Fast Fourier transform (FFT)
The Fast Fourier Transform (FFT) is a mathematical algorithm that efficiently analyzes and measures frequency ranges of signals, vibrations, and other waveforms. By converting a set of equally spaced data samples into a single sequence, the FFT significantly reduces the computational effort required to calculate the discrete Fourier transform (DFT) and its inverse. This process allows for an examination of how time and frequency relate within sample data, making it invaluable for analyzing large datasets. The algorithm has roots in the work of mathematician Joseph Fourier, who developed methods for breaking down complex functions into simpler sinusoidal components.
The FFT is widely used across various fields, including telecommunications, where it helps in managing signal integrity and data transmission efficiency. It also finds applications in image compression, such as JPEG files, and in disciplines like economics, engineering, and physics for pattern recognition in time series data. Additionally, the FFT is employed to analyze seismic activity, enabling better understanding and engineering solutions to withstand earthquakes. Overall, the FFT serves as a powerful tool for transforming and interpreting data represented as waveforms, with impacts across science and technology.
Fast Fourier transform (FFT)
A fast Fourier transform (FFT) is any mathematical function that is used to analyze and measure the frequency ranges of signals, vibrations, and other types of waveforms with great efficiency and speed. It can be used to summarize large amounts of data. The FFT offers a quick means of mathematically determining both the discrete Fourier transform (DFT) of a sequence and its inverse. The DFT converts a group of equally spaced samples of data into one single sequence that is the same length as the samples used to make the composite sequence. It can be used to examine the relationship between time and frequency in sample data. Without FFT algorithms, these calculations would have to be measured using convolution, which requires much greater effort and time. The FFT vastly reduces the number of computations necessary to determine the DFT.
Brief History
The Fourier transform can be traced back to French mathematician Joseph Fourier. Orphaned at the age of nine years old, he was raised in the Benedictine Order and considered joining the priesthood. However, his growing passion for mathematics and science won out, and he enrolled in college, where he demonstrated great potential in a broad category of sciences. In 1798, he served as Napoleon Bonaparte's science advisor on his invasion of Egypt. In 1801, he was recalled to France to serve as the administrator for a region near Grenoble under the orders of Napoleon. Although he served effectively, he filled the position only reluctantly. In Grenoble, Fourier dedicated himself to the study of the transfer of heat. His resulting findings were published in his 1807 paper, "On the Propagation of Heat in Solid Bodies."
In this work, Fourier established a method of representing all mathematic functions as a collection of simpler trigonometric series. His suggestion of representing functions so that they may be divided into their individual components is called the Fourier analysis, in his honor. The inverse of this—that is, the addition of a group of functions to determine their cumulative function—is called the Fourier synthesis. These ideas are in turn an extension of the Fourier series, which allows data to be summarized as a function of sine waves.
These ideas had been theorized by German mathematician Carl Friedrich Gauss in 1805 during his research into the orbits of asteroids. However, he was unable to implement his ideas. James W. Cooley and John Tukey developed the most commonly used FFT algorithm in 1965.
Overview
The FFT is used to measure the sequences of a waveform. A waveform is a collection of data points shown on a graph where the x-axis is an expression of time or frequency and the y-axis indicates another selected unit of measurement that is a function of amplitude. For instance, in measuring how stock prices have changed over time, the calendar (a measurement of time) is represented on the x-axis, and the value of the stocks (a unit of amplitude) is indicated on the y-axis. The waveform allows data analysts to study information by using a visual aid.
The FFT is itself a function of the Fourier transform. The Fourier transform is generally used to study waveforms that are periodic, meaning that the waves repeat both forward and backward for infinity. However, some analysts have sought to use FFT to study aperiodic waveforms like the stock market, which does not repeat infinitely in this manner.
The Fourier transform is used to examine the components of a waveform and deconstruct them so that the waveform becomes the sum of a series of simple individual sinusoids (or sine waves). A sinusoid is a waveform whose curves measure wavelengths. A sinusoid has a repeating pattern of waves, although these waves have peaks and valleys (such as the weekly highs and lows in a stock market). These wavelengths can be measured by calculating the distance between its repeating curves. The sinusoid shows the variation between the extreme points. Earthquakes, tides, sound, light, radio waves, and other aspects of the natural and scientific worlds all have waves that may be indicated by graphs composed of sinusoids and can be subject to the Fourier transform. Sinusoids may have varying levels of amplitude, period, phase, or any combination of these elements.
An example of a natural Fourier transform may be found in the light spectrum, in which white light is split into its many constituent colors after passing through a prism. Another example may be found in the process of hearing. In this case, the inner ear acts as a Fourier transform, quickly processing various acoustic signals into distinct sounds that the brain can understand.
The FFT allows people studying large amounts of information a way of representing data expressed as a waveform in another way. Using the FFT, a waveform can be separated into its individual components—that is, the building blocks that compose the sinusoid. In the case of earthquakes, this may be the speed and amplitude of each measured seismic vibration. In the example of the stock market, these components may be individual stocks measured over time. The FFT may also be used in the reverse, that is, to create a waveform using many pieces of information. The Fourier transform and FFT are particularly useful in finding patterns within various time series (a series of data points recorded over a period). It can further be used to remove, add, or multiply single waves (or some other individual measurement of data) to filter the collective waveform to assess data based on different types of criteria. Data is recorded in this way in many fields, including economics, engineering, astronomy, chemistry, oceanography, physics, and other areas of applied science.
Applications
The Fourier transform is a key component of analyzing how telecommunication signals behave when passed through any type of filter. For instance, if too much data is transmitted through a signal or channel, it will become corrupted, resulting in unusable, indecipherable content on the other end. The FFT can be used to efficiently review the maximum allowable data through a particular channel. The FFT is therefore a valuable means of determining frequencies in satellite dishes and Wi-Fi antennas.
The FFT enables the file size of pictures to be reduced through JPEG image compression. It has been applied toward architectural codes so that buildings may withstand the most powerful seismic waves. The FFT was used to send radio waves and radar signals to map the surface of Venus. It also has applications in finance, in which it can be used to present a way to study real-time price movements, and in aerospace engineering, in which it is used to review the vibrations of a plane's wingtip.
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