Sampling Theorem (Shannon's Theorem)
The Fundamental Theorem Underpinning the Digitization of Analog Signals
1. Statement of the Theorem
The sampling theorem provides the theoretical foundation for representing and processing continuous-time signals in digital form. It was rigorously formulated in the context of information theory by C. E. Shannon (1949), and is therefore also known as Shannon's theorem. However, earlier results leading to the sampling theorem had been independently obtained by E. T. Whittaker (1915), H. Nyquist (1928), and V. A. Kotelnikov (1933). In Japan, Isao Someya (染谷勲, 1949) independently derived the theorem at the same time as Shannon, systematizing it—including extensions of the sampling theorem—in his book Hakei Densou (波形傳送, "Waveform Transmission"). Due to Japan's academic isolation in the post-war period, with no communication with the West, this is recognized as a completely independent discovery. Internationally it is most often called the Nyquist–Shannon theorem, while the mathematical literature has settled on the Whittaker–Kotelnikov–Shannon (WKS) theorem, naming the earlier contributors. Longer forms that include Someya are also used, but those normally add Raabe as well: Whittaker–Kotelnikov–Raabe–Shannon–Someya. In Japan it is also known as the Shannon–Someya theorem.
Theorem (Sampling Theorem).
Let $f(t)$ be a continuous-time signal that is band-limited, meaning its Fourier transform $F(f)$ vanishes identically for $|f| > B$, where $B$ is the highest frequency [Hz] present in the signal. If the sampling frequency $f_s$ satisfies
$$f_s > 2B$$then $f(t)$ can be perfectly reconstructed from the sample values $\{f(nT)\}_{n \in \mathbb{Z}}$, where $T = 1/f_s$ is the sampling interval.
Caution: Critical Sampling at $f_s = 2B$
At equality $f_s = 2B$ (exactly the Nyquist rate), a signal component at frequency $B$ may not be correctly reconstructed. With sampling interval $T = 1/(2B)$, the samples of a general sinusoid $A\cos(2\pi Bt + \phi)$ at frequency $B$ are
$$A\cos\!\left(2\pi B \cdot \dfrac{n}{2B} + \phi\right) = A\cos(n\pi + \phi)$$Applying the addition formula:
$$= A\bigl[\cos(n\pi)\cos\phi - \underbrace{\sin(n\pi)}_{=\,0}\sin\phi\bigr] = A(-1)^n\cos\phi$$Since $\sin(n\pi) = 0$, regardless of the phase $\phi$, the sin component vanishes and only the cos component survives. For example, if $\phi = 0$ (pure cosine), the sample values are $A(-1)^n$, which correctly captures the signal; but if $\phi = \pi/2$ (pure sine), all sample values are zero. In other words, the odd-function component (sin component) at the Nyquist frequency is fundamentally unrecoverable.
Intuitively, the theorem states: "For a strictly band-limited signal, if every sample is obtained exactly and the reconstruction is ideal, then sampling faster than twice the highest frequency present loses no information." It is not a statement about real measurements, where the samples are finite in number, finite in precision and noisy.
1.1 Nyquist Rate vs. Nyquist Frequency
Two important concepts related to the sampling theorem are distinguished as follows:
| Term | Definition | Meaning |
|---|---|---|
| Nyquist rate | $2B$ | The critical sampling rate for a signal of bandwidth $B$. Guaranteeing perfect reconstruction requires $f_s > 2B$ (see the note below) |
| Nyquist frequency | $f_N = f_s / 2$ | The highest frequency that can be represented at a given sampling frequency |
The sampling theorem condition $f_s > 2B$ is equivalent to stating that the Nyquist frequency $f_N = f_s/2$ exceeds the signal's highest frequency $B$.
2. Proof Outline via Fourier Analysis
The core of the proof is the fact that sampling in the time domain corresponds to periodic repetition of the spectrum in the frequency domain.
2.1 Properties of the Dirac Delta Function
The proof relies on the following properties of the Dirac delta function $\delta(t)$.
Property 1 (Sifting Property). For any continuous function $f(t)$,
$$f(t)\,\delta(t - t_0) = f(t_0)\,\delta(t - t_0)$$Intuitively this holds because $\delta(t - t_0)$ is zero for $t \neq t_0$, so $f(t)$ can be replaced by its value at $t = t_0$. ($\delta$ is not an ordinary function but a distribution, so strictly the identity means that both sides agree when integrated against any test function.) Integrating both sides, the delta function extracts only the value at $t = t_0$ from the integrand:
$$\int_{-\infty}^{\infty} f(t)\,\delta(t - t_0)\,dt = f(t_0)$$Property 2 (Fourier Transform of the Delta Function). Here we define the Fourier transform using ordinary frequency $f$ [Hz]: $\displaystyle F(f) = \int_{-\infty}^{\infty} g(t)\,e^{-j2\pi ft}\,dt$. Then
$$\mathcal{F}[\delta(t - t_0)] = \int_{-\infty}^{\infty} \delta(t - t_0)\,e^{-j2\pi ft}\,dt = e^{-j2\pi f t_0}$$In particular, when $t_0 = 0$, $\mathcal{F}[\delta(t)] = 1$ — a constant Fourier-transform amplitude over all frequencies. Since $\delta$ is not a finite-energy signal, the word "energy" is best avoided here.
Property 3 (Fourier Transform of the Dirac Comb). The Fourier transform of a Dirac comb with period $T$ is a Dirac comb with period $f_s = 1/T$:
$$\mathcal{F}\!\left[\sum_{n=-\infty}^{\infty} \delta(t - nT)\right] = f_s \sum_{k=-\infty}^{\infty} \delta(f - k f_s)$$Derivation of Property 3
The Dirac comb $\delta_T(t) = \displaystyle\sum_{n=-\infty}^{\infty} \delta(t - nT)$ is periodic with period $T$, so it can be expanded as a Fourier series:
$$\delta_T(t) = \sum_{n=-\infty}^{\infty} C_n \, e^{j 2\pi n f_s t}$$To find the Fourier coefficients $C_n$: on the interval $-T/2 \leq t \leq T/2$, $\delta_T(t) = \delta(t)$, so
$$C_n = \dfrac{1}{T}\int_{-T/2}^{T/2} \delta_T(t)\, e^{-j2\pi n f_s t}\, dt = \dfrac{1}{T}\int_{-T/2}^{T/2} \delta(t)\, e^{-j2\pi n f_s t}\, dt$$By Property 1 (the sifting property), $\displaystyle\int \delta(t)\, g(t)\, dt = g(0)$, so
$$C_n = \dfrac{1}{T}\, e^{-j2\pi n f_s \cdot 0} = \dfrac{1}{T}$$That is, $C_n = 1/T$ for all $n$. Therefore
$$\delta_T(t) = \dfrac{1}{T}\sum_{n=-\infty}^{\infty} e^{j 2\pi n f_s t}$$The Fourier transform of each term is $\mathcal{F}[e^{j2\pi f_0 t}] = \delta(f - f_0)$, so
$$\mathcal{F}[\delta_T(t)] = \dfrac{1}{T}\sum_{n=-\infty}^{\infty} \delta(f - n f_s) = f_s \sum_{n=-\infty}^{\infty} \delta(f - n f_s) \qquad \square$$2.2 Mathematical Representation of Sampling
To make the analysis tractable, sampling is idealised as impulse sampling: sampling a continuous signal $f(t)$ at intervals $T = 1/f_s$ is represented as the distribution obtained by multiplying $f(t)$ by a Dirac comb. This is a mathematical model only; it is not the discrete sequence an ADC produces, nor its zero-order-hold output. Applying Property 1 to each term:
$$f_\delta(t) = f(t) \cdot \sum_{n=-\infty}^{\infty} \delta(t - nT) = \sum_{n=-\infty}^{\infty} f(nT)\, \delta(t - nT)$$The subscript is $\delta$ (the signal multiplied by the delta train). Writing the sampled signal as $f_s(t)$ would clash with $f_s$, the symbol already used for the sampling frequency.
The second equality uses $f(t)\,\delta(t - nT) = f(nT)\,\delta(t - nT)$ (Property 1). That is, the sampled signal $f_\delta(t)$ consists of delta functions at each sample point $t = nT$, each with amplitude $f(nT)$.
2.3 Effect in the Frequency Domain
We compute the Fourier transform of $f_\delta(t)$. In the ordinary-frequency Fourier transform, multiplication in the time domain corresponds directly to convolution in the frequency domain ($\mathcal{F}[g \cdot h] = G * H$). Therefore
$$F_\delta(f) = F(f) * \left[f_s \sum_{k=-\infty}^{\infty} \delta(f - k f_s)\right]$$where Property 3 has been used. Since convolution with a delta function is a shift (the integral form of Property 1):
$$F_\delta(f) = f_s \sum_{k=-\infty}^{\infty} F(f - k f_s)$$That is, the original spectrum $F(f)$ is repeated at intervals of $f_s$.
2.4 Principle of Recovery
When $f_s > 2B$ holds, the repeated spectral copies do not overlap. Applying an ideal low-pass filter whose cutoff $f_c$ satisfies $B < f_c < f_s - B$ (for instance $f_c = f_s/2$) therefore isolates the $k = 0$ term. What that isolates, however, is not $F(f)$ itself but $f_s F(f)$ — note the factor $f_s$ in the expression above. Recovering the original $F(f)$ requires a passband gain of $T = 1/f_s$. This is the essence of the sampling theorem (the gain is worked through in Step 2 of Section 3.1).
3. Reconstruction Formula (Sinc Interpolation)
Writing the "application of an ideal low-pass filter" described above in the time domain yields the Whittaker–Shannon interpolation formula:
Reconstruction Formula.
$$f(t) = \sum_{n=-\infty}^{\infty} f(nT)\, \operatorname{sinc}\!\left(\dfrac{t - nT}{T}\right)$$where $T = 1/f_s$ is the sampling interval and $\operatorname{sinc}(x) = \dfrac{\sin(\pi x)}{\pi x}$ for $x \neq 0$, extended continuously by $\operatorname{sinc}(0) = 1$, is the normalized sinc function.
This formula interpolates between sample points by superposing sinc functions weighted by each sample value $f(nT)$. The sinc function is the impulse response of the ideal low-pass filter, and satisfies $\operatorname{sinc}(0) = 1$ and $\operatorname{sinc}(n) = 0$ for any nonzero integer $n$. Therefore, at $t = mT$, only the $n = m$ term contributes $f(mT) \cdot 1$, while all other terms vanish, so the original values are exactly reproduced at the sample points.
3.1 Derivation of the Reconstruction Formula
Derivation
Step 1: Spectrum of the sampled signal.
As shown in Section 2, the spectrum of the sampled signal is
$$F_\delta(f) = f_s \sum_{k=-\infty}^{\infty} F(f - k f_s)$$When $f_s > 2B$, only the $k = 0$ term contributes in the range $|f| < f_s/2$, so
$$F_\delta(f) = f_s \, F(f) \qquad (|f| < f_s/2)$$Step 2: Recovery via ideal low-pass filter.
Apply an ideal low-pass filter $H(f)$ with cutoff frequency $f_s/2$ and gain $T = 1/f_s$. The gain $T$ is needed because Step 1 shows $F_\delta(f) = f_s F(f)$, so we must multiply by $1/f_s = T$ to recover the original spectrum $F(f)$:
$$H(f) = \begin{cases} T & (|f| \leq f_s/2) \\ 0 & (|f| > f_s/2) \end{cases}$$Writing the recovered spectrum as $\hat{F}(f)$,
$$\hat{F}(f) = H(f) \cdot F_\delta(f) = T \cdot f_s \, F(f) = F(f) \qquad (|f| < f_s/2)$$Since $F(f) = 0$ for $|f| > B$ and $f_s/2 > B$, $\hat{F}$ agrees with $F$ at every frequency.
Step 3: Transform back to the time domain.
Taking the inverse Fourier transform of $H(f) \cdot F_\delta(f)$, multiplication becomes convolution in the time domain:
$$f(t) = h(t) * f_\delta(t)$$We compute the inverse Fourier transform of $H(f)$ (the impulse response of the ideal LPF):
\begin{align} h(t) &= \int_{-f_s/2}^{f_s/2} T \, e^{j2\pi ft}\,df \\ &= T \left[\dfrac{e^{j2\pi ft}}{j2\pi t}\right]_{-f_s/2}^{f_s/2} \\ &= \dfrac{T}{j2\pi t}\left(e^{j\pi f_s t} - e^{-j\pi f_s t}\right) \end{align}Applying Euler's formula $e^{j\theta} - e^{-j\theta} = 2j\sin\theta$,
\begin{align} h(t) &= \dfrac{T}{j2\pi t} \cdot 2j\sin(\pi f_s t) \\ &= T \cdot \dfrac{\sin(\pi f_s t)}{\pi t} \end{align}Substituting $f_s = 1/T$ and simplifying,
$$h(t) = \dfrac{\sin(\pi t/T)}{\pi t/T} = \operatorname{sinc}\!\left(\dfrac{t}{T}\right)$$Convolving with $f_\delta(t) = \sum_n f(nT)\,\delta(t - nT)$, using the definition of convolution and Property 1:
\begin{align} f(t) &= h(t) * f_\delta(t) = \int_{-\infty}^{\infty} h(t - \tau)\, f_\delta(\tau)\, d\tau \\ &= \int_{-\infty}^{\infty} h(t - \tau) \sum_{n=-\infty}^{\infty} f(nT)\,\delta(\tau - nT)\, d\tau \\ &= \sum_{n=-\infty}^{\infty} f(nT) \underbrace{\int_{-\infty}^{\infty} h(t - \tau)\,\delta(\tau - nT)\, d\tau}_{= h(t - nT)} \\ &= \sum_{n=-\infty}^{\infty} f(nT)\, \operatorname{sinc}\!\left(\dfrac{t - nT}{T}\right) \qquad \square \end{align}3.2 Convergence Conditions
Band-limiting is essential for the sinc series to reproduce the original signal: a continuous function that is not band-limited is not recovered by this series, however square-summable its samples may be. For the series above to converge pointwise to the signal $f$, it suffices that $f$ be band-limited (writing the frequency variable as $\xi$ to keep it apart from the signal: $F(\xi) = 0$ for $|\xi| > B$) and that $f \in L^2(\mathbb{R})$ ($L^2$ convergence is guaranteed by Plancherel's theorem). Under these assumptions $f$ is automatically continuous with $\sum_n |f(nT)|^2 < \infty$, and the series converges uniformly.
4. Aliasing
When $f_s < 2B$—that is, when the sampling frequency is insufficient—the spectral copies in the frequency domain overlap. This is called aliasing. The critical case $f_s = 2B$, where the copies merely touch at the endpoints, is treated separately from this overlap (see the note in Section 1).
When aliasing occurs, frequency components above the Nyquist frequency $f_s/2$ appear as spurious low-frequency components below $f_s/2$. Specifically, a signal component at frequency $f_0$ is observed at the frequency $|f_0 - k f_s|$ for the integer $k$ that brings it into $[0,\, f_s/2]$. To capture the direction as well, fold the signed value $f_0 - k f_s$ into $(-f_s/2,\, f_s/2]$ (Section 4.1).
4.1 Intuitive Examples of Aliasing
Video example: The phenomenon where helicopter rotors or car wheels appear to rotate slowly in reverse in movies and on television is a classic example of temporal aliasing. Because the camera's frame rate (sampling frequency) is insufficient for the temporal frequency of the blade pattern (rotation rate × number of blades), the apparent frequency folds back below the Nyquist frequency. Depending on the sign of the folded value, the motion can look like a slow reverse rotation or a slow forward one.
Fluorescent light and fan example: Fluorescent lights flicker 100 times per second (East Japan, 50 Hz mains) or 120 times per second (West Japan, 60 Hz mains). This flickering acts as a kind of strobe (temporal sampling), so when a fan's blade frequency is close to the flicker frequency, the blades appear to rotate slowly or in reverse. The apparent pattern frequency is the signed value $f_p - k f_s$ folded into $(-f_s/2,\, f_s/2]$, and a negative value means reverse rotation. For example, if a 3-blade fan rotates at 27 revolutions per second (1620 rpm), the blade pattern frequency is $f_p = 27 \times 3 = 81$ Hz. Under East Japan fluorescent lighting (100 Hz sampling) this gives $81 - 100 = -19$ Hz. That is the frequency of the blade pattern, so dividing by the number of blades, the fan appears to turn backwards at about $19/3 \approx 6.3$ revolutions per second. If the speed increases to 35 rps ($f_p = 35 \times 3 = 105$ Hz), the value becomes $105 - 100 = +5$ Hz, so the fan now appears to creep forward at about $5/3 \approx 1.7$ revolutions per second. With LED lighting, which either does not flicker or flickers at very high frequency, this phenomenon is much less likely to occur.
5. Practical Considerations
5.1 Anti-Aliasing Filter
Real-world signals are often not strictly band-limited. Therefore, an anti-aliasing filter (an analog low-pass filter) is inserted before sampling to remove components above the Nyquist frequency. This ensures that the input signal to the ADC (analog-to-digital converter) satisfies the band-limiting condition.
Ideally, a rectangular (brick-wall) filter characteristic is desired, but this is physically unrealizable. In practice, approximate low-pass filters such as Butterworth or Chebyshev filters are used. Since a transition band (the frequency range between the passband edge and the Nyquist frequency) is needed, $f_s$ is set somewhat higher than $2B$.
Audio example (A/B comparison): The clips below play the opening of Tchaikovsky's "March" from The Nutcracker, synthesized by additive sawtooth synthesis (a sum of sine harmonics whose $k$-th amplitude is $1/k$). Both use the same sampling frequency $f_s = 8\text{ kHz}$ (Nyquist frequency $4\text{ kHz}$) but are generated differently. The right version keeps only the harmonics below the Nyquist frequency (equivalent to passing through an anti-aliasing filter), while the left version is not band-limited: a harmonic $k f_0$ above $4\text{ kHz}$ (where $f_0$ is the fundamental) folds back to $|k f_0 - m f_s|$ for the appropriate integer $m$, turning into a muddy sound that destroys the piece's harmony.
Not band-limited (aliased)
high harmonics fold back and clash
Band-limited (no aliasing)
harmonics below Nyquist only
5.2 Oversampling
Oversampling means using a sampling frequency significantly higher than the Nyquist rate. Oversampling offers the following advantages:
- Anti-aliasing filter design becomes easier (the transition band can be made wider)
- Under the usual model that treats quantization error as white noise, the quantization noise is spread over a wider bandwidth, improving the in-band SNR
- Combined with noise shaping ($\Sigma\Delta$ modulation), high effective resolution can be achieved
For example, audio $\Sigma\Delta$ ADCs employ 64× or 128× oversampling and, even with a 1-bit or few-bit internal quantizer, obtain a high in-band signal-to-noise ratio and effective resolution by combining it with noise shaping.
Note that a 24-bit output word length is not the same thing as 24 effective bits (ENOB). Reaching a true 24-bit ENOB in a real audio ADC is extremely demanding.
5.3 Practical Reconstruction
The theoretical reconstruction formula (sinc interpolation) requires an infinitely long sinc function, making it impractical to use directly. At the output of a DAC (digital-to-analog converter), a zero-order hold (staircase output) followed by a reconstruction filter (analog low-pass filter) is the standard approach. In the digital domain, the standard approach is to approximate ideal sinc interpolation with finite-length FIR filters, windowed-sinc interpolation, and polyphase filters. Depending on the application, polynomial and spline interpolation are used as well (image resampling, for instance).
6. Applications
6.1 Digital Audio (CD: 44.1 kHz)
The human audible range is approximately 20 Hz – 20 kHz. The sampling theorem puts the boundary at twice that, 40 kHz, but to guarantee perfect reconstruction including the 20 kHz component itself, the condition $f_s > 2B$ adopted in this article requires going above 40 kHz. The direct reason 44.1 kHz was chosen is historical: compatibility with the scheme used to record early PCM digital audio onto video tape recorders. Under NTSC, 245 active lines/field × 3 samples/line × 60 fields/s = 44,100; under PAL, 294 active lines/field × 3 samples/line × 50 fields/s = 44,100 — the same figure either way. Technically it also works out well: the Nyquist frequency lands at 22.05 kHz, leaving a transition band between the 20 kHz audible limit and the anti-aliasing/reconstruction filters.
Modern high-resolution audio uses sampling rates of 96 kHz or 192 kHz. The advantages usually cited are a wider usable signal bandwidth, more room for the transition bands of analog and digital filters, and lower latency in some processing chains. The claim that "finer sample spacing means finer time resolution" is not correct from the standpoint of the sampling theorem: for a signal band-limited to $B$, the waveform between samples is already determined continuously by sinc interpolation, so the 22.7 µs sample spacing of 44.1 kHz is not a quantum of time.
6.2 Digital Imaging
Viewing an image as a two-dimensional spatial signal, the reasoning of the sampling theorem carries over to spatial frequencies. Real images, however, are not strictly band-limited, and the sampling grid, the lens and the sensor array all have an effect. What sets the spatial sampling frequency is the sensor's pixel pitch; the optical MTF (modulation transfer function) of the lens governs how strongly each spatial frequency reaches the sensor. Spatial frequency components above the Nyquist frequency set by the pixel pitch appear as moiré (spatial aliasing). With a Bayer array the effective sampling structure differs per colour component, so the real picture is more involved. In digital cameras, an optical low-pass filter (OLPF) serves as the anti-aliasing mechanism.
6.3 ADC (Analog-to-Digital Converter)
In ADC design, the sampling theorem governs the relationship between input bandwidth and sampling clock. Successive approximation register (SAR) ADCs typically operate near the Nyquist rate, while $\Sigma\Delta$ ADCs combine substantial oversampling with decimation to achieve high resolution. In communications, bandpass sampling can digitize an RF signal directly at a sampling frequency below twice its highest frequency. The minimum usable rate is tied to the signal bandwidth, but bandwidth alone does not determine it: the centre frequency and band edges must also sit correctly relative to $f_s$ so that the aliased spectral copies do not overlap.
7. Frequently Asked Questions (FAQ)
Q1. What is the Sampling Theorem (Shannon's theorem)?
The sampling theorem states that a band-limited signal (one containing frequency components only up to $B$ Hz) can be perfectly reconstructed from its samples if the sampling frequency satisfies $f_s > 2B$. Reconstruction uses the sinc interpolation formula. This theorem is fundamental to all analog-to-digital conversion in digital audio, image processing, communications engineering, and beyond.
Q2. What is aliasing?
Aliasing is a phenomenon that occurs when sampling under the condition $f_s < 2B$. Spectral copies overlap in the frequency domain, and components above the Nyquist frequency fold back into lower frequencies. Once aliasing has occurred, it cannot be removed by post-processing. Prevention methods include applying an anti-aliasing filter (low-pass filter) before sampling, or using a sufficiently high sampling frequency.
Q3. Why is the CD sampling rate 44.1 kHz?
The direct reason is historical: compatibility with the scheme used to record early PCM digital audio onto video tape recorders. Technically it also places the Nyquist frequency at 22.05 kHz, leaving a transition band between the 20 kHz audible limit and the anti-aliasing filter. See Section 6.1 for details.
Summary
The key points of the sampling theorem are summarized below.
| Item | Content |
|---|---|
| Assumption | The signal $f(t)$ is band-limited (writing the frequency variable as $\xi$, its Fourier transform satisfies $F(\xi) = 0$ for $|\xi| > B$) |
| Theorem | If $f_s > 2B$, then $f(t)$ can be reconstructed exactly from the samples $\{f(nT)\}_{n \in \mathbb{Z}}$, where $T = 1/f_s$ |
| Nyquist rate | $2B$ (the critical sampling rate). A band-limited signal of finite energy can still be reconstructed at $f_s = 2B$, but a sinusoidal component sitting exactly at $\pm B$ is lost. To cover that case as well, $f_s > 2B$ is required |
| Reconstruction | $f(t) = \displaystyle\sum_{n=-\infty}^{\infty} f(nT)\, \operatorname{sinc}\!\left(\frac{t - nT}{T}\right)$ (Whittaker–Shannon interpolation formula) |
| Aliasing | When $f_s$ is too low, the replicated spectra overlap in the frequency domain and high-frequency content folds back into the low-frequency band |
| In practice | Band-limit the signal beforehand with an anti-aliasing filter, and set $f_s$ somewhat above $2B$ to allow for the filter transition band |
Points to remember
- The inequality is kept strict in order to cover a sinusoidal component sitting exactly at $\pm B$; for band-limited signals of finite energy alone, $f_s = 2B$ already suffices. At $f_s = 2B$, the samples of a sinusoid at frequency $B$ become $A(-1)^n\cos\phi$, so all that can be read off is the single number $A\cos\phi$ — neither the amplitude nor the phase is determined on its own; for $\phi = \pi/2$ every sample is $0$.
- In the frequency domain, sampling amounts to replicating the spectrum at intervals of $f_s$. Aliasing is simply the overlap of these replicas.
- $\operatorname{sinc}$ is the impulse response of an ideal low-pass filter, with $\operatorname{sinc}(0) = 1$ and $\operatorname{sinc}(n) = 0$ for every nonzero integer $n$. This is why the original values are reproduced exactly at the sample points.
- The theorem goes by several names: the Nyquist–Shannon theorem internationally, the Whittaker–Kotelnikov–Shannon (WKS) theorem in the mathematical literature, and in Japan the Shannon–Someya theorem.
References
- C. E. Shannon, "Communication in the Presence of Noise," Proceedings of the IRE, vol. 37, no. 1, pp. 10–21, 1949.
- I. Someya (染谷勲), Hakei Densou (波形傳送, "Waveform Transmission"), Shukyosha, Tokyo, 1949. (Koushuu Kagaku Ronsou, vol. 3) — National Diet Library catalog
- H. Ogawa, "A memorial tribute to Isao Someya, 1915–2007," Sampling Theory in Signal and Image Processing, vol. 7, no. 3, pp. 227–228, 2008.
- A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing, 3rd ed., Pearson, 2010. (Ch. 4, "Sampling of Continuous-Time Signals")
- Wikipedia: Nyquist–Shannon sampling theorem
- Wikipedia: Aliasing