---
title: "Angle modulation; single tone FM time and frequency domain"
subject: "Communication Systems"
chapter: 3
key: "semester_6/communication_systems:angle-mod-single-tone-fm-time-freq"
source: https://www.bctengineer.com/q/semester_6/communication_systems/angle-mod-single-tone-fm-time-freq
site: BCT Engineering
---

# Angle modulation; single tone FM time and frequency domain

Communication Systems · Chapter 3: Angle Modulation

- Asked in: 2075 Baisakh (CS I), 2+3+3 marks
- Video: [Single Tone Frequency Modulation Explained: Frequency Deviation, Modulation Index & Deviation Ratio](https://www.youtube.com/watch?v=T4FjXxkt-HE) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/angle-mod-single-tone-fm-time-freq
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3

## Question

What is angle modulation? Find the time domain and frequency domain expression of single tone modulated FM signal.

## Short answer

Angle modulation: carrier angle varies with message, amplitude constant. FM: θ_i=2π f_c t+2π k_f∫ m dt. For m(t)=A_m cos 2π f_m t, s(t)=A_c cos(2π f_c t+β sin 2π f_m t), β=Δf/f_m=k_f A_m/f_m. Frequency domain: S(f)=A_c/2 Σ J_n(β)[δ(f-f_c-n f_m)+δ(f+f_c+n f_m)], sidebands at f_c±n f_m.

## Answer

#### Angle modulation

**Angle modulation** is modulation in which the angle (phase) of the carrier is varied according to the message while its amplitude stays constant:

$$
s(t) = A_c\cos\theta_i(t)
$$

It has two forms: **phase modulation** (PM), $\theta_i = \omega_ct + k_pm(t)$, and **frequency modulation** (FM), $\theta_i = \omega_ct + 2\pi k_f\int m(\tau)d\tau$.

#### Time-domain expression of single-tone FM

Let $m(t) = A_m\cos2\pi f_mt$. Instantaneous frequency:

$$
f_i(t) = f_c + k_fA_m\cos2\pi f_mt = f_c + \Delta f\cos2\pi f_mt
$$

Instantaneous phase:

$$
\theta_i(t) = 2\pi\int_0^tf_i(\tau)d\tau = 2\pi f_ct + \frac{\Delta f}{f_m}\sin2\pi f_mt
$$

So

$$
s(t) = A_c\cos(2\pi f_ct + \beta\sin2\pi f_mt), \qquad \beta = \frac{\Delta f}{f_m} = \frac{k_fA_m}{f_m}
$$

#### Frequency-domain expression

Write $s(t) = \text{Re}[A_ce^{j\beta\sin\omega_mt}e^{j\omega_ct}]$. The periodic term has Fourier series

$$
e^{j\beta\sin\omega_mt} = \sum_{n=-\infty}^{\infty}J_n(\beta)e^{jn\omega_mt}, \quad J_n(\beta) = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{j(\beta\sin x - nx)}dx
$$

Therefore

$$
s(t) = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c + nf_m)t]
$$

Taking the Fourier transform:

$$
S(f) = \frac{A_c}{2}\sum_{n=-\infty}^{\infty}J_n(\beta)\left[\delta(f - f_c - nf_m) + \delta(f + f_c + nf_m)\right]
$$

The spectrum has a carrier of amplitude $A_cJ_0(\beta)$ and sideband pairs at $f_c \pm nf_m$ with amplitudes $A_c|J_n(\beta)|$. Total power is $A_c^2/2$, and practical bandwidth is $2(\beta + 1)f_m$.

## Check yourself

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

1. What is varied in angle modulation?
   A. The amplitude of the carrier
   B. The phase/angle of the carrier, with constant amplitude
   C. Only the carrier power
   D. The message bandwidth
   Answer: B. PM and FM both vary the carrier angle. (Key point: Angle modulation)

2. For m(t) = Am cos 2pi fm t, what is the FM phase?
   A. 2pi fc t + Delta f cos 2pi fm t
   B. 2pi fc t + fm sin 2pi Delta f t
   C. 2pi fc t + (Delta f/fm) sin 2pi fm t
   D. 2pi fc t + (fm/Delta f) sin 2pi fm t
   Answer: C. Integrating fi = fc + Delta f cos(2pi fm t) gives the phase with beta = Delta f/fm. (Key point: Instantaneous phase)

3. What is the spectrum of single-tone FM?
   A. Carrier Ac J0(beta) plus sideband pairs at fc +- n fm with Ac|Jn(beta)|
   B. Carrier and one sideband pair only
   C. Single line at fc
   D. Lines at n fm with equal amplitude
   Answer: A. Bessel-expansion gives an infinite set of lines spaced fm apart. (Key point: Frequency domain)

4. What is the practical bandwidth quoted?
   A. 2 beta
   B. fm / beta
   C. (beta + 1) / 2 fm
   D. 2(beta + 1) fm
   Answer: D. This is Carson's rule form. (Key point: Practical bandwidth)

## Related in Communication Systems

- [Single tone DSB-FC AM in time and frequency domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asingle-tone-dsbfc-time-freq) — Communication Systems, chapter 2
- [DSB-FC single tone expression and modulation index](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbfc-single-tone-index) — Communication Systems, chapter 2
- [SQNR of delta modulation for a single-tone signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adm-sqnr-single-tone) — Communication Systems, chapter 4
- [Single tone FM in Bessel coefficients; practical bandwidth](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-bessel-practical-bandwidth) — Communication Systems, chapter 3
- [Single tone FM in Bessel functions; average power](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-bessel-average-power) — Communication Systems, chapter 3
- [Need for modulation; single tone AM expressions](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amodulation-need-am-single-tone) — Communication Systems, chapter 2, asked 2×

## Related in other subjects

- [Single tone FM parameters](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afm-single-tone-numeric) — Data Communication, chapter 5
- [Modulation; AM and FM modulated waves](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Amodulation-am-fm-wave) — Basic Electronics Engineering, chapter 5
- [Frequency modulation](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afm-explain) — Data Communication, chapter 5