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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.

A bandlimited function has Fourier transform supported in a bounded frequency interval. An integrable bandlimited function has a bounded continuous representative when its Fourier transform is integrable, and that representative extends analytically using the compact-frequency integral. Scaling arguments scales its frequency support inversely, while multiplying two such functions adds their support intervals through convolution.

For a nonzero nonnegative integrable bandlimited function $f$ with frequency support in $[-\Delta ,\Delta ]$, the displayed construction is even, nonnegative and normalizable. Each factor has half-width frequency support, so their product retains width $\Delta$. Boundedness gives ${\int }_T^{\infty }w(t)dt\le 2C\Vert f{\Vert }_{\infty }{\int }_{T/2}^{\infty }f(u)du$. Thus a positive-time tail estimate becomes a two-sided estimate after evenization, with a rescaled decay constant. A simple average with $f(-t)$ would not establish this from an uncontrolled negative tail.

Choose a real even smooth compactly supported Fourier transform $\eta$ equal to one near zero and let $K$ be its inverse Fourier transform. Repeated integration by parts makes $K$ a Schwartz function. It is real, integrable and square-integrable, with unit integral. Moment differentiation of the Fourier transform gives $\int x^{j}K(x)dx=i^j{\eta }^{(j)}(0)=0$ for every positive integer $j$. Such a kernel for density estimation therefore satisfies arbitrary finite-order moment cancellation, and necessarily takes negative values. If the transform plateau contains the frequency support of a bandlimited function $f$, the convolution theorem proves $K\ast f=f$.

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]

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.

Digital acquisition converts a continuous voltage signal into timed numerical samples. A practical chain is a suitably buffered and amplified sensor signal, an analogue anti-aliasing filter, a sampling or sample-and-hold stage, an analog-to-digital converter, and a clocked data-storage system. Choose the gain so that the interesting variation uses the available input range without clipping. Finite input impedance can load a sensor, so an appropriate buffer or operational amplifier may be needed.

Sampling makes time discrete. If the conditioned signal is confined below frequency $f_{\max }$, choose a sampling rate $f_s>2f_{\max }$, allowing a transition band for the physical filter. The Nyquist–Shannon sampling theorem permits ideal reconstruction of a suitably bandlimited function from exact samples. Without filtering, components separated by integer multiples of $f_s$ have identical samples; in particular real sinusoids at $f$ and $f_s-f$ can be indistinguishable. This aliasing cannot generally be undone by later digital filtering. Oversampling can ease filter design, but does not replace the requirement to suppress unwanted high-frequency content before sampling.

Amplitude quantization is separate from sampling. A uniform $b$-bit converter covering voltage span $V_{\mathrm{span}}$ has step $q\simeq V_{\mathrm{span}}/2^b$. Rounding an in-range input gives the uniform-quantization error bound $|ϵ|\le q/2$. If the rounding errors can additionally be modeled as uniform and uncorrelated, their standard deviation is $q/\sqrt{12}$. This statistical estimate is not a guarantee for a constant input or a coherent periodic signal. More bits increase resolution, but accuracy also depends on voltage-reference calibration, offset, gain error, nonlinearity and electronic noise. Out-of-range signals clip and invalidate the rounding bound.

Clock quality matters too. Timing jitter $\delta t$ gives a voltage error approximately $(dV/dt)\delta t$; for a sinusoid its scale grows with $2\pi f{V}_p{\sigma }_t$. A clock calibration error biases measured frequencies and time intervals. Simultaneous multichannel signals need known relative timing, and the sampling and analogue settling times must be appropriate to the signal bandwidth.

For $n_c$ channels the raw recording rate is approximately $n_c{f}_{s}b$ bits per second, before overhead, so storage and sustained transfer speed must prevent missing data. Store units, channel identities, timestamps, sampling rate, input ranges and calibration information along with the readings. Preserve the original samples when applying digital filters or other processing. Averaging can improve precision for independent fluctuating errors, but cannot restore aliased frequencies, clipped peaks or a fixed rounding bias. Digital recording makes reproducible analysis and transmission possible when its sampling, amplitude and timing limitations are all controlled.

For a Schwartz function on the real numbers, its periodization $P_f$ is a smooth periodic function of period of a function one. Rapid decay gives uniform convergence on $[0,1]$ of the defining series and each differentiated series. Its $k$th Fourier coefficient isThe change of variable $u=x+n$ uses $e^{2\pi ikn}=1$. Rapidly decreasing Fourier coefficients give the absolutely and uniformly convergent Fourier series $P_f(x)={\sum }_k\hat{f}(k)e^{2\pi ikx}$. Evaluation at $x=0$ proves the Poisson summation formula.^[ref]

The same identity explains aliasing in the Nyquist–Shannon sampling theorem: applying the Poisson summation formula to the Schwartz function $x\mapsto f(x)e^{-2\pi i\xi x}$ givesThe sampled spectrum is periodic in frequency, and shifted copies of the continuous Fourier transform add together. When those copies overlap, the samples cannot separate their contributions. The Nyquist–Shannon sampling theorem adds a bandlimited function hypothesis to make reconstruction possible.

In spatial sampling of a wave, a detector records a wave at positions $x_n=np$ separated by pitch $p>0$. Angular wavenumbers differing by $2\pi m/p$, with integers $m$, give identical samples of a complex Fourier mode:This is spatial aliasing. To distinguish arbitrary sinusoids within a prescribed band, require $|k|<\pi /p$, equivalently $|{\nu }_x|<1/(2p)$, the Nyquist spatial frequency. Thus $p<{\lambda }_{\min }/2$ for the shortest allowed wavelength. At equality a sine can vanish at every sample, so phase information can be lost. The Nyquist–Shannon sampling theorem states the precise reconstruction conditions for a bandlimited function; a finite scan introduces further limitations.

The synthetic two-tone snapshot iswhere both ${\nu }_j$ count cycles per metre. The component wavelengths are $0.50\mathrm{m}$ and $0.125\mathrm{m}$. With $p=0.05\mathrm{m}$, both lie below the Nyquist spatial frequency of $10$ cycles per metre. With $p=0.10\mathrm{m}$, the $8$-cycle component aliases to the $2$-cycle component: at every $x_n=n/10$, $\cos (2\pi \cdot 8x_n)=\cos (2\pi \cdot 2x_n)$. The sampled sum is therefore indistinguishable from $1.5\cos (4\pi x)$, although the continuous curves differ between samples. Drawing lines between sample points cannot resolve this ambiguity.

Rounding a voltage to the nearest level in a uniform grid of spacing $q$ gives a quantization error of magnitude at most $q/2$, provided the input is within the converter's range. This follows because each grid point represents the interval bounded by the midpoints to its two neighbors. Clipping invalidates the bound.

The bound concerns recorded samples, whereas the Nyquist–Shannon sampling theorem concerns reconstruction from exact samples of a bandlimited function. If a reconstruction is $\tilde{v}(t)={\sum }_n{Q}_q(v_n)w_n(t)$, its error is ${\sum }_n{ϵ}_n{w}_n(t)$, so a general bound involves $(q/2){\sum }_n|w_n(t)|$, not simply $q/2$. Thus sampling bandwidth and per-sample rounding precision are distinct specifications. Under an additional uniform independent-error model, the quantization-noise standard deviation is $q/\sqrt{12}$, an rms statement rather than a worst-case reconstruction guarantee.