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In mathematics, the Fourier transform ... whose amplitude and phase are given by the analysis formula: = / / (). The actual Fourier series is the ...
The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms. [8] In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum.
Fourier transform (bottom) is zero except at discrete points. The inverse transform is a sum of sinusoids called Fourier series. Center-right: Original function is discretized (multiplied by a Dirac comb) (top). Its Fourier transform (bottom) is a periodic summation of the original transform.
The Fourier transform is also part of Fourier analysis, but is defined for functions on . Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic.
An example FFT algorithm structure, using a decomposition into half-size FFTs A discrete Fourier analysis of a sum of cosine waves at 10, 20, 30, 40, and 50 Hz. A fast Fourier transform (FFT) is an algorithm that computes the Discrete Fourier Transform (DFT) of a sequence, or its inverse (IDFT).
For example, using the Fourier transform, a sound wave, such as human speech, can be broken down into its component tones of different frequencies, each represented by a sine wave of a different amplitude and phase. The response of a system, as a function of frequency, can also be described by a complex function.
The lower right corner depicts samples of the DTFT that are computed by a discrete Fourier transform (DFT). The utility of the DTFT is rooted in the Poisson summation formula, which tells us that the periodic function represented by the Fourier series is a periodic summation of the continuous Fourier transform: [b]
The six arrows represent the first six terms of the Fourier series of a square wave. The two circles at the bottom represent the exact square wave (blue) and its Fourier-series approximation (purple). (Odd) harmonics of a 1000 Hz square wave Graph showing the first 3 terms of the Fourier series of a square wave