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

基本信号运算包括时间反转、尺度变换、移位和幅度变换。
时间反转是将连续时间信号关于时间零点处的垂直轴进行镜像，通过将“t”替换为负的“t”来实现。对于所考虑的信号，其结果以图形方式展示。
时间缩放通过将“t”替换为“at”来实现信号在时间上的压缩或扩展，其中“a”为常数。若该常数的绝对值大于1，则信号被压缩；若小于1，则信号被扩展。
该常数的负值会同时引起时间反转以及压缩或扩展。这可以通过所考虑的信号进行图形化表示。
连续时间信号的时间平移通过将“t”替换为“t − t0”来实现，其中“t0”为常数。当该常数为正时，信号将被延迟并向右平移；当该常数为负时，信号将被提前并向左平移。
连续时间信号的幅度变换具有如下一般形式，其中“A”和“B”为常数。图中显示了一个示例信号的幅度变换。

信号的基本操作

Basic Operations on Signals

基本信号操作包括时间反转、时间缩放、时间移位和振幅变换。这些操作是信号处理和分析的基础。

时间反转在 t=0 处将连续时间信号镜像到垂直轴。这是通过将 t 替换为 −t 来实现的。例如，如果考虑信号 x(t)，则时间反转信号为 x(−t)。此操作可以图形表示，显示镜像信号。

<img alt="Figure1" src="/files/ftp_upload/16044/16044_PT%20Figure_1.svg" style="height: 150px;">

时间缩放通过将 t 替换为 at 来压缩或扩展时间信号，其中 a 是常数。如果 ∣a∣&gt;1，则信号压缩；如果 ∣a∣&lt;1，则信号扩展。 a 的负值会引起时间反转和压缩或扩展。此操作对信号的图形表示展示了这些效果。

连续时间信号的时间移位是通过将 t 替换为 t−t_0 来完成的，其中 t_0 是常数。正 t_0 会延迟并将信号向右移动，而负 t_0 会提前信号并将其向左移动。例如，移位 t_0 的信号 x(t) 表示为 x(t−t_0)。

连续时间信号的幅度变换通常采用 A*x(t)+B 的形式，其中 A 和 B 是常数。这些变换会缩放幅度，也可以垂直移动幅度。从图形上看，信号的振幅变换显示了按 A 缩放和按 B 移位的效果。

这些基本操作（时间反转、缩放、移位和振幅变换）是操纵和分析信号的重要工具，允许在信号处理应用中进行各种调整和修改。

基本信号运算包括时间反转、尺度变换、移位和幅度变换。  
时间反转将连续时间信号关于...进行镜像

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Introduction to Signals and Systems

信号与系统导论

KEY TERMS AND CONCEPTS

关键术语与概念

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