Name: continuous and discrete system models
Uploaded: 2024-09-26T13:16:54.000Z
Duration: 1 min 31 s
Channel: JoVE

למערכות זמן רציף יש אותות קלט ופלט רציפים, כאשר הזמן נמדד ברציפות, והם מוגדרים בדרך כלל על ידי משוואות דיפרנציאליות או אלגבריות.
לדוגמה, במעגל RC, הקשר בין מתח כניסה למתח יציאה מבוטא באמצעות משוואה דיפרנציאלית הנגזרת באמצעות חוק אוהם ויחסי מתח-זרם של הקבל.
מערכות זמן בדידות מעבדות אותות קלט ופלט במרווחי זמן ספציפיים והתנהגותן מתוארת על ידי משוואות הפרשים במופעי זמן שונים. דוגמה למערכת כזו היא מודל פשוט ליתרה בחשבון בנק מחודש לחודש.
סוג נוסף של מערכת הוא משתנה בזמן ומשתנה בזמן.
למערכת המשתנה בזמן יש פרמטרים המשתנים עם הזמן. מערכות רציפות, משתנות זמן, מתוארות על ידי משוואות דיפרנציאליות עם מקדמים תלויי זמן, בעוד שמערכות בדידות, משתנות זמן, מתוארות על ידי משוואות הפרש עם פרמטרים תלויי זמן.
מערכות אינווריאנטיות בזמן מייצרות פלט שכאשר אות הקלט מוסט בזמן, משתנה באופן זהה מבלי לשנות את מאפייני המערכת.
מעגל RC הוא משתנה בזמן אם ערכי ההתנגדות והקיבוליות נשארים קבועים. אם אלה משתנים, המערכת הופכת למשתנה בזמן מכיוון שתוצאות הניסוי ישתנו בהתאם למועד עריכת הניסוי.

סיווג מערכות-ii

<div style="direction: rtl;">
 מערכות בזמן רציף כוללות אותות קלט ופלט רציפים, כאשר הזמן נמדד ברציפות. מערכות אלו מוגדרות בדרך כלל על ידי משוואות דיפרנציאליות או אלגבריות. לדוגמה, במעגל RC (נגדי-קבלי), הקשר בין מתח הכניסה למתח היציאה מבוטא באמצעות משוואה דיפרנציאלית שנגזרת מחוק אוהם ומשוואת הקבל:
 
 <img alt="Equation1" src="/files/ftp_upload/16046/16046_PT%20EQUATION_1.svg" style="text-align: center; height: 30px;">
 מערכות בזמן בדיד כוללות אותות קלט ופלט בנקודות זמן ספציפיות, מוגדרות ברגעים מובחנים על ידי משוואות הפרש. דוגמה לכך היא מודל פשוט של יתרת חשבון בנק מחודש לחודש, המיוצג כך:
 
 <img alt="Equation2" src="/files/ftp_upload/16046/16046_PT%20EQUATION_2.svg" style="text-align: center; height: 14px;">
 כאשר B[n] היא היתרה בחודש n, D[n] היא ההפקדה ו- W[n] היא המשיכה.
 
 ניתן גם לסווג מערכות כמערכות משתנות בזמן או קבועות בזמן. מערכת משתנה בזמן היא מערכת שהפרמטרים שלה משתנים לאורך זמן. מערכות בזמן רציף משתנות בזמן מתוארות על ידי משוואות דיפרנציאליות משתנות בזמן, בעוד שמערכות בזמן בדיד משתנות בזמן מתוארות על ידי משוואות הפרש משתנות בזמן.
 
 מערכת קבועה בזמן מציגה תזוזה זהה באותות הפלט כאשר אות הקלט מוזז בזמן. מבחינה מתמטית, אם y(t) הוא הפלט עבור קלט x(t), אז עבור מערכת קבועה בזמן, הפלט עבור קלט מוזז, y(t−t_0) יהיה x(t−t_0). לדוגמה, מעגל RC הוא מערכת קבועה בזמן אם ערכי הנגד \(R\) והקבל \(C\) נשארים קבועים. אם ערכים אלו משתנים, המערכת הופכת למשתנה בזמן, כלומר התוצאות ישתנו בהתאם למתי מבוצע הניסוי.
 
 הבנת הבחנות אלו בין מערכות בזמן רציף וזמן בדיד, כמו גם בין מערכות קבועות בזמן ומשתנות בזמן, היא חיונית לניתוח ותכנון מגוון רחב של מערכות הנדסיות.
 
</div>

סיווג מערכות-II

למערכות זמן רציף יש אותות קלט ופלט רציפים, כאשר הזמן נמדד ברציפות, והם מוגדרים בדרך כלל על ידי אותות דיפרנציאליים או אלגבריים ...

Education

JoVE Core

Electrical Engineering

הנדסה חשמלית

Unit 1

יחידה 1

Introduction to Signals and Systems

מבוא לאותים ומערכות

KEY TERMS AND CONCEPTS

מונחים ומושגים מרכזיים

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 inputs and outputs are defined at specific time steps, and they are described by difference equations.

A bank account balance from month to month is a simple discrete-time model. In that example, B[n] is the balance at month n, D[n] is the deposit, and W[n] is the withdrawal. Systems are also grouped as time-varying or time-invariant. A time-varying system has parameters that change with time, while a time-invariant system keeps the same shift in output when the input is shifted.

For a time-invariant system, if y(t) is the output for input x(t), then y(t − t0) is the output for x(t − t0). An RC circuit is time-invariant when the resistance R and capacitance C stay constant. If those values change, the system becomes time-variant. Then the result depends on when the experiment is done. Knowing these categories helps in analyzing and designing many engineering systems.

Continuous and Discrete System Models

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

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-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 continuous-time and discrete-time systems differ, plus how time-varying and time-invariant behavior is identified using RC circuits and bank account examples.

Continuous and Discrete System Models | 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

Continuous and Discrete System Models

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