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Nyquist–Shannon sampling theorem Nyquistův–Shannonův vzorkovací teorém Teorema de muestreo de Nyquist-Shannon Теорема Котельникова Teoremo pri specimenado Teorema de mostratge de Nyquist-Shannon Bemonsteringstheorema van Nyquist-Shannon Теорема відліків Віттекера — Найквіста — Котельникова — Шеннона Teorema da amostragem de Nyquist–Shannon 采样定理 Nyquist-Shannons samplingsteorem مبرهنة نايكويست وشانون 표본화 정리 Théorème d'échantillonnage Twierdzenie o próbkowaniu 標本化定理 Nyquist–Shannonen laginketa teorema Nyquist-Shannon-Abtasttheorem Teorema del campionamento di Nyquist-Shannon
rdfs:comment
, Nyquist-Shannonen laginketa teorema eta arteko lokarria da. Teoremak laginketa burutzean sortutako laginek denbora jarraituko seinalearen informazio guztia gorde dezaten bete beharreko baldintza ezartzen du. Teorema maiztasun banda mugatua duten seinale jarraituentzat baliagarria da soilik. Теоре́ма ві́дліків Вітте́кера — На́йквіста — Коте́льникова — Ше́ннона (теорема відліків) свідчить, що якщо неперервний сигнал x(t) має спектр, обмежений частотою Fmax, то його можна однозначно і без втрат відновити за дискретними відліками, узятими з частотою fдискр=2*Fmax, або, по-іншому, за відліками, взятими з періодом Tдискр=. Теорему відліків можна сформулювати обернено: Для того, щоб відновити сигнал за його відліками без втрат, необхідно, щоб частота дискретизації була хоча б удвічі більшою за максимальну частоту первинного неперервного сигналу. Fд ≥ 2Fmax. Nyquist-Shannons samplingsteorem, även kallad Nyquistteoremet, Shannonteoremet eller samplingsteoremet, talar om med vilken frekvens man måste mäta en vågrörelse med hjälp av sampling för att kunna återskapa signalen. Teoremet går i grova drag ut på att man måste, för att undvika fel, sampla med en frekvens som är minst dubbla signalens bandbredd annars blir resultatet av mätningen lägre än signalens verkliga frekvens. Twierdzenie o próbkowaniu, twierdzenie Nyquista–Shannona – fundamentalne twierdzenie teorii informacji, telekomunikacji oraz cyfrowego przetwarzania sygnałów opisujące matematyczne podstawy procesów próbkowania sygnałów oraz ich rekonstrukcji: Z sygnału dyskretnego złożonego z próbek danego sygnału ciągłego można wiernie odtworzyć sygnał 표본화 정리(標本化定理, 영어: sampling theorem) 또는 나이퀴스트-섀넌 표본화 정리(영어: Nyquist-Shannon sampling theorem)는 원거리 통신과 신호 처리를 다루는 정보이론의 기본이 되는 원리이다. Laŭ la teoremo pri specimenado (teoremo de Nyquist–Shannon–Kotelnikov, teoremo de Whittaker–Shannon–Kotelnikov, teoremo de Whittaker–Nyquist–Kotelnikov–Shannon, aŭ pli simple teoremo de Nyquist): preciza rekreo de eblas, se signalo havas maksimuman frekvencon kaj la (frekvenco de Nyquist, ) estas pli ol dufoje la maksimuma frekvenco. El teorema de mostratge de Nyquist-Shannon, també conegut com a teorema de mostratge de Whittaker-Nyquist-Kotelnikov-Shannon, criteri de Nyquist o teorema de Nyquist, és un teorema fonamental de la teoria de la informació, d'especial interès en les telecomunicacions. Aquest teorema va ser formulat en forma de conjectura per primer cop per Harry Nyquist l'any 1928 (Certain topics in telegraph transmission theory), i va ser demostrat formalment per Claude E. Shannon l'any 1949 (Communication in the presence of noise). La intenció del suec Harry Nyquist en formular aquest teorema era la d'obtenir una enregistrament digital de qualitat i també es pot conèixer amb el nom de condició de Nyquist. Si es fa un mostreig a un baix valor, hi ha una possibilitat que el senyal original no estigui únicam Het bemonsteringstheorema van Nyquist-Shannon is de stelling in de informatietheorie dat wanneer een analoog signaal naar een tijddiscreet signaal wordt geconverteerd, de bemonsteringsfrequentie minstens tweemaal zo hoog moet zijn als de hoogste in het signaal aanwezige frequentie om het origineel zonder fouten te kunnen reproduceren. De helft van de bemonsteringsfrequentie is de nyquistfrequentie. Anders gezegd, voor een foutloze reproductie na bemonstering mag het analoge signaal geen frequenties bevatten hoger dan de nyquistfrequentie. De tijd tussen de bemonsteringen is de nyquistinterval. Het is genoemd naar Harry Nyquist die dit theorema in 1928 bewees. Теоре́ма Коте́льникова (в англоязычной литературе — теорема Найквиста — Шеннона, теорема отсчётов) — фундаментальное утверждение в области цифровой обработки сигналов, связывающее непрерывные и дискретные сигналы и гласящее, что «любую функцию , состоящую из частот от 0 до , можно непрерывно передавать с любой точностью при помощи чисел, следующих друг за другом менее чем через секунд». При доказательстве теоремы взяты ограничения на спектр частот , где . El teorema de muestreo de Nyquist-Shannon, también conocido como teorema de muestreo de Whittaker-Nyquist-Kotelnikov-Shannon o bien teorema de Nyquist, es un teorema fundamental de la teoría de la información, de especial interés en las telecomunicaciones. Este teorema fue formulado en forma de conjetura por primera vez por Harry Nyquist en 1928 (Certain topics in telegraph transmission theory), y fue demostrado formalmente por Claude E. Shannon en 1949 (Communication in the presence of noise). In elettronica e telecomunicazioni, il teorema del campionamento di Nyquist-Shannon o semplicemente teorema del campionamento, il cui nome si deve a Harry Nyquist e Claude Shannon, è un risultato di notevole rilevanza nell'ambito della teoria dei segnali. Il teorema, comparso per la prima volta nel 1949 in un articolo di C. E. Shannon, dovrebbe chiamarsi Whittaker-Nyquist-Kotelnikov-Shannon (WNKS), secondo l'ordine cronologico di chi ne dimostrò versioni via via più generalizzate. O teorema da amostragem de Nyquist–Shannon, também conhecido simplesmente como teorema de Nyquist, é fundamental no campo da teoria da informação, particularmente na área de telecomunicações e processamento de sinais. Amostrar é o processo no qual se converte um sinal (por exemplo, uma função contínua no tempo ou espaço) em uma sequência numérica (uma função discreta no tempo ou espaço). A versão de Shannon do teorema é: em que é a maior frequência em Hertz do sinal em questão. The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth. Das Nyquist-Shannon-Abtasttheorem, auch nyquist-shannonsches Abtasttheorem und in neuerer Literatur auch WKS-Abtasttheorem (für Whittaker, Kotelnikow und Shannon) genannt, ist ein grundlegendes Theorem der Nachrichtentechnik, Signalverarbeitung und Informationstheorie. Wladimir Kotelnikow formulierte das Abtasttheorem 1933. Die Veröffentlichung in einem sowjetischen Konferenzbericht wurde im Osten seit den 1950er Jahren referenziert, blieb aber allgemein im Westen bis in die 1980er weitgehend unbekannt. Unabhängig von Kotelnikow formulierte es Claude Elwood Shannon 1948 als Ausgangspunkt seiner Theorie der maximalen Kanalkapazität, d. h. der maximalen Bitrate in einem frequenzbeschränkten, rauschbelasteten Übertragungskanal. Nyquistův–Shannonův vzorkovací teorém (také Shannonův teorém, Nyquistův teorém, Kotělnikovův teorém, Nyquistův–Shannonův teorém, Shannonův–Nyquistův–Kotělnikovův teorém, apod.) je fyzikální tvrzení o tom, že „přesná rekonstrukce spojitého, frekvenčně omezeného signálu z jeho vzorků je možná tehdy, pokud byla vzorkovací frekvence vyšší než dvojnásobek nejvyšší harmonické složky vzorkovaného signálu.“ 標本化定理(ひょうほんかていり、英: sampling theorem)またはサンプリング定理は、連続的な信号(アナログ信号)を離散的な信号(デジタル信号)へと変換する際に元の信号に忠実であるにはどの程度の間隔で標本化(サンプリング)すればよいかを示す、情報理論の定理である。 مبرهنة شانون ونايكويست أو مبرهنة شانون ونايكويست في الاستعيان (بالإنجليزية: Nyquist–Shannon sampling theorem)‏ هي من أهم المبرهنات في التقنيات الرقمية الحديثة والعلوم المتصلة بها مثل المعالجة الرقمية للإشارة والمعلوماتية ونظرية المعلومات. تعود المبرهنة إلى جهد العالمين كلود شانون وهاري نايكست. وممن اشتهر في هذه المبرهنة العالم العراقي الأمريكي عبدالجبار جري. 采样定理是数字信号处理领域的重要定理。定理內容是连续信号(通常称作“模拟信号”)与离散信号(通常称作“数字信号”)之间的一个基本桥梁。它确定了信号带宽的上限,或能捕获连续信号的所有信息的离散采样信号所允许的采样频率的下限。 严格地说,定理仅适用于具有傅里叶变换的一类数学函数,即频率在有限区域以外为零(参照图1)。离散时间傅里叶变换(泊松求和公式的一种形式)提供了实际信号的解析延拓,但只能近似该条件。直观上我们希望,当把连续函数化为采样值(叫做“样本”)的离散序列并插值到连续函数中,结果的保真度取决于原始采样的密度(或采样率)。采样定理介绍了对带宽限制的函数类型来说保真度足够完整的采样率的概念;在采样过程中"信息"实际没有损失。定理用函数的带宽来表示采样率。定理也导出了一个数学上理想的原连续信号的重构公式。 该定理没有排除一些并不满足采样率准则的特殊情况下完整重构的可能性。(参见下文,以及壓縮感知。) 奈奎斯特–香农采样定理的名字是为了紀念哈里·奈奎斯特和克劳德·香农。该定理及其在插值理论中的原型曾被奥古斯丁-路易·柯西、埃米尔·博雷尔、雅克·阿达马、、、弗拉基米尔·亚历山德罗维奇·科捷利尼科夫等人发现或研究。所以它还叫做奈奎斯特–香农–科捷利尼科夫定理、惠特克–香农–科捷利尼科夫定理、惠特克–奈奎斯特–科捷利尼科夫–香农定理及插值基本定理。 Le théorème d'échantillonnage, dit aussi théorème de Shannon ou théorème de Nyquist-Shannon, établit les conditions qui permettent l'échantillonnage d'un signal de largeur spectrale et d'amplitude limitées. La connaissance de plus de caractéristiques du signal permet sa description par un nombre inférieur d'échantillons, par un processus d'acquisition comprimée.
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Het bemonsteringstheorema van Nyquist-Shannon is de stelling in de informatietheorie dat wanneer een analoog signaal naar een tijddiscreet signaal wordt geconverteerd, de bemonsteringsfrequentie minstens tweemaal zo hoog moet zijn als de hoogste in het signaal aanwezige frequentie om het origineel zonder fouten te kunnen reproduceren. De helft van de bemonsteringsfrequentie is de nyquistfrequentie. Anders gezegd, voor een foutloze reproductie na bemonstering mag het analoge signaal geen frequenties bevatten hoger dan de nyquistfrequentie. De tijd tussen de bemonsteringen is de nyquistinterval. Het is genoemd naar Harry Nyquist die dit theorema in 1928 bewees. Als een signaal bemonsterd wordt en er komen frequenties in voor hoger dan de nyquistfrequentie, resulteert dit in een teruggevouwen signaal waarvan de frequentie beneden de nyquistfrequentie is. Zo zal een signaal met een frequentie van 12 kHz dat op 20 kHz bemonsterd wordt, hetzelfde lijken als de bemonstering van een 8 kHz ingangssignaal, de 12 kHz vouwt om de 10 kHz Nyquistfrequentie. Dit geeft een probleem als het ingangssignaal componenten met zowel 8 kHz als 12 kHz bevat; in dat geval is uit het bemonsterde signaal niet meer op te maken wat van de 8 kHz en 12 kHz afkomstig is. Deze fout, c.q. vervorming van het signaal, wordt aliasing genoemd. Om dit te voorkomen, moet het bemonsteringssysteem voorzien zijn van een anti-aliasing filter. Dit is een analoog laagdoorlaat-filter dat signalen met frequenties hoger dan de nyquistfrequentie uit het ingangssignaal verwijdert. Voor bepaalde toepassingen kan ook bewust gebruik worden gemaakt van dit vouweffect om de frequentie van een signaal naar beneden te brengen. Het door Nyquist opgestelde theorema houdt geen rekening met ruis na conversie naar het discrete domein. Claude Shannon breidde de theorie in 1949 uit door wel rekening te houden met de beperking veroorzaakt door ruis. Hij stelde de wet van Shannon-Hartley op voor de maximale informatiecapaciteit van een bandbreedtegelimiteerd kanaal met ruis. Hoewel het bemonsteringstheorema van Nyquist-Shannon meestal in één adem met digitale informatie wordt genoemd, is het geldig in alle systemen waar bemonsterd wordt. In technisch Nederlands wordt voor bemonstering ook wel het woord samplen gebruikt, naar het Engelse sampling. Das Nyquist-Shannon-Abtasttheorem, auch nyquist-shannonsches Abtasttheorem und in neuerer Literatur auch WKS-Abtasttheorem (für Whittaker, Kotelnikow und Shannon) genannt, ist ein grundlegendes Theorem der Nachrichtentechnik, Signalverarbeitung und Informationstheorie. Wladimir Kotelnikow formulierte das Abtasttheorem 1933. Die Veröffentlichung in einem sowjetischen Konferenzbericht wurde im Osten seit den 1950er Jahren referenziert, blieb aber allgemein im Westen bis in die 1980er weitgehend unbekannt. Unabhängig von Kotelnikow formulierte es Claude Elwood Shannon 1948 als Ausgangspunkt seiner Theorie der maximalen Kanalkapazität, d. h. der maximalen Bitrate in einem frequenzbeschränkten, rauschbelasteten Übertragungskanal. Das Abtasttheorem besagt, dass ein auf bandbegrenztes Signal aus einer Folge von äquidistanten Abtastwerten exakt rekonstruiert werden kann, wenn es mit einer Frequenz von größer als abgetastet wurde. O teorema da amostragem de Nyquist–Shannon, também conhecido simplesmente como teorema de Nyquist, é fundamental no campo da teoria da informação, particularmente na área de telecomunicações e processamento de sinais. Amostrar é o processo no qual se converte um sinal (por exemplo, uma função contínua no tempo ou espaço) em uma sequência numérica (uma função discreta no tempo ou espaço). A versão de Shannon do teorema é: "Seja um sinal, limitado em banda, e seu intervalo de tempo dividido em partes iguais, de forma que se obtenham intervalos tais que, cada subdivisão compreenda um intervalo com período segundos, onde é menor do que , e se uma amostra instantânea é tomada arbitrariamente de cada subintervalo, então o conhecimento da amplitude instantânea de cada amostra somado ao conhecimento dos instantes em que é tomada a amostra de cada subintervalo contém toda a informação do sinal original." em que é a maior frequência em Hertz do sinal em questão. O teorema é, muitas vezes, chamado de Teorema da amostragem de Shannon, ou Nyquist-Shannon-Kotelnikov, Whittaker-Shannon-Kotelnikov, Whittaker-Nyquist-Kotelnikov-Shannon, WKS e etc. Também é muitas vezes chamado simplesmente de Teorema da Amostragem. Pode-se concluir então, que o teorema mostra que um sinal analógico, limitado em banda, que foi amostrado, pode ser perfeitamente recuperado a partir de uma sequência infinita de amostras, se a taxa de amostragem for maior que amostras por segundo, em que é a maior frequência do sinal original. Porém, se um sinal contiver uma componente exatamente em hertz, e amostras espaçadas de exatamente segundos, não se consegue recuperar totalmente o sinal. Interpretações mais recentes do teorema são cuidadosas ao excluir a condição de igualdade; isso é, a condição de que não contém frequências maiores ou iguais a ; tal condição é equivalente à exceção prevista por Shannon, quando uma função inclui uma componente estável senoidal exatamente na frequência . O teorema assume uma idealização de qualquer situação do mundo real, uma vez que o mesmo só se aplica a sinais que são amostrados para tempo infinito; Um sinal x(t) limitado em tempo não pode ser perfeitamente limitado em banda. A recuperação perfeita do modelo idealizado é matematicamente possível, mas é somente uma aproximação de sinais do mundo real, embora na prática seja uma aproximação muito boa. O teorema também leva a uma fórmula para a reconstrução do sinal original. A prática do teorema leva ao entendimento do aliasing que ocorre quando o sistema amostrador não satisfaz as condições do teorema مبرهنة شانون ونايكويست أو مبرهنة شانون ونايكويست في الاستعيان (بالإنجليزية: Nyquist–Shannon sampling theorem)‏ هي من أهم المبرهنات في التقنيات الرقمية الحديثة والعلوم المتصلة بها مثل المعالجة الرقمية للإشارة والمعلوماتية ونظرية المعلومات. تعود المبرهنة إلى جهد العالمين كلود شانون وهاري نايكست. وممن اشتهر في هذه المبرهنة العالم العراقي الأمريكي عبدالجبار جري. The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth. Strictly speaking, the theorem only applies to a class of mathematical functions having a Fourier transform that is zero outside of a finite region of frequencies. Intuitively we expect that when one reduces a continuous function to a discrete sequence and interpolates back to a continuous function, the fidelity of the result depends on the density (or sample rate) of the original samples. The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are band-limited to a given bandwidth, such that no actual information is lost in the sampling process. It expresses the sufficient sample rate in terms of the bandwidth for the class of functions. The theorem also leads to a formula for perfectly reconstructing the original continuous-time function from the samples. Perfect reconstruction may still be possible when the sample-rate criterion is not satisfied, provided other constraints on the signal are known (see below and compressed sensing). In some cases (when the sample-rate criterion is not satisfied), utilizing additional constraints allows for approximate reconstructions. The fidelity of these reconstructions can be verified and quantified utilizing Bochner's theorem. The name Nyquist–Shannon sampling theorem honours Harry Nyquist and Claude Shannon, but the theorem was also previously discovered by E. T. Whittaker (published in 1915) and Shannon cited Whittaker's paper in his work. The theorem is thus also known by the names Whittaker–Shannon sampling theorem, Whittaker–Shannon, and Whittaker–Nyquist–Shannon, and may also be referred to as the cardinal theorem of interpolation. Теоре́ма ві́дліків Вітте́кера — На́йквіста — Коте́льникова — Ше́ннона (теорема відліків) свідчить, що якщо неперервний сигнал x(t) має спектр, обмежений частотою Fmax, то його можна однозначно і без втрат відновити за дискретними відліками, узятими з частотою fдискр=2*Fmax, або, по-іншому, за відліками, взятими з періодом Tдискр=. Теорему відліків можна сформулювати обернено: Для того, щоб відновити сигнал за його відліками без втрат, необхідно, щоб частота дискретизації була хоча б удвічі більшою за максимальну частоту первинного неперервного сигналу. Fд ≥ 2Fmax. Теорема відліків розглядає ідеальний випадок, коли сигнал почався нескінченно давно й ніколи не закінчиться, а також не має в часовій характеристиці точок розриву. Саме це означає поняття «спектр, обмежений частотою Fmax». Реальні сигнали скінченні в часі і, звичайно, мають у часовій характеристиці розриви, відповідно їхній спектр нескінченний. У такому випадку повне відновлення сигналу неможливе і з теореми відліків випливають 2 наслідки: * Будь-який аналоговий сигнал можна відновити з якою завгодно точністю за його дискретними відліками, взятими з частотою , де — максимальна частота, якою обмежений спектр реального сигналу. * Якщо максимальна частота в сигналі перевищує половину частоти дискретизації, то способу відновити сигнал з дискретного в аналоговий без спотворення не існує. Теорему сформулював Гаррі Найквіст 1928 року в праці «Certain topics in telegraph transmission theory». 1933 року подібні дані опублікував В. О. Котельников у праці «Про пропускну здатність ефіру і дроту в електрозв'язку». Теорема є однією з основоположних тверджень у теорії й техніці цифрового зв'язку. Le théorème d'échantillonnage, dit aussi théorème de Shannon ou théorème de Nyquist-Shannon, établit les conditions qui permettent l'échantillonnage d'un signal de largeur spectrale et d'amplitude limitées. La connaissance de plus de caractéristiques du signal permet sa description par un nombre inférieur d'échantillons, par un processus d'acquisition comprimée. Nyquist-Shannons samplingsteorem, även kallad Nyquistteoremet, Shannonteoremet eller samplingsteoremet, talar om med vilken frekvens man måste mäta en vågrörelse med hjälp av sampling för att kunna återskapa signalen. Teoremet går i grova drag ut på att man måste, för att undvika fel, sampla med en frekvens som är minst dubbla signalens bandbredd annars blir resultatet av mätningen lägre än signalens verkliga frekvens. Teoremet har sitt namn efter Claude Shannon och Harry Nyquist, som ungefär vid samma tidpunkt beskrev vilka krav som ställs på det antal mätpunkter man behöver per tidsenhet för att kunna återskapa en signal. El teorema de mostratge de Nyquist-Shannon, també conegut com a teorema de mostratge de Whittaker-Nyquist-Kotelnikov-Shannon, criteri de Nyquist o teorema de Nyquist, és un teorema fonamental de la teoria de la informació, d'especial interès en les telecomunicacions. Aquest teorema va ser formulat en forma de conjectura per primer cop per Harry Nyquist l'any 1928 (Certain topics in telegraph transmission theory), i va ser demostrat formalment per Claude E. Shannon l'any 1949 (Communication in the presence of noise). La intenció del suec Harry Nyquist en formular aquest teorema era la d'obtenir una enregistrament digital de qualitat i també es pot conèixer amb el nom de condició de Nyquist. Si es fa un mostreig a un baix valor, hi ha una possibilitat que el senyal original no estigui únicament definit pel nostre senyal mostrejat. Si això passa, no es té cap garantia que el senyal estigui correctament reconstruït. Per aquest motiu es va crear el teorema de Nyquist, que diu el següent: Теоре́ма Коте́льникова (в англоязычной литературе — теорема Найквиста — Шеннона, теорема отсчётов) — фундаментальное утверждение в области цифровой обработки сигналов, связывающее непрерывные и дискретные сигналы и гласящее, что «любую функцию , состоящую из частот от 0 до , можно непрерывно передавать с любой точностью при помощи чисел, следующих друг за другом менее чем через секунд». При доказательстве теоремы взяты ограничения на спектр частот , где . In elettronica e telecomunicazioni, il teorema del campionamento di Nyquist-Shannon o semplicemente teorema del campionamento, il cui nome si deve a Harry Nyquist e Claude Shannon, è un risultato di notevole rilevanza nell'ambito della teoria dei segnali. Definisce la minima frequenza, detta frequenza di Nyquist (o anche cadenza di Nyquist), necessaria per campionare un segnale analogico senza perdere informazioni, e per poter quindi ricostruire il segnale analogico tempo continuo originario. In particolare, il teorema afferma che, data una funzione la cui trasformata di Fourier sia nulla al di fuori di un certo intervallo di frequenze (ovvero un segnale a banda limitata), nella sua conversione analogico-digitale la minima frequenza di campionamento necessaria per evitare aliasing e perdita di informazione nella ricostruzione del segnale analogico originario (ovvero nella riconversione digitale-analogica) deve essere maggiore del doppio della sua frequenza massima. Il teorema, comparso per la prima volta nel 1949 in un articolo di C. E. Shannon, dovrebbe chiamarsi Whittaker-Nyquist-Kotelnikov-Shannon (WNKS), secondo l'ordine cronologico di chi ne dimostrò versioni via via più generalizzate. Nyquistův–Shannonův vzorkovací teorém (také Shannonův teorém, Nyquistův teorém, Kotělnikovův teorém, Nyquistův–Shannonův teorém, Shannonův–Nyquistův–Kotělnikovův teorém, apod.) je fyzikální tvrzení o tom, že „přesná rekonstrukce spojitého, frekvenčně omezeného signálu z jeho vzorků je možná tehdy, pokud byla vzorkovací frekvence vyšší než dvojnásobek nejvyšší harmonické složky vzorkovaného signálu.“ 표본화 정리(標本化定理, 영어: sampling theorem) 또는 나이퀴스트-섀넌 표본화 정리(영어: Nyquist-Shannon sampling theorem)는 원거리 통신과 신호 처리를 다루는 정보이론의 기본이 되는 원리이다. 標本化定理(ひょうほんかていり、英: sampling theorem)またはサンプリング定理は、連続的な信号(アナログ信号)を離散的な信号(デジタル信号)へと変換する際に元の信号に忠実であるにはどの程度の間隔で標本化(サンプリング)すればよいかを示す、情報理論の定理である。 El teorema de muestreo de Nyquist-Shannon, también conocido como teorema de muestreo de Whittaker-Nyquist-Kotelnikov-Shannon o bien teorema de Nyquist, es un teorema fundamental de la teoría de la información, de especial interés en las telecomunicaciones. Este teorema fue formulado en forma de conjetura por primera vez por Harry Nyquist en 1928 (Certain topics in telegraph transmission theory), y fue demostrado formalmente por Claude E. Shannon en 1949 (Communication in the presence of noise). El teorema trata del muestreo, que no debe ser confundido o asociado con la cuantificación, proceso que sigue al de muestreo en la digitalización de una señal y que, al contrario del muestreo, no es reversible (se produce una pérdida de información en el proceso de cuantificación, incluso en el caso ideal teórico, que se traduce en una distorsión conocida como error o ruido de cuantificación y que establece un límite teórico superior a la relación señal-ruido). Dicho de otro modo, desde el punto de vista del teorema, las muestras discretas de una señal son valores exactos que aún no han sufrido redondeo o truncamiento alguno sobre una precisión determinada, es decir, aún no han sido cuantificadas. Laŭ la teoremo pri specimenado (teoremo de Nyquist–Shannon–Kotelnikov, teoremo de Whittaker–Shannon–Kotelnikov, teoremo de Whittaker–Nyquist–Kotelnikov–Shannon, aŭ pli simple teoremo de Nyquist): preciza rekreo de eblas, se signalo havas maksimuman frekvencon kaj la (frekvenco de Nyquist, ) estas pli ol dufoje la maksimuma frekvenco. Twierdzenie o próbkowaniu, twierdzenie Nyquista–Shannona – fundamentalne twierdzenie teorii informacji, telekomunikacji oraz cyfrowego przetwarzania sygnałów opisujące matematyczne podstawy procesów próbkowania sygnałów oraz ich rekonstrukcji: Z sygnału dyskretnego złożonego z próbek danego sygnału ciągłego można wiernie odtworzyć sygnał Jest podstawową zasadą pozwalającą przekształcać sygnał ciągły w czasie (często nazywany „sygnałem analogowym”) w sygnał dyskretny (często nazywany „sygnałem cyfrowym”). Ustanawia warunek dla częstotliwości próbkowania, która pozwala dyskretnej sekwencji próbek (cyfrowych) na przechwytywanie wszystkich informacji z sygnału ciągłego (analogowego) o skończonej szerokości pasma – częstotliwość Nyquista. , Nyquist-Shannonen laginketa teorema eta arteko lokarria da. Teoremak laginketa burutzean sortutako laginek denbora jarraituko seinalearen informazio guztia gorde dezaten bete beharreko baldintza ezartzen du. Teorema maiztasun banda mugatua duten seinale jarraituentzat baliagarria da soilik. 采样定理是数字信号处理领域的重要定理。定理內容是连续信号(通常称作“模拟信号”)与离散信号(通常称作“数字信号”)之间的一个基本桥梁。它确定了信号带宽的上限,或能捕获连续信号的所有信息的离散采样信号所允许的采样频率的下限。 严格地说,定理仅适用于具有傅里叶变换的一类数学函数,即频率在有限区域以外为零(参照图1)。离散时间傅里叶变换(泊松求和公式的一种形式)提供了实际信号的解析延拓,但只能近似该条件。直观上我们希望,当把连续函数化为采样值(叫做“样本”)的离散序列并插值到连续函数中,结果的保真度取决于原始采样的密度(或采样率)。采样定理介绍了对带宽限制的函数类型来说保真度足够完整的采样率的概念;在采样过程中"信息"实际没有损失。定理用函数的带宽来表示采样率。定理也导出了一个数学上理想的原连续信号的重构公式。 该定理没有排除一些并不满足采样率准则的特殊情况下完整重构的可能性。(参见下文,以及壓縮感知。) 奈奎斯特–香农采样定理的名字是为了紀念哈里·奈奎斯特和克劳德·香农。该定理及其在插值理论中的原型曾被奥古斯丁-路易·柯西、埃米尔·博雷尔、雅克·阿达马、、、弗拉基米尔·亚历山德罗维奇·科捷利尼科夫等人发现或研究。所以它还叫做奈奎斯特–香农–科捷利尼科夫定理、惠特克–香农–科捷利尼科夫定理、惠特克–奈奎斯特–科捷利尼科夫–香农定理及插值基本定理。
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