Name: fast fourier transform for signal analysis
Uploaded: 2024-09-26T13:17:04.000Z
Duration: 1 min 10 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

Fast Fourier Transform

התמרת פורייה מהירה (באנגלית: Fast Fourier Transform, FFT) הוא אלגוריתם חישובי לחישוב התמרת פורייה בדידה על ידי פירוק החישובים למקטעים קטנים יותר הניתנים לניהול.
חישוב DFT של N-point דורש N כפלים מרוכבים בריבוע, בעוד שאלגוריתם FFT דורש רק N על שתיים ובסיס שני כפל לוגריתם N, מה שמציע ביצועים מהירים משמעותית.
ככל ש-N גדל, ה-FFT הופך מהיר ויעיל יותר על ידי הפחתת מספר הפעולות מהסולם הריבועי ללוגריתמי.
הוא משתמש בתכונות סימטריה ומחזוריות וממזער חישובים וכפלים מיותרים.
התמרת פורייה ההפוכה והמהירה, IFFT, משחזרת את האות המקורי מייצוג תחום התדר שלו ביעילות חישובית משופרת.
הוא נפוץ בעיבוד אותות ותמונות, וממלא גם תפקיד חיוני בתקשורת אלחוטית, במחקר מדעי ובניתוח נתונים.

התמרת פורייה מהירה (fft)

<div style="direction: rtl;">
 התמרת פורייה מהירה (FFT) היא אלגוריתם חישובי שנועד לחשב את התמרת פורייה הבדידה (DFT) בצורה יעילה. על ידי פירוק החישובים לחלקים קטנים יותר וקלים יותר לניהול, התמרת FFT מפחיתה באופן משמעותי את המורכבות החישובית. חישוב ישיר של DFT עם N נקודות דורש N_2 הכפלות מרוכבות, בעוד שאלגוריתם ה-FFT דורש רק N/2)​log_2​N) כפליים, ומעניק ביצועים מהירים בהרבה.
 
 היעילות החישובית של התמרת FFT בולטת במיוחד כאשר N גדל. התמרת FFT מפחיתה את מספר הפעולות מהיקף ריבועי להיקף לוגריתמי, ובכך משפרת את המהירות והיעילות. האלגוריתם מנצל את תכונות הסימטריה והמחזוריות הגלומות בהתמרת פורייה כדי למזער חישובים מיותרים, מה שמפחית באופן משמעותי את מספר ההכפלות הנדרשות.
 
 התמרת פורייה ההפוכה המהירה (IFFT) חשובה באותה מידה, ומשחזרת את האות המקורי מייצוגו בתחום התדר. התמרת IFFT שומרת על היעילות החישובית של FFT, ומבטיחה שההמרה חזרה למישור הזמן תתבצע במהירות ובדיוק. תכונה זו חשובה במגוון יישומים, כולל עיבוד אותות וניתוח נתונים.
 
 התמרת FFT משמשת רבות בעיבוד אותות לניתוח אותות אודיו, ומספקת תובנות לגבי רכיבי התדר של הצליל. בעיבוד תמונה, היא מסייעת במשימות כמו סינון ושיפור תמונה. בנוסף, ל-FFT תפקיד חשוב בתקשורת אלחוטית, שם היא מסייעת באפנון ובפענוח אותות. במחקר מדעי, התמרת FFT משמשת לעיבוד נתונים ניסיוניים, ובניתוח נתונים היא מסייעת בזיהוי דפוסים ומגמות בתוך מאגרי נתונים גדולים.
 
 לסיכום, התמרת FFT היא כלי שאין לו תחליף במגוון תחומים, ומספקת אמצעי רב עוצמה לניתוח ולעיבוד אותות בצורה יעילה. יכולתה להמיר נתונים בין מישורי הזמן והתדר, יחד עם יעילותה החישובית, הופכת אותה לאבן יסוד בעיבוד וניתוח אותות מודרני.
 
</div>

התמרת פורייה מהירה (FFT)

התמרת פורייה המהירה, FFT, היא אלגוריתם חישובי לחישוב התמרת פורייה בדידה על ידי פירוק החישובים לקטנים יותר, ...

Education

JoVE Core

Electrical Engineering

הנדסה חשמלית

Unit 1

יחידה 1

The Fourier Transform

התמרה פורייה

KEY TERMS AND CONCEPTS

מונחים ומושגים מרכזיים

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 of operations from a quadratic scale to a logarithmic one.

The FFT uses symmetry and periodicity in the Fourier Transform to avoid repeated calculations. The Inverse Fast Fourier Transform, or IFFT, does the reverse job. It rebuilds the original signal from its frequency-domain form and does so with the same kind of computational efficiency.

Because of this speed, the FFT is useful in many kinds of signal processing and data analysis. It helps study audio signals by showing their frequency components. It also supports image processing tasks such as filtering and image enhancement, and it plays a role in wireless communication for modulation and demodulation. In scientific research and data analysis, the FFT helps process experimental data and find patterns and trends in large datasets.

Fast Fourier Transform for Signal Analysis

fast-fourier-transform-for-signal-analysis

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

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...

fast fourier transform

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 Fast Fourier Transform speeds up DFT calculations, supports the IFFT, and helps analyze audio, images, wireless signals, and data.

Fast Fourier Transform for 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

Travelling Waves

A wave is a disturbance that propagates from its source, repeating itself periodically, and is typically associated with simple harmonic motion. ...

Wave Parameters

The simplest mechanical waves are associated with simple harmonic motion and repeat themselves for several cycles. These simple harmonic waves can be ...

Equations of Wave Motion

Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a ...

Graphing the Wave Function

Consider the wave equation for a sinusoidal wave moving in the positive x-direction. The wave equation is a function of both position and time. From the ...

/files/media/science-education/science-education-thumbs/14721.jpg

Velocity and Acceleration of a Wave

A wave propagates through a medium with a constant speed, known as a wave velocity. It is different from the speed of the particles of the medium, which ...

Speed of a Transverse Wave

The speed of a wave depends on the characteristics of the medium. For example, in the case of a guitar, the strings vibrate to produce the sound. The ...

/files/media/science-education/science-education-thumbs/14722.jpg

Kinetic and Potential Energy of a Wave

All forms of waves carry energy; this is directly visualized in nature. For instance, the waves of earthquakes are so intense that they can shake huge ...

Energy and Power of a Wave

The total energy associated with a wavelength is the sum of the potential energy and the kinetic energy. The average rate of energy transfer associated ...

Propagation of Waves

When a wave propagates from one medium to another, part of it may get reflected in the first medium, and part of it may get transmitted to the second ...

Sound Waves

Sound waves can be thought of as fluctuations in the pressure of a medium through which they propagate. Since the pressure also makes the medium&#39;s ...

Sound as Pressure Waves

Sound waves, which are longitudinal waves, can be modeled as the displacement amplitude varying as a function of the spatial and temporal coordinates. As ...

/files/media/science-education/science-education-thumbs/14724.jpg

Speed of Sound in Solids and Liquids

Most solids and liquids are incompressible&#8212;their densities remain constant throughout. In the presence of an external force, the molecules tend to ...

Speed of Sound in Gases

The speed of sound in a gaseous medium depends on various factors. Since gases constitute molecules that are free to move, they are highly compressible. ...

Doppler Effect - I

The Doppler effect and Doppler shift were named after the Austrian physicist and mathematician Christian Johann Doppler in 1842, who conducted experiments ...

Doppler Effect - II

The Doppler effect has several practical, real-world applications. For instance, meteorologists use Doppler radars to interpret weather events based on ...

Shock Waves

While deriving the Doppler formula for the observed frequency of a sound wave, it is assumed that the speed of sound in the medium is greater than the ...

Echo

The human ear cannot distinguish between two sources of sound if they happen to reach within a specific time interval, typically 0.1 seconds apart. More ...

/files/media/science-education/science-education-thumbs/14729.jpg

Electromagnetic Waves

James Clerk Maxwell formulated a single theory combining all the electric and magnetic effects scientists knew during that time, calling the phenomena his ...

Generating Electromagnetic Radiations

The German physicist Heinrich Hertz (1857&#8211;1894) was the first to generate and detect certain types of electromagnetic waves in the laboratory. ...

Electromagnetic Wave Equation

Maxwell&#39;s equations for electromagnetic fields are related to source charges, either static or moving. These fields act on a test charge, whose ...

propagation-speed-of-electromagnetic-waves

Propagation Speed of Electromagnetic Waves

Electromagnetic waves are consistent with Ampere&#39;s law. Assuming there is no conduction current Ampere&#39;s law is given as:


Consider a plane ...

Electromagnetic Waves in Matter

Electromagnetic waves can travel in the vacuum as well as in matter. For example light, which is an electromagnetic wave, can travel through air, water, ...

Energy Carried By Electromagnetic Waves

Anyone who has used a microwave oven knows there is energy in electromagnetic waves. Sometimes, this energy is obvious, such as in the summer sun&#39;s ...

The Wave Nature of Light

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

Electromagnetic waves are categorized according to their wavelengths and frequencies, giving the electromagnetic spectrum. These waves are classified as ...

Classification of Signals

Signals are categorized as Continuous-time or discrete-time, Periodic or aperiodic, Analog or digital, and Causal or noncausal.
A continuous-time signal ...

Discrete Fourier Transform

Consider a vibration sensor that continuously captures data in the form of a continuous time-dependent signal.
However, in reality, the sensor can only ...

Discrete-Time Fourier Series

The Discrete-Time Fourier Series is a counterpart to the Fourier-series expansion of continuous-time periodic signals.
Calculating the expansion ...

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.
Multiplying ...

Properties of Fourier Transform I

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

Signal and System

Signals are sets of data or information.
One-dimensional signals include single-variable functions, such as the variations in air pressure over time in ...

Methods of Documentation VII: EMR

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

The Fast Fourier Transform, FFT, is a computational algorithm for calculating the Discrete Fourier Transform by breaking the calculations into smaller, ...

reconstruction-of-signal-using-interpolation

Reconstruction of Signal using Interpolation

Consider a continuous signal and its sampled counterpart with a sampling period T.
Applying a zero-order hold to the sampled signal results in a piecewise ...

Downsampling

Consider a sampled sequence with zero values between sampling instants. Replace it by taking every N-th value of the sampled sequence.
The original and ...

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.
The ...

Upsampling

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

The Discrete Fourier Transform (DFT) analyzes the frequency content of discrete-time signals.
It maps the N-sampled discrete time-domain sequence to its ...

Feedback control systems

Feedback control systems are categorized as linear or nonlinear, time-varying or time-invariant, and classified by signal types as continuous or ...

Properties of the z-Transform I

Certain properties provide a solid foundation for analyzing discrete-time systems using the Z-transform.
Considering two discrete-time signals, the ...

Magnetic Resonance Imaging

Magnetic resonance imaging (MRI) is a noninvasive medical imaging technique based on a phenomenon of nuclear physics discovered in the 1930s, in which ...

Evaluate questions about the advantages of using digital transmission and storage of information. [Clarification Statement: Examples of advantages could include that digital information is stable because it can be stored reliably in computer memory, transferred easily, and copied and shared rapidly. Disadvantages could include issues of easy deletion, security, and theft.]

Interference and Superposition of Waves

When two waves of the same nature occur in the same region simultaneously, they result in interference. Interference of waves implies that the net effect ...

Sound Waves: Interference

Sound waves can be modeled either as longitudinal waves, wherein the molecules of the medium oscillate around an equilibrium position, or as pressure ...

Interference: Path Lengths

Consider two sources of sound, that may or may not be in phase, emitting waves at a single frequency, and consider the frequencies to be the same.
Two ...

/files/media/science-education/science-education-thumbs/14727.jpg

concept-of-resonance-and-its-characteristics

Concept of Resonance and its Characteristics

If a driven oscillator needs to resonate at a specific frequency, then very light damping is required. An example of light damping includes playing piano ...

/files/media/science-education/science-education-thumbs/12772.jpg

Sound Waves: Resonance

Resonance is produced depending on the boundary conditions imposed on a wave. Resonance can be produced in a string under tension with symmetrical ...

Reflection of Waves

When a wave travels from one medium to another, it gets reflected at the boundary of the second medium. A common example of this is when a person yells at ...

Standing Waves

Sometimes waves do not seem to move; rather, they just vibrate in place. Unmoving waves can be seen on the surface of a glass of milk kept in a ...

/files/media/science-education/science-education-thumbs/12783.jpg

Modes of Standing Waves - I

A close look at earthquakes provides evidence for the conditions appropriate for resonance, standing waves, and constructive and destructive interference. ...

/files/media/science-education/science-education-thumbs/12784.jpg

Modes of Standing Waves: II

The starting point for expressing the modes of standing waves is understanding the boundary conditions that the waves must follow. The boundary conditions ...

/files/media/science-education/science-education-thumbs/14723.jpg

Beats

The study of music provides many examples of the superposition of waves and the constructive and destructive interference that occurs. Very few examples ...

/files/media/science-education/science-education-thumbs/14728.jpg

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
...

Electromagnetic Fields

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

Photoelectric Effect

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

Plane Electromagnetic Waves I

The existence of combined electric and magnetic fields that propagate through space as electromagnetic (EM) waves is the most significant prediction of ...

Plane Electromagnetic Waves II

Consider a plane wavefront traveling in position x-direction with a constant speed. This wavefront can be utilized to obtain the relationship between ...

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

The electromagnetic spectrum consists of all the types of electromagnetic radiation arranged according to their frequency and wavelength. Each of the ...

Biological Effects of Radiation

All radioactive nuclides emit high-energy particles or electromagnetic waves. When this radiation encounters living cells, it can cause heating, break ...

Intensity Of Electromagnetic Waves

The energy transport per unit area per unit time, or the Poynting vector, gives the energy flux of an electromagnetic wave at any specific time. For a ...

When light of a particular wavelength strikes a metal surface, electrons are emitted. This is called the photoelectric effect. The minimum frequency of ...

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

Fast Fourier Transform for Signal Analysis

Description

Transcript

More videos by Standards