Durch die Verfilmungen der "Millennium"-Trilogie von Stieg Larsson wurde der schwedische Schauspieler Michael Nyqvist weltberühmt. Michael Nyqvist wurde vor allem durch die "Millennium"-Verfilmungen bekannt. In Hollywood war er als charismatischer Bösewicht begehrt. Der ausgebildete Theaterschauspieler Michael Nyqvist gehörte zum festen Ensemble des Königlichen Dramatischen Theaters (Dramaten) in Stockholm, wo er.
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Rolf Åke Mikael Nyqvist war ein schwedischer Film- und Theaterschauspieler. Rolf Åke Mikael Nyqvist [ˈmikaɛl ˈnʏˌkvist] (auch Michael Nyqvist; * 8. November in Stockholm; † Juni ebenda) war ein schwedischer Film-. Als Journalist Mikael Blomqvist in den "Millennium"-Filmen wurde Michael Nyqvist weltberühmt. Später übernahm der Schwede auch Rollen in. Zum Tod von Mikael Nyqvist:Bis in die verwahrlostesten Winkel menschlicher Widerwärtigkeiten. Wem immer etwas fehlt, der ist immer auf der. Michael Nyqvist wurde vor allem durch die "Millennium"-Verfilmungen bekannt. In Hollywood war er als charismatischer Bösewicht begehrt. Mit Michael Nyqvist verliert Schweden einen seiner vielen international anerkannten Schauspieler. Das nordische Land hat einen hohen. Der schwedische Schauspieler ("Millennium"-Trilogie von Stieg Larsson) starb nach langer Krankheit an Lungenkrebs. Mikael Nyqvist (†56) lebt nicht mehr. Der.
Rolf Åke Mikael Nyqvist [ˈmikaɛl ˈnʏˌkvist] (auch Michael Nyqvist; * 8. November in Stockholm; † Juni ebenda) war ein schwedischer Film-. Durch die Verfilmungen der "Millennium"-Trilogie von Stieg Larsson wurde der schwedische Schauspieler Michael Nyqvist weltberühmt. Mit Michael Nyqvist verliert Schweden einen seiner vielen international anerkannten Schauspieler. Das nordische Land hat einen hohen.
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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.
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 "solution" to higher sampling in the spatial domain for this case would be to move closer to the shirt, use a higher resolution sensor, or to optically blur the image before acquiring it with the sensor using an optical low-pass filter.
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.
The top image is what happens when the image is downsampled without low-pass filtering: aliasing results. The sampling theorem applies to camera systems, where the scene and lens constitute an analog spatial signal source, and the image sensor is a spatial sampling device.
Each of these components is characterized by a modulation transfer function MTF , representing the precise resolution spatial bandwidth available in that component.
When the optical image which is sampled by the sensor device contains higher spatial frequencies than the sensor, the under sampling acts as a low-pass filter to reduce or eliminate aliasing.
When the area of the sampling spot the size of the pixel sensor is not large enough to provide sufficient spatial anti-aliasing , a separate anti-aliasing filter optical low-pass filter may be included in a camera system to reduce the MTF of the optical image.
Instead of requiring an optical filter, the graphics processing unit of smartphone cameras performs digital signal processing to remove aliasing with a digital filter.
Digital filters also apply sharpening to amplify the contrast from the lens at high spatial frequencies, which otherwise falls off rapidly at diffraction limits.
The sampling theorem also applies to post-processing digital images, such as to up or down sampling. Effects of aliasing, blurring, and sharpening may be adjusted with digital filtering implemented in software, which necessarily follows the theoretical principles.
That sort of ambiguity is the reason for the strict inequality of the sampling theorem's condition. As discussed by Shannon: .
A similar result is true if the band does not start at zero frequency but at some higher value, and can be proved by a linear translation corresponding physically to single-sideband modulation of the zero-frequency case.
That is, a sufficient no-loss condition for sampling signals that do not have baseband components exists that involves the width of the non-zero frequency interval as opposed to its highest frequency component.
See Sampling signal processing for more details and examples. The corresponding interpolation function is the impulse response of an ideal brick-wall bandpass filter as opposed to the ideal brick-wall lowpass filter used above with cutoffs at the upper and lower edges of the specified band, which is the difference between a pair of lowpass impulse responses:.
Other generalizations, for example to signals occupying multiple non-contiguous bands, are possible as well. Even the most generalized form of the sampling theorem does not have a provably true converse.
That is, one cannot conclude that information is necessarily lost just because the conditions of the sampling theorem are not satisfied; from an engineering perspective, however, it is generally safe to assume that if the sampling theorem is not satisfied then information will most likely be lost.
The sampling theory of Shannon can be generalized for the case of nonuniform sampling , that is, samples not taken equally spaced in time.
The Shannon sampling theory for non-uniform sampling states that a band-limited signal can be perfectly reconstructed from its samples if the average sampling rate satisfies the Nyquist condition.
The general theory for non-baseband and nonuniform samples was developed in by Henry Landau. In the late s, this work was partially extended to cover signals of when the amount of occupied bandwidth was known, but the actual occupied portion of the spectrum was unknown.
In particular, the theory, using signal processing language, is described in this paper. Note that minimum sampling requirements do not necessarily guarantee stability.
The Nyquist—Shannon sampling theorem provides a sufficient condition for the sampling and reconstruction of a band-limited signal. When reconstruction is done via the Whittaker—Shannon interpolation formula , the Nyquist criterion is also a necessary condition to avoid aliasing, in the sense that if samples are taken at a slower rate than twice the band limit, then there are some signals that will not be correctly reconstructed.
However, if further restrictions are imposed on the signal, then the Nyquist criterion may no longer be a necessary condition. A non-trivial example of exploiting extra assumptions about the signal is given by the recent field of compressed sensing , which allows for full reconstruction with a sub-Nyquist sampling rate.
Specifically, this applies to signals that are sparse or compressible in some domain. As an example, compressed sensing deals with signals that may have a low over-all bandwidth say, the effective bandwidth EB , but the frequency locations are unknown, rather than all together in a single band, so that the passband technique does not apply.
In other words, the frequency spectrum is sparse. Traditionally, the necessary sampling rate is thus 2 B. Using compressed sensing techniques, the signal could be perfectly reconstructed if it is sampled at a rate slightly lower than 2 EB.
With this approach, reconstruction is no longer given by a formula, but instead by the solution to a linear optimization program.
Another example where sub-Nyquist sampling is optimal arises under the additional constraint that the samples are quantized in an optimal manner, as in a combined system of sampling and optimal lossy compression.
For stationary Gaussian random signals, this lower bound is usually attained at a sub-Nyquist sampling rate, indicating that sub-Nyquist sampling is optimal for this signal model under optimal quantization.
The sampling theorem was implied by the work of Harry Nyquist in ,  in which he showed that up to 2 B independent pulse samples could be sent through a system of bandwidth B ; but he did not explicitly 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 sine integral ; this bandlimiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a Küpfmüller filter but seldom so in English.
The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Kotelnikov published similar results in ,  as did the mathematician E.
Whittaker in ,  J. Whittaker in ,  and Gabor in "Theory of communication". In , the Eduard Rhein Foundation awarded Kotelnikov their Basic Research Award "for the first theoretically exact formulation of the sampling theorem".
In and , Claude E. Shannon published - 16 years after Vladimir Kotelnikov - the two revolutionary articles in which he founded the information theory.
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 Jerri  and by Lüke.
Raabe, an assistant to Küpfmüller, proved the theorem in his Ph. Meijering  mentions several other discoverers and names in a paragraph and pair of footnotes:.
As pointed out by Higgins , the sampling theorem should really be considered in two parts, as done above: the first stating the fact that a bandlimited function is completely determined by its samples, the second describing how to reconstruct the function using its samples.
Both parts of the sampling theorem were given in a somewhat different form by J. Whittaker [, , ] and before him also by Ogura [, ]. They were probably not aware of the fact that the first part of the theorem had been stated as early as by Borel .
However, he appears not to have made the link . In later years it became known that the sampling theorem had been presented before Shannon to the Russian communication community by Kotel'nikov .
In more implicit, verbal form, it had also been described in the German literature by Raabe . Several authors [33, ] have mentioned that Someya  introduced the theorem in the Japanese literature parallel to Shannon.
In the English literature, Weston  introduced it independently of Shannon around the same time. However, the paper of Cauchy does not contain such a statement, as has been pointed out by Higgins .
To avoid confusion, perhaps the best thing to do is to refer to it as the sampling theorem, "rather than trying to find a title that does justice to all claimants" .
Exactly how, when, or why Harry Nyquist had his name attached to the sampling theorem remains obscure.
The term Nyquist Sampling Theorem capitalized thus appeared as early as in a book from his former employer, Bell Labs ,  and appeared again in ,  and not capitalized in In , Blackman and Tukey cited Nyquist's article as a reference for the sampling theorem of information theory ,  even though that article does not treat sampling and reconstruction of continuous signals as others did.
Their glossary of terms includes these entries:. This explains Nyquist's name on the critical interval, but not on the theorem.
Similarly, Nyquist's name was attached to Nyquist rate in by Harold S. Black :. According to the OED , this may be the origin of the term Nyquist rate.
In Black's usage, it is not a sampling rate, but a signaling rate. From Wikipedia, the free encyclopedia. Main article: Aliasing.
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