Name: signal transformations in time and amplitude
Uploaded: 2024-09-26T13:16:54.000Z
Duration: 1 min 22 s
Channel: JoVE

Grundlegende Signaloperationen sind Zeitumkehr, Skalierung, Verschiebung und Amplitudentransformationen.
Die Zeitumkehr spiegelt ein kontinuierliches Zeitsignal um die vertikale Achse bei einer Zeit gleich Null, die durch Ersetzen von "t" durch negatives "t" erreicht wird. Für ein betrachtetes Signal werden die Ergebnisse grafisch dargestellt.
Die Zeitskalierung komprimiert oder erweitert ein Signal in der Zeit, indem "t" durch "at" ersetzt wird, wobei "a" konstant ist. Wenn die Größe dieser Konstante größer als 1 ist, wird das Signal komprimiert. Wenn er kleiner als 1 ist, wird er erweitert.
Ein negativer Wert dieser Konstante induziert sowohl eine Zeitumkehr als auch eine Kompression oder Expansion. Dies kann anhand des betrachteten Signals grafisch dargestellt werden.
Die Zeitverschiebung eines kontinuierlichen Zeitsignals erfolgt durch Ersetzen von 't' durch 't − t0', wobei 't0' konstant ist. Eine positive Konstante verzögert und verschiebt das Signal vom Ursprung nach rechts, während eine negative Konstante das Signal vorschiebt und nach links verschiebt.
Die Amplitudentransformationen eines zeitkontinuierlichen Signals nehmen die allgemeine Form an, wobei "A" und "B" Konstanten sind. Die Grafik zeigt eine Amplitudentransformation eines beispielhaften Signals.

grundlegende operationen an signalen

Zu den grundlegenden Signaloperationen gehören Zeitumkehr, Zeitskalierung, Zeitverschiebung und Amplitudentransformationen. Diese Operationen sind grundlegend für die Signalverarbeitung und -analyse.

Die Zeitumkehr spiegelt ein kontinuierliches Zeitsignal an der vertikalen Achse bei t=0. Dies wird erreicht, indem t durch −t ersetzt wird. Wenn beispielsweise ein Signal x(t) betrachtet wird, ist das zeitumgekehrte Signal x(−t). Diese Operation kann grafisch dargestellt werden, indem das gespiegelte Signal angezeigt wird.

<img alt="Figure1" src="/files/ftp_upload/16044/16044_PT%20Figure_1.svg" style="height: 150px;">

Die Zeitskalierung staucht oder dehnt ein Signal zeitlich, indem t durch αt ersetzt wird, wobei α eine Konstante ist. Wenn ∣α∣&gt;1, wird das Signal komprimiert; wenn ∣α∣&lt;1, dehnt es sich aus. Ein negativer Wert von α ​​bewirkt sowohl eine Zeitumkehr als auch eine Kompression oder Expansion. Eine grafische Darstellung dieser Operation an einem Signal veranschaulicht diese Effekte.

Die Zeitverschiebung eines kontinuierlichen Signals erfolgt durch Ersetzen von t durch t−t_0, wobei t_0 eine Konstante ist. Ein positiver t_0 verzögert und verschiebt das Signal nach rechts, während ein negativer t_0 das Signal vorschiebt und nach links verschiebt. Beispielsweise wird ein um t_0 verschobenes Signal x(t) als x(t−t_0) dargestellt.

Amplitudentransformationen eines kontinuierlichen Signals haben im Allgemeinen die Form Ax(t)+B, wobei A und B Konstanten sind. Diese Transformationen skalieren die Amplitude und können sie auch vertikal verschieben. Grafisch zeigt eine Amplitudentransformation eines Signals die Wirkung der Skalierung um A und der Verschiebung um B.

Diese grundlegenden Operationen – Zeitumkehr, Skalierung, Verschiebung und Amplitudentransformationen – sind wichtige Werkzeuge zur Handhabung und Analyse von Signalen und ermöglichen verschiedene Anpassungen und Änderungen in Signalverarbeitungsanwendungen.

Grundlegende Operationen an Signalen

Grundlegende Signaloperationen sind Zeitumkehr, Skalierung, Verschiebung und Amplitudentransformationen.
Die Zeitumkehr spiegelt ein kontinuierliches Zeitsignal über die ...

Education

Lehre

JoVE Core

Electrical Engineering

Elektrotechnik

Unit 1

Einheit 1

Introduction to Signals and Systems

Einführung in Signale und Systeme

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

Basic Operations on Signals

signal transformations in time and amplitude

Signal transformations in time and amplitude are key tools in signal processing. They include time reversal, time scaling, time shifting, and amplitude changes. These operations help students describe how a continuous-time signal changes when it is mirrored, stretched, moved, or resized.

Time reversal flips a continuous-time signal about the vertical axis at t = 0. It is done by replacing t with -t, so x(t) becomes x(-t). A graph of the signal shows this mirrored shape clearly.

Time scaling changes how fast or slow a signal appears in time by replacing t with at, where a is a constant. If |a| &gt; 1, the signal compresses. If |a| &lt; 1, the signal expands. A negative value of a causes time reversal as well as compression or expansion.

Time shifting moves a continuous-time signal left or right by replacing t with t - t0, where t0 is a constant. A positive t0 delays the signal and shifts it to the right. A negative t0 advances the signal and shifts it to the left. Amplitude transformations use the form Ax(t) + B, where A and B are constants. These changes scale the signal’s height and can shift it up or down vertically.

Signal Transformations in Time and Amplitude

signal-transformations-in-time-and-amplitude

signal and system

Signals and Systems in Everyday Data

Signals and systems help describe how data moves through the world. A signal x(t) is a set of data or a time function that represents a variable of interest. It can carry information about atmospheric temperature, humidity, human voice, television images, a dog’s bark, or birdsongs. A signal can also depend on more than one independent variable. An image, for example, depends on horizontal and vertical position, so it can be treated as a two-dimensional signal. In this course, the focus is on...

classification of signals

Signal Types by Time and Amplitude

Signals in signal processing are grouped by time behavior, amplitude, and whether they start before or after time zero. These signal classes help students describe and compare signals before they are analyzed or processed. A continuous-time signal has a value at every instant in time. A discrete-time signal has values only at specific moments, often written as x(n), where n is an integer. Discrete-time signals often come from systems with naturally discrete values, such as digital audio...

energy and power signals

Power and Energy in Electrical Signals

Power and energy in electrical signals are measured from voltage and current across a resistor. These signals help describe how much energy a signal carries and how much power it delivers over time. For a continuous-time signal, total energy is found by integrating the square of the signal's magnitude over a chosen time interval. Time-averaged power is then the total energy divided by the length of that interval. For a discrete-time signal, total energy is found by summing the squares of the...

even and odd signals

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...

basic continuous time signals

Step, Impulse, and Ramp Signals

Step, impulse, and ramp signals are the main basic continuous-time signals. They are also called singularity functions. These signals are known for discontinuities or for derivatives that are discontinuous. The unit step function, written as u(t), is zero for negative time values and one for positive time values. It has a jump at t = 0. In many cases, it can model an abrupt change, such as the step voltage that appears when a car ignition key is turned. The derivative of the unit step...

rectangular and triangular pulse function

rectangular-and-triangular-pulse-function

Modeling Pulses with Rectangles and Triangles

Rectangular and triangular pulse functions are basic signal shapes used to model waveforms in signal processing. The rectangular pulse has a flat top and is centered at the origin with a height of one unit. It is described by two parameters: T, which gives the center location on the time axis, and τ, which sets the pulse duration. A rectangular pulse can also represent a real signal with a chosen amplitude and length. For example, a pulse with a 5 V amplitude, a 3-second duration, and a center...

exponential and sinusoidal signals

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...

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...

basic operations on signals

Signal transformations in time and amplitude are key tools in signal processing. They include time reversal, time scaling, time shifting, and amplitude changes. These operations help students describe how a continuous-time signal changes when it is mirrored, stretched, moved, or resized. Time reversal flips a continuous-time signal about the vertical axis at t = 0. It is done by replacing t with -t, so x(t) becomes x(-t). A graph of the signal shows this mirrored shape clearly. Time scaling...

classification of systems-i

System Properties: Linearity, Causality, Memory

System properties such as linearity, causality, and memory help describe how a system responds to input. These ideas are useful for studying and designing systems in engineering. A system can be checked by looking at how its output changes with different inputs. Linearity means the input-output relationship follows homogeneity and additivity. Homogeneity says that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Additivity means the response...

classification of systems-ii

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 time reversal, time scaling, time shifting, and amplitude transformations change continuous-time signals in signal processing and graph form.

Signal Transformations in Time and Amplitude | 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

Signal Transformations in Time and Amplitude

Description

Transcript

Tags