---
title: "FM numerical: 102.4 MHz carrier, 75 kHz deviation"
subject: "Communication Systems"
chapter: 3
key: "semester_6/communication_systems:fm-102-4mhz-75khz"
source: https://www.bctengineer.com/q/semester_6/communication_systems/fm-102-4mhz-75khz
site: BCT Engineering
---

# FM numerical: 102.4 MHz carrier, 75 kHz deviation

Communication Systems · Chapter 3: Angle Modulation

- Asked in: 2075 Baisakh (CS I), 3+3+3 marks
- Video: [5. Frequency Modulation Examples | GATE FM Examples in Analog Communication by Engineering Funda](https://www.youtube.com/watch?v=tBAxCMNyoBw) — 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/fm-102-4mhz-75khz
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3

## Question

A 102.4 MHz carrier signal is frequency modulated by a 5 kHz sine wave. The resultant FM signal has frequency deviation of 75 kHz. Now, determine the followings: (a) carrier swing of FM signal, (b) the bandwidth occupied by FM signal and (c) modulation index.

## Short answer

Carrier swing = 150 kHz; bandwidth by Carson's rule = 2(75+5) = 160 kHz; modulation index = 75 kHz / 5 kHz = 15.

## Answer

**Given:** $f_c = 102.4$ MHz, $f_m = 5$ kHz, $\Delta f = 75$ kHz.

#### (a) Carrier swing

The carrier frequency swings between $f_c - \Delta f = 102.325$ MHz and $f_c + \Delta f = 102.475$ MHz:

$$
\text{Carrier swing} = 2\Delta f = 2 \times 75 = 150\ \text{kHz}
$$

#### (b) Bandwidth

By Carson's rule:

$$
B_T = 2(\Delta f + f_m) = 2(75 + 5) = 160\ \text{kHz}
$$

(Equivalently, Carson's rule keeps sidebands up to $n = \beta + 1 = 16$, so $B = 2 \times 16 \times 5 = 160$ kHz. A Bessel-table (1%) estimate would be somewhat wider, but Carson's rule is the standard answer.)

#### (c) Modulation index

$$
\beta = \frac{\Delta f}{f_m} = \frac{75}{5} = 15
$$

**Answer:** carrier swing = 150 kHz; bandwidth = 160 kHz; $\beta = 15$.

## Check yourself

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

1. What is the carrier swing?
   A. 2 x Delta f = 150 kHz
   B. Delta f = 75 kHz
   C. fc + Delta f = 102.475 MHz
   D. Delta f / fm = 15 kHz
   Answer: A. Swing is the total frequency excursion from fc - Delta f to fc + Delta f. (Key point: Carrier swing)

2. Using Carson's rule, what is the bandwidth?
   A. 2 x 75 = 150 kHz
   B. 2(75 + 5) = 160 kHz
   C. 2 x 5 = 10 kHz
   D. 75 + 5 = 80 kHz
   Answer: B. B = 2(Delta f + fm). (Key point: Bandwidth)

3. What is beta?
   A. 5/75
   B. 75 x 5 = 375
   C. 75/5 = 15
   D. 102.4 MHz / 5 kHz
   Answer: C. beta = Delta f / fm. (Key point: Modulation index)

## Related in Communication Systems

- [FM numerical: 10 kHz baseband, 100 MHz carrier, 75 kHz deviation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-10khz-100mhz-75khz) — Communication Systems, chapter 3
- [FM numerical: 5cos(18849.55t), 9 kHz/V — deviation, swing, Carson](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-5cos18849-peak-carson) — Communication Systems, chapter 3
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- [FM numerical: 5cos(18849.55t), 9 kHz/V, 25 V, 88.7 MHz (five parts)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-5cos18849-five-parts) — Communication Systems, chapter 3, asked 3×
- [FM numerical: m(t) = 5cos(18849.55t), 9 kHz/V](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-numerical-5cos18849) — Communication Systems, chapter 3, asked 3×
- [Armstrong FM modulator: find n1 and n2](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aarmstrong-n1-n2-numerical) — Communication Systems, chapter 3

## Related in other subjects

- [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