Name: dtfs for periodic signal analysis
Uploaded: 2024-09-26T13:17:01.000Z
Duration: 1 min 20 s
Channel: JoVE

K12 Standards

Next Generation Science Standards

High School

Physical Science

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

Waves and Their Applications in Technologies for Information Transfer

HS-PS4-2

Discrete-Time Fourier Series

سلسلة فورييه المنفصلة هي نظير لتوسيع سلسلة فورييه للإشارات الدورية في الوقت المستمر.
يتضمن حساب معاملات التمدد في DTFS المجاميع بدلا من التكاملات. يمكن تحديد استجابة النظام الخطي الثابت زمنيا لإشارة دورية زمنيا منفصلة من خلال عملية خطوة بخطوة.
أولا ، يجب حساب DTFS لإشارة الإدخال. يتم حساب استجابة الخرج لكل مصطلح DTFS باستخدام استجابة تردد النظام. أخيرا ، يتم تلخيص هذه النتائج للحصول على إجمالي إشارة الإخراج.
الإشارة الدورية في الوقت المستمر لها فترة ذات ترددات دائرية وزاوية ، ممثلة بسلسلة فورييه أسية معقدة. وبالمثل ، فإن الإشارة الدورية ذات الوقت المنفصل لها تردد زاوي أساسي.
يتكون توسع DTFS من حدود محدودة ، على عكس سلسلة فورييه للوقت المستمر ، والتي تتكون من عدد لا حصر له من المصطلحات.
في معالجة الإشارات الرقمية ، يساعد DTFS في فحص العينات الدورية ، وتحديد ترددات محددة ، وتصفية الضوضاء غير المرغوب فيها بشكل فعال.

متسلسلة فورييه المتقطعة-الزمنية

<div style="direction: rtl;">
 تُعدُّ سلسلة فورييه المنفصلة الزمن مفهومًا أساسيًا في معالجة الإشارات، حيث تعمل كنظير للزمن المنفصل لسلسلة فورييه المتصلة الزمن. وهي تسمح بتمثيل وتحليل الإشارات الدورية المنفصلة الزمن من حيث مكونات ترددها. وعلى عكس نظيرتها المستمرة، التي تستخدم التكاملات، فإن حساب معاملات توسيع سلسلة فورييه المنفصلة الزمن ينطوي على عمليات جمع بسبب الطبيعة المنفصلة للإشارة.
 
 بالنسبة لإشارة دورية منفصلة الزمن x[n] ذات فترة N_0، يتم حساب معاملات سلسلة فورييه المنفصلة الزمن X[k] باستخدام الصيغة:
 
 <img alt="Equation1" src="/files/ftp_upload/16065/16065_PT%20EQUATION_1.svg" style="text-align: center; height: 48px;">
 حيث k=0,1,2,...,N_0−1. تمثل هذه المعاملات X[k] الإشارة في مجال التردد، وتلتقط سعة وطور كل مكوّن تردد.
 
 لتحديد استجابة نظام خطي مستقل زمنيًا (LTI) لإشارة دورية منفصلة الوقت، يتم اتباع نهج منهجي:
 
 <ol style="display: flex; flex-direction: column; gap: 10px;">
 <li style="text-align: right;">حساب DTFS لإشارة الإدخال: احسب معاملات DTFS X[k] لإشارة الإدخال x[n].
 </li>
 <li style="text-align: right;">احسب استجابة الإخراج لكل مصطلح DTFS: استخدم استجابة التردد للنظام H(e^(jΩ)) لتحديد الإخراج لكل مكون تردد. معاملات DTFS الناتجة Y[k] معطاة بواسطة
 <img alt="Equation2" src="/files/ftp_upload/16065/16065_PT%20EQUATION_2.svg" style="text-align: center; height: 36px;">
 حيث Ω =2πk/N
 
 </li>
 <li style="text-align: right;">جمع الاستجابات: أخيرًا، اجمع مساهمات جميع مصطلحات DTFS للحصول على إجمالي إشارة الإخراج في مجال الوقت.
 </li>
 </ol>
 في الإشارات المستمرة، يتم تعريف الدورية فيما يتعلق بفترة T، والتي تتوافق مع الترددات الدائرية والزاوية. بالنسبة لإشارة زمنية منفصلة، ​​ترتبط الدورية بتردد زاوي أساسي Ω_k =2πk/N_0 ، حيث N_0 هي فترة الإشارة المنفصلة. إن توسع DTFS محدود، ويتكون من N مصطلحات، على النقيض من السلسلة اللانهائية في سلسلة فورييه المستمرة.
 
 يلعب DTFS دورًا حاسمًا في معالجة الإشارات الرقمية (DSP)، وخاصة في تحليل ومعالجة الإشارات الدورية المستمدة من البيانات المأخوذة من العينات. إنه مفيد في مهام مثل تحديد ترددات معينة داخل الإشارات الصوتية، وتعزيز أو قمع مكونات تردد معينة، وتصفية الضوضاء غير المرغوب فيها. من خلال تحويل الإشارات إلى مجال التردد، يسهل DTFS تحليل ومعالجة الإشارات بكفاءة، مما يتيح تحسين الأداء في تطبيقات مختلفة مثل الاتصالات السلكية واللاسلكية، وهندسة الصوت، وأنظمة التحكم.
 
</div>

سلسلة فورييه المنفصلة هي نظير لتوسيع سلسلة فورييه للإشارات الدورية في الوقت المستمر.
حساب التوسع ...

Education

JoVE Core

Electrical Engineering

الهندسة الكهربائية

Unit 1

الوحدة الأولى

Fourier Series

سلاسل فورير

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

dtfs for periodic signal analysis

Discrete-Time Fourier Series (DTFS) helps analyze periodic signals in discrete time by breaking them into frequency components. It is the discrete-time counterpart of the continuous-time Fourier series. Instead of using integrals, DTFS uses summations because the signal is discrete.

For a discrete-time periodic signal x[n] with period N0, the DTFS coefficients X[k] are found from the DTFS formula. The index k runs from 0 to N0 − 1. These coefficients describe the signal in the frequency domain and show the amplitude and phase of each frequency component.

DTFS is also useful for finding the response of a linear time-invariant (LTI) system to a periodic input. First, compute the DTFS coefficients of the input signal x[n]. Then use the system frequency response H(e^jΩ) to find the output for each frequency term. The output coefficients Y[k] are determined from that frequency response, and the total time-domain output is found by summing all DTFS terms.

In discrete-time signals, periodicity is tied to the fundamental angular frequency Ωk = 2πk/N0, where N0 is the signal period. The DTFS expansion has a finite number of terms, unlike the infinite series used for continuous-time Fourier series. This makes DTFS an important tool in digital signal processing for studying sampled periodic signals, identifying frequencies in audio, and helping enhance or suppress frequency components. It is also used to filter unwanted noise and support applications in telecommunications, audio engineering, and control systems.

DTFS for Periodic Signal Analysis

trigonometric fourier series

Fourier Coefficients and Harmonic Terms

Fourier series describe periodic functions by writing them as sums of sine and cosine waves. The trigonometric Fourier series uses a function with period T and represents it with simple harmonic terms. This approach helps break a complex repeating signal into parts that are easier to study. In the standard form, the series includes a0, an, and bn. The term a0 gives the average value of the function over one period. The coefficients an and bn show how much each cosine and sine term contributes...

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properties of fourier series i

Fourier Series Properties in Signal Processing

Fourier series properties help describe periodic signals in signal processing and communications. A Fourier series breaks a signal into sine and cosine parts. These properties make it easier to study how signals behave and how they can be handled in real systems. Linearity is one of the most important properties. If two periodic signals are added together or combined in a linear way, the new signal has Fourier coefficients that are the same linear combination of the original coefficients. This...

properties of fourier series ii

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Fourier series change in useful ways when a signal is time-scaled or when it has symmetry. Time scaling changes the fundamental frequency, but the Fourier series coefficients stay the same. This idea helps show how a signal is represented over time in signal processing. Symmetry also makes Fourier series easier to work with. A function is even if f(t) = f(-t). For even functions, the sine terms disappear because sine is an odd function, and the integral of an odd function over a symmetric...

parseval's theorem

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Parseval's theorem links signal power to Fourier series coefficients. For a periodic function, it says the average power over one period equals the sum of the squared magnitudes of its complex Fourier coefficients. This makes the theorem useful for studying how energy is distributed in a signal. The theorem also works for the trigonometric form of the Fourier series. In that form, a function is written using sine and cosine terms. The Fourier coefficients can be related to the trigonometric...

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Fourier series can represent a periodic signal as an infinite sum of complex exponentials. For practical use, that infinite series is cut off after a finite number of terms. The result is a partial sum that gives a workable approximation of the original signal. Truncating the series creates a challenge near discontinuities. The approximation develops the Gibbs phenomenon, which is the set of persistent oscillations and overshoots around a jump in the signal. These high-frequency ripples do not...

discrete-time fourier series

Discrete-Time Fourier Series (DTFS) helps analyze periodic signals in discrete time by breaking them into frequency components. It is the discrete-time counterpart of the continuous-time Fourier series. Instead of using integrals, DTFS uses summations because the signal is discrete. For a discrete-time periodic signal x[n] with period N0, the DTFS coefficients X[k] are found from the DTFS formula. The index k runs from 0 to N0 − 1. These coefficients describe the signal in the frequency domain...

key terms and concepts

Learn how DTFS represents discrete-time periodic signals, finds frequency components, and helps analyze LTI system responses in digital signal processing.

DTFS for Periodic Signal Analysis | 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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HS-PS4-5

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HS-PS4-4

DTFS for Periodic Signal Analysis

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