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Frequency modulation carries a message by changing the instantaneous frequency of a carrier wave. In the ideal model the wave amplitude stays constant: information changes the rate at which the wave phase advances. Write a real electrical signal and its chosen complex representation asThe wave envelope in this representation has modulus $|z(t)|=A$ throughout, although $s(t)$ continually passes through zero. A smaller instantaneous value of $s(t)$ therefore does not mean a smaller wave amplitude. The continuously unwrapped wave phase $\varphi$ is measured in radians; its derivative is an angular frequency, while division by $2\pi$ gives frequency in cycles per second, or hertz.

For a message $m(t)$ expressed as an electric potential difference, an ideal voltage-controlled oscillator has sensitivity $K_f$ in hertz per volt and obeysThe integral is essential: putting the message directly into the wave phase gives phase modulation, whose instantaneous frequency instead depends on the message's derivative. Amplitude modulation changes the multiplying wave amplitude. These are different ways to encode the same kind of message signal.

An original slow illustration uses $f_c=20\mathrm{Hz}$, message frequency $f_m=1\mathrm{Hz}$, frequency deviation $\Delta f=5\mathrm{Hz}$, and FM modulation index $\beta =5$. With zero initial wave phase,Here $t$ is in seconds. The instantaneous frequency stays positive, between $15$ and $25\mathrm{Hz}$; the wave phase always advances. The oscillations bunch up where $f_i$ is large and spread out where it is small, while the wave envelope remains at $A$.

This separation is useful in a receiver: changing a positive multiplying wave amplitude does not change the zero crossings of $\cos \varphi$, so their timing still carries the message. Noise that moves those crossings, a vanishing wave amplitude, and frequency response distortion can still damage it. The surprising beauty of FM is that a constant wave envelope can conceal a rich frequency spectrum: the components interfere to change timing instead of the envelope's height.

The New Jersey Institute of Technology's Lab 2: FM Modulation, Demodulation and Bessel Zeros, Part I, prepares a voltage-controlled oscillator in low-frequency mode. A frequency counter reads its output while a buffered constant electric potential difference calibrates its sensitivity. Sweeping the control voltage checks linearity; the calibrated gain is then held fixed.^[ref]

An audio oscillator replaces the constant control voltage. An oscilloscope displays the signal and its FFT frequency spectrum together. Zero modulation supplies a carrier reference. Increasing message amplitude reveals sidebands, compares their heights with the carrier, and locates a carrier minimum. These readouts test the spectrum and infer frequency deviation. This is a student procedure, not a report of measured results.^[ref]

A carrier minimum is a minimum of one Fourier coefficient, not disappearance of the whole signal. In the ideal model the first positive zero of $J_0$ is approximately $\beta =2.4048256$, while the wave envelope remains $A$. The lab's introductory special-cases paragraph agrees with this rounded value; step T12 mistakenly repeats the equal-sideband value for the carrier zero.^[ref]

The interpretation also has practical limits. A finite recording limits frequency resolution; inadequate sampling can cause aliasing; and nonlinear VCO response can introduce extra components. Keeping the gain fixed and first checking linearity helps distinguish those effects from a change in FM modulation index. The diagrams above are independent model calculations, not laboratory measurements.

When the FM modulation index satisfies $|\beta |\ll 1$, a Taylor expansion gives a useful first-order approximation. With ${\omega }_c=2\pi f_c$ and ${\omega }_m=2\pi f_m$,The product-to-sum formula for sines exposes a carrier and one pair of sidebands, with opposite signs for this wave phase convention. Higher sidebands are neglected, not identically absent. For $\beta =0.1$, the predicted normalized first-sideband magnitude $0.05$ is close to the original exact evaluation $J_1(0.1)\approx 0.0499375$. The neglected waveform term is bounded in absolute value by ${\beta }^2/2=0.005$, using the second-derivative bound for the cosine. Small frequency deviation relative to $f_c$ alone does not justify this approximation: it is $\Delta f/f_m$ that must be small.

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: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.^[ref]

Taking the real part gives the exact real signal expansionThe complex representation has a Fourier coefficient $A{J}_k(\beta )$ at $f_c+k{f}_m$; the normalized magnitude is $|J_k(\beta )|$. The identity $J_{-k}(\beta )=(-1{)}^k{J}_k(\beta )$ 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 $J_0(5)\approx -0.17760$, $J_1(5)\approx -0.32758$, $J_5(5)\approx 0.26114$ and $J_6(5)\approx 0.13105$. 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 $n{f}_m$, $n>0$, 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.

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:At $\beta =5$, the sum through $|k|\le 6$ 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.