--- title: "Bessel function properties; FM spectrum components" subject: "Communication Systems" chapter: 3 key: "semester_6/communication_systems:bessel-fm-spectrum" source: https://www.bctengineer.com/q/semester_6/communication_systems/bessel-fm-spectrum site: BCT Engineering --- # Bessel function properties; FM spectrum components Communication Systems · Chapter 3: Angle Modulation - Asked in: 2076 Bhadra (CS I), 2+6 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/bessel-fm-spectrum - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3 ## Question What are the properties of Bessel function? Show that a FM signal consists infinite number of cosine signal components centered at frequencies fc + n fm, where fc = carrier frequency, fm = message frequency and n = 0, ±1, ±2, ±3, .... ## Short answer Bessel functions: J_{-n}=(-1)^n J_n, total power ∑J_n^2=1, negligible for n>β+1. Single-tone FM s(t)=A_c cos(2πf_ct+β sin 2πf_mt) expands as A_c Σ J_n(β) cos[2π(f_c+n f_m)t] because e^{jβ sin ω_mt} has Fourier series coefficients J_n(β), giving infinite sidebands at f_c ± n f_m. ## Answer #### Properties of Bessel functions $J_n(\beta)$ 1. $J_{-n}(\beta) = (-1)^nJ_n(\beta)$: negative-order coefficients equal positive ones, with sign change for odd $n$. 2. $\sum_{n=-\infty}^{\infty}J_n^2(\beta) = 1$: total FM power is constant. 3. For small $\beta$: $J_0(\beta) \approx 1$, $J_1(\beta) \approx \beta/2$, $J_n(\beta) \approx 0$ for $n \ge 2$ (NBFM). 4. For a given $\beta$, $J_n(\beta)$ becomes very small when $n > \beta + 1$, so only a finite number of sidebands are significant. 5. $J_n(\beta)$ is oscillatory; $J_0(\beta) = 0$ at β = 2.405, 5.52, 8.65, ... (carrier disappears). #### FM wave as a sum of cosines at $f_c + nf_m$ Single-tone FM: $$ s(t) = A_c\cos(2\pi f_ct + \beta\sin2\pi f_mt), \qquad \beta = \frac{\Delta f}{f_m} $$ Write it using the complex envelope: $$ s(t) = \text{Re}\left[A_ce^{j\beta\sin\omega_mt}e^{j\omega_ct}\right] $$ $\tilde s(t) = A_ce^{j\beta\sin\omega_mt}$ is periodic in $T_m = 1/f_m$, so $$ \tilde s(t) = \sum_{n=-\infty}^{\infty}c_ne^{jn\omega_mt}, \qquad c_n = f_m\int_{-1/2f_m}^{1/2f_m}A_ce^{j\beta\sin\omega_mt}e^{-jn\omega_mt}dt $$ Put $x = \omega_mt$: $$ c_n = \frac{A_c}{2\pi}\int_{-\pi}^{\pi}e^{j(\beta\sin x - nx)}dx = A_cJ_n(\beta) $$ Hence $$ s(t) = \text{Re}\left[\sum_{n=-\infty}^{\infty}A_cJ_n(\beta)e^{j2\pi(f_c + nf_m)t}\right] = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c + nf_m)t] $$ This proves that the FM signal is a sum of **infinitely many cosine components** at $f_c + nf_m$, $n = 0, \pm1, \pm2, \dots$, with amplitudes $A_cJ_n(\beta)$. Spectrum: $$ S(f) = \frac{A_c}{2}\sum_{n}J_n(\beta)[\delta(f - f_c - nf_m) + \delta(f + f_c + nf_m)] $$ ```text | | | | | | | | | | | | | | --+-----+-----+-----+-----+-----+-----+--> f -3 -2 -1 0 +1 +2 +3 <- n (line n is at f = fc + n fm; lines spaced fm apart, heights A_c|J_n(beta)|) ``` Although infinite in theory, sidebands with $|n| > \beta + 1$ are negligible, giving Carson's bandwidth $2(\beta + 1)f_m$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which statement about Bessel coefficients is correct? A. The sum of J_n^2(beta) over all n equals 1, so FM power is constant B. The sum of J_n(beta) equals 1 C. J_n is non-zero only for n <= 1 D. J_0(beta) always increases with beta Answer: A. Sum of squares equals 1, giving total power Ac^2/2 regardless of beta. (Key point: Bessel property) 2. Why can the FM envelope exp(j beta sin wm t) be expanded as a Fourier series? A. It is real and even B. It is periodic with period 1/fm C. It has finite energy D. It is zero outside one period Answer: B. A periodic function has Fourier coefficients c_n, which turn out to be J_n(beta). (Key point: Fourier series idea) 3. What does the expansion show about the FM spectrum? A. Only lines at fc +- fm B. Lines at n fc with amplitude Ac cos(n) C. Lines at fc + n fm with amplitudes Ac J_n(beta) D. A continuous flat spectrum Answer: C. s(t) = Ac sum J_n(beta) cos[2pi(fc + n fm)t]. (Key point: Resulting spectrum) 4. Why is the infinite spectrum still of finite practical bandwidth? A. Sidebands vanish exactly beyond n = 1 B. The carrier absorbs all higher sidebands C. Filters in the modulator remove them D. J_n(beta) is negligible for n > beta + 1, giving B = 2(beta+1)fm Answer: D. Higher-order Bessel coefficients become very small. 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