Nyquist–Shannon sampling theorem

The Nyquist–Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing.

The theorem is commonly called Shannon's sampling theorem, and is also known as Nyquist–Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov, Whittaker–Nyquist–Kotelnikov–Shannon, WKS, etc., sampling theorem. In addition to its mathematical originator E. T. Whittaker, and its American engineering originators Claude Shannon and Harry Nyquist, it is also attributed to the Russian engineering originator V. A. Kotelnikov and sometimes to its German engineering originators Karl Küpfmüller and H. Raabe. It is often referred to as simply the sampling theorem. See the historical background section.

Sampling is the process of converting a signal (e.g., a function of continuous time or space) into a numeric sequence (a function of discrete time or space). The theorem states conditions under which the samples represent no loss of information and can therefore be used to reconstruct the original signal with arbitrarily good fidelity. It states that unambiguous reconstruction is possible if the signal is bandlimited and the sampling frequency is greater than twice the signal bandwidth.

Introduction

A signal that is bandlimited is constrained in terms of how fast it can change and therefore how much detail it can convey in between discrete moments of time. The sampling theorem means that the discrete samples are a complete representation of the signal if the bandwidth is less than half the sampling rate, which is referred to as the Nyquist frequency. Frequency components that are at or above the Nyquist frequency are subject to a phenomenon called aliasing, which is undesirable in most applications. The severity of the problem depends on the relative strength of the aliased components.

To formalize these concepts, let $x(t)\,$ represent a real-valued continuous-time signal and let $X(f)\,$ represent its unitary Fourier transform (to the domain of ordinary frequency, Hz). That is:

X(f)  =  \mathcal{F} \{  x(t)  \}   =  \int_{-\infty}^{\infty} x(t) \ e^{-j 2 \pi f t} \ dt

Suppose $x(t)\,$ is bandlimited, such that the highest nonzero $X(f)\,$ is at frequency $f_H\,$ (as in the figure). That is:

X(f) = 0 \quad \mbox{ for all } |f| > f_H \,

Then the condition for alias-free sampling at rate $f_s\,$ (in samples per second) is:

f_s > 2 f_H\,

(where

2 f_H\,

is called the Nyquist rate for this signal)

or equivalently:

f_H < f_s /2\,

(where

f_s /2\,

is called the Nyquist frequency of this sampling system)

The time between successive samples is referred to as the sampling interval. It is given by:

T = \frac{1}{f_s} \,

And the samples of $x(t)\,$ are denoted by:

x

^* \equiv x(nT), \quad n\in\mathbb{Z}\, (integers)

The sampling theorem also states a procedure for obtaining the original $x(t)\,$ from its samples $x$ ^*\,, which works exactly when the above condition is satisfied.

The sampling process

From a signal processing perspective, the theorem describes two processes; a sampling process, in which a continuous time signal is converted to a discrete time signal, and a reconstruction process, in which the continuous signal is recovered from the discrete signal.

Let us assume that the continuous signal varies over time and that the sampling process is done by simply measuring the continuous signal's value every T seconds, which is called the sampling interval (in practice, the sampling interval is typically quite small, on the order of milliseconds or even microseconds). This results in a sequence of numbers which can be said to represent the original signal in one way or another. Let us call the elements of this sequence samples. Notice that each sample is associated to the specific point in time where it was measured. Notice also that 1/T can be interpreted as a sampling frequency, which is often represented by the symbol f_s and measured in samples per second, or equivalently, hertz.

Let us also assume that the reconstruction process is done by somehow interpolating a continuous/analog signal from the samples.

A very practical question would be to ask: under what circumstances is it possible to reconstruct the original signal completely and exactly (perfect reconstruction)?

The answer is provided by the sampling theorem, which can be interpreted as two parts:

The prodecure: Each sample should be multiplied by a particular function, called a sinc function. The frequency of the zero-crossings of the sinc function is time-scaled to equal $f_s\,$ , and the location of the sinc-function's central point is shifted to the time of that sample. All of these shifted and scaled functions are then added together to recover the original signal. Recall that a sinc-function is continuous, which means that the result of this operation is indeed a continuous signal. This procedure represents the Whittaker–Shannon interpolation formula.

The condition: The signal obtained from this reconstruction process will have no frequencies higher than one-half the sampling frequency. This reconstructed signal will match the original signal if the original signal contained no frequencies equal to or above half the sampling frequency; that is, if the sampling frequency exceeds twice the highest frequency in the original signal. This condition is sometimes called the Nyquist criterion or the Raabe condition.

Note that if the original signal contains a frequency component exactly equal to one-half the sampling rate, the condition is not satisfied, and the resulting reconstructed signal may or may not have a component at that frequency, which will in general not match the original component.

This reconstruction with sinc functions is not the only, or necessarily best, reconstruction, but it is uniquely the one that satisfies the stated property. If the original signal is bandlimited to some other band away from zero (DC) frequency, then other procedures and conditions can apply to make an appropriate generalization of the sampling theorem to that situation.

Practical considerations

A few practical conclusions can be drawn from the theorem:

If it is known that the signal which we sample has a certain highest frequency f_H, the theorem gives us a lower bound on the sampling frequency to assure perfect reconstruction. This minimum value of the sampling frequency is called the critical frequency or Nyquist rate, denoted f_N.

If instead the sampling frequency is known, the theorem gives us an upper bound for frequency components of the signal to allow for perfect reconstruction.

Both of these cases imply that the signal to be sampled should be bandlimited, i.e., any component of this signal which has a frequency above a certain bound should be zero, or at least sufficiently close to zero to allow us to neglect its influence on the resulting reconstruction. In the first case the condition of bandlimitation of the sampled signal can be accomplished by assuming a model of the signal which can be analysed in terms of the frequency components it contains, e.g., sounds which are made by a speaking human normally contains very small frequency components at or above 5 kHz and it is then sufficient to sample such an audio signal with a sampling frequency of at least 10 kHz. For the second case, we have to assure that the sampled signal is bandlimited such that frequency components at or above half of the sampling frequency can be neglected. This is usually accomplished by means of a suitable low-pass filter.

In practice, neither of the two statements of the sampling theorem described above can be completely satisfied, and neither can the reconstruction formula be precisely implemented. The reconstruction process which involves the sinc-functions can be described as ideal. It cannot be realized in practice since it implies that each sample contributes to the reconstructed signal at almost all time points. Instead some type of approximations of the sinc-functions which are truncated to limited intervals have to be used. The error which corresponds to the sinc-function approximation is referred to as interpolation error. Furthermore, in practice the sampled signal can never be exactly bandlimited. This means that even if an ideal reconstruction could be made, the reconstructed signal would not be exactly the sampled signal. The error which corresponds to the failure of bandlimitation is referred to as aliasing.

The sampling theorem does not say what happens when the conditions and procedures are not exactly met, but its proof suggests an analytical framework in which the non-ideality can be studied. Someone who is going to design a system which deals with sampling and reconstruction processes needs a thorough understanding of the signal to be sampled, in particular its frequency content, the sampling frequency, how the signal is reconstructed in terms of interpolation, and the requirement for the total reconstruction error, including aliasing and interpolation error. These properties and parameters have to be carefully tuned in order to obtain a useful system.

Aliasing

If the sampling condition is not satisfied, then frequencies will overlap (see the proof below); that is, frequencies above half the sampling rate will be reconstructed as, and interfere with, frequencies below half the sampling rate. The resulting distortion is called aliasing; the reconstructed signal is said to be an alias of the original signal, in the sense that it has the same set of sample values.

To prevent or reduce aliasing, two things can be done:

Increase the sampling rate, to above twice some or all of the frequencies that are aliasing.
Introduce an anti-aliasing filter or make the anti-aliasing filter more stringent.

The anti-aliasing filter is to restrict the bandwidth of the signal to satisfy the sampling condition. Such a restriction works in theory, but is not precisely satisfiable in reality, because real filters will always allow some leakage of high frequencies. However, the leakage energy can be small enough that the aliasing effects are negligible.

Application to multivariable signals and images

The sampling theorem is usually formulated for functions of a single variable. Consequently, the theorem is directly applicable to time-dependent signals and is normally formulated in that context. However, the sampling theorem can be extended in a straightforward way to functions of arbitrarily many variables. Grayscale images, for example, are often represented as two-dimensional arrays (or matrices) of real numbers representing the relative intensities of pixels (picture elements) located at the intersections of row and column sample locations. As a result, images require two independent variables, or indices, to specify each pixel uniquely — one for the row, and one for the column.

Color images typically consist of a composite of three separate grayscale images, one to represent each of the three primary colors — red, green, and blue, or RGB for short. Other colorspaces using 3-vectors for colors include HSV, LAB, XYZ, etc. Some colorspaces such as cyan, magenta, yellow, and black (CMYK) may represent color by four dimensions. All of these are treated as vector-valued functions over a two-dimensional sampled domain.

Similar to one-dimensional discrete-time signals, images can also suffer from aliasing if the sampling resolution, or pixel density, is inadequate. For example, a digital photograph of a striped shirt with high frequencies (in other words, the distance between the stripes is small), can cause aliasing of the shirt when it is sampled by the camera's image sensor. The aliasing appears as a Moiré pattern. The "solution" to higher sampling in the spatial domain for this case would be to move closer to the shirt or use a higher resolution sensor.

Another example is shown to the right in the brick patterns. The top image shows the effects when the sampling theorem's condition is not satisfied. When software rescales an image (the same process that creates the thumbnail shown in the lower image) it, in effect, runs the image through a low-pass filter first and then downsamples the image to result in a smaller image that does not exhibit the Moiré pattern. The top image is what happens when the image is downsampled without low-pass filtering: aliasing results.

The top image was created by zooming out in GIMP and then taking a screenshot of it. The likely reason that this causes a banding problem is that the zooming feature simply downsamples without low-pass filtering (probably for performance reasons) since the zoomed image is for on-screen display instead of printing or saving.

The application of the sampling theorem to images should not be made without care. For example, the sampling process in any standard image sensor (CCD or CMOS camera) is relatively far from the ideal sampling which would measure the image intensity at a single point. Instead these devices have a relatively large sensor area at each sample point in order to obtain sufficient amount of light. Also, it is not obvious that the analog image intensity function which is sampled by the sensor device is bandlimited. It should be noted, however, that the non-ideal sampling is itself a type of low-pass filter, although far from one that ideally removes high frequency components. Despite images having these problems in relation to the sampling theorem, the theorem can be used to describe the basics of down and up sampling of images.

Downsampling

When a signal is downsampled, the sampling theorem can be invoked via the artifice of resampling a hypothetical continuous-time reconstruction. The Nyquist criterion must still be satisfied with respect to the new lower sampling frequency in order to avoid aliasing. To meet the requirements of the theorem, the signal must usually pass through a low-pass filter of appropriate cutoff frequency as part of the downsampling operation. This low-pass filter, which prevents aliasing, is called an anti-aliasing filter.

Critical frequency

The Nyquist rate is defined as twice the bandwidth of the continuous-time signal. It should be noted that the sampling frequency must be strictly greater than the Nyquist rate of the signal to achieve unambiguous representation of the signal. This constraint is equivalent to requiring that the system's Nyquist frequency (also known as critical frequency, and equal to half the sample rate) be strictly greater than the bandwidth of the signal. If the signal contains a frequency component at precisely the Nyquist frequency then the corresponding component of the sample values cannot have sufficient information to reconstruct the Nyquist-frequency component in the continuous-time signal because of phase ambiguity. In such a case, there would be an infinite number of possible and different sinusoids (of varying amplitude and phase) of the Nyquist-frequency component that are represented by the discrete samples.

As an example, consider this family of signals at the critical frequency:

x(t) = \frac{1}{\cos(\theta)} \cos(2 \pi \frac{f_s}{2} t + \theta) = \cos(2 \pi \frac{f_s}{2} t) + A\sin(2 \pi \frac{f_s}{2} t)\

Where the samples:

x

^* = x(n/f_s) = \cos(\pi n) = (-1)^n \

are in every case just alternating –1 and +1 for any phase θ or sine component amplitude A; thus there is no way to determine both the amplitude and the phase of the continuous-time sinusoid x(t) that x^* was sampled from. This ambiguity is the reason for the strict inequality of the sampling condition.

Mathematical basis for the theorem

The Nyquist-Shannon sampling and reconstruction theorem asserts that, given a bandlimited continuous-time signal, x(t), that is uniformly sampled at a sufficient rate, even if all of the information in the signal between samples is discarded, there remains sufficient information in the samples so that the original continuous-time signal can be mathematically reconstructed from only those samples. To prove this, a mathematical representation of the uniform sampled signal that effectively discards the information between samples must be constructed:

x_s(t) = x(t)\cdot \left(T\cdot \Delta_T(t) \right) \

x(t) is the signal before sampling.

x_s(t) is the resulting sampled signal

Δ_T(t) is the sampling operator called the Dirac comb and, being periodic with period T, can be expressed as a Fourier series:

T\cdot \Delta_T(t)\,

\equiv T\cdot \sum_{n=-\infty}^{\infty} \delta(t - nT) \

= T\cdot \frac{1}{T}\sum_{k=-\infty}^{\infty} e^{i 2 \pi k t/T} \

= \sum_{k=-\infty}^{\infty} e^{i 2 \pi k f_s t} \

f_s = 1/T is the sampling frequency and is the fundamental frequency of the periodic function Δ_T(t).

δ(t-nT) is a dirac impulse delayed to time nT.

It is clear that Δ_T(t) takes a value of zero except for values of t that are at the sampling instances, nT, for integer n. Equally clear that x_s(t) takes on zero values for all t except for the sampling instances nT. Multiplying x(t) by Δ_T(t) effectively discards all of the information between sampling instances and retains information only at the sampling instances nT. x_s(t) can be represented in terms of the samples:

x_s(t)\,

= x(t) \cdot T \sum_{n=-\infty}^{\infty} \delta(t - nT) \

= T \sum_{n=-\infty}^{\infty} x(t)\cdot \delta(t - nT) \

= T \sum_{n=-\infty}^{\infty} x(nT) \cdot \delta(t - nT) \

= \sum_{n=-\infty}^{\infty} T\cdot x(nT) \cdot \delta(t - nT) \

Where x(nT) are the samples. As shown above, x_s(t) can also be written as:

x_s(t)\,

= x(t) \cdot \sum_{k=-\infty}^{\infty} e^{i 2 \pi k f_s t} \

= \sum_{k=-\infty}^{\infty} x(t) \cdot e^{i 2 \pi k f_s t} \

Using the frequency shifting property of the continuous Fourier transform,

X_s(f)\,

\equiv \mathcal{F} \left \{ x_s(t) \right \} \

= \mathcal{F} \left \{ \sum_{k=-\infty}^{\infty} x(t) \cdot e^{i 2 \pi k f_s t}  \right \} \

= \sum_{k=-\infty}^{\infty} \mathcal{F} \left \{ x(t) \cdot e^{i 2 \pi k f_s t}  \right \} \

= \sum_{k=-\infty}^{\infty} X(f - k f_s) \

where X(f) is the Fourier transform of x(t). This says that the spectrum of the signal being sampled is shifted and repeated forever at integral multiples of the sampling frequency, f_s. Now constrain x(t) or X(f) to be bandlimited to f_H (i.e. X(f) = 0 for all |f| > f_H) and consider what condition would allow no overlap of the tails of adjacent images X(f):

"right tail of k^th image of X(f)" < "left tail of (k+1)^th image"

k f_s + f_H\,

< (k+1) f_s - f_H = k f_s + f_s - f_H \

f_H\,

< f_s - f_H \

2 f_H\,

< f_s = \frac{1}{T} \

With that condition satisfied, there is no overlap of images and X(f) (and thus x(t)) can be reconstructed from X_s(f) (or x_s(t)) by low pass filtering out all of the images of X(f) in X_s(f) except the original image at the baseband. To do that, f_s > 2f_H (to prevent overlap) and the frequency response of the reconstruction filter H(f) must be:

H(f) = \mathrm{rect} \left(\frac{f}{f_s} \right) = \begin{cases}1 & |f| < \frac{f_s}{2} \\ 0 & |f| \ge \frac{f_s}{2} \end{cases}

With H(f) so defined, it is clear that

X(f) = H(f) \cdot X_s(f) \

and the spectrum of the original signal that was sampled is recovered from the spectrum of the sampled signal. This means, in the time domain, that the original signal that was sampled is recovered from the sampled signal. This is the sampling half of the Nyquist-Shannon sampling and reconstruction theorem. It says that the sampling frequency, f_s, must be strictly greater than twice the bandwidth, f_H, of the continuous-time signal, x(t), for no information to be lost (or "aliased"). The reconstruction half follows.

The impulse response of the reconstruction filter is the inverse Fourier transform of H(f):

h(t)\,

= \mathcal{F}^{-1} \left \{ H(f) \right \} \

= \int_{-\infty}^{\infty} H(f) e^{i 2 \pi f t} \,df \

= \int_{-\infty}^{\infty} \mathrm{rect} \left(\frac{f}{f_s} \right) e^{i 2 \pi f t} \,df \

= \int_{-f_s /2}^{f_s /2} e^{i 2 \pi f t} \,df \

= \frac{1}{i 2 \pi t} e^{i 2 \pi f t}\bigg >_{-f_s /2}^{f_s /2} \

= \frac{1}{\pi t} \frac{\left( e^{i \pi f_s t} - e^{-i \pi f_s t} \right)}{2 i} \

= \frac{\sin(\pi f_s t)}{\pi t}  \

= f_s\cdot \mathrm{sinc}(f_s t) \

, in terms of the normalized sinc function

This function is the impulse response of the reconstruction filter that has for its input, the sampled signal x_s(t):

x_s(t) = \sum_{n=-\infty}^{\infty} T\cdot x(nT)\cdot \delta(t - nT) \

x_s(t) is no more than a collection of dirac impulses, δ(t-nT), each delayed to the time of their sampling instance, nT and weighted by a value proportional to the continuous-time signal that was sampled at that instance, x(nT). Since the reconstruction filter is a linear, time-invariant system, each impulse at time nT generates its own impulse response delayed to the same time, and the output of the reconstruction filter is the sum of outputs driven by each weighted impulse separately. For each input impulse, the component of the output is the impulse response delayed by the same delay of that input impulse, h(t-nT), and weighted by the same scaling factor attached to that input impulse, T•x(nT). That is, the output of the reconstruction filter is:

x(t)\,

= h(t) * x_s(t) \

= h(t) * \sum_{n=-\infty}^{\infty} T \cdot x(nT)\cdot \delta(t - nT)  \

= \sum_{n=-\infty}^{\infty} x(nT)\cdot T\cdot \left * \delta(t - nT)\right \

= \sum_{n=-\infty}^{\infty} x(nT)\cdot  \frac{1}{f_s} \cdot h(t - nT) \

= \sum_{n=-\infty}^{\infty} x(nT)\cdot \mathrm{sinc} \left( f_s (t - nT) \right) \

= \sum_{n=-\infty}^{\infty} x(nT)\cdot \mathrm{sinc} \left( \frac{t - nT}{T} \right) \

This shows explicitly how the samples of the original continuous-time function, x(nT), are combined to reconstruct that function, x(t), at times that are between the samples when such data was discarded in the sampling process and is known as the Whittaker–Shannon interpolation formula. This completes the reconstruction half of the Nyquist-Shannon sampling and reconstruction theorem.

In summary, assuming an adequate sampling rate, f_s > 2f_H, the spectrum of the original input, X(f), is identical to the baseband image of the sampled signal, X_s(f), and the ideal reconstruction filter simply discards all of the other images in X_s(f), where k≠0, leaving the baseband image untouched.

When considering the effect of practical reconstruction (dirac impulses are decidedly not practical), the question one asks is "what device, perhaps an LTI system, must be placed between the ideal sampled signal x_s(t), and the practical signal outputted (before the final reconstruction filter removes the images other than the baseband image) by some device such as a digital-to-analog converter?" This is the basis for modeling and understanding the effect of the zero-order hold on the most common practical means of reconstruction of the sampled signal.

Concise summary of sampling and the Fourier transform

There is no actual device that produces the infinite-valued samples implied by the Dirac comb model of sampling. The finite-valued samples, {x(nT)}, are not a function of continuous time, and their continuous ^* Fourier transform is undefined.
To use that analysis tool, a continuous-time function is contrived conceptually (not actually nor numerically) by using the samples to modulate the "teeth" of a Dirac comb function, which does have a continuous-time Fourier transform (not within the strict definition that requires square integrable functions, but in the generalization that allows Schwartz distributions).
- The transform of the ^* modulated comb, $X_s(f)\,$ , is related to the transform of the physical waveform, $X(f)\,$ , via a superposition of shifted copies (which is equivalent to convolution with a frequency-domain Dirac comb); this superposition viewpoint leads to an understanding of aliasing and ways to mitigate it, such as lowpass filtering and/or increasing the sample-rate.
- The Fourier transform view also reveals that the sample-rate can be higher than twice the highest frequency, with no ill effect. Undersampling, which causes aliasing, is not a reversible operation. Oversampling may be inefficient or wasteful, but it is also reversible, meaning that no information is actually lost.

Undersampling

When sampling a non-baseband signal, the theorem must be restated as follows. Let

0 be the lower and higher boundaries of a frequency band and W = f_\mathrm{H} - f_\mathrm{L} be the bandwidth. Then there is a non-negative integer

N with

N \le  { f_\mathrm{L}  \over W } <   (N+1) \le  { f_\mathrm{H} \over W }

In addition, we define the remainder r as

r := f_\mathrm{L}-NW\in

^*.

Any real-valued signal x(t) with a spectrum limited to this frequency band, that is with

X(f)= \mathcal{F} \{ x \}(f)  = 0

for

|f| \,

outside the interval

^* \,,

is uniquely determined by its samples obtained at a sampling rate of $f_\mathrm{s}$ , if this sampling rate satisfies one of the following conditions:

$2\left(W+\frac{nW+r}{N-n+1}\right)\le f_\mathrm{s}\le 2\left(W+\frac{nW+r}{N-n}\right)$ for one value of n = { 0, 1, ..., N-1 }

- OR -

$2f_\mathrm{H}\le f_\mathrm{s}$ .

If $N>0$ , then the first conditions result in a sampling rate less than the Nyquist frequency $2f_\mathrm{H}$ obtained from the upper bound of the spectrum. If the so obtained sampling rates are still too high, the intuitive sampling-by-taking-values has to be replaced by sampling-by-taking-scalar-products, as is (implicitly) the case in Frequency-division multiplexing.

Example: Consider FM radio to illustrate the idea of undersampling.

In the US, FM radio operates on the frequency band from

f_L

= 88 MHz to

f_H

= 108 MHz. The bandwidth is given by

W = f_H - f_L = 108 \ \mathrm{MHz} - 88 \ \mathrm{MHz} = 20 \ \mathrm{MHz}

The sampling conditions are satisfied for

N < 4.4 = { 88 \ \mathrm{MHz} \over 20 \ \mathrm{MHz} } < N+1

Therefore

N=4, r=8 MHz and

n = 0, 1, 2, 3

The value

n=0

gives the lowest sampling frequencies interval

43.2\ \mathrm{MHz} and this is a scenario of undersampling.

Note that when undersampling a real-world signal, the sampling circuit must be fast enough to capture the highest signal frequency of interest. Theoretically, each sample should be taken during an infinitesimally short interval, but this is not practically feasible. Instead, the sampling of the signal should be made in a short enough interval that it can represent the instantaneous value of the signal with the highest frequency. This means that in the FM radio example above, the sampling circuit must be able to capture a signal with a frequency of 108 MHz, not 43.2 MHz. Thus, the sampling frequency may be only a little bit greater than 43.2 MHz, but the input bandwidth of the system must be at least 108 MHz.

If the theorem is misunderstood to mean twice the highest frequency, then the required sampling rate would be assumed to be greater than the Nyquist-frequency 216 MHz.

While this does satisfy the last condition on the sampling rate, it is grossly oversampled.

Note that if the FM radio band is sampled, e.g., at

43.2\ \mathrm{MHz}, then a band-pass filter is required for the anti-aliasing filter .

In certain problems, the frequencies of interest are not an interval of frequencies, but perhaps some more interesting set F of frequencies. Again, the sampling frequency must be proportional to the size of F. For instance, certain domain decomposition methods fail to converge for the 0th frequency (the constant mode) and some medium frequencies. Then the set of interesting frequencies would be something like 10 Hz to 100 Hz, and 110 Hz to 200 Hz. In this case, one would need to sample at a data rate of 360 Hz — i.e. at a sampling rate of 20 Hz with 18 real values in each sample — not 400 Hz, to fully capture these signals.

As we have seen, the normal condition for reversible sampling is that $X(f) = 0\,$ outside the interval: $\left$ ^*

And the reconstructive interpolation function is $\operatorname{sinc} \left( t/T \right)$ .

To accommodate undersampling, the generalized condition is that $X(f) = 0\,$ outside the union

\left^* \cup\left^* for some

N\in\N

which includes the normal condition as case N=0.

And the corresponding interpolation function is:

(N+1)\operatorname{sinc} \left(\frac{(N+1)t}T\right) - N\operatorname{sinc} \left( \frac{Nt}T \right)

Historical background

The sampling theorem was implied by the work of Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory"), in which he showed that up to 2B independent pulse samples could be sent through a system of bandwidth B; but he did not explicity consider the problem of sampling and reconstruction of continuous signals. About the same time, Karl Küpfmüller showed a similar result, and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step response Integralsinus; this bandlimiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a Küpfmüller filter (but seldom in English).

The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Shannon in 1949 ("Communication in the presence of noise"). V. A. Kotelnikov published similar results in 1933 ("On the transmission capacity of the 'ether' and of cables in electrical communications", translation from the Russian), as did the mathematician E. T. Whittaker in 1915 ("Expansions of the Interpolation-Theory," "Theorie der Kardinalfunktionen"), J. M. Whittaker in 1935 ("Interpolatory function theory"), and Gabor in 1946 ("Theory of communication").

Others who have independently discovered or played roles in the development of the sampling theorem have been discussed in several historical articles, for example by JerriJerri, Abdul J., "The Shannon Sampling Theorem—Its Various Extensions and Applications: A Tutorial Review," Proceedings of the IEEE, 65:1565–1595, Nov. 1977. and by LükeLüke, Hans Dieter, "The Origins of the Sampling Theorem," IEEE Communications Magazine, pp.106–108, April 1999.. For example, Lüke points out that H. Raabe, an assistant to Küpfmüller, proved the theorem in his 1939 Ph.D. dissertation; the term Raabe condition came to be associated with the criterion for unambiguous representation (sampling rate greater than twice the bandwidth).

Historical references

References

E. T. Whittaker, "On the Functions Which are Represented by the Expansions of the Interpolation Theory," Proc. Royal Soc. Edinburgh, Sec. A, vol.35, pp.181-194, 1915
H. Nyquist, "Certain topics in telegraph transmission theory," Trans. AIEE, vol. 47, pp. 617-644, Apr. 1928 Reprint as classic paper in: ''Proc. IEEE, Vol. 90, No. 2, Feb 2002.
Karl Küpfmüller, "Utjämningsförlopp inom Telegraf- och Telefontekniken," ("Transients in telegraph and telephone engineering"), Teknisk Tidskrift, no. 9 pp.153-160 and 10 pp.178-182, 1931. [http://runeberg.org/tektid/1931e/0182.html
V. A. Kotelnikov, "On the carrying capacity of the ether and wire in telecommunications," Material for the First All-Union Conference on Questions of Communication, Izd. Red. Upr. Svyazi RKKA, Moscow, 1933 (Russian). (english translation, PDF)
C. E. Shannon, "Communication in the presence of noise", Proc. Institute of Radio Engineers, vol. 37, no.1, pp. 10-21, Jan. 1949. Reprint as classic paper in: Proc. IEEE, Vol. 86, No. 2, (Feb 1998)
J. R. Higgins: Five short stories about the cardinal series, Bulletin of the AMS 12(1985)
Michael Unser: Sampling-50 Years after Shannon, Proc. IEEE, vol. 88, no. 4, pp. 569-587, April 2000

External links

Learning by Simulations Interactive simulation of the effects of inadequate sampling
Undersampling and an application of it
Sampling Theory For Digital Audio

Digital signal processing | Information theory | Mathematical theorems | Claude Shannon

Gourt Home::Gourt :: Nyquist–Shannon sampling theorem