---
title: "Single tone FM in Bessel coefficients; practical bandwidth"
subject: "Communication Systems"
chapter: 3
key: "semester_6/communication_systems:fm-bessel-practical-bandwidth"
source: https://www.bctengineer.com/q/semester_6/communication_systems/fm-bessel-practical-bandwidth
site: BCT Engineering
---

# Single tone FM in Bessel coefficients; practical bandwidth

Communication Systems · Chapter 3: Angle Modulation

- Asked in: 2069 Bhadra (CS I), 6+4 marks
- Video: [Carson's Rule in FM (Calculating Bandwidth of FM Signal) - Carson Rule [HD]](https://www.youtube.com/watch?v=5AXJ6TrWFzQ) — 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/fm-bessel-practical-bandwidth
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3

## Question

Find the time domain expression for single tone FM modulated wave, in terms of Bessel coefficients. Derive the expression for estimating practical bandwidth of a FM signal.

## Answer

#### Single-tone FM in terms of Bessel coefficients

For $m(t) = A_m\cos 2\pi f_m t$:

$$
s(t) = A_c\cos\left[2\pi f_c t + \beta\sin 2\pi f_m t\right] = \text{Re}\left\{A_c\, e^{j2\pi f_c t}\, e^{j\beta\sin 2\pi f_m t}\right\}
$$

The term $e^{j\beta\sin 2\pi f_m t}$ is periodic with period $1/f_m$, so it can be written as a complex Fourier series:

$$
e^{j\beta\sin 2\pi f_m t} = \sum_{n=-\infty}^{\infty} c_n\, e^{j2\pi n f_m t}
$$

with coefficients (put $x = 2\pi f_m t$)

$$
c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} e^{j(\beta\sin x - nx)}\,dx = J_n(\beta)
$$

which is the **Bessel function of the first kind** of order $n$. Substituting:

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

So the FM wave has a carrier component $A_cJ_0(\beta)$ and infinitely many side frequencies at $f_c \pm nf_m$ with amplitudes $A_cJ_n(\beta)$, using $J_{-n}(\beta) = (-1)^n J_n(\beta)$.

Properties: $\sum_n J_n^2(\beta) = 1$, so total power $= A_c^2/2$, independent of $\beta$. The carrier amplitude $J_0(\beta)$ falls as $\beta$ increases and is zero at $\beta = 2.405, 5.52, \ldots$

#### Practical bandwidth of FM

Although the spectrum is infinite, $J_n(\beta)$ becomes very small when $n > \beta$. For a fixed $\beta$, $J_n(\beta)$ decreases rapidly once $n$ exceeds about $\beta + 1$. Keeping the significant sidebands up to $n_{max}$:

$$
B = 2\, n_{max}\, f_m
$$

**Carson's rule:** taking $n_{max} = \beta + 1$ (which keeps about 98% of the power):

$$
B_T = 2(\beta + 1)f_m = 2(\Delta f + f_m)
$$

Two limiting cases support this:

- $\beta \gg 1$ (WBFM): $B_T \approx 2\beta f_m = 2\Delta f$ (the carrier sweeps over $\pm\Delta f$).
- $\beta \ll 1$ (NBFM): only $J_0$ and $J_1$ are significant, so $B_T \approx 2f_m$.

**1% rule (universal curve):** keep all sidebands with $|J_n(\beta)| > 0.01$; this gives a slightly larger bandwidth, e.g. for $\beta = 5$, $n_{max} = 8$ and $B = 16 f_m$ versus Carson's $12 f_m$.

For a general message with highest frequency $W$, use the deviation ratio $D = \Delta f/W$: $B_T = 2(\Delta f + W) = 2W(D+1)$.

## Check yourself

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

1. What does the Bessel form s(t) = Ac sum Jn(beta) cos[2π(fc + n fm)t] show?
   A. FM has only one pair of sidebands
   B. FM has a carrier Ac J0(beta) and infinitely many sidebands at fc +/- n fm with amplitudes Ac Jn(beta)
   C. FM has no carrier
   D. FM has sidebands only at fc + fm
   Answer: B. Expanding the periodic exponential as a Fourier series gives Bessel-function coefficients for carrier and every side frequency. (Key point: Bessel expansion)

2. Since the sum of Jn^2(beta) equals 1, what is the total FM power?
   A. Ac^2/2, independent of beta
   B. Ac^2 (1 + beta^2)/2
   C. Ac^2 beta/2
   D. It increases with beta
   Answer: A. Total power is Ac^2/2 for all beta; modulation only redistributes power between carrier and sidebands. (Key point: Constant power)

3. Taking nmax = beta + 1, what bandwidth results?
   A. 2 beta fm
   B. 2(beta + 1) fm = 2(delta f + fm)
   C. (beta + 1) fm
   D. 2 beta^2 fm
   Answer: B. B = 2 nmax fm = 2(beta + 1) fm = 2(delta f + fm), keeping about 98% of the power. (Key point: Carson's rule)

4. What does Carson's rule give for beta << 1 and for beta >> 1?
   A. 2fm for large beta; 2 delta f for small beta
   B. 2fm for both
   C. About 2fm for NBFM and about 2 delta f for WBFM
   D. 2 delta f for both
   Answer: C. For small beta only J0 and J1 matter so B ~ 2fm; for large beta B ~ 2 beta fm = 2 delta f. (Key point: Limiting cases)

## Related in Communication Systems

- [Single-tone FM in terms of Bessel coefficients](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asingle-tone-fm-bessel) — Communication Systems, chapter 3, asked 5×
- [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
- [Angle modulation; single tone FM time and frequency domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aangle-mod-single-tone-fm-time-freq) — Communication Systems, chapter 3
- [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
- [NBFM time domain expression; NBFM to WBFM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anbfm-time-domain-wbfm) — Communication Systems, chapter 3
- [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

## 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