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Fouriertransformation für zeitdiskrete Signale 이산시간 푸리에 변환 Transformada de Fourier de senyal discret Transformada de Fourier de tempo discreto Trasformata di Fourier a tempo discreto Discrete-time Fourier transform 离散时间傅里叶变换 離散時間フーリエ変換 Discrete-time Fourier transform
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In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function. Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calcul 離散時間フーリエ変換（英: Discrete-time Fourier transform、DTFT）はフーリエ変換の一種。したがって、通常時間領域の関数を周波数領域に変換する。ただし、DTFTでは元の関数は離散的でなければならない。そのような入力は連続関数の標本化によって生成される。 DTFTの周波数領域の表現は常に周期的関数である。したがって1つの周期に必要な情報が全て含まれるため、DTFTを「有限な」周波数領域への変換であるということもある。 Em matemática, a transformada de Fourier de tempo discreto (DTFT) é uma transformada integral estreitamente relacionada com a transformada de Fourier e com a transformada Z. A DTFT difere da transformada de Fourier ao aplicar-se a funções cuja variável independente é discreta (descontínua), e não contínua, como é o caso da transformada de Fourier. A DTFT não deve ser confundida com a transformada discreta de Fourier (DFT), que pode ser considerada como um seu caso especial, que aparece numa situação muito comum: quando a função original é periódica. 在数学中，离散时间傅里叶变换（DTFT，Discrete-time Fourier Transform）是傅里叶分析的一种形式，适用于连续函数的均匀间隔采样。离散时间是指对采样间隔通常以时间为单位的离散数据（样本）的变换。仅根据这些样本，它就可以产生原始连续函数的连续傅里叶变换的的以频率为变量的函数。在采样定理所描述的一定理论条件下，可以由DTFT完全恢复出原来的连续函数，因此也能从原来的离散样本恢复。DTFT本身是频率的连续函数，但可以通过离散傅里叶变换（DFT）很容易计算得到它的离散样本（参见对DTFT采样），而DFT是迄今为止现代傅里叶分析最常用的方法。 这两种变换都是可逆的。离散时间傅里叶逆变换得到的是原始采样数据序列。离散傅里叶逆变换是原始序列的周期求和。快速傅里叶变换（FFT）是用于计算DFT的一个周期的算法，而它的逆变换会产生一个周期的离散傅里叶逆变换。 Die Fouriertransformation für zeitdiskrete Signale, auch als englisch discrete-time Fourier transform, abgekürzt DTFT bezeichnet, ist eine lineare Transformation aus dem Bereich der Fourier-Analysis. Sie bildet ein unendliches, zeitdiskretes Signal auf ein kontinuierliches, periodisches Frequenzspektrum ab, welches auch als Bildbereich bezeichnet wird. Die DTFT ist mit der Diskreten Fourier-Transformation (DFT) verwandt, welche mit diskreten Zeitsignalen und diskreten Spektren arbeitet. Die DTFT unterscheidet sich von der DFT darin, dass sie ein kontinuierliches Spektrum bildet, welches sich, unter Umständen, als abschnittsweise geschlossener mathematischer Ausdruck angeben lässt. Wie auch die DFT bildet die DTFT im Bildbereich ein periodisch fortgesetztes Frequenzspektrum, welches als Spi In matematica, la trasformata di Fourier a tempo discreto, spesso abbreviata con DTFT (acronimo del termine inglese Discrete-Time Fourier Transform), è una trasformata che a partire da un segnale discreto ne fornisce una descrizione periodica nel dominio della frequenza, analogamente alla trasformata di Fourier tradizionale (definita per funzioni continue). Si tratta di un caso particolare della trasformata zeta: La Transformada de Fourier de senyal discret (DTFT, acrònim anglès de Discret Time Fourier Transform) és la transformada de Fourier aplicada a un senyal discret creat a partir s'un senyal continu. Després d'efectuar la transformada de Fourier s'obté una funció en la freqüència que és un sumatori periòdic de la transformada de Fourier del senyal continu original. Aquesta transformada de Fourier es pot realitzar amb DFT (Discret Fourier Transform) de forma ràpida. La transformada inversa DTFT també és viable. 이산시간 푸리에 변환(Discrete-time Fourier transform, DTFT)은 푸리에 변환의 일종이다. 따라서, 시간 도메인 영역의 함수를 주파수 도메인의 함수로 변환한다. DTFT에서 변환전의 원래 함수는 이산적인 값의 수열인데, 이러한 은 연속함수의 샘플링에 의하여 생성된다. DTFT의 주파수 도메인에서의 표현은 항상 주기적인 함수이다. 따라서 하나의 주기에 필요한 정보가 모두 포함되어 있으므로 DTFT를 "유한" 주파수 영역으로의 변환으로 부르기도 한다. De discrete-time Fourier transform (of DTFT) maakt deel uit van de familie van de fouriertransformaties. Hij transformeert een functie van een discrete-tijdsvariabele , met , naar een continu, periodiek spectrum .
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離散時間フーリエ変換（英: Discrete-time Fourier transform、DTFT）はフーリエ変換の一種。したがって、通常時間領域の関数を周波数領域に変換する。ただし、DTFTでは元の関数は離散的でなければならない。そのような入力は連続関数の標本化によって生成される。 DTFTの周波数領域の表現は常に周期的関数である。したがって1つの周期に必要な情報が全て含まれるため、DTFTを「有限な」周波数領域への変換であるということもある。 De discrete-time Fourier transform (of DTFT) maakt deel uit van de familie van de fouriertransformaties. Hij transformeert een functie van een discrete-tijdsvariabele , met , naar een continu, periodiek spectrum . Die Fouriertransformation für zeitdiskrete Signale, auch als englisch discrete-time Fourier transform, abgekürzt DTFT bezeichnet, ist eine lineare Transformation aus dem Bereich der Fourier-Analysis. Sie bildet ein unendliches, zeitdiskretes Signal auf ein kontinuierliches, periodisches Frequenzspektrum ab, welches auch als Bildbereich bezeichnet wird. Die DTFT ist mit der Diskreten Fourier-Transformation (DFT) verwandt, welche mit diskreten Zeitsignalen und diskreten Spektren arbeitet. Die DTFT unterscheidet sich von der DFT darin, dass sie ein kontinuierliches Spektrum bildet, welches sich, unter Umständen, als abschnittsweise geschlossener mathematischer Ausdruck angeben lässt. Wie auch die DFT bildet die DTFT im Bildbereich ein periodisch fortgesetztes Frequenzspektrum, welches als Spiegelspektrum bezeichnet wird. Im Gegensatz zur DFT besitzt die DTFT nur eine geringe Bedeutung in praktischen Anwendungen wie der digitalen Signalverarbeitung, primärer Anwendungsbereich liegt bei der theoretischen Signalanalyse. La Transformada de Fourier de senyal discret (DTFT, acrònim anglès de Discret Time Fourier Transform) és la transformada de Fourier aplicada a un senyal discret creat a partir s'un senyal continu. Després d'efectuar la transformada de Fourier s'obté una funció en la freqüència que és un sumatori periòdic de la transformada de Fourier del senyal continu original. Aquesta transformada de Fourier es pot realitzar amb DFT (Discret Fourier Transform) de forma ràpida. La transformada inversa DTFT també és viable. In matematica, la trasformata di Fourier a tempo discreto, spesso abbreviata con DTFT (acronimo del termine inglese Discrete-Time Fourier Transform), è una trasformata che a partire da un segnale discreto ne fornisce una descrizione periodica nel dominio della frequenza, analogamente alla trasformata di Fourier tradizionale (definita per funzioni continue). Si tratta di un caso particolare della trasformata zeta: che si ottiene ponendo ( è inteso come angolo). Dal momento che , la trasformata di Fourier a tempo discreto è la valutazione della trasformata zeta sul cerchio unitario nel piano complesso. 이산시간 푸리에 변환(Discrete-time Fourier transform, DTFT)은 푸리에 변환의 일종이다. 따라서, 시간 도메인 영역의 함수를 주파수 도메인의 함수로 변환한다. DTFT에서 변환전의 원래 함수는 이산적인 값의 수열인데, 이러한 은 연속함수의 샘플링에 의하여 생성된다. DTFT의 주파수 도메인에서의 표현은 항상 주기적인 함수이다. 따라서 하나의 주기에 필요한 정보가 모두 포함되어 있으므로 DTFT를 "유한" 주파수 영역으로의 변환으로 부르기도 한다. In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function. Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the discrete Fourier transform (DFT) (see ), which is by far the most common method of modern Fourier analysis. Both transforms are invertible. The inverse DTFT is the original sampled data sequence. The inverse DFT is a periodic summation of the original sequence. The fast Fourier transform (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT. Em matemática, a transformada de Fourier de tempo discreto (DTFT) é uma transformada integral estreitamente relacionada com a transformada de Fourier e com a transformada Z. A DTFT difere da transformada de Fourier ao aplicar-se a funções cuja variável independente é discreta (descontínua), e não contínua, como é o caso da transformada de Fourier. A DTFT não deve ser confundida com a transformada discreta de Fourier (DFT), que pode ser considerada como um seu caso especial, que aparece numa situação muito comum: quando a função original é periódica. Funções discretas são sequências de valores, que aparecem quando se amostra uma função contínua em intervalos definidos. Assim, a DTFT encontra muitas aplicações em áreas como cálculo numérico e controle digital. A função transformada é sempre periódica. Uma vez que um período da função já exibe toda a informação contida na função, pode-se dizer que a DTFT é uma representação da função original em um domínio da frequência finito. A DTFT é dual, no sentido de Pontryagin, à série de Fourier, que faz a transformação inversa, ou seja, produz uma representação de uma função periódica no tempo em um domínio discreto de frequências . 在数学中，离散时间傅里叶变换（DTFT，Discrete-time Fourier Transform）是傅里叶分析的一种形式，适用于连续函数的均匀间隔采样。离散时间是指对采样间隔通常以时间为单位的离散数据（样本）的变换。仅根据这些样本，它就可以产生原始连续函数的连续傅里叶变换的的以频率为变量的函数。在采样定理所描述的一定理论条件下，可以由DTFT完全恢复出原来的连续函数，因此也能从原来的离散样本恢复。DTFT本身是频率的连续函数，但可以通过离散傅里叶变换（DFT）很容易计算得到它的离散样本（参见对DTFT采样），而DFT是迄今为止现代傅里叶分析最常用的方法。 这两种变换都是可逆的。离散时间傅里叶逆变换得到的是原始采样数据序列。离散傅里叶逆变换是原始序列的周期求和。快速傅里叶变换（FFT）是用于计算DFT的一个周期的算法，而它的逆变换会产生一个周期的离散傅里叶逆变换。
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