Bessel function properties; FM spectrum components

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:

$$S(f) = \frac{A_c}{2}\sum_{n}J_n(\beta)[\delta(f - f_c - nf_m) + \delta(f + f_c + nf_m)]$$

              |           |
        |     |     |     |     |
  |     |     |     |     |     |     |
--+-----+-----+-----+-----+-----+-----+--> 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 $2(\beta + 1)f_m$.