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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.

Gear vibration order expresses the frequency of a measured vibration component relative to a specified shaft's rotation rate:Here $f$ is frequency in hertz and $n$ is shaft speed in revolutions per minute. The dimensionless order is cycles per shaft revolution. It must name the reference shaft: the same gear mesh frequency has different orders relative to the two gears in one mesh. Order tracking follows such rotation-related components as speed changes.^[ref]

For a24-tooth input at1500 revolutions per minute, $f_r=25\mathrm{Hz}$ and mesh frequency is600Hz, so mesh order is24. Its60-tooth mate turns at10Hz, giving mesh order60 relative to that shaft. If the input rises to1800 revolutions per minute, mesh frequency becomes720Hz but input-relative mesh order remains24. An unrelated structural natural frequency at600Hz would instead move from order24 to order20 as reference speed increases.

Angular resampling makes the distinction explicit. Let $\theta (t)=2\pi {\int }_0^t{f}_r(u)du$ be shaft angle. A tooth-periodic illustrative signal $a(t)=A\cos [N\theta (t)]$ can spread across many fixed-frequency bins during acceleration, while as a function of angle it remains $a(\theta )=A\cos (N\theta )$, with order $N$. Enough angle samples are needed to resolve the desired tooth-related oscillations. This derivation describes the resampling approach; not every order tracking algorithm uses that implementation.

Rotation-synchronous amplitude modulation can produce neighboring modulation sidebands: a24th-order component modulated once per input revolution has orders23 and25. Modulation associated with another shaft need not give integer spacing in the chosen reference. Orders identify kinematic relationships, while amplitudes still depend on loads, mounting and frequency response. A high mesh order or a sideband is therefore not itself a diagnosis of damaged teeth.^[ref]

A gear vibration spectrum describes how a measured gearbox motion signal is distributed over frequency. An accelerometer on a housing supplies a response spectrum; a rotary encoder supplies a rotation reference. Gear mesh frequency predicts tooth-passage components, but their observed amplitudes also depend on applied loads, sensor direction and the mechanical frequency response. A response peak alone therefore does not identify a structural natural frequency.^[ref]

For an original example, a24-tooth gear at1500 revolutions per minute rotates at $f_r=25\mathrm{Hz}$ and has mesh frequency $f_m=600\mathrm{Hz}$. Consider an illustrative acceleration signal with amplitude modulation:The product-to-sum identity gives a600Hz component of amplitude $A$ and modulation sidebands at575 and625Hz, each of amplitude $\mu A/2$. For $A=1m/{\mathrm{s}}^2$ and $\mu =0.20$, each sideband has peak amplitude $0.10m/{\mathrm{s}}^2$. Gear vibration order expresses these frequencies relative to the chosen shaft: the central order is24 and the neighboring orders23 and25. These positions follow from the constructed signal, without requiring a damaged tooth.

A primary apparatus example makes that ambiguity concrete. NASA's2015 transmission experiment compared rotating accelerometers on a pinion shaft with stationary housing sensors. The rotating radial channel showed sidebands at1854 and2060Hz around a1957Hz mesh frequency, spaced by the103Hz shaft rate. The report derives this shift from the rotating measurement direction and discusses sums and differences of tangential channels. A measurement-frame effect can therefore create sidebands as well as a mechanical defect.^[ref]

A useful test records speed, load, mounting and sensor orientation with each spectrum. If speed changes appreciably within one record, fixed-frequency lines can spread; comparisons should then account for the changing rotation rate. Record duration also controls Fourier frequency resolution. Sideband changes are evidence to interpret alongside a baseline and the test conditions, rather than a unique diagnosis from one spectral peak.

Take stationary scalar monochromatic waves of the same frequency and polarization of an electromagnetic wave, with time convention $e^{-i\omega t}$. Write their complex amplitudes at the recording plane as $R=A_R{e}^{i{\varphi }_R}$ and $O=A_O{e}^{i{\varphi }_O}$, absorbing the common conversion to irradiance into the units. An exposure short enough to avoid fringe motion but long compared with an optical period recordsThe two cross terms are where relative wave phase enters the measured light intensity. Blocking the holographic reference wave leaves $|O{|}^2$ and removes them. Loss of mutual optical coherence, incompatible polarization of an electromagnetic wave or motion that averages the relative wave phase similarly reduces fringe contrast. These are useful controls for distinguishing a hologram from an ordinary brightness record. One local cosine value does not uniquely determine a wave phase; known reference structure and separation of the cross terms supply the additional information in the example below.

Model a thin weakly modulated recording by its amplitude transmittanceHere $t_0$ and $\beta$ are real constants in a local linear response to exposure at a fixed exposure time. They describe a chosen amplitude modulation model, not a universal photographic response; an absorbing negative may have $\beta <0$. Keep $0\le t\le 1$ for this passive real-transmission model. Ignore material thickness, saturation, shrinking, clipping and changes of wavelength. Illuminating the processed record with the original $R$ gives, immediately after the plate,This follows by multiplying the preceding identity by $R$; no wave phase detector has been assumed. If $|R|$ is uniform, the desired term reproduces $O$ up to a constant. A nonuniform $|R{|}^2$ weights the recovered field. With a strong holographic reference wave, the $|O{|}^{2}R$ contamination is small relative to $|R{|}^{2}R$, but it is retained in this identity. Subsequent diffraction propagates each term: reproduction at the plate is the starting boundary field, not yet a focused image.

The schematic uses a plane wave reference and one optical point source to separate the functions. The top row records $R+O$; the bottom removes the object and illuminates the developed hologram with $R$. Dashed extensions identify the desired virtual image; the conjugate component converges toward a real image. All three outgoing components coexist. The coloured paths distinguish terms, not different wavelengths; distances and strengths are schematic. This functional drawing is not a reconstruction of the historical apparatus.

Apply the product-to-sum formula $2\cos u\cos v=\cos (u+v)+\cos (u-v)$ to the amplitude modulation:The amplitude modulation sidebands have angular frequencies $\omega +{\omega }_m$ and $\omega -{\omega }_m$, each with amplitude $b/2$ in this real cosine convention. The carrier wave has angular frequency $\omega$ and amplitude $a$.

This identity is exact for the supplied space-time expression. A physical qualification is needed if every term is meant to be a freely propagating electromagnetic wave with speed of light $c$: the dispersion relation requires $k_{\pm }=(\omega \pm {\omega }_m)/c$, whereas the prescribed modulation at every position leaves all three wavenumbers equal to $k=\omega /c$. A freely propagating modulated wave instead has envelope $a+b\cos ({\omega }_m(t-x/c))$; its analogous expansion has $k_{\pm }$. For example, at $x=t=0$ the prescribed expression has $A_{tt}-c^2{A}_{xx}=-b{\omega }_m^2$, which is nonzero for nontrivial modulation. The given formula can describe modulation imposed throughout space or the temporal signal at one position.

Radio communication conveys information using radio waves coupled into and out of space by antennas. Amplitude modulation and frequency modulation vary a carrier wave; wireless telegraphy can instead encode symbols in transmission intervals. The useful separation is between the rapid electrical oscillation and the slower message timing. A receiver needs both frequency selection and a way to turn its selected signal into a readable output.

Single-sideband modulation transmits one of the two translated message bands generated by amplitude modulation. For a real signal, the transmitted negative-frequency band is the complex conjugate of the positive-frequency band. Removing the redundant band reduces the required signal bandwidth. The carrier wave may also be suppressed; its retention or removal must be specified separately.