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The FFT is used in digital recording, sampling, additive synthesis and pitch correction software. [45] The FFT's importance derives from the fact that it has made working in the frequency domain equally computationally feasible as working in the temporal or spatial domain. Some of the important applications of the FFT include: [15] [46]
The fast Fourier transform (FFT) is an algorithm for computing the DFT. Definition ... the Fourier transform is rotation by 90° in the time–frequency domain, ...
The spectrum of a chirp pulse describes its characteristics in terms of its frequency components. This frequency-domain representation is an alternative to the more familiar time-domain waveform, and the two versions are mathematically related by the Fourier transform.
By a derivation similar to Eq.1, there is an analogous theorem for sequences, such as samples of two continuous functions, where now denotes the discrete-time Fourier transform (DTFT) operator. Consider two sequences u [ n ] {\displaystyle u[n]} and v [ n ] {\displaystyle v[n]} with transforms U {\displaystyle U} and V {\displaystyle V} :
In signal processing, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains simultaneously, using various time–frequency representations. Rather than viewing a 1-dimensional signal (a function, real or complex-valued, whose domain is the real line) and some transform (another function ...
The frequency axis has units of FFT "bins" when the window of length N is applied to data and a transform of length N is computed. For instance, the value at frequency 1 / 2 "bin" is the response that would be measured in bins k and k + 1 to a sinusoidal signal at frequency k + 1 / 2 . It is relative to the maximum possible ...
The inverse Fourier transform converts the frequency-domain function back to the time-domain function. A spectrum analyzer is a tool commonly used to visualize electronic signals in the frequency domain. A frequency-domain representation may describe either a static function or a particular time period of a dynamic function (signal or system).
While the ordinary DFT corresponds to a periodic signal in both time and frequency domains, = / produces a signal that is anti-periodic in frequency domain (+ =) and vice versa for = /. Thus, the specific case of a = b = 1 / 2 {\displaystyle a=b=1/2} is known as an odd-time odd-frequency discrete Fourier transform (or O 2 DFT).