Name: signal symmetry and decomposition
Uploaded: 2024-09-26T13:16:53.000Z
Duration: 1 min 17 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

Even and Odd Signals

Ein gerades Signal, entweder in kontinuierlicher Zeit oder diskreter Zeit, ist definiert als ein Signal, das seiner eigenen zeitumgekehrten Version entspricht.
Gerade Signale sind symmetrisch um die vertikale Achse, d.h. das Signal für negative Zeitwerte spiegelt das Signal für positive Zeitwerte.
Ein Signal wird als ungerade bezeichnet, wenn es dem Minus seines zeitumgekehrten Gegenstücks entspricht.
Ungerade Signale erkennt man daran, dass sie antisymmetrisch um die vertikale Achse angeordnet sind.
Jedes zeitkontinuierliche Signal kann als Kombination aus geraden und ungeraden Komponenten ausgedrückt werden.
Hier ist die erste Komponente auf der rechten Seite eine gerade Funktion, während die zweite Komponente ungerade ist.
Ein komplexes Signal kann mit Hilfe von konjugierten Symmetrien in gerade und ungerade Komponenten zerlegt werden.
Das Produkt aus einer geraden und einer ungeraden Funktion ist eine ungerade Funktion. Das Multiplizieren von zwei Funktionen desselben Typs, entweder gerade oder ungerade, führt zu einer geraden Funktion.
Schließlich ergibt das Addieren oder Subtrahieren von zwei geraden Funktionen eine gerade Funktion, und in ähnlicher Weise ergibt das Addieren oder Subtrahieren von zwei ungeraden Funktionen eine ungerade Funktion.

gerade und ungerade signale

Ein gerades Signal, egal ob in kontinuierlicher oder diskreter Zeit, wird durch seine Symmetrie mit seiner zeitumgekehrten Version definiert. Mathematisch wird dies wie folgt dargestellt:

<img alt="Equation1" src="/files/ftp_upload/16039/16039_PT%20EQUATION_1.svg" style="text-align: center; height: 14px;">

für kontinuierliche Signale und

<img alt="Equation2" src="/files/ftp_upload/16039/16039_PT%20EQUATION_2.svg" style="text-align: center; height: 14px;">

für diskrete Signale. Gerade Signale weisen Symmetrie um die vertikale Achse auf, was bedeutet, dass das Signal für negative Zeitwerte das für positive Zeitwerte widerspiegelt.

Im Gegensatz dazu wird ein Signal als ungerade bezeichnet, wenn es nicht mit seinem zeitumgekehrten Gegenstück übereinstimmt, dargestellt durch

<img alt="Equation3" src="/files/ftp_upload/16039/16039_PT%20EQUATION_3.svg" style="text-align: center; height: 14px;">

für kontinuierliche Signale und

<img alt="Equation4" src="/files/ftp_upload/16039/16039_PT%20EQUATION_4.svg" style="text-align: center; height: 14px;">

für diskrete Signale. Ungerade Signale sind durch ihre asymmetrische Natur um die vertikale Achse gekennzeichnet.

Jedes kontinuierliche Signal kann als Kombination aus geraden und ungeraden Komponenten ausgedrückt werden. Diese Zerlegung ergibt sich aus:

<img alt="Equation5" src="/files/ftp_upload/16039/16039_PT%20EQUATION_5.svg" style="text-align: center; height: 14px;">

Hier ist der erste Term auf der rechten Seite eine gerade Funktion, während der zweite Term eine ungerade Funktion ist.

Komplexe Signale können auch mithilfe konjugierter Symmetrien in gerade und ungerade Komponenten zerlegt werden. Das Produkt einer geraden Funktion und einer ungeraden Funktion ergibt eine ungerade Funktion. Darüber hinaus ergibt die Multiplikation zweier Funktionen desselben Typs – entweder beide gerade oder beide ungerade – eine gerade Funktion.

Darüber hinaus folgen die algebraischen Operationen Addition und Subtraktion bestimmten Regeln: Das Addieren oder Subtrahieren zweier gerader Funktionen ergibt eine gerade Funktion, und das Addieren oder Subtrahieren zweier ungerader Funktionen ergibt eine ungerade Funktion. Diese Eigenschaften sind grundlegend für die Signalanalyse und ermöglichen die Zerlegung und Vereinfachung komplexer Signale in übersichtliche Komponenten.

Gerade und ungerade Signale

Ein gerades Signal, entweder in kontinuierlicher Zeit oder diskreter Zeit, ist definiert als ein Signal, das seiner eigenen zeitumgekehrten Version entspricht.
Sogar Signale sind ...

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Unit 1

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KERNBEGRIFFE UND KONZEPTE

signal symmetry and decomposition

Signal symmetry helps classify continuous-time and discrete-time signals as even or odd. An even signal matches its time-reversed version. It mirrors around the vertical axis, so the values at negative time match the values at positive time.

An odd signal does not match its time-reversed counterpart. It has antisymmetry about the vertical axis. These symmetry rules are used in both continuous-time and discrete-time signal analysis.

Any continuous-time signal can be split into even and odd parts. The decomposition writes the signal as a sum of one even function and one odd function. Complex signals can also be separated into even and odd components by using conjugate symmetry.

These signal types also follow clear algebra rules. The product of an even function and an odd function is odd. The product of two functions of the same type, whether both even or both odd, is even. Addition and subtraction also stay consistent: even with even gives even, and odd with odd gives odd. These patterns make complex signals easier to analyze and simplify.

Signal Symmetry and Decomposition

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Signals and Systems in Everyday Data

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even and odd signals

Signal symmetry helps classify continuous-time and discrete-time signals as even or odd. An even signal matches its time-reversed version. It mirrors around the vertical axis, so the values at negative time match the values at positive time. An odd signal does not match its time-reversed counterpart. It has antisymmetry about the vertical axis. These symmetry rules are used in both continuous-time and discrete-time signal analysis. Any continuous-time signal can be split into even and odd...

basic continuous time signals

Step, Impulse, and Ramp Signals

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rectangular and triangular pulse function

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Modeling Pulses with Rectangles and Triangles

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basic discrete time signals

Discrete-Time Signal Basics and Shapes

Discrete-time signals include several basic sequences used in signal processing. The unit step sequence is 1 for zero and positive values of the integer n. A graph of eight sample points shows it as a step that begins at n = 0 and stays constant after that. The unit impulse, also called the sample sequence, is 0 for every n except at n = 0, where it equals 1. This sequence is written as δ(n). It is the first difference of the unit step sequence, and the unit step sequence u(n) is the...

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Signal Transformations in Time and Amplitude

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classification of systems-i

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Continuous and Discrete System Models

Continuous-time and discrete-time systems are two common ways to model signals and their outputs. In continuous-time systems, both input and output signals change smoothly as time moves continuously. These systems are often written with differential or algebraic equations. An RC circuit is one example of a continuous-time system. Its input and output voltage can be related by a differential equation based on Ohm’s law and the capacitor relation. Discrete-time systems work differently. Their...

key terms and concepts

Learn how even and odd signals are identified by symmetry, how any signal can be split into even and odd parts, and the rules for combining them.

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Signal Symmetry and Decomposition

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