Single-tone FM in terms of Bessel coefficients

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