Name: dft for finding signal frequencies
Uploaded: 2024-09-26T13:17:04.000Z
Duration: 1 min 15 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 Fourier Transform

Si consideri un sensore di vibrazione che acquisisce continuamente i dati sotto forma di un segnale continuo dipendente dal tempo.
Tuttavia, in realtà, il sensore può registrare solo un numero finito di vibrazioni come punti dati discreti a intervalli di tempo specifici.
Qui, la trasformata discreta di Fourier, DFT, può essere utilizzata per analizzare le componenti di frequenza delle vibrazioni.
La DFT scompone qualsiasi segnale in una somma di semplici onde sinusoidali e coseno, per le quali è possibile misurare la frequenza, l'ampiezza e la fase.
Nello spettro di ampiezza DFT, il segnale ottenuto dal sensore può essere rappresentato come il grafico a barre corrispondente alle quattro onde sinusoidali.
L'altezza della barra, dopo la normalizzazione, è l'ampiezza del corrispondente segnale sinusoidale nel dominio del tempo.
Matematicamente, la DFT è rappresentata come una somma finita del prodotto del segnale nel dominio del tempo con un esponente complesso dipendente dalla frequenza k.
Se X in funzione di k è grande per un certo k, indica che il segnale di vibrazione ha forti componenti di frequenza a quelle frequenze.

trasformata discreta di fourier

La Trasformata discreta di Fourier (DFT) è uno strumento fondamentale nell'elaborazione del segnale, che estende la trasformata di Fourier a tempo discreto valutando i segnali discreti a intervalli di frequenza uniformemente distanziati. Questa trasformazione converte una sequenza finita di campioni del dominio del tempo in componenti di frequenza, ciascuna rappresentante sinusoidi complesse ordinate per frequenza. La DFT traduce queste sequenze nel dominio della frequenza, indicando in modo efficace l'ampiezza e la fase di ciascuna componente di frequenza presente nel segnale.

Una delle proprietà chiave della DFT è la sua linearità. Questa proprietà implica che la DFT di una somma di sequenze è uguale alla somma delle loro singole DFT. Un'altra proprietà importante è lo spostamento temporale. Quando una sequenza viene spostata nel dominio del tempo, la sua DFT subisce uno spostamento di fase corrispondente.

Lo spostamento di frequenza nel dominio del tempo determina lo spostamento degli indici della DFT. Se una sequenza viene moltiplicata per un esponenziale complesso, la sua DFT viene spostata di conseguenza nel dominio della frequenza. L'inversione temporale, che inverte la sequenza nel dominio del tempo, influisce sulla simmetria della DFT. Se una sequenza viene invertita, i componenti della DFT vengono riordinati e coniugati.

La proprietà di coniugazione afferma che se una sequenza viene coniugata, anche i componenti della DFT vengono coniugati e riordinati. Il teorema di convoluzione è particolarmente potente, perché semplifica il processo di convoluzione nel dominio del tempo in una semplice moltiplicazione nel dominio della frequenza.

Grazie alla sua natura periodica, la DFT è ampiamente usata nelle applicazioni di elaborazione del segnale per la transizione tra i domini del tempo e della frequenza. Questa periodicità deriva dal processo di campionamento intrinseco nella DFT, rendendola uno strumento versatile per l'analisi e la manipolazione dei segnali. La capacità di semplificare operazioni complesse e fornire chiare informazioni sui componenti di frequenza di un segnale sottolinea l'importanza della DFT in varie attività di elaborazione del segnale.

Trasformata discreta di Fourier

Si consideri un sensore di vibrazione che acquisisce continuamente i dati sotto forma di un segnale continuo dipendente dal tempo.
Tuttavia, in realtà, il sensore può solo ...

Education

Didattica

JoVE Core

Electrical Engineering

Ingegneria elettrica

Unit 1

Unità 1

The Fourier Transform

La trasformata di Fourier

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

dft for finding signal frequencies

The Discrete Fourier Transform (DFT) turns a finite set of time-domain samples into frequency components. It is a key tool in signal processing because it evaluates a discrete signal at evenly spaced frequency intervals. The result shows the magnitude and phase of each complex sinusoid in the signal.

The DFT also follows several important properties. It is linear, so the DFT of a sum of sequences equals the sum of their individual DFTs. When a sequence shifts in time, its DFT changes by a phase shift. If the sequence is shifted in frequency in the time domain, the DFT indices shift as well.

Other DFT rules describe how different changes affect the frequency domain. Multiplying a sequence by a complex exponential shifts its DFT in frequency. Time reversal flips a sequence in time and changes the symmetry of the DFT, while conjugation produces conjugated and reordered DFT components. The convolution theorem is especially useful because it turns convolution in the time domain into multiplication in the frequency domain.

Because the DFT is periodic, it is widely used to move between time and frequency views of a signal. That periodicity comes from the sampling process built into the transform. These features make the DFT useful for analyzing and manipulating signals in many signal-processing tasks.

DFT for Finding Signal Frequencies

continuous -time fourier transform

From Fourier Series to Fourier Transform

The Fourier transform extends Fourier series ideas to nonperiodic functions. A Fourier series breaks a periodic signal into a sum of sines and cosines. When the period becomes infinite, that discrete sum changes into a continuous integral. A pulse-train waveform shows this change clearly. With a finite period, the repeating rectangular pulses can be represented by a Fourier series. As the period grows without bound, the waveform becomes a single isolated pulse, and the description shifts to...

basic signals of fourier transform

Fourier Transform Signals in Time and Frequency

The Fourier Transform links time-domain signals to frequency-domain signals. It is a key tool in signal processing. It helps show how a waveform can be described by frequency parts instead of only by time changes. The sinc function is one of the basic signals in this topic. It is written as sinc(x) = sin(πx)/(πx). It equals 1 when x is 0, and it has even symmetry about the y-axis. In the frequency domain, it appears as the Fourier transform of a rectangular pulse. A rectangular pulse has a...

properties of fourier transform i

Fourier Transform in Radio Broadcasting

Fourier Transform properties help radio broadcasting manage signals in the time and frequency domains. These properties are useful for sending multiple audio channels at once, changing clip speed, adding broadcast delays, adjusting audio frequencies, and demodulating signals. They make it easier to transmit and receive audio clearly. When broadcasters combine several audio signals, the Fourier Transform simplifies the process. If f(t) and g(t) have Fourier Transforms F(ω) and G(ω), then the...

properties of fourier transform ii

Fourier Transform Rules in Signal Analysis

The Fourier Transform helps analyze signals by linking the time domain and the frequency domain. Its properties show how a signal changes when it is shifted, differentiated, or combined with another signal. These rules are important in signal processing, radio broadcasting, and audio work. The frequency shifting property shows that moving a signal in frequency causes a phase change in time. If x(t) has a Fourier Transform X(f), then multiplying x(t) by e^j2πf0t gives a transform of X(f−f0).

parseval's theorem for fourier transform

Signal Energy in Time and Frequency

Parseval's theorem links signal energy in the time domain and the frequency domain. In signal processing, it shows that the same energy can be calculated in either view. This makes it an important idea for comparing time-based and frequency-based analysis. To find signal energy, the standard approach uses a resistor value of 1 ohm. In that case, power equals the square of the signal's voltage or current. This gives a simple way to relate power to energy before moving into Fourier analysis. <br...

discrete-time fourier transform

DTFT and Frequency Analysis of Digital Signals

The Discrete-Time Fourier Transform (DTFT) shows how a discrete-time signal can be analyzed in the frequency domain. It takes a signal from the time domain and reveals its frequency components and spectral features. In the DTFT, the integral used for continuous-time signals is replaced by a summation to match the discrete form of the signal. The DTFT spectrum X(Ω) is periodic with a period of 2π. This periodicity means that X(Ω) can be written as a Fourier series, which helps with analysis and...

properties of dtft i

DTFT Signal Properties and Frequency Effects

Discrete-Time Fourier Transforms (DTFTs) help describe discrete-time signals in the frequency domain. The main DTFT properties include linearity, time shifting, frequency shifting, time reversal, conjugation, and time scaling. These rules make it easier to analyze and change signals in signal processing. Linearity is the most basic property. If two discrete-time signals are multiplied by constants a and b and then added together, the DTFT of the result is the weighted sum of the two individual...

properties of dtft ii

DTFT Rules for Differentiation and Energy

The DTFT has useful rules for frequency differentiation, convolution, accumulation, and energy checks. These rules help students see how a discrete-time signal changes in the frequency domain. For frequency differentiation, start with a DTFT pair and differentiate both sides with respect to ω. After multiplying by j, the result gives the DTFT of n x[n]. This property is useful for studying how the slope of a spectrum relates to the signal in time. Convolution brings out another key DTFT rule.

discrete fourier transform

The Discrete Fourier Transform (DFT) turns a finite set of time-domain samples into frequency components. It is a key tool in signal processing because it evaluates a discrete signal at evenly spaced frequency intervals. The result shows the magnitude and phase of each complex sinusoid in the signal. The DFT also follows several important properties. It is linear, so the DFT of a sum of sequences equals the sum of their individual DFTs. When a sequence shifts in time, its DFT changes by a...

fast fourier transform

Fast Fourier Transform for Signal Analysis

The Fast Fourier Transform (FFT) is a computational algorithm that makes the Discrete Fourier Transform (DFT) faster to calculate. It works by splitting the calculation into smaller parts. This reduces the amount of work needed to move data between the time domain and the frequency domain. A direct N-point DFT needs N² complex multiplications. The FFT lowers that cost to about (N/2)log₂N multiplications. That difference becomes more important as N gets larger, because the FFT cuts the number...

key terms and concepts

Learn how the Discrete Fourier Transform converts sampled signals into frequency components and how shifting, reversal, and convolution affect the result.

DFT for Finding Signal Frequencies | 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

Travelling Waves

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Wave Parameters

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Equations of Wave Motion

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Graphing the Wave Function

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Kinetic and Potential Energy of a Wave

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Energy and Power of a Wave

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Doppler Effect - I

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Energy Carried By Electromagnetic Waves

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The nature of light has been a subject of inquiry since antiquity. In the seventeenth century, Isaac Newton performed experiments with lenses and prisms ...

Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling in various media. [Clarification Statement: Examples of data could include electromagnetic radiation traveling in a vacuum and glass, sound waves traveling through air and water, and seismic waves traveling through the earth.] [Assessment Boundary: Assessment is limited to algebraic relationships and describing those relationships qualitatively.]

HS-PS4-1

The Electromagnetic Spectrum

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A continuous-time signal ...

Consider a vibration sensor that continuously captures data in the form of a continuous time-dependent signal.
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Sampling Theorem

Consider the continuous-time signal x(t), and a train of impulses where Ts,&#xA0; is the sampling interval, and fs, is the sampling frequency.
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Properties of Fourier Transform I

In radio broadcasting, Fourier Transform properties are applied in simultaneous multi-channel transmission, adjusting audio clip speeds, live broadcast ...

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Methods of Documentation VII: EMR

Electronic Medical Records (EMRs) primarily center around electronically documenting patients&#39; health information within a single healthcare ...

Fast Fourier Transform

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reconstruction-of-signal-using-interpolation

Reconstruction of Signal using Interpolation

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Design Example

An engineer designs a circuit for a speaker system with a 2 kHz crossover frequency requirement, considering the speaker&#39;s resistance of 8 ohms.
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Consider a decimated signal with a reduced frequency range due to its lower sampling rate.
Insert zeros between each sample to upsample, that introduces ...

Properties of DTFT I

Consider two discrete-time signals, each with their respective Discrete-Time Fourier Transforms DTFTs.
The signals are first multiplied by constants a and ...

Relation of DFT to z-Transform

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It maps the N-sampled discrete time-domain sequence to its ...

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Certain properties provide a solid foundation for analyzing discrete-time systems using the Z-transform.
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Two ...

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Concept of Resonance and its Characteristics

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Sound Waves: Resonance

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Modes of Standing Waves - I

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Beats

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Interference and Diffraction

Source: Yong P. Chen, PhD, Department of Physics &#38; Astronomy, College of Science, Purdue University, West Lafayette, IN
Interference and diffraction ...

Reflection and Refraction

Source: Derek Wilson,&#160;Asantha Cooray, PhD, Department of Physics &#38; Astronomy, School of Physical Sciences, University of California, Irvine, CA
...

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Electric fields generated by static charges, often referred to as electrostatic fields, are characteristically different from electric fields created by ...

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Evaluate the claims, evidence, and reasoning behind the idea that electromagnetic radiation can be described either by a wave model or a particle model, and that for some situations one model is more useful than the other. [Clarification Statement: Emphasis is on how the experimental evidence supports the claim and how a theory is generally modified in light of new evidence. Examples of a phenomenon could include resonance, interference, diffraction, and photoelectric effect.] [Assessment Boundary: Assessment does not include using quantum theory.]

HS-PS4-3

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Evaluate the validity and reliability of claims in published materials of the effects that different frequencies of electromagnetic radiation have when absorbed by matter. [Clarification Statement: Emphasis is on the idea that photons associated with different frequencies of light have different energies, and the damage to living tissue from electromagnetic radiation depends on the energy of the radiation. Examples of published materials could include trade books, magazines, web resources, videos, and other passages that may reflect bias.] [Assessment Boundary: Assessment is limited to qualitative descriptions.]

HS-PS4-4

DFT for Finding Signal Frequencies

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