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

# Single-tone FM in terms of Bessel coefficients

Communication Systems · Chapter 3: Angle Modulation

- Asked in: 2075 Bhadra (CS I), 8 marks; 2072 Asoj (CS I), 6 marks; 2071 Bhadra (CS I), 6 marks; 2070 Magh (CS I), 6 marks; 2068 Bhadra (CS I), 8 marks (asked 5 times)
- Video: [Bessel Functions And the Spectrum of FM Signals | Engineering & Math Insights || Lec-42](https://www.youtube.com/watch?v=8gVXtFuigFI) — Engg. & Math Insights
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/single-tone-fm-bessel
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3

## Question

Derive the expression for single tone modulated FM signal in terms of Bessel coefficients.

## Short answer

The single-tone FM signal s(t)=A_c cos(ω_c t+β sin ω_m t) expands via Bessel functions to s(t)=A_c Σ_{n=-∞}^∞ J_n(β) cos[(ω_c+n ω_m)t]. The spectrum has a carrier at f_c (amplitude A_c J_0(β)) and infinite sidebands at f_c±n f_m with amplitudes A_c|J_n(β)|; total power remains constant.

## Answer

In **frequency modulation**, the instantaneous frequency of the carrier varies linearly with the message. For a single tone, the FM wave can be expanded with Bessel functions into a carrier and an infinite set of sidebands.

#### Instantaneous frequency and phase

Let $m(t) = A_m\cos\omega_mt$ and $c(t) = A_c\cos\omega_ct$. The instantaneous frequency is

$$
f_i(t) = f_c + k_fm(t) = f_c + k_fA_m\cos\omega_mt = f_c + \Delta f\cos\omega_mt
$$

where $k_f$ is the frequency sensitivity (Hz/V) and $\Delta f = k_fA_m$ is the peak frequency deviation. The phase is

$$
\theta_i(t) = 2\pi\int_0^t f_i(\tau)\,d\tau = \omega_ct + \frac{\Delta f}{f_m}\sin\omega_mt = \omega_ct + \beta\sin\omega_mt
$$

with modulation index $\beta = \Delta f/f_m$. So

$$
s(t) = A_c\cos(\omega_ct + \beta\sin\omega_mt)
$$

#### Complex envelope and Fourier series

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

The term $\tilde s(t) = A_ce^{j\beta\sin\omega_mt}$ is periodic with period $T_m = 1/f_m$, so it has a Fourier series:

$$
\tilde s(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_mt}
$$

$$
c_n = \frac{1}{T_m}\int_{-T_m/2}^{T_m/2} A_c e^{j\beta\sin\omega_mt}e^{-jn\omega_mt}dt
$$

Put $x = \omega_mt$ ($dt = dx/\omega_m$, limits $-\pi$ to $\pi$):

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

where $J_n(\beta)$ is the **Bessel function of the first kind of order n**:

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

#### Final expression

$$
\tilde s(t) = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)e^{jn\omega_mt}
$$

$$
s(t) = \text{Re}\left[A_c\sum_n J_n(\beta)e^{j(\omega_c + n\omega_m)t}\right] = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[(\omega_c + n\omega_m)t]
$$

Expanded, using $J_{-n}(\beta) = (-1)^nJ_n(\beta)$:

$$
\begin{aligned}
s(t) &= A_cJ_0(\beta)\cos\omega_ct \\
&\quad + A_cJ_1(\beta)[\cos(\omega_c+\omega_m)t - \cos(\omega_c-\omega_m)t] \\
&\quad + A_cJ_2(\beta)[\cos(\omega_c+2\omega_m)t + \cos(\omega_c-2\omega_m)t] + \dots
\end{aligned}
$$

#### Conclusions

1. The spectrum has a carrier at $f_c$ (amplitude $A_cJ_0(\beta)$) and infinite sidebands at $f_c \pm nf_m$ with amplitudes $A_c|J_n(\beta)|$.
2. Odd-order lower sidebands have reversed sign.
3. The carrier amplitude depends on $\beta$ and becomes zero when $J_0(\beta) = 0$ (β = 2.405, 5.52, ...).
4. Since $\sum J_n^2(\beta) = 1$, total power is constant: $P = A_c^2/2$.
5. Only about $\beta + 1$ sidebands on each side are significant, giving Carson's rule $B_T \approx 2(\beta+1)f_m = 2(\Delta f + f_m)$.

## Check yourself

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

1. For m(t) = Am cos ωm t, what is the FM wave?
   A. Ac cos(ωc t + β cos ωm t)
   B. Ac[1 + β sin ωm t] cos ωc t
   C. Ac cos(ωc t + β sin ωm t)
   D. Ac cos(ωc t + β ωm t)
   Answer: C. Integrating the instantaneous frequency gives phase ωc t + (Δf/fm) sin ωm t, with β = Δf/fm. (Key point: Phase of single-tone FM)

2. What are the Fourier series coefficients of the complex envelope Ac e^{jβ sin ωm t}?
   A. cn = Ac Jn(β)
   B. cn = Ac β^n
   C. cn = Ac sin(nβ)
   D. cn = Ac/n
   Answer: A. The coefficient integral equals the Bessel function of the first kind of order n, so cn = Ac Jn(β). (Key point: Fourier coefficients)

3. What does s(t) = Ac Σ Jn(β) cos[(ωc + nωm)t] say about the spectrum?
   A. Only two sidebands exist
   B. A carrier Ac J0(β) plus infinite sidebands at fc ± n fm with amplitudes Ac|Jn(β)|
   C. A carrier of fixed amplitude Ac plus sidebands
   D. Sidebands only at odd multiples of fm
   Answer: B. The n = 0 term is the carrier with amplitude Ac J0(β); other n give sidebands at fc ± n fm. (Key point: Resulting spectrum)

4. Why is FM total power constant, and what is the bandwidth rule?
   A. Σ Jn^2(β) = 1, so P = Ac^2/2; Carson: BT ≈ 2(Δf + fm)
   B. Σ Jn(β) = 1, so P = Ac^2; BT = 2fm
   C. J0(β) is always 1; BT = 2Δf
   D. Sidebands cancel the carrier; BT = fm
   Answer: A. The sum of squares of Bessel coefficients is 1, so power is Ac^2/2 independent of β. Carson's rule gives about 2(Δf + fm). (Key point: Constant power and bandwidth)

## Related in Communication Systems

- [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
- [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
- [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
- [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
- [FM expression after change of modulating signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-new-waveform-500hz-2khz) — 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