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Aliasing is the inability to distinguish different continuous frequencies from sampled values. The identity above says that frequencies separated by an integer multiple of the sampling rate $f_s$ give identical samples of a complex Fourier mode. For real cosines, changing the sign of a frequency also gives the same waveform, explaining the $2$-versus-$8$ example at rate $10$. The Nyquist–Shannon sampling theorem prevents this ambiguity within its specified class of bandlimited functions. Before sampling, unwanted higher frequencies must be suppressed if they would violate that assumption; once two waveforms give the same samples, those samples cannot identify which was present.

For integers $m,q,n$, adding $qN$ to a frequency index changes the complex exponential by $e^{2\pi iqn}=1$. Thus different continuous Fourier modes become identical discrete Fourier modes on the sampling grid. This is aliasing, caused by sampling before any discrete Fourier transform is computed.

For $N=8$, compare the continuous functions $\cos (2\pi mt)$ and $\sin (2\pi mt)$ with $m=1$ and $m=9$, sampled at $t=n/8$, $n=0,\dots ,7$. With $a=\sqrt{2}/2$, both cosines give exactly the first row and both sines give exactly the second:The nine-cycle sine agrees in sign as well as magnitude with the one-cycle sine: the frequency indices differ by $8$, rather than by a sign reversal. Consequently, either cosine produces the same discrete Fourier transform, $X_1=X_7=4$ with all other coefficients zero; either sine produces $X_1=-4i$, $X_7=4i$, with all other coefficients zero.

An observed low frequency therefore does not establish that physical high frequencies are absent. To exclude such alternatives, the Nyquist–Shannon sampling theorem requires a specified class of bandlimited functions. For arbitrary pure sinusoids with frequencies at most $B$, choose sampling rate $f_s>2B$; equality needs care because $\sin (2\pi (f_s/2)t)$ vanishes at every $t=n/f_s$. The finite-energy L2 space version of the Nyquist–Shannon sampling theorem permits equality under its own hypotheses. In either case, exact continuous reconstruction generally requires the entire infinite sample sequence; eight observed samples alone do not supply it. Demanet derives the grid aliasing condition as a difference by an integer multiple of the sampling angular frequency.^[ref]

An anti-aliasing filter suppresses signal components outside the intended acquisition band before sampling. For voltage recording it is an analogue filter ahead of the analog-to-digital converter, with a transition band allowed below half the sampling rate. Once two frequencies have become indistinguishable through aliasing, a digital filter generally cannot recover the lost distinction. The Nyquist–Shannon sampling theorem concerns an already bandlimited function; a physical filter approximates that prerequisite.

For incoherent imaging with an unobstructed circular aperture of diameter $D$, the intensity-image spatial-frequency cutoff is $D/(\lambda f)$ in the focal plane. The Nyquist–Shannon sampling theorem therefore gives the displayed pitch limit. The Rayleigh criterion instead gives a point-image separation $1.22\lambda f/D$. Consequently two samples per Rayleigh separation and sampling the full optical bandwidth differ by the factor $1.22$. Neither sampling estimate guarantees useful contrast at arbitrarily low signal-to-noise ratio.

Diffraction-limited resolution is a scale for distinguishing details with a specified ideal diffraction response and a specified criterion. For two equally bright, mutually incoherent optical point sources imaged through a uniformly transmitting, unobscured circular optical pupil, the Rayleigh criterion places the centre of one Airy disk at the first zero of the other. At small angles in air this separation isHere $D$ is pupil diameter, $\lambda$ is wavelength in air, and $j_{1,1}$ is the first positive zero of the order-one Bessel function of the first kind. This is the radius to the first zero, not the diameter of the central spot. Fraunhofer diffraction gives the radial peak-normalized light intensityThe removable value at zero follows from $J_1(q)\sim q/2$. The pupil Fourier transform reduces to a radial integral by circular symmetry, which produces the Bessel function of the first kind; its first zero supplies the factor $1.22$.^[ref]

In the usual scalar circular-pupil approximation for a fluorescence microscope, the corresponding object-plane Rayleigh criterion isHere ${\lambda }_0$ is the vacuum emission wavelength, and the object-side numerical aperture already includes the medium's refractive index. Do not divide ${\lambda }_0$ by $n$ again. For small cone angles in air, $\mathrm{NA}\simeq D/(2f)$ and image-plane radius $f{\theta }_{\mathrm{R}}$ agrees with this form. The assumptions include a single wavelength, ideal optical focus, and mutually incoherent emitters. High numerical aperture can require a vector description with polarization of an electromagnetic wave and emitter orientation; confocal microscopy has its own excitation-and-detection response. The same $0.61$ formula therefore does not describe every microscope or illumination method.^[ref]

In this original model, ${\lambda }_0=550\mathrm{nm}$ and numerical apertures $0.25$ and $0.50$ give first-zero radii $1.342\mu \mathrm{m}$ and $0.671\mu \mathrm{m}$. Each point spread function is divided by its own peak to compare shape. The pupil axes are transverse wave vector components multiplied by ${\lambda }_0/(2\pi )$, so pupil radius equals numerical aperture. A wider pupil accepts a wider range of transverse wave vectors, producing a narrower image. Increasing magnification alone enlarges the same spot and does not reduce its object-plane radius.

The Rayleigh criterion is a useful visual convention, not an absolute information barrier. For equal incoherent sources with total detected radiant flux $F$ and separation vector $\mathbf{s}$, the ideal image isBelow the Rayleigh criterion this still differs from $Fp(\mathbf{x})$; for small $\mathbf{s}$, symmetry cancels the linear change and the first change is quadratic in separation. Estimating two unknown source positions is then increasingly sensitive to noise, background, relative brightness and uncertainty in the point spread function. Localizing one isolated source and recovering an arbitrary extended object are different inference tasks. Measurements that separate optical modes can also extract separation information differently from direct image light intensity measurement.^[ref]

There remains a real limit for arbitrary images in this model: the optical transfer function is zero above the transmitted spatial frequency band. Its cutoff for scalar incoherent imaging is ${\nu }_c=2\mathrm{NA}/{\lambda }_0$ in object-plane cycles per unit length. The Nyquist–Shannon sampling theorem therefore suggests object-referred detector pixel pitch $p_{\mathrm{pix}}\le {\lambda }_0/(4\mathrm{NA})$, with finer sampling providing margin. Finite detector pixels also integrate over their area, suppressing contrast; coarse sampling can cause aliasing. Finer sampling cannot alone recover spatial frequencies removed by the optics, and Fourier deconvolution amplifies noise where the optical transfer function is small. Photon shot noise, read noise, optical aberrations and defocus can all worsen practical discrimination. A useful resolution claim therefore reports the object model, criterion, wavelength, numerical aperture, sampling and signal-to-noise ratio in photon counting.

A discrete-time signal assigns a value to each of a discrete collection of times, often indexed by the integers. Regular sampling at rate $f_s$ gives $x[n]=x(n/f_s)$, so a sinusoid with normalized frequency $f$ cycles per sample has physical frequency $f{f}_s$. Discreteness of time does not itself require the amplitudes to be quantized; rounding the values is an additional operation. The Nyquist–Shannon sampling theorem gives conditions for recovering a band-limited continuous signal from its samples.

The Nyquist–Shannon sampling theorem says that a bandlimited function can be recovered exactly from equally spaced samples, provided the sampling rate is high enough. It is the bridge between a continuous waveform and a sequence of numbers: the missing values between samples are determined by the restriction on its frequencies.

Here is a precise finite-energy version. Let $f\in L^2(\mathbb{R})$, meaning ${\int }_{\mathbb{R}}|f(t){|}^{2}dt<\infty$, and suppose its Fourier transform, defined through the Plancherel theorem, is zero outside $[-2\pi B,2\pi B]$. Thus $B>0$ is the largest allowed ordinary frequency, in cycles per second, rather than an angular frequency. Set $T=1/f_s$, where $f_s$ is the number of samples per second. If $f_s\ge 2B$, the continuous representative of $f$ is recovered byThe quotient is the normalized sinc function, with value one when its argument is zero. Each shifted sinc function is one at its own sampling time and zero at every other sampling time. Its coefficient is therefore exactly the required sample value. For this L2 space formulation, the reconstruction converges in the L2 space norm and uniformly in $t$. The critical rate $f_s=2B$ is allowed for this class; choosing $f_s>2B$ leaves a gap between the occupied frequencies and the sampling boundary.

For a numerical example, a bandlimited function with no frequencies above $3$ cycles per second can be reconstructed from its samples every $0.1$ seconds: $f_s=10>6$. Exact reconstruction uses the whole sequence indexed by $\mathbb{Z}$. A finite recording and a finite sum of sinc functions introduce truncation error; joining adjacent samples by straight lines is generally a different reconstruction.

The obstruction is aliasing. At sampling rate $f_s=10$, the two distinct waveformshave identical samples at $t=n/10$, becauseThe samples alone cannot tell whether the waveform oscillates twice or eight times per second. If we know in advance that all frequencies are below $5$, the $8$-cycle waveform is excluded and the ambiguity disappears. These pure cosines are an illustrative aliasing example, rather than members of the finite-energy L2 space class in the stated theorem.

Exactly at a sampling boundary, the function class matters. At rate $f_s$, the sine $\sin (2\pi (f_s/2)t)$ has zero values at every $n/f_s$. It is a nonzero waveform indistinguishable from zero on that grid, but has infinite energy and is excluded by the finite-energy hypothesis above. This explains why a rule intended to cover arbitrary pure sinusoids should demand a strict inequality. For spatial data, replace seconds by distance: the Nyquist spatial frequency is $1/(2p)$ for sample pitch $p$.

The proof idea is short: the samples are Fourier coefficients of the Fourier transform restricted to a sampling-sized frequency interval. Because a bandlimited function has no contribution outside that interval, its Fourier series determines the whole Fourier transform; Fourier inversion then gives the sinc function expansion. See sampling expansion by periodic Fourier projection for this connection.

Claude E. Shannon's Communication in the Presence of Noise (1949), Section II, Theorem 1 and equation (7), gives the sampling statement and reconstruction using shifted sinc functions in a communication setting. Shannon explicitly notes earlier mathematical forms and credits Nyquist with emphasizing the sampling interval in telegraphy.^[ref]

Atomic-resolution cryo-electron microscopy became possible by retaining high-frequency structural information through the whole experiment: specimen preparation, electron optics, detection and computation. The decisive advances improved the information obtained per damaging electron and made it possible to align and average many weak images without averaging away structural differences. A short de Broglie wavelength had always been available; the practical limit was preserving and extracting the information carried by those electrons.

First, vitrification preserves hydrated biological material in a thin noncrystalline layer of water. Rapid freezing avoids the strong periodic scattering and disruption associated with ordinary ice crystals, while cryogenic imaging slows some damage processes. The specimen still suffers radiation damage, and a thicker layer adds background and multiple scattering. Reproducible thin ice, clean grids, stable supports and reduced mechanical drift are therefore prerequisites. The specimen must remain structurally intact and occupy enough orientations for a three-dimensional reconstruction. Adsorption at the air–water interface, preferred orientation and variable thickness can defeat an otherwise excellent microscope. Sample optimization should include independent checks of assembly integrity and activity where feasible.

Second, direct electron detection greatly improved the useful signal relative to earlier indirect cameras. In a counting detector, individual electron events can be localized and recorded without treating the deposited signal merely as a continuous intensity. detective quantum efficiency measures how efficiently the detector preserves the input signal-to-noise ratio as a function of spatial frequency. High efficiency matters particularly because the permissible exposure is limited. At excessive event rates, overlapping events cause coincidence losses, so faster readout permits useful counting at practical collection rates. A nominally smaller camera pixel alone cannot replace a detector that preserves high-frequency information.

Fast detectors also turn an exposure into a sequence of movie frames. Aligning those frames corrects specimen drift and beam-induced movement; local correction handles different movements across the imaged area. Uncorrected movement preferentially smears fine features. dose weighting then combines early and later frames according to the information they retain: later exposure can help low-frequency alignment even after fine structural details have been damaged. ^[ref] exemplifies this computational use of detector movies. Frame alignment cannot restore chemistry already destroyed by radiation damage, and extremely noisy individual frames require constrained motion estimates.

Third, stable and coherent electron optics increase contrast transfer at the finest spacings. A narrow source energy spread reduces chromatic aberration, because electrons with different energies otherwise focus differently. A cold field emission gun provides one route; a monochromator provides another. An energy filter removes many inelastically scattered electrons that otherwise contribute background. This does not remove all damage or every unwanted scattering event. Improved mechanical and electrical stability, reproducible lens settings, accurate coma-free alignment and calibrated magnification make the information more consistent across images.

The 2020 apoferritin reconstructions illustrate the final hardware step. One study combined a cold source, improved energy filtering and a fast direct camera with advanced correction to obtain a map at about $1.22\text{A˚}$. Another used an instrument with monochromation and aberration correction to reach about $1.25\text{A˚}$.^[ref][ref] The two instruments demonstrate complementary ways of preserving coherent high-frequency information. These unusually rigid, symmetric particles are favorable test specimens; their resolutions do not imply that a flexible cellular assembly will reach the same limit.

Fourth, modern reconstruction treats the noisy images statistically. single-particle analysis estimates how many randomly oriented projections correspond to a common three-dimensional structure. Bayesian inference and maximum likelihood estimation permit uncertainty in image orientations and class membership to be handled rather than committing every weak image to a single exact alignment from the beginning. ^[ref] provides a concrete implementation. Two-dimensional classes reject many damaged or overlapping particles; three-dimensional classification can separate different compositions or conformations. Appropriate symmetry increases the effective information, but an incorrect symmetry assumption can erase a real asymmetric feature.

Refinement increasingly corrects individual particle movements and optical imperfections. Per-particle defocus estimates accommodate height differences in ice. Beam tilt, higher-order aberrations and anisotropic magnification need correction when their accumulated phase errors become important. The contrast transfer function changes the magnitude and sign of Fourier components; combining images recorded with different defocus values recovers information suppressed at particular transfer-function zeros. Near atomic spacings, accounting for Ewald sphere curvature can further improve reconstruction. Such corrections require adequate signal and calibration; simply adding more adjustable parameters to weak data risks fitting noise.

Automation and fast computation make these methods practical. Stable automated acquisition and beam-image-shift strategies collect large image sets; parallel computation permits repeated alignment, classification and refinement. More particles improve the average only if they carry consistent signal. Under an ideal independent-noise model, the amplitude signal-to-noise ratio improves in proportion to $\sqrt{N}$; a systematic alignment error or persistent structural heterogeneity will not disappear by this scaling. For a target spacing $d$ and real-space pixel size $a$, Nyquist–Shannon sampling theorem requires $a\le d/2$. For $d=1.2\text{A˚}$ this gives $a\le 0.6\text{A˚}$ per pixel, a necessary sampling condition rather than a guarantee of such resolution.

Validation is part of the technological achievement. Split particles into independent halves and refine them independently; calculate their Fourier shell correlation, commonly reporting the spacing at the $0.143$ criterion for independently refined half-maps. Shared high-resolution references, poorly handled masks or correlated data can inflate agreement, so independence and mask correction matter. Inspect directional coverage and local resolution, compare models against held-out map information, and check stereochemistry. The global resolution is a summary: flexible loops, bound lipids and minor components may be much worse than the ordered core.

Finally, distinguish an atomistic model from atomically resolved data. At ordinary near-atomic resolution, side chains and backbone features can support building an atomic model while individual neighboring chemical atoms are not all separated. At true atomic resolution, many individual atomic peaks become distinguishable and some hydrogen contributions may be detectable under favorable conditions. cryo-electron microscopy reconstructs scattering electrostatic potential, so interpretation of very fine features is not identical to interpreting X-ray crystallography electron-density maps. The breakthrough was the combined improvement of specimen quality, counting movies, optical coherence and statistical reconstruction, with independent validation showing that the fine detail was supported by the data.

Use $\hat{f}(\lambda )=\int f(t)e^{-i\lambda t}dt$ and interpret the regularity assumption to include $f\in L^2(\mathbb{R})$ with the continuous representative furnished by Fourier inversion. Write $F=\hat{f}$. The bandwidth assumption implies $F\in L^1$ as well as $L^2$, by the Cauchy-Schwarz inequality on its finite support, sois continuous and defined at every $t$. The integer samples are precisely the Fourier coefficients of $F$ with the sign convention appropriate to the basis $e^{-in\lambda }$:The completeness and Parseval identity for the exponential orthonormal basis on $[-\pi ,\pi ]$ give $F={\sum }_{n\in \mathbb{Z}}f(n)e^{-in\lambda }$ in $L^2$ andThe last equality is the Plancherel theorem. Apply the inverse integral to the finite symmetric Fourier partial sums. Sincewe obtain the Nyquist–Shannon sampling theorem in this normalization:where the quotient is one at $t=n$. This is a sampling expansion by periodic Fourier projection, not merely an almost-everywhere inversion. Indeed, the inverse integral of an error $G\in L^2[-\pi ,\pi ]$ is bounded, uniformly in $t$, by $\Vert G{\Vert }_2/\sqrt{2\pi }$. Hence the symmetric finite sampling sums converge uniformly on all of $\mathbb{R}$. Moreover, the Parseval identity applied to $e^{it\lambda }$ givesThe Cauchy-Schwarz inequality therefore makes the sampling series absolutely convergent at every $t$, with its absolute tail bounded uniformly by $({\sum }_{|n|>N}|f(n){|}^2{)}^{1/2}$. Thus the integer samples recover the function everywhere.

For the nonuniqueness beyond this bandwidth, fix $0<{ϵ}_0<\min (ϵ,\pi )$ and a nonzero smooth function $\Phi$ compactly supported in $(-{ϵ}_0,{ϵ}_0)$. Let $h$ be its inverse Fourier transform and putThe function $h$, and hence $g$, is a Schwartz function. The two shifted spectral supports are disjoint, so $\hat{g}\ne 0$, but its support is contained strictly inside $(-\pi -ϵ,\pi +ϵ)$. Every integer sample of $g$ is zero. Therefore $g$ and the zero function have identical samples but are distinct, providing a strictly supercritical sampling alias counterexample for every positive bandwidth enlargement.

For $p^2+q^2>1$, $m=i\sqrt{p^2+q^2-1}$ and the Weyl plane-wave representation contains evanescent waves. Their amplitudes decay exponentially with the distance from the scattering region. Excluding them removes high transverse spatial frequencies, including subwavelength information. Near-field measurements can in principle detect them, but back-propagating their decay exponentially amplifies measurement errors.

For propagating data, the Fourier transfer satisfies $|{\mathbf{k}}_s-{\mathbf{k}}_i|\le 2k$. One illumination still samples only the Ewald sphere, so this frequency bound does not imply access to every mode inside that ball. The lack of a volumetric spectrum is the separate nonuniqueness described in the previous part.

The Born approximation requires the incident field to remain a good approximation inside the scatterer. A useful sufficient smallness estimate for a weak, extended inclusion is small accumulated phase $kD|n-1|\ll 1$, alongside weak reflection and multiple scattering. Small pointwise contrast alone can fail for a thick inclusion or a resonance. The scalar Helmholtz and background-medium assumptions, and the extra linearization of $n^2-1$, also belong to the model's validity conditions.

Numerically, a finite aperture windows the measured fields and limits angular and Fourier resolution. Discrete sampling must satisfy the Nyquist–Shannon sampling theorem for the retained transverse bandwidth; spacings $\Delta x,\Delta y\lesssim \pi /K_{\max }$ avoid its simplest aliasing. Finite discrete Fourier transforms also introduce periodic continuation, truncation, interpolation and window errors. Noise, unresolved evanescent modes and near-grazing modes with $m\simeq 0$ limit reliable recovery. Refining the grid cannot by itself fill the missing three-dimensional Fourier data.

The optical pupil mapped onto the deformable mirror has actuator pitch approximately $s=D/N$. A sinusoidal wavefront error must have at least two actuator intervals per period to avoid aliasing, by the Nyquist–Shannon sampling theorem. ThusIf the $N$ actuator centres include both endpoints of the optical pupil diameter, the literal pitch is $D/(N-1)$ and the corresponding expression is $2D/(N-1)$; the paper uses the customary large-$N$ pitch convention. At the exact Nyquist boundary one quadrature can be poorly sampled, and real influence functions and actuator coupling reduce useful correction near the cutoff. The result is an ideal sampling bound, not a guarantee of arbitrary correction at its endpoint.

The projected actuator pitch is $p=D/N$ in the question's sampling convention. A sinusoidal wavefront error needs at least two samples per spatial period, by the Nyquist–Shannon sampling theorem. Thus the shortest limiting correctable scale at the primary isThe Nyquist spatial frequency is $N/(2D)$ cycles per unit length. Exactly at that limiting frequency some phases are poorly sampled; actual deformable mirror performance also depends on actuator influence functions, so this is an ideal bandwidth limit.

For each beam in vacuum, its magnetic field is ${\mathbf{B}}_j={\hat{\mathbf{k}}}_j\times {\mathbf{E}}_j/c$. Write the two complex scalar contributions, including the beam splitter factor, as $u_1=f(t-t_1)/\sqrt{2}$ and $u_2=f(t-t_2)/\sqrt{2}$. Since their electric fields are both along $z$,The optical-cycle and ensemble-averaged Poynting vector isThe mean individual light intensities are equal, soThus energy flows along the bisector of the two beam directions, crossing a detector plane normal to $x$. Such a sensor can in principle record the variation across its $y$ coordinate. Its actual normal energy flux is $I(y)/\sqrt{2}$ in the normalization used above; this fixed factor does not change interferometric visibility.

There is a practical resolution condition in addition to the direction of energy flow. The interference fringe spacing $\lambda /\sqrt{2}$ is very small, so direct pixels would need spacing no larger than $\lambda /(2\sqrt{2})$ as required by the Nyquist–Shannon sampling theorem, or optical magnification and sufficient angular aperture to preserve both incident directions. An ordinary optical camera with much larger effective pixels averages the interference fringes. The Poynting-vector argument establishes the allowed measurement geometry, not guaranteed resolution with any optical camera.

With $D=5\mathrm{mm}$, the circular-aperture diffraction limit is ${\theta }_R=244\lambda$ when $\lambda$ is in metres. No wavelength is specified, so a unique numerical diffraction angle cannot be deduced. Taking representative visible light at $\lambda =550\mathrm{nm}$ givesFor a roughly square receptor arrangement with surface density $n=10^5{\mathrm{mm}}^{-2}$, the characteristic receptor pitch is $s\simeq n^{-1/2}=3.16\mathrm{micrometres}$. With the retina $f_{\mathrm{eye}}=17\mathrm{mm}$ behind the lens, that pitch subtendsThis is an order-of-magnitude estimate of the point-separation scale from retinal sampling. A periodic intensity pattern needs at least two samples per period under the Nyquist–Shannon sampling theorem, giving a corresponding period angle of about $3.72\times 10^{-4}\mathrm{rad}$ or $1.28\mathrm{arcmin}$. A different receptor packing changes the pitch by an order-one geometric factor; density alone does not specify a full sampling lattice or a contrast threshold.

At a $5\mathrm{mm}$ pupil and the chosen visible wavelength, the optical diffraction limit is somewhat finer than the receptor pitch. The supplied model is therefore not purely diffraction limited: retinal sampling is comparable to, and slightly more restrictive than, ideal aperture diffraction. Real optical aberrations and neural processing can further reduce resolution. Different wavelengths or a smaller pupil can change which effect dominates.

At focal length $f=80\mathrm{m}$, the stroke maps to a focal-plane width $w_i=fs/H=12\mu \mathrm{m}$. A practical estimate with two detector samples across that stroke isThe required sensor extent in that direction is $f(500\mathrm{m})/H=0.20\mathrm{m}$. One $12\mu \mathrm{m}$ pixel per stroke would give about $1.67\times 10^4$ pixels across, but cannot reliably encode a stroke's position and contrast as the image moves relative to the pixel grid.

For a precise interpretation of “fully resolve” as sampling the entire diffraction-limited incoherent image, use the optical intensity-transfer cutoff ${\nu }_c=D/(\lambda f)$. The Nyquist–Shannon sampling theorem requires $1/(2p)\ge {\nu }_c$. With the circular mirror chosen in 10d,This stricter value retains the full circular-aperture spatial bandwidth; the $6\mu \mathrm{m}$ and $3.3\times 10^4$ estimates use two samples per specified feature instead. An engineering design would provide further sampling and contrast margin. The distinction states the sampling convention rather than treating the order-of-magnitude estimates as contradictory.

The Rayleigh criterion gives a separation in the sensor plane $\delta =f\Delta \theta =2.44\mu \mathrm{m}$, using the $5\mathrm{mm}$ image distance. Counting one square pixel per such spatial resolution element givesThis is an estimate of the number of optically distinguishable elements. To sample all the spatial information, specify a signal sampling criterion as well. For incoherent imaging through an ideal circular aperture, the optical transfer function has cutoff spatial frequency ${\nu }_c=D/(\lambda f)$. The Nyquist–Shannon sampling theorem requires a pixel pitchA common intermediate rule, two pixels per Rayleigh separation, gives about $8.1$ million pixels; it is a heuristic and does not quite sample the full ideal optical transfer function bandwidth. Thus about $2$ million is the elementary spatial resolution element count, while $12$ million is the ideal monochrome Nyquist–Shannon sampling theorem answer to “full use.” Color-filter layout and finite pixel area would impose further practical conditions.

Uniform samples at $t_n=n/f_s$ cannot distinguish frequencies separated by integer multiples of $f_s$:For a real cosine, opposite frequencies also coincide. Thus an unknown high-frequency component can appear in the sampled record at its folded frequency $f_{\mathrm{alias}}\in [0,f_s/2]$. For example, 2 Hz and 8 Hz cosines give identical samples at 10 samples per second. This ambiguity is aliasing, rather than uniform quantization error.

The Nyquist–Shannon sampling theorem permits reconstruction of an ideally bandlimited function of bandwidth $B$ when $f_s>2B$. Use an analogue anti-aliasing filter before conversion and allow a transition band between the desired passband and the frequencies that fold into it. Sampling faster relaxes the filter requirement. Digital filtering after undersampling cannot separate already identical sample sequences. Finite observation time, clock jitter, noise and imperfect filter rejection impose further practical limits.

For an experimental-noise interpretation of Nyquist's theorem, a resistor $R$ in thermal equilibrium at temperature $T$ has one-sided open-circuit voltage-noise spectral densityin the classical range $hf\ll k_{B}T$, over an approximately rectangular bandwidth $\Delta f$. Across a matched load, half the open-circuit voltage appears, giving available noise power $k_{B}T\Delta f$. This is thermal agitation, present without a mean current, and is distinct from shot noise. A real readout weights the spectrum with its transfer function: $⟨v_{\mathrm{out}}^2⟩={\int }_0^{\infty }{S}_V(f)|H(f){|}^{2}df$. Cooling, reduced bandwidth and suitable impedance/noise matching reduce measured noise; the resistor's noise does not simply disappear in an ideal amplifier model.^[ref]

The bare name also has a sampling interpretation. The Nyquist–Shannon sampling theorem permits exact reconstruction of a bandlimited signal from sufficiently fast uniform samples. The practical condition is $f_s>2f_{\max }$, with an anti-aliasing filter before sampling; otherwise distinct continuous frequencies produce identical samples. Ideal reconstruction uses shifted sinc functions. Critical-rate equality needs an appropriate signal class and can fail for unrestricted edge-frequency sinusoids with unknown phase. Sampling fast enough prevents aliasing, but does not remove thermal noise. This distinction covers the source's unspecified use of “Nyquist's theorem”.^[ref]

Ideal spectrum. Write $P_i(f)=C/(f^2+f_c^2)$, with $C>0$. The overdamped harmonic-trap power spectrum is a Lorentzian line shape:Thus its log-log plot is horizontal at low frequency and approaches slope $-2$ at high frequency, changing around $f_c$. The left panel below compares two characteristic frequencies with the same illustrative $C$; a common numerator is explicitly assumed for this sketch. Their high-frequency tails coincide and their low-frequency plateaux differ as $f_c^{-2}$.

The right panel plots all nine original data points, including the very-low-frequency excess and the highest-frequency suppression. It also shows the ideal curve inferred only from the requested intermediate range.

Graphical corner estimate. Between roughly $1$ and $100\mathrm{Hz}$ the measured power spectral density is approximately the plateau $P_0=10^{-6}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}$. At $10^4$ and $10^5\mathrm{Hz}$,so the descending asymptote is $C/f^2$ with $C\simeq 1{\mathrm{V}}^2\mathrm{Hz}$. The graphical corner frequency of a Lorentzian spectrum is where these asymptotes intersect:Independently, the measured value $P_m(1000\mathrm{Hz})=5\times 10^{-7}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}=P_0/2$ is exactly the half-plateau condition $P_i(f_c)=P_0/2$. Both graphical readings agree. The inferred ideal curve is approximately

Low-frequency excess and recording duration. The two lowest-frequency measurements increase approximately as $1/f$, whereas the ideal trapped oscillator tends to a constant plateau. This resembles flicker noise or slow apparatus drift superposed on the motion. The transimpedance amplifier, detector or environment could contribute; the spectrum alone does not identify the source uniquely. It would be incorrect to attribute that whole excess to the ideal Brownian oscillator.

A finite record of duration $T$ has ordinary Fourier frequency resolution of order $1/T$. Resolving the lowest tabulated frequency $0.01\mathrm{Hz}$ therefore requires at leastThis is a formal lower bound corresponding to about one cycle; a reliable noise-spectrum estimate normally needs a substantially longer record and averaging.

Sampling. The largest reported frequency is $f_{\max }=10^6\mathrm{Hz}$. The Nyquist–Shannon sampling theorem therefore gives the Nyquist lower boundIn practice choose a rate above $2\mathrm{MS}{\mathrm{s}}^{-1}$, with an analogue anti-aliasing filter suppressing unmeasured content above half the sampling rate. Exact equality cannot recover an arbitrary pure sinusoid precisely at the boundary for every phase; it is the limiting lower-bound answer rather than a comfortable practical sampling design.

High-frequency departure and amplifier design. Extrapolating the fitted $f^{-2}$ tail to $10^6\mathrm{Hz}$ predicts approximately $10^{-12}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}$, while the measured value is $10^{-15}{\mathrm{V}}^2\mathrm{H}{\mathrm{z}}^{-1}$. The observed drop from $10^5$ to $10^6\mathrm{Hz}$ has effective logarithmic slope $-5$, much steeper than the ideal $-2$.

A detector or amplifier bandwidth limit, or intentional filtering before digitization, can explain such suppression. For a linear filter the spectral density transformation under a linear filter is $P_{\mathrm{out}}=|H{|}^2{P}_{\mathrm{in}}$, so the last point corresponds approximately to an additional relative voltage gain $\sqrt{10^{-3}}=0.0316$, if attributed entirely to the readout transfer function. One additional point does not determine a unique filter order or identify the circuit responsible.

Rolling off an amplifier's gain beyond the useful signal band reduces integrated high-frequency noise, suppresses interference, and helps prevent aliasing in the analogue-to-digital converter. Limiting high-frequency loop gain can also improve feedback stability and suppress unwanted oscillation. The cutoff must retain the frequencies needed for the measurement; excessive filtering would distort the desired fluctuation spectrum itself.

In this digital-sampling context, Nyquist's theorem means the Nyquist–Shannon sampling theorem. A continuous signal with no Fourier content outside $|f|\le B$ can be reconstructed from uniformly spaced samples taken sufficiently fast. The familiar practical condition isHere $f_N$ is the Nyquist frequency, while $2B$ is the Nyquist rate required by that signal bandwidth. A finite-energy bandlimited formulation allows equality; a rule intended to include arbitrary pure sinusoids at the boundary must be strict, because $\sin (\pi f_{s}t)$ vanishes at every sampling time $t=n/f_s$.

The mechanism is visible in the Fourier transform. Sampling by a time comb creates shifted copies of the original spectrum, separated by $f_s$:If those copies do not overlap, an ideal reconstruction filter selects one copy and recovers the original waveform. Equivalently, ideal interpolation uses the normalized sinc function:Each interpolation function is one at its own sample and zero at all other sample times. Exact reconstruction presumes the bandwidth condition, an ideal sampling clock and the full sample sequence.

If the spectrum extends beyond half the sampling rate, different continuous frequencies become indistinguishable: this is aliasing. A sinusoid at frequency $f$ is indistinguishable on the sampling grid from frequencies differing by integer multiples of $f_s$, with the usual sign reflection for real signals. For example, $10\mathrm{kHz}$ sampled at $8\mathrm{kS}{\mathrm{s}}^{-1}$ aliases to $2\mathrm{kHz}$; a later digital filter cannot tell which signal generated those samples.

Use an analogue anti-aliasing filter before the analogue-to-digital converter and choose a sampling rate leaving room for its finite transition band. Faster sampling can relax that analogue filtering problem, but simply increasing the number of samples in a fixed-duration record does not improve the independent Fourier-bin spacing beyond approximately $1/T$. A long recording is needed for low-frequency resolution, while a small sampling interval is needed for high-frequency coverage. Finite records also introduce windowing and spectral leakage.

Digitization adds amplitude quantization as well as time sampling. An ideal $N$-bit converter spanning voltage range $V_{\mathrm{span}}$ has step $\Delta =V_{\mathrm{span}}/2^N$. When quantization error may be modeled as independent and uniform across one step, its rms size is $\Delta /\sqrt{12}$; small coherent signals need not obey that white-noise approximation. Sampling rate controls temporal ambiguity, recording duration controls frequency resolution, and converter resolution controls amplitude discretization.