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

La serie di Fourier a tempo discreto è una controparte dell'espansione della serie di Fourier dei segnali periodici a tempo continuo.
Il calcolo dei coefficienti di dilatazione del DTFS comporta sommatorie piuttosto che integrali. La risposta di un sistema lineare tempo-invariante a un segnale periodico a tempo discreto può essere determinata attraverso un processo passo-passo.
Innanzitutto, è necessario calcolare il DTFS del segnale di ingresso. La risposta di uscita a ciascun termine DTFS viene calcolata utilizzando la risposta in frequenza del sistema. Infine, questi risultati vengono sommati per ottenere il segnale di uscita totale.
Un segnale periodico in tempo continuo ha un periodo con frequenze circolari e angolari, rappresentate da una serie di Fourier complesso-esponenziale. Allo stesso modo, un segnale periodico a tempo discreto ha una frequenza angolare fondamentale.
L'espansione DTFS consiste di termini finiti, a differenza della serie di Fourier per il tempo continuo, che consiste di un numero infinito di termini.
Nell'elaborazione del segnale digitale, il DTFS aiuta a esaminare campioni periodici, individuare frequenze specifiche e filtrare efficacemente il rumore indesiderato.

serie di fourier a tempo discreto

La serie di Fourier a tempo discreto (DTFS) è un concetto fondamentale nell'elaborazione del segnale, che fa da controparte a tempo discreto della serie di Fourier a tempo continuo. Consente la rappresentazione e l'analisi dei segnali periodici a tempo discreto in termini di componenti di frequenza. A differenza della controparte continua, che usa gli integrali, il calcolo dei coefficienti di espansione DTFS comporta sommatorie dovute alla natura discreta del segnale.

Per un segnale periodico a tempo discreto x[n] con un periodo N_0, i coefficienti DTFS X[k] vengono calcolati utilizzando la formula:

<img alt="Equation1" src="/files/ftp_upload/16065/16065_PT%20EQUATION_1.svg" style="text-align: center; height: 48px;">

Dove k=0,1,2,…,N_0−1. Questi coefficienti X[k] rappresentano il segnale nel dominio della frequenza, catturando l'ampiezza e la fase di ciascuna componente di frequenza.

Per determinare la risposta di un sistema lineare tempo-invariante (LTI) a un segnale periodico a tempo discreto, si segue un approccio sistematico:

<ol style="padding-left: 35px; margin: 0;">
	<li>Calcolare il DTFS del segnale di ingresso: si calcolano i coefficienti DTFS X[k] per il segnale di ingresso X[n].</li>
	<li>Calcolare la risposta di uscita per ogni termine DTFS: prendere la risposta in frequenza del sistema H(e^(jΩ)) per determinare l'uscita per ogni componente di frequenza. I coefficienti DTFS di uscita Y[k] sono dati da: </li>
 <img alt="Equation2" src="/files/ftp_upload/16065/16065_PT%20EQUATION_2.svg" style="text-align: center; height: 36px;"> 
 dove Ω = 2πk/N
	<li>Sommare le risposte: infine, bisogna sommare i contributi di tutti i termini DTFS per ottenere il segnale di uscita totale nel dominio del tempo.</li>
</ol>

Nei segnali a tempo continuo, la periodicità è definita rispetto a un periodo T, corrispondente a frequenze circolari e angolari. Per un segnale a tempo discreto, la periodicità è associata ad una frequenza angolare fondamentale Ω_k = 2πk/N_0, dove N_0 è il periodo del segnale discreto. L'espansione DTFS è finita, costituita da termini N, in contrasto con la serie infinita nelle serie di Fourier a tempo continuo.

DTFS svolge un ruolo cruciale nell'elaborazione del segnale digitale (DSP), in particolare nell'analisi e nella manipolazione di segnali periodici derivati da dati campionati. È fondamentale nelle attività come l'identificazione di frequenze specifiche all'interno dei segnali audio, il potenziamento o la soppressione di determinati componenti di frequenza e il filtraggio del rumore indesiderato. Trasformando i segnali nel dominio della frequenza, la DTFS facilita l'analisi e l'elaborazione efficienti del segnale, consentendo prestazioni migliorate in varie applicazioni come le telecomunicazioni, l’ingegneria audio e i sistemi di controllo.

Serie di Fourier a tempo discreto

La serie di Fourier a tempo discreto è una controparte dell'espansione della serie di Fourier dei segnali periodici a tempo continuo.
Calcolo dell'espansione ...

Education

Didattica

JoVE Core

Electrical Engineering

Ingegneria elettrica

Unit 1

Unità 1

Fourier Series

Serie di Fourier

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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...

exponential fourier series

Fourier Series in Audio Signal Processing

Fourier series are used in audio signal processing to break complex sounds into simpler sinusoidal parts. This helps explain how musical notes and other periodic signals can be analyzed and rebuilt from frequency components. The exponential Fourier series writes a periodic signal as a sum of complex exponentials at positive and negative harmonic frequencies. These harmonic terms capture the repeated pattern in the signal. Euler's identity connects the exponential form to cosine and sine, which...

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

Fourier Series Symmetry and Time Scaling

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

Parseval's Theorem in Signal Power

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...

convergence of fourier series

Fourier Series Truncation and Gibbs Ripples

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.

electrical engineering

HS-PS4-5

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HS-PS4-4

DTFS for Periodic Signal Analysis

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