Name: complex exponentials in signal behavior
Uploaded: 2024-09-26T13:16:53.000Z
Duration: 1 min 18 s
Channel: JoVE

K12 Standards

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South Carolina College and Career Ready Science Standards

High School

Physics

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Waves and Their Applications in Technologies for Information Transfer

P-PS4-5

Exponential and Sinusoidal Signals

Die Exponentialfunktion dient als wesentliches Werkzeug zur Charakterisierung von Wellenformen, die schnell ansteigen und abfallen. Diese Funktion wird über die Konstanten m und A definiert. Wenn beide Konstanten reell sind, kann die Funktion grafisch dargestellt werden.
Ein komplexes Exponential wird erreicht, wenn die Konstante m rein imaginär ist. Sie ist periodisch, wenn sie eine Größenordnung der Einheit beibehält.
Ein zeitkontinuierliches sinusförmiges Signal kann in Bezug auf Frequenz und Zeitraum ausgedrückt werden.
Die Eulersche Beziehung kann verwendet werden, um das sinusförmige Signal als periodische komplexe Exponentielle mit der gleichen Grundfrequenz auszudrücken.
In ähnlicher Weise kann die komplexe Exponentialfunktion in Form von sinusförmigen Signalen ausgedrückt werden, die alle die gleiche Grundfrequenz haben.
Zum Beispiel kann die Summe zweier komplexer Exponentielle als Produkt aus einem einzigen komplexen Exponential und einer einzelnen Sinuskurve geschrieben werden.
Sowohl sinusförmige als auch komplexe Exponentialsignale werden häufig verwendet, um die Energieerhaltung in einem mechanischen System zu beschreiben, in dem eine Masse, die über eine Feder mit einem stationären Träger verbunden ist, eine einfache harmonische Bewegung zeigt.

exponentialfunktionen und sinusförmige signale

Die Exponentialfunktion ist entscheidend für die Charakterisierung von Wellenformen, die schnell ansteigen und abfallen. Diese kontinuierliche Exponentialfunktion wird mithilfe von Exponentialtermen mit den Konstanten a und A definiert. Wenn beide Konstanten real sind, wird die Funktion wie folgt dargestellt:

<img alt="Equation1" src="/files/ftp_upload/16042/16042_PT%20EQUATION_1.svg" style="height: 16px;">

und kann grafisch dargestellt werden, um exponentielles Wachstum oder Abfallen zu zeigen. Wenn die Konstante a rein imaginär ist, ist das Ergebnis eine komplexe Exponentialfunktion, ausgedrückt als:

<img alt="Equation2" src="/files/ftp_upload/16042/16042_PT%20EQUATION_2.svg" style="height: 16px;">

wobei j die imaginäre Einheit w_0 die Winkelfrequenz ist. Diese Funktion ist periodisch, wenn sie eine Größe von 1 beibehält.

Ein kontinuierliches sinusförmiges Signal kann anhand von Frequenz und Zeitperiode beschrieben werden. Die Eulersche Formel ermöglicht es, das sinusförmige Signal als periodische komplexe Exponentialfunktionen mit derselben Grundperiode auszudrücken. Somit wird ein sinusförmiges Signal wie folgt dargestellt:

<img alt="Equation3" src="/files/ftp_upload/16042/16042_PT%20EQUATION_3.svg" style="height: 14px;">

kann mit komplexen Exponentialfunktionen wie folgt umgeschrieben werden:

<img alt="Equation4" src="/files/ftp_upload/16042/16042_PT%20EQUATION_4.svg" style="height: 28px;">

In ähnlicher Weise kann die komplexe Exponentialfunktion in Form von sinusförmigen Signalen ausgedrückt werden, die alle dieselbe Grundperiode haben. Beispielsweise kann die Summe zweier komplexer Exponentialfunktionen als Produkt einer einzelnen komplexen Exponentialfunktion und einer einzelnen Sinuskurve geschrieben werden, wie beispielsweise:

<img alt="Equation5" src="/files/ftp_upload/16042/16042_PT%20EQUATION_5.svg" style="height: 16px;">

Sowohl sinusförmige als auch komplexe exponentielle Signale werden häufig verwendet, um die Energieerhaltung in mechanischen Systemen zu beschreiben, wie beispielsweise einer Masse, die über eine Feder mit einem stationären Träger verbunden ist und eine einfache harmonische Bewegung aufweist. Diese Signale bilden eine Grundlage für die Analyse des Schwingungsverhaltens und der Resonanzphänomene in solchen Systemen.

Exponentialfunktionen und sinusförmige Signale

Die Exponentialfunktion dient als wesentliches Werkzeug zur Charakterisierung von Wellenformen, die schnell ansteigen und abfallen. Diese Funktion wird über die Schaltfläche ...

Education

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JoVE Core

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Elektrotechnik

Unit 1

Einheit 1

Introduction to Signals and Systems

Einführung in Signale und Systeme

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

complex exponentials in signal behavior

Exponential and sinusoidal signals are used to describe how waveforms change over time. The continuous-time exponential function uses constants α and A. When both are real, it shows exponential growth or decay. Graphs of these signals make the rise or drop easy to see.

When α is purely imaginary, the signal becomes a complex exponential. In this form, j is the imaginary unit and ω0 is the angular frequency. A complex exponential is periodic when its magnitude is unity. That periodic behavior links it closely to sinusoidal motion.

A continuous-time sinusoidal signal can be described by its frequency and time period. Using Euler’s relation, a sinusoid can be written as a set of periodic complex exponentials with the same fundamental period. The same idea also works in reverse, so a complex exponential can be expressed in terms of sinusoidal signals.

The sum of two complex exponentials can also be rewritten in a simpler form. It becomes the product of one complex exponential and one sinusoid. These signal forms are widely used to model energy conservation in mechanical systems, such as a mass attached to a fixed support by a spring. In that setting, they help describe simple harmonic motion, oscillation, and resonance.

Complex Exponentials in Signal Behavior

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even and odd signals

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rectangular and triangular pulse function

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exponential and sinusoidal signals

Exponential and sinusoidal signals are used to describe how waveforms change over time. The continuous-time exponential function uses constants α and A. When both are real, it shows exponential growth or decay. Graphs of these signals make the rise or drop easy to see. When α is purely imaginary, the signal becomes a complex exponential. In this form, j is the imaginary unit and ω0 is the angular frequency. A complex exponential is periodic when its magnitude is unity. That periodic behavior...

basic discrete time signals

Discrete-Time Signal Basics and Shapes

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classification of systems-i

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key terms and concepts

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Complex Exponentials in Signal Behavior

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