Name: z-transform rules for sum and convolution
Uploaded: 2024-09-26T13:17:10.000Z
Duration: 1 min 16 s
Channel: JoVE

Die Eigenschaft von Accumulation wird abgeleitet, indem die akkumulierte Summe ausgedrückt und die Zeitverschiebungseigenschaft angewendet wird, um die Z-Transformation aufzulösen.
Sie besagt, dass die Summierung eines zeitdiskreten Signals ein weiteres Signal erzeugt, dessen Z-Transformation der Z-Transformation des Originalsignals multipliziert mit z über z minus 1 entspricht.
Die Faltungseigenschaft zeigt, dass die Faltung zweier Signale im Zeitbereich zum Produkt ihrer Z-Transformationen im Frequenzbereich führt.
Dies gilt sowohl für kausale als auch für nicht-kausale Signale.
Das Anwenden der Zeitverschiebungseigenschaft auf die Zeitbereichsgleichung hilft bei der Überprüfung der Faltungseigenschaft.
Der Anfangswertsatz setzt den Anfangswert eines Signals in Beziehung zu seiner Z-Transformation. Für ein Signal x[n] ist der Anfangswert der Grenzwert von X(z), wenn z sich unendlich nähert.
In ähnlicher Weise besagt der Schlusswertsatz, dass der Endwert der Grenzwert von 1 minus dem Kehrwert von z multipliziert mit X(z) ist, wenn z sich eins nähert.
Sie gilt nur, wenn x im Unendlichen existiert und alle Pole innerhalb eines Einheitskreises liegen, außer bei z gleich eins.

eigenschaften der z-transformation ii

Die Eigenschaft der Akkumulation in der Signalverarbeitung wird abgeleitet, indem die akkumulierte Summe eines zeitdiskreten Signals analysiert und die Zeitverschiebung verwendet wird, um seine Z-Transformation zu bestimmen. Dieses Prinzip zeigt, dass die Z-Transformation des summierten Signals durch einen Multiplikationsfaktor mit der Z-Transformation des ursprünglichen Signals in Beziehung steht.

Darüber hinaus zeigt die Faltungseigenschaft an, dass die Faltung zweier Signale im Zeitbereich dem Produkt ihrer Z-Transformationen im Frequenzbereich entspricht. Diese Eigenschaft gilt sowohl für kausale als auch für nichtkausale Signale. Die Faltungseigenschaft kann bestätigt werden, indem die Zeitverschiebung auf die entsprechende Zeitbereichsgleichung angewendet wird.

Der Anfangswertsatz stellt eine Verbindung zwischen dem Anfangswert eines Signals und seiner Z-Transformation her. Für ein gegebenes Signal kann der Anfangswert erhalten werden, indem die Z-Transformation ausgewertet wird, wenn sich die Variable Null nähert. Dieser Satz ist besonders nützlich, um die Startbedingungen eines Systems aus seiner Z-Transformation zu bestimmen.

Umgekehrt bestimmt der Endwertsatz den Endwert eines Signals, indem er dessen Z-Transformation untersucht, wenn sich die Variable Eins nähert. Dieser Satz ist nur anwendbar, wenn das Signal im Unendlichen weiter existiert und alle Pole der Z-Transformation innerhalb des Einheitskreises liegen, außer an dem Punkt, an dem die Variable Eins ist.

Diese Eigenschaften sind für die Analyse und den Entwurf zeitdiskreter Systeme von entscheidender Bedeutung. Durch die Verwendung der Akkumulations-, Faltungs-, Anfangswert- und Endwertsätze kann das Verhalten zeitdiskreter Signale und Systeme im Z-Bereich effektiv untersucht werden. Die Beherrschung dieser Eigenschaften ermöglicht die Handhabung und Transformation von Signalen und hilft bei der Erstellung von Filtern und Steuerungssystemen, die im zeitdiskreten Bereich funktionieren.

Eigenschaften der Z-Transformation II

Die Eigenschaft von Accumulation wird abgeleitet, indem die akkumulierte Summe ausgedrückt und die Zeitverschiebungseigenschaft angewendet wird, um die Z-Transformation aufzulösen.
Darin heißt es ...

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Electrical Engineering

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z-Transform

z-Transformation

KEY TERMS AND CONCEPTS

SCHLÜSSELBEGRIFFE UND KONZEPTE

Properties of the z-Transform II

z-transform rules for sum and convolution

The z-transform rules for sum and convolution help analyze discrete-time signals in signal processing. These rules show how changes in the time domain connect to the z-domain, which is the transform domain used for discrete-time systems.

The accumulation property comes from the accumulated sum of a discrete-time signal. By applying the time-shifting property, the z-transform of the summed signal can be found from the z-transform of the original signal. The result is a simple multiplicative factor that links the two signals.

The convolution property shows that convolution in the time domain becomes multiplication of z-transforms in the frequency domain. This property applies to both causal signals, which depend on present and past values, and noncausal signals, which can also include future values. It can be verified by using the time-shifting property on the time-domain expression.

The initial value theorem connects a signal’s starting value to its z-transform. To find that initial value, evaluate the z-transform as the variable approaches zero. This is useful for finding the starting conditions of a system from its z-transform.

The final value theorem gives the ending value of a signal from its z-transform. It is found by examining the z-transform as the variable approaches one. This theorem works only if the signal still exists at infinity and the poles of the z-transform are inside the unit circle, except at the point where the variable equals one.

These properties are important for analyzing and designing discrete-time systems. Accumulation, convolution, the initial value theorem, and the final value theorem help describe signal behavior in the z-domain. They also support the design of filters and control systems that operate in the discrete-time domain.

Z-Transform Rules for Sum and Convolution

z-transform-rules-for-sum-and-convolution

definition of z-transform

z-Transform for Discrete-Time Signals

The z-transform is a key math tool for analyzing discrete-time signals and systems. It is the discrete-time counterpart of the Laplace transform used for continuous-time systems. In signal and systems analysis, it works alongside the discrete-time Fourier transform. The z-transform changes a discrete-time signal into a power series with a complex variable, z. Each term in the series represents one sampling instant of the signal. This form makes it easier to study the signal in detail and solve...

region of convergence

Z-Transform Convergence and Stability

The z-transform and its Region of Convergence (ROC) are key ideas in discrete-time signals and systems. The z-transform is a mathematical tool for analyzing signals that change at separate time steps. Its values do not always converge, so the ROC identifies the range of complex numbers z where the transform is valid. The ROC can appear in different shapes in the complex plane. It may lie inside a circle, outside a circle, or within an annulus, which is the space between two circles. For an...

properties of the z-transform i

Z-Transform Rules for Signals

The z-transform rules for signals help students analyze discrete-time systems in digital signal processing. These rules turn time-domain changes into simpler expressions in the z-domain, which makes signal work easier to follow. Linearity is one key rule. It means the z-transform of a linear combination of two discrete-time signals equals the same linear combination of their individual z-transforms. This is useful when signals are added together or superimposed. Time-shifting is another...

properties of the z-transform ii

The z-transform rules for sum and convolution help analyze discrete-time signals in signal processing. These rules show how changes in the time domain connect to the z-domain, which is the transform domain used for discrete-time systems. The accumulation property comes from the accumulated sum of a discrete-time signal. By applying the time-shifting property, the z-transform of the summed signal can be found from the z-transform of the original signal. The result is a simple multiplicative...

inverse z-transform by partial fraction expansion

inverse-z-transform-by-partial-fraction-expansion

Inverse z-Transform Using Poles and Residues

The inverse z-transform converts a function from the z-domain back to the time domain. One useful approach is the partial fraction method. It breaks a complex function into simpler fractions with distinct coefficients. The process starts by identifying the poles of the function. A pole is a value that makes the function grow without a finite limit. The function is then written in terms of these poles, and each pole adds one term to the partial fraction form. Next, the coefficients for each...

difference equation solution using z-transform

difference-equation-solution-using-z-transform

Solving Discrete-Time Responses with z-Transform

The z-transform helps solve discrete-time systems that are written as linear difference equations. These equations are common in digital signal processing and control systems. For a higher-order difference equation, you need the input signal and the initial conditions up to one term less than the order of the equation. The method also makes delayed signals easier to handle. A delay in time becomes a shift in the z-domain. An advance in time is shifted in the opposite direction. This link...

relation of dft to z-transform

DFT Through the Unit Circle

The Discrete Fourier Transform (DFT) shows how a discrete-time signal breaks into frequency parts. It takes N samples from the time domain and gives N values in the frequency domain. Each value represents one frequency component. The z-transform helps explain this process. A z-transform represents a discrete sequence in the complex frequency domain by summing sequence terms multiplied by powers of a complex number. For causal sequences, which start at a fixed time and move forward, the sum can...

key terms and concepts

Learn how accumulation, convolution, and the initial and final value theorems connect discrete-time signals in the z-domain for system analysis and design.

Z-Transform Rules for Sum and Convolution | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Z-Transform Rules for Sum and Convolution

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