= Single-tone FM sidebands
{c}
{title2=$f_k=f_c+kf_m$}
A single-tone <FM> <signal> has components displaced from the <carrier wave> <frequency> by integer multiples of the message <frequency>. These are temporal sidebands; the <spatial sidebands> used to perturb a spatial pattern are a related but different setting. To find the temporal components, apply the <Jacobi-Anger expansion> to the complex representation:
$$
e^{\mathrm i\beta\sin\theta}=\sum_{k=-\infty}^{\infty}J_k(\beta)e^{\mathrm ik\theta},\qquad
z(t)=A\sum_{k=-\infty}^{\infty}J_k(\beta)e^{2\pi\mathrm i(f_c+kf_m)t}.
$$
Here $J_k$ is a <Bessel function of the first kind>. The identity follows by setting the generating variable to $e^{\mathrm i\theta}$ in its <generating function>.<NIST Digital Library of Mathematical Functions>[Chapter 10, §10.12, equation 10.12.1 and the substitution described in the section notes]{ref}
Taking the <real part> gives the exact real <signal> expansion
$$
s(t)=A\sum_{k=-\infty}^{\infty}J_k(\beta)\cos\bigl(2\pi(f_c+kf_m)t\bigr).
$$
The complex representation has a <Fourier coefficient> $AJ_k(\beta)$ at $f_c+kf_m$; the normalized magnitude is $|J_k(\beta)|$. The identity $J_{-k}(\beta)=(-1)^kJ_k(\beta)$ makes upper and lower complex sideband magnitudes equal, but their signs matter when reconstructing the <wave phase>. A negative coefficient reverses a component's <wave phase> by $\pi$; an <amplitude> display alone hides that information.
For $\beta=5$, original evaluation gives $J_0(5)\approx-0.17760$, $J_1(5)\approx-0.32758$, $J_5(5)\approx0.26114$ and $J_6(5)\approx0.13105$. Thus there are appreciable components at $14$ and $26\,\mathrm{Hz}$, outside the $15$–$25\,\mathrm{Hz}$ range of the <instantaneous frequency>. There is no contradiction: <instantaneous frequency> describes the local rate of the combined <wave phase>, while the <frequency spectrum> describes a global <superposition> of constant-<frequency> components.
For the real <signal>, negative <frequencies> must also be accounted for: a <cosine> at $-f$ equals one at $f$. When $f_c/f_m=20$, the normalized positive-<frequency> <cosine> coefficient at $nf_m$, $n>0$, is exactly $J_{n-20}(\beta)+J_{-n-20}(\beta)$. The second term is extremely small over the plotted range, but the figure includes it. The chosen complex representation need not have a <frequency spectrum> supported only at positive <frequencies>, even when its specified <instantaneous frequency> is everywhere positive.
\Image[/frequency-modulation-sideband-amplitude-comparison.png]
{title=FM sideband amplitudes for three modulation indices at fixed carrier and message frequencies}
{description=Original one-sided cosine-amplitude spectra for $f_c=20\,\mathrm{Hz}$ and $f_m=1\,\mathrm{Hz}$. Shading marks the instantaneous-frequency range, not a spectral cutoff. The middle modulation index is near the first zero of $J_0$; all three modeled envelopes have amplitude $A$. Only a finite set of lines is displayed.}
For a nonzero <FM modulation index>, the ideal <frequency spectrum> has infinitely many sidebands. Many become tiny, but drawing only visible lines does not make the exact <signal bandwidth> finite. A useful engineering width must state a threshold or a retained-power fraction. For the complex representation, <Parseval's identity> gives a simple check:
$$
\frac{1}{2\pi}\int_0^{2\pi}\bigl|e^{\mathrm i\beta\sin\theta}\bigr|^2\,d\theta
=\sum_{k=-\infty}^{\infty}|J_k(\beta)|^2=1.
$$
At $\beta=5$, the sum through $|k|\leq6$ is approximately $0.99356$. Retaining those complex components captures over $99\%$ of their squared magnitudes and spans $12\,\mathrm{Hz}$ from $14$ to $26\,\mathrm{Hz}$. This is a stated truncation criterion, not an exact cutoff or a universal bandwidth rule. It also explains why a diminished carrier component need not imply a diminished envelope: its contribution has been redistributed among sidebands.
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