---
title: "Short note: Phase modulation"
subject: "Communication Systems"
chapter: 3
key: "semester_6/communication_systems:short-note-phase-modulation"
source: https://www.bctengineer.com/q/semester_6/communication_systems/short-note-phase-modulation
site: BCT Engineering
---

# Short note: Phase modulation

Communication Systems · Chapter 3: Angle Modulation

- Asked in: 2065 Kartik (CS I), 4 marks
- Video: [Phase Modulation (Basics, Definition and Waveform) [HD]](https://www.youtube.com/watch?v=OQfPdhVfShw) — Engineering Made Easy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/short-note-phase-modulation
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3

## Question

Write a short note on phase modulation.

## Short answer

Phase modulation varies the instantaneous carrier phase linearly with the message: s(t)=A_c cos(2π f_c t + k_p m(t)). The frequency deviation depends on dm/dt. For a tone, the modulation index β_p=k_p A_m is independent of f_m. PM has constant envelope and can be generated by differentiating the message before angle modulation.

## Answer

**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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In phase modulation, what varies linearly with the message?
   A. Carrier amplitude
   B. Instantaneous phase of the carrier
   C. Instantaneous frequency
   D. Carrier power
   Answer: B. PM: s(t) = Ac cos[2π fc t + kp m(t)], so the phase follows m(t) while amplitude stays constant. (Key point: PM definition)

2. In PM, the instantaneous frequency deviation is proportional to what?
   A. The message m(t)
   B. The integral of m(t)
   C. The rate of change dm/dt
   D. The message amplitude squared
   Answer: C. fi = fc + (kp/2π) dm/dt, so deviation depends on the rate of change of the message. (Key point: Frequency in PM)

3. For a single-tone message, how does PM deviation vary as fm increases?
   A. Stays constant
   B. Decreases
   C. The PM index increases
   D. The index kp Am stays fixed but delta f = kp Am fm increases
   Answer: D. beta_p = kp Am is independent of fm, so the frequency deviation kp Am fm grows with fm. (Key point: PM index vs fm)

4. How can PM be obtained using an FM modulator?
   A. Pass the message through an integrator first
   B. Pass the message through a differentiator first
   C. Pass the message through a limiter first
   D. Square the message first
   Answer: B. PM of m(t) equals FM of dm/dt; the reverse (integrator plus phase modulator) gives FM. (Key point: FM-PM relation)

## Related in Communication Systems

- [Differential coding; DPSK modulation and detection](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adifferential-coding-dpsk) — Communication Systems, chapter 7
- [DSB-FC vs DSB-SC; phase shift method for SSB](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbfc-dsbsc-ssb-phase-shift) — Communication Systems, chapter 2
- [General FM expression; FM using phase modulator](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-general-expression-phase-modulator) — Communication Systems, chapter 3