---
title: "FM numerical: 5cos(18849.55t), 9 kHz/V — deviation, swing, Carson"
subject: "Communication Systems"
chapter: 3
key: "semester_6/communication_systems:fm-5cos18849-peak-carson"
source: https://www.bctengineer.com/q/semester_6/communication_systems/fm-5cos18849-peak-carson
site: BCT Engineering
---

# FM numerical: 5cos(18849.55t), 9 kHz/V — deviation, swing, Carson

Communication Systems · Chapter 3: Angle Modulation

- Asked in: 2071 Bhadra (CS I), 2×4 marks
- Video: [PS4: Finding the frequency deviation, swing, bandwidth and power](https://www.youtube.com/watch?v=eic3hsHW-BM) — Dr Waleed Al-Nuaimy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/fm-5cos18849-peak-carson
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3

## Question

A modulating signal m(t) = 5 cos 18849.55t is applied to an FM modulator that has a frequency sensitivity of 9 kHz/V. Compute (i) peak frequency deviation, (ii) modulation index, (iii) frequency swing and (iv) Carson's bandwidth.

## Short answer

Peak frequency deviation is kf*Am = 9*5 = 45 kHz; modulation index is 45/3 = 15; frequency swing is 2*45 = 90 kHz; Carson bandwidth is 2(45+3) = 96 kHz.

## Answer

**Given:** $m(t) = 5\cos(18849.55\,t)$, $k_f = 9$ kHz/V.

So $A_m = 5$ V and

$$
f_m = \frac{\omega_m}{2\pi} = \frac{18849.55}{2\pi} = 3000\ \text{Hz} = 3\ \text{kHz}
$$

#### (i) Peak frequency deviation

$$
\Delta f = k_f A_m = 9\ \text{kHz/V} \times 5\ \text{V} = 45\ \text{kHz}
$$

#### (ii) Modulation index

$$
\beta = \frac{\Delta f}{f_m} = \frac{45\ \text{kHz}}{3\ \text{kHz}} = 15
$$

#### (iii) Frequency swing

The carrier swings from $f_c - \Delta f$ to $f_c + \Delta f$:

$$
\text{Swing} = 2\Delta f = 2 \times 45 = 90\ \text{kHz}\ (\pm 45\ \text{kHz})
$$

#### (iv) Carson's bandwidth

$$
B_T = 2(\Delta f + f_m) = 2(45 + 3) = 96\ \text{kHz}
$$

**Answer:** $\Delta f = 45$ kHz, $\beta = 15$, swing = 90 kHz, $B_T = 96$ kHz.

## Check yourself

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

1. For m(t) = 5 cos(18849.55 t), what is the modulating frequency?
   A. 18849.55 Hz
   B. 3 kHz
   C. 5 kHz
   D. 118 kHz
   Answer: B. fm = 18849.55/(2π) = 3000 Hz. (Key point: Find fm)

2. With kf = 9 kHz/V and Am = 5 V, what is the peak frequency deviation?
   A. 9 kHz
   B. 14 kHz
   C. 45 kHz
   D. 1.8 kHz
   Answer: C. delta f = kf Am = 9 x 5 = 45 kHz. (Key point: Peak deviation)

3. What are the modulation index and frequency swing?
   A. beta = 3, swing = 45 kHz
   B. beta = 15, swing = 45 kHz
   C. beta = 5, swing = 90 kHz
   D. beta = 15, swing = 90 kHz
   Answer: D. beta = 45/3 = 15 and swing = 2 delta f = 90 kHz. (Key point: Modulation index and swing)

4. What is Carson's bandwidth for this signal?
   A. 96 kHz
   B. 90 kHz
   C. 48 kHz
   D. 6 kHz
   Answer: A. BT = 2(delta f + fm) = 2(45 + 3) = 96 kHz. (Key point: Carson's bandwidth)

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