier plus Bessel-weighted sidebands. The difference is hidden in how the modulation index relates to message parameters. For FM, mf = kf Am / fm, which decreases with increasing fm. For PM, mp = kp Am, which is constant regardless of fm. GATE problems exploit this by changing the message frequency and asking how the modulation index or bandwidth changes.

Practical Understanding

PM is directly used in digital phase-shift keying (PSK), where discrete phase values represent binary or multilevel data. BPSK, QPSK, and higher-order PSK schemes are all forms of PM applied to digital data. In analog systems, PM is less common than FM but serves as the theoretical basis for generating FM indirectly through Armstrong's indirect FM method.

In Armstrong's indirect FM method, the message signal is first integrated to convert it from m(t) to integral(m) and then applied to a phase modulator. Since phase modulation with an integrated message produces the same instantaneous frequency as FM, the output is an FM signal. This technique was historically important for generating stable FM from crystal-controlled oscillators that could not be directly frequency-modulated.

Given:
PM system with phase sensitivity kp = 2 rad/V
Message: m(t) = 3 cos(2*pi*2000*t), Am = 3 V, fm = 2 kHz

Why this formula applies:
For PM, mp = kp * Am (modulation index) and
frequency deviation depends on fm unlike FM.

Formula:
mp = kp * Am
Peak frequency deviation delta_f = kp * Am * fm
(using rad/V form: delta_f in Hz = mp * fm / 1)

Substitution:
mp = 2 rad/V * 3 V = 6 rad
delta_f = mp * fm = 6 * 2000 Hz

Calculation:
mp = 6 rad (modulation index)
delta_f = 12000 Hz = 12 kHz

Final Answer:
PM modulation index mp = 6 rad, frequency deviation = 12 kHz.
Note: if fm were doubled to 4 kHz, mp stays 6 but delta_f would double to 24 kHz.

Exam Tip: The key FM vs PM distinction is: in FM, mf = kf*Am/fm (decreases with fm). In PM, mp = kp*Am (constant with fm). If a GATE question changes message frequency and asks what happens to modulation index, this distinction gives the answer immediately.

PM versus FM: Core Differences

• In FM, instantaneous frequency varies with m(t). In PM, instantaneous phase varies with m(t) directly.
• FM modulation index mf = kf Am / fm decreases as message frequency fm increases.
• PM modulation index mp = kp Am is constant regardless of message frequency fm.
• In FM, frequency deviation delta_f = kf Am is independent of fm. In PM, delta_f = kp Am fm increases with fm.
• FM signal can be generated from PM by integrating the message before applying to a PM modulator.
• PM signal can be generated from FM by differentiating the message before applying to an FM modulator.
• Both PM and FM are forms of angle modulation with constant amplitude and Bessel function spectra for sinusoidal messages.

Quick Revision

• PM equation: s(t) = Ac cos[2pi fc t + kp m(t)]. Phase deviation: phi(t) = kp m(t).
• PM modulation index: mp = kp Am (radians). Independent of message frequency fm.
• PM instantaneous frequency: fi(t) = fc + (kp/2pi) dm/dt. Frequency tracks derivative of message.
• PM frequency deviation delta_f = kp Am fm -- increases with fm, unlike FM.
• GATE trap: In FM, if fm doubles, mf halves. In PM, if fm doubles, mp stays the same but delta_f doubles.
• FM from PM: integrate message then phase-modulate. PM from FM: differentiate message then frequency-modulate.
• PSK in digital communications is the practical application of phase modulation to discrete data symbols.