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

# Single tone FM in Bessel functions; average power

Communication Systems · Chapter 3: Angle Modulation

- Asked in: 2074 Bhadra (CS I), 6+2 marks
- Video: [L 47 | Properties of Bessel Function | Angle Modulation| Frequency Modulation | Analog Communication](https://www.youtube.com/watch?v=jI7q8ios9gA) — Dopamine
- 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-average-power
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3

## Question

Derive the time domain expression of single tone FM in terms of Bessel's function Jn(β). Use the result to find the expression of average power of FM signal whose significant Bessel's coefficients are taken from '−n' to '+n'.

## Short answer

s(t) = A_c Σ_n J_n(β) cos[(ω_c+nω_m)t]. Average power with sidebands up to ±n is (A_c²/2)[J_0²+2 Σ_{k=1}^n J_k²]; total power from all sidebands is A_c²/2.

## Answer

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

For $m(t) = A_m\cos\omega_mt$, the instantaneous frequency is $f_i = f_c + k_fA_m\cos\omega_mt$, so the phase is $\omega_ct + \beta\sin\omega_mt$ with $\beta = k_fA_m/f_m = \Delta f/f_m$:

$$
s(t) = A_c\cos(\omega_ct + \beta\sin\omega_mt) = \text{Re}\left[A_ce^{j\beta\sin\omega_mt}e^{j\omega_ct}\right]
$$

$e^{j\beta\sin\omega_mt}$ is periodic with period $1/f_m$; its Fourier series is

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

(substituting $x = \omega_mt$). Therefore

$$
s(t) = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[(\omega_c + n\omega_m)t]
$$

i.e. a carrier $A_cJ_0(\beta)$ at $f_c$ and sidebands $A_cJ_n(\beta)$ at $f_c + nf_m$.

#### Average power with components from −n to +n

Each component is a cosine of amplitude $A_cJ_k(\beta)$; the components are at different frequencies, so their powers add (1 Ω load):

$$
P_n = \sum_{k=-n}^{n}\frac{A_c^2J_k^2(\beta)}{2} = \frac{A_c^2}{2}\left[J_0^2(\beta) + 2\sum_{k=1}^{n}J_k^2(\beta)\right]
$$

using $J_{-k}^2 = J_k^2$. When all components are included, $\sum_{k=-\infty}^{\infty}J_k^2(\beta) = 1$, so

$$
P = \frac{A_c^2}{2}
$$

the same as the unmodulated carrier. The fraction of power within $\pm n$ sidebands is $J_0^2 + 2\sum_{1}^{n}J_k^2$. Example: for $\beta = 1$, $n = 2$ gives $0.7652^2 + 2(0.4401^2 + 0.1149^2) = 0.9992$, i.e. 99.9% of the power.

## Check yourself

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

1. What amplitudes do the FM components have?
   A. All components Ac
   B. Ac / n
   C. Ac sin(n beta)
   D. Carrier Ac J0(beta), sidebands Ac Jn(beta) at fc + n fm
   Answer: D. The expansion is s(t) = Ac sum Jn(beta) cos[(wc + n wm)t]. (Key point: Amplitude of components)

2. What is the power using components from -n to +n?
   A. (Ac^2/2)[J0^2 + 2 sum_{k=1..n} Jk^2]
   B. (Ac^2/2)[J0^2 + sum_{k=1..n} Jk^2]
   C. Ac^2[J0 + 2 sum Jk]
   D. (Ac^2/2)[J0 + 2 sum Jk]^2
   Answer: A. Powers of different-frequency cosines add and J_-k^2 = J_k^2. (Key point: Power with n pairs)

3. What is the total power when all components are included?
   A. Ac^2/2 x beta
   B. Ac^2/2, same as the unmodulated carrier
   C. Ac^2/4
   D. It grows without bound
   Answer: B. Sum of Jk^2 over all k is 1. (Key point: Total power)

4. For beta = 1 and n = 2, what fraction of power is contained?
   A. About 77%
   B. About 58%
   C. About 99.9%
   D. About 50%
   Answer: C. 0.7652^2 + 2(0.4401^2 + 0.1149^2) = 0.9992. (Key point: Example check)

## 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 coefficients; practical bandwidth](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-bessel-practical-bandwidth) — 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
- [Bessel function properties; FM spectrum components](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Abessel-fm-spectrum) — Communication Systems, chapter 3

## 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