Name: dtft rules for differentiation and energy
Uploaded: 2024-09-26T13:17:04.000Z
Duration: 1 min 24 s
Channel: JoVE

Considérons une paire de transformée de Fourier en temps discret (DTFT), dérivez les deux côtés par rapport à Ω, puis multipliez par j.
Le membre de droite devient la transformée de Fourier de nx[n], illustrant la propriété de différenciation de fréquence.
 Appliquez la convolution temporelle discrète aux paires DTFT, modifiez l’ordre des sommations sur le côté droit et appliquez la propriété de décalage temporel.
La substitution des équations donne la propriété de convolution temporelle.
Multipliez les signaux du domaine bitemporel et échangez l’ordre d’intégration et de sommation du côté droit.
Cela correspond à une convolution périodique dans le domaine fréquentiel, mise à l’échelle par l’inverse de la période, démontrant la convolution fréquentielle.
La propriété d’accumulation se concentre sur la sommation d’un signal à temps discret.
Le DTFT du signal accumulé a le DTFT d’origine mis à l’échelle par un facteur exponentiel plus un terme de fonction delta, indiquant des composantes périodiques à des multiples de 2π.
L’énergie du signal Ex est la somme des normes au carré dans le domaine temporel. Ceci est égal à l'intégrale des grandeurs au carré de sa DTFT, démontrant la relation de Parseval.

propriétés de la transformée de fourier à temps discret ii

Properties of DTFT II

Dans l’étude du traitement du signal à temps discret, la compréhension des propriétés de la transformée de Fourier à temps discret (TFtd) est cruciale pour l’analyse et la manipulation des signaux dans le domaine fréquentiel. Plusieurs propriétés, notamment la différenciation de fréquence, la convolution, l’accumulation et la relation de Parseval, offrent des outils puissants pour l’analyse du signal.

La propriété de différenciation de fréquence est illustrée en considérant une paire TFtd et en différenciant les deux côtés par rapport à ω. En multipliant par j (l’unité imaginaire), le côté droit devient la transformée de Fourier de nx[n]. Mathématiquement, si X(e^(jω)) est la TFtd de x[n], alors j(d/dω(X(e^(jω)))) est la TFtd de nx[n]. Cette propriété est utile pour trouver les caractéristiques de fréquence liées à la pente du spectre du signal.

En appliquant la convolution à temps discret aux paires de TFtd, on observe une autre propriété importante. En changeant l'ordre de sommation du côté droit et en appliquant la propriété de décalage temporel, on arrive à la propriété de convolution temporelle. Celle-ci stipule que la convolution de deux signaux dans le domaine temporel correspond à la multiplication de leurs TFtd dans le domaine fréquentiel. Inversement, la multiplication de deux signaux dans le domaine temporel conduit à une convolution périodique de leurs TFtd dans le domaine fréquentiel, échelonnée par l'inverse de la période.

La propriété d'accumulation se concentre sur la sommation d'un signal à temps discret au fil du temps. La transformée de Fourier à temps discret (TFtd) de ce signal accumulé est liée à la TFtd du signal d'origine mais est modifiée par un facteur d'échelle exponentiel. De plus, il existe un terme qui inclut une fonction delta, qui introduit des composantes périodiques à des intervalles de 2π dans le domaine fréquentiel. Cette propriété met en évidence comment l'accumulation dans le domaine temporel affecte la représentation dans le domaine fréquentiel, ce qui conduit à des caractéristiques périodiques.

La relation de Parseval est un résultat clé qui relie l'énergie d'un signal dans le domaine temporel à sa représentation dans le domaine fréquentiel. Plus précisément, l'énergie totale du signal x[n], qui est la somme des carrés des amplitudes dans le domaine temporel, est égale à l'intégrale des carrés des amplitudes de sa TFtd. Cette relation est fondamentale pour analyser la puissance et l'énergie du signal dans les deux domaines.

Ces propriétés améliorent collectivement la capacité d'analyse, de conception et de compréhension des systèmes à temps discret, ce qui les rend indispensables dans le traitement du signal numérique.

Propriétés de la transformée de Fourier à temps discret II

Considérons une paire de transformée de Fourier en temps discret (DTFT), dérivez les deux côtés par rapport à Ω, puis multipliez par j.
Le côté droit ...

Education

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Electrical Engineering

Génie électrique

Unit 1

Unité 1

The Fourier Transform

La transformée de Fourier

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dtft rules for differentiation and energy

The DTFT has useful rules for frequency differentiation, convolution, accumulation, and energy checks. These rules help students see how a discrete-time signal changes in the frequency domain.

For frequency differentiation, start with a DTFT pair and differentiate both sides with respect to ω. After multiplying by j, the result gives the DTFT of n x[n]. This property is useful for studying how the slope of a spectrum relates to the signal in time.

Convolution brings out another key DTFT rule. When two signals are convolved in time, their DTFTs multiply in frequency. The reverse is also true in a periodic form: multiplying two signals in time leads to a periodic convolution of their DTFTs, scaled by the inverse of the period.

The accumulation property looks at the DTFT of a signal after its values are summed over time. The transformed result includes an exponential scaling factor, along with a delta-function term. That delta term adds periodic components at intervals of 2π in the frequency domain.

Parseval's relation connects signal energy in both domains. The total energy of x[n], found by summing the squared magnitudes in time, equals the integral of the squared magnitude of its DTFT. Together, these DTFT properties make it easier to analyze, design, and understand discrete-time systems in digital signal processing.

DTFT Rules for Differentiation and Energy

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properties of dtft ii

The DTFT has useful rules for frequency differentiation, convolution, accumulation, and energy checks. These rules help students see how a discrete-time signal changes in the frequency domain. For frequency differentiation, start with a DTFT pair and differentiate both sides with respect to ω. After multiplying by j, the result gives the DTFT of n x[n]. This property is useful for studying how the slope of a spectrum relates to the signal in time. Convolution brings out another key DTFT rule.

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Learn DTFT properties for frequency differentiation, convolution, accumulation, and Parseval's relation. See how time-domain changes appear in frequency.

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electrical engineering

DTFT Rules for Differentiation and Energy

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