Name: discrete-time signal basics and shapes
Uploaded: 2024-09-26T13:16:53.000Z
Duration: 1 min 16 s
Channel: JoVE

Последовательность единичных шагов выражается как 1 для нуля и положительных значений целого числа 'n'. Это можно графически отобразить с помощью набора из восьми точек выборки.
Единичный импульс или последовательность выборок математически выражается для всех дискретных значений n. Это ноль для всех значений 'n', кроме нуля.
Единичный импульс — это первая разница единичного шага, в то время как единичный шаг — это кумулятивная сумма единичной выборки. Эта корреляция наглядно демонстрируется с помощью графика.
Единичная импульсная последовательность может эффективно дискретизировать значение сигнала при n, равном нулю, поскольку оно не равно нулю только в этой точке.
Единичная линейная последовательность демонстрирует линейное нарастание значения с увеличением числа выборок. Для последовательности из 12 выборок на единичном пандусе график показывает линейное увеличение амплитуды с числом выборки.
Синусоидальная последовательность определяется ее амплитудными и фазовыми параметрами.
Экспоненциальная последовательность определяется с помощью комплексных чисел, при этом экспоненциально затухающие и увеличивающиеся последовательности представлены на графике.

Основные сигналы,  дискретные по времени

Basic Discrete Time Signals

Последовательность единичных шагов определяется как 1 для нулевых и положительных значений целого числа n. Эту последовательность можно графически отобразить с помощью набора из восьми точек выборки, показывая ступенчатую функцию, начинающуюся с n=0 и остающуюся постоянной после этого.

Последовательность единичных импульсов или выборок математически выражается как ноль для всех значений n, за исключением n=0, где она равна единице. Последовательность единичных импульсов, обозначаемая как δ(n), является первой разностью последовательности единичных шагов, в то время как последовательность единичных шагов u(n) является кумулятивной суммой последовательности единичных импульсов. Эту связь можно наглядно продемонстрировать с помощью графика, подчеркивающего, как единичный импульс может эффективно дискретизировать значение сигнала при n=0.

Последовательность функций единичного линейного роста демонстрирует линейное увеличение значения с увеличением номера выборки. Например, последовательность из 12 выборок на единичном наклонном участке будет демонстрировать линейное увеличение амплитуды с каждым номером выборки, представленное графически в виде прямой линии.

Синусоидальная последовательность определяется ее параметрами амплитуды и фазы. Эта последовательность может быть представлена ​​как,

<img alt="Equation1" src="/files/ftp_upload/16043/16043_PT%20EQUATION_1.svg" style="text-align: center; height: 14px;">

где A — амплитуда, ω — угловая частота, а φ — фаза.

Экспоненциальная последовательность определяется с использованием комплексных чисел, с экспоненциально затухающими и возрастающими последовательностями, представленными на графике. Экспоненциально затухающая последовательность может быть записана как,

<img alt="Equation2" src="/files/ftp_upload/16043/16043_PT%20EQUATION_2.svg" style="text-align: center; height: 14px;">

а экспоненциально возрастающая последовательность выражается как,

<img alt="Equation3" src="/files/ftp_upload/16043/16043_PT%20EQUATION_3.svg" style="text-align: center; height: 14px;">

где A — начальная амплитуда, а α — положительная константа. Эти последовательности имеют основополагающее значение для анализа различных приложений обработки сигналов благодаря своим уникальным свойствам и поведению.

Последовательность единичных шагов выражается как 1 для нуля и положительных значений целого числа 'n'. Это можно графически отобразить с помощью набора из восьми ...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

Introduction to Signals and Systems

Введение в сигналы и системы

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 cumulative sum of the unit impulse sequence. A graph can show how the impulse samples the signal value at n = 0.

The unit ramp sequence increases linearly as the sample number increases. For example, a 12-sample ramp rises by the same amount at each sample, so its graph appears as a straight line. This makes the ramp easy to recognize on a plot.

A sinusoidal sequence is set by its amplitude and phase. In the expression for this sequence, A is the amplitude, ω is the angular frequency, and Φ is the phase. An exponential sequence uses complex numbers. It can describe both exponentially decaying and exponentially increasing sequences. The decaying form and the increasing form are both shown on graphs, with A as the initial amplitude and α as a positive constant. These basic sequences are important because they help describe many signal-processing applications.

Discrete-Time Signal Basics and Shapes

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 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

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 the key discrete-time signal shapes used in signal processing, including the unit step, impulse, ramp, sinusoidal, and exponential sequences.

Discrete-Time Signal Basics and Shapes | 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

Discrete-Time Signal Basics and Shapes

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