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

単位ステップシーケンスは、ゼロと整数 'n' の正の値に対して 1 として表されます。これは、8つのサンプルポイントのセットを使用してグラフィカルに表示できます。
単位インパルスまたはサンプルシーケンスは、nのすべての離散値について数学的に表されます。ゼロを除くすべての 'n' 値に対して 0 です。
単位インパルスは単位ステップの最初の差であり、単位ステップは単位サンプルの累積合計です。この相関関係は、グラフで視覚的に示されています。
単位インパルスシーケンスは、ゼロに等しいnの信号値を効果的にサンプリングできます。これは、そのポイントでのみゼロであるためです。
単位ランプシーケンスは、サンプル数の増加に伴って値の線形エスカレーションを示します。ユニットランプ上の12個のサンプルのシーケンスの場合、グラフはサンプル番号とともに振幅が線形に増加していることを示しています。
正弦波シーケンスは、その振幅パラメータと位相パラメータによって定義されます。
指数数列は複素数を使用して定義され、指数関数的に減衰するシーケンスと増加するシーケンスがグラフで表されます。

基本的な離散時間信号

Basic Discrete Time Signals

単位ステップ列は、整数 n のゼロおよび正の値に対して 1 として定義されます。この列は、8 つのサンプル箇所のセットを使用してグラフで表すことができます。これは n=0 から始まり、その後は一定となるステップ関数を示します。

単位インパルスまたはサンプル列は、n=0 のときを除いてすべての n 値に対して 0 として数学的に表現され、n=0 のときは 1 になります。δ(n) で示される単位インパルス列は、単位ステップ列の最初の差であり、単位ステップ列 u(n) は、単位インパルス列の累積和です。この関係は、単位インパルスが n=0 で信号値を効果的にサンプリングする方法を強調して、グラフで視覚的に示せます。

単位ランプ列は、サンプル数の増加に伴って値が直線的に増加します。たとえば、単位ランプ上の 12 個のサンプルの列は、サンプル番号ごとに振幅が直線的に増加し、グラフでは直線で示されます。

正弦波列は、振幅と位相のパラメータによって定義されます。この列は、次のように示すことができます。

<img alt="Equation1" src="/files/ftp_upload/16043/16043_PT%20EQUATION_1.svg" style="height: 14px;">

ここで、A は振幅、ω は角周波数、Φ は位相です。

指数列は複素数を使用して定義され、指数的に減少する列と増加する列がグラフ上に示されます。指数的に減少する列は、次のように記述できます。

<img alt="Equation2" src="/files/ftp_upload/16043/16043_PT%20EQUATION_2.svg" style="height: 14px;">

一方、指数的に増加する列は、次のように示されます。

<img alt="Equation3" src="/files/ftp_upload/16043/16043_PT%20EQUATION_3.svg" style="height: 14px;">

ここで、A は初期振幅、α は正の定数です。これらの列は、その独自の特性と挙動により、さまざまな信号処理のアプリケーションを分析する上で不可欠です。

単位ステップシーケンスは、ゼロと整数 'n' の正の値に対して 1 として表されます。これは、8つのセットを使用してグラフィカルに表示することができます...

Education

JoVE Core

Electrical Engineering

電気工学

Unit 1

ユニット1

Introduction to Signals and Systems

信号とシステム概論

KEY TERMS AND CONCEPTS

主要用語と概念

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 電気工学 は、電気回路の基本概念と回路解析テクニックを網羅したアニメーションビデオシリーズです。オペアンプと回路変換および 1 次・ 2 次回路の概念を網羅しています。実験を撮影した実写動画では、設計例や現代社会での応用が含まれており、複雑な概念も分かりやすく学習いただけます。

electrical engineering

Discrete-Time Signal Basics and Shapes

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