What are Fourier transforms and where are they applied?
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◆The Fourier transform is important in mathematics,engineering, and the physical sciences. Its discrete counterpart, the Discrete Fourier Transform (DFT), which is normally computed using the so-called Fast Fourier Transform (FFT), has revolutionized modern society, as it is ubiquitous in digital electronics and signal processing. Radio astronomers are particularly avid users of Fourier transforms
Fourier transforms are key components in data processing (e.g., periodicity searches) andinstruments (e.g., antennas, receivers, spectrometers), and they are the corner-stores of interferometry and aperture synthesis.The Fourier transform is a reversible, linear transform with many important properties. For any function f(x) (which in astronomy is usually real valued, but f(x) may be complex), the Fourier transform can be denoted F(s), where the product of x and s is dimensionless. Often x is a measure of time t (i.e., the time-domain signal) and so s corresponds to inverse time, or frequency V (i.e., the frequency-domain signal).
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The Fourier Transform :- is a mathematical technique that transforms a function of time, x(t), to a function of frequency, X(ω). ... Likewise, we can derive the Inverse Fourier Transform (i.e., the synthesis equation) by starting with the synthesis equation for the Fourier Series (and multiply and divide by T).
The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, nonperiodic function by a continuous superposition or integral of complex exponentials.
In precise, the Fourier transform (FT) decomposes a function (often a function of the time, or a signal) into its constituent frequencies. ... The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time.
The Fourier Transform is an important image processing tool which is used to decompose an image into its sine and cosine components. The output of the transformation represents the image in the Fourier or frequency domain, while the input image is the spatial domain equivalent.
The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics