Name: complex exponentials in signal behavior
Uploaded: 2024-09-26T13:16:53.000Z
Duration: 1 min 18 s
Channel: JoVE

K12 Standards

south-carolina-college-and-career-ready-science-standards

South Carolina College and Career Ready Science Standards

High School

Physics

waves-and-their-applications-in-technologies-for-information-transfer

Waves and Their Applications in Technologies for Information Transfer

P-PS4-5

Exponential and Sinusoidal Signals

تعمل الوظيفة الأسية كأداة أساسية لتوصيف الأشكال الموجية التي ترتفع وتتحلل بسرعة. يتم تعريف هذه الوظيفة باستخدام الثوابت m و A. عندما يكون كلا الثامنين حقيقيين ، يمكن تمثيل الدالة بيانيا.
يتم تحقيق الأس المعقد عندما يكون الثابت m خياليا بحتا. إنه دوري إذا حافظ على حجم من الوحدة.
يمكن التعبير عن الإشارة الجيبية ذات الوقت المستمر من حيث التردد والفترة الزمنية.
يمكن استخدام علاقة أويلر للتعبير عن الإشارة الجيبية كأس مركب دوري بنفس التردد الأساسي.
وبالمثل ، يمكن التعبير عن الدالة الأسية المعقدة من حيث الإشارات الجيبية ، وكلها تشترك في نفس التردد الأساسي.
على سبيل المثال ، يمكن كتابة مجموع اثنين من الأس المركب كحاصل ضرب أسي معقد واحد وجيوب واحد.
يتم استخدام كل من الإشارات الأسية الجيبية والمعقدة على نطاق واسع لوصف الحفاظ على الطاقة في نظام ميكانيكي حيث تظهر الكتلة ، المتصلة بدعامة ثابتة عبر زنبرك ، حركة توافقية بسيطة.

الإشارات الأُسِّية والجيبية

<div style="direction: rtl;">
تعتبر الدالة الأسية بالغة الأهمية لوصف الأشكال الموجية التي ترتفع وتنخفض بسرعة. يتم تعريف هذه الدالة الأسية المستمرة في الزمن باستخدام مصطلحات أسية ذات ثابتين α وA. عندما يكون كلا الثابتين حقيقيين، يتم تمثيل الدالة على النحو التالي،

<img alt="Equation1" src="/files/ftp_upload/16042/16042_PT%20EQUATION_1.svg" style="text-align: center; height: 16px;">
ويمكن تصويرها بيانياً لإظهار النمو أو التضاؤل الأسي. عندما يكون الثابت α تخيليًا بحتًا، تكون النتيجة أسية مركبة، يتم التعبير عنها على النحو التالي،

<img alt="Equation2" src="/files/ftp_upload/16042/16042_PT%20EQUATION_2.svg" style="text-align: center; height: 16px;">
حيث j هي الوحدة التخيلية وω_0 هو التردد الزاوي. تكون هذه الدالة دورية إذا حافظت على مقدار واحد.

يمكن وصف إشارة جيبية مستمرة في الزمن من حيث التردد والفترة الزمنية. تسمح علاقة أويلر بالتعبير عن الإشارة الجيبية على أنها أسية معقدة دورية لها نفس الفترة الأساسية. وبالتالي، يتم تمثيل الإشارة الجيبية على النحو التالي،

<img alt="Equation3" src="/files/ftp_upload/16042/16042_PT%20EQUATION_3.svg" style="text-align: center; height: 14px;">
يمكن إعادة كتابة الإشارة الجيبية باستخدام الأسية المعقدة على النحو التالي،

<img alt="Equation4" src="/files/ftp_upload/16042/16042_PT%20EQUATION_4.svg" style="text-align: center; height: 28px;">
وبالمثل، يمكن التعبير عن الدالة الأسية المعقدة من حيث الإشارات الجيبية، التي تشترك جميعها في نفس الفترة الأساسية. على سبيل المثال، يمكن كتابة مجموع أسي معقدين على أنه حاصل ضرب أسية معقدة واحدة وجيب واحد، على سبيل المثال،

<img alt="Equation5" src="/files/ftp_upload/16042/16042_PT%20EQUATION_5.svg" style="text-align: center; height: 16px;">
يتم استخدام كل من الإشارات الجيبية والأسية المعقدة على نطاق واسع لوصف الحفاظ على الطاقة في الأنظمة الميكانيكية، مثل الكتلة المتصلة بدعامة ثابتة عبر زنبرك، والتي تظهر حركة توافقية بسيطة. توفر هذه الإشارات الأساس لتحليل السلوك التذبذبي وظواهر الرنين في مثل هذه الأنظمة.

</div>

تعمل الوظيفة الأسية كأداة أساسية لتوصيف الأشكال الموجية التي ترتفع وتتحلل بسرعة. يتم تعريف هذه الوظيفة باستخدام ...

Education

JoVE Core

Electrical Engineering

الهندسة الكهربائية

Unit 1

الوحدة الأولى

Introduction to Signals and Systems

مقدمة في الإشارات والأنظمة

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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 links it closely to sinusoidal motion.

A continuous-time sinusoidal signal can be described by its frequency and time period. Using Euler’s relation, a sinusoid can be written as a set of periodic complex exponentials with the same fundamental period. The same idea also works in reverse, so a complex exponential can be expressed in terms of sinusoidal signals.

The sum of two complex exponentials can also be rewritten in a simpler form. It becomes the product of one complex exponential and one sinusoid. These signal forms are widely used to model energy conservation in mechanical systems, such as a mass attached to a fixed support by a spring. In that setting, they help describe simple harmonic motion, oscillation, and resonance.

Complex Exponentials in Signal Behavior

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

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 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

Explore how exponential, complex exponential, and sinusoidal signals describe growth, decay, periodic motion, and spring-mass oscillations in high school science.

Complex Exponentials in Signal Behavior | 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.

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Complex Exponentials in Signal Behavior

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