Single tone FM in Bessel coefficients; practical bandwidth

Single-tone FM in terms of Bessel coefficients

For $m(t) = A_m\cos 2\pi f_m t$:

$$s(t) = A_c\cos\left[2\pi f_c t + \beta\sin 2\pi f_m t\right] = \text{Re}\left\{A_c\, e^{j2\pi f_c t}\, e^{j\beta\sin 2\pi f_m t}\right\}$$

The term $e^{j\beta\sin 2\pi f_m t}$ is periodic with period $1/f_m$, so it can be written as a complex Fourier series:

$$e^{j\beta\sin 2\pi f_m t} = \sum_{n=-\infty}^{\infty} c_n\, e^{j2\pi n f_m t}$$

with coefficients (put $x = 2\pi f_m t$)

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

which is the Bessel function of the first kind of order $n$. Substituting:

$$s(t) = A_c\sum_{n=-\infty}^{\infty} J_n(\beta)\cos\left[2\pi(f_c + n f_m)t\right]$$

So the FM wave has a carrier component $A_cJ_0(\beta)$ and infinitely many side frequencies at $f_c \pm nf_m$ with amplitudes $A_cJ_n(\beta)$, using $J_{-n}(\beta) = (-1)^n J_n(\beta)$.

Properties: $\sum_n J_n^2(\beta) = 1$, so total power $= A_c^2/2$, independent of $\beta$. The carrier amplitude $J_0(\beta)$ falls as $\beta$ increases and is zero at $\beta = 2.405, 5.52, \ldots$

Practical bandwidth of FM

Although the spectrum is infinite, $J_n(\beta)$ becomes very small when $n > \beta$. For a fixed $\beta$, $J_n(\beta)$ decreases rapidly once $n$ exceeds about $\beta + 1$. Keeping the significant sidebands up to $n_{max}$:

$$B = 2\, n_{max}\, f_m$$

Carson's rule: taking $n_{max} = \beta + 1$ (which keeps about 98% of the power):

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

Two limiting cases support this:

• $\beta \gg 1$ (WBFM): $B_T \approx 2\beta f_m = 2\Delta f$ (the carrier sweeps over $\pm\Delta f$).
• $\beta \ll 1$ (NBFM): only $J_0$ and $J_1$ are significant, so $B_T \approx 2f_m$.

1% rule (universal curve): keep all sidebands with $|J_n(\beta)| > 0.01$; this gives a slightly larger bandwidth, e.g. for $\beta = 5$, $n_{max} = 8$ and $B = 16 f_m$ versus Carson's $12 f_m$.

For a general message with highest frequency $W$, use the deviation ratio $D = \Delta f/W$: $B_T = 2(\Delta f + W) = 2W(D+1)$.