Fast Fourier transformA fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT), or its inverse (IDFT), of a sequence. A Fourier transform converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies.
Fourier transformMathematical transform that expresses a function of time as a function of frequencyThe Fourier transform applied to the waveform of a C majorpianochord (with logarithmic horizontal (frequency) axis). The first three peaks on the left correspond to the fundamental frequencies of the chord (C, E, G).
Window functionFunction used in signal processingA popular window function, the Hann window. Most popular window functions are similar bell-shaped curves.In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval. Typically, window functions are symmetric around the middle of the interval, approach a maximum in the middle, and taper away from the middle.
List of Fourier-related transformsThis is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinusoidal and are therefore strongly localized in the frequency spectrum. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single frequency component.
SpectrogramA spectrogram is a visual representation of the spectrum of frequencies of a signal as it varies with time. When applied to an audio signal, spectrograms are sometimes called sonographs, voiceprints, or voicegrams. When the data are represented in a 3D plot they may be called waterfall displays. Spectrograms are used extensively in the fields of music, linguistics, sonar, radar, speech processing, seismology, ornithology, and others.
Quantization (signal processing)Process of mapping a continuous set to a countable setThe simplest way to quantize a signal is to choose the digital amplitude value closest to the original analog amplitude. This example shows the original analog signal (green), the quantized signal (black dots), the signal reconstructed from the quantized signal (yellow) and the difference between the original signal and the reconstructed signal (red).
Nyquist frequencyIn signal processing, the Nyquist frequency (or folding frequency) is a characteristic of a sampler, which converts a continuous function or signal into a discrete sequence. It is named after Harry Nyquist. For a given sampling rate (samples per second), the Nyquist frequency (cycles per second) is the frequency whose cycle-length (or period) is twice the interval between samples, thus 0.5 cycles/sample. For example, audio CDs have a sampling rate of 44,100 samples/second.
Hann functionThe Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing or hanning. The function, with length L {\displaystyle L} and amplitude 1 / L , {\displaystyle 1/L,} is given by: w 0 ( x ) ≜ { 1 L ( 1 2 + 1 2 cos ( 2 π x L ) ) = 1 L cos 2 ( π x L ) , | x | ≤ L / 2 0 , | x | > L / 2 } .
Instantaneous phase and frequencyIn 1922 John Renshaw Carson defined the instantaneous frequency of a signal "as the time derivative of the signal's phase angle." In frequency modulation, instantaneous frequency describes the frequency varying above and below the carrier frequency, at the audio tone frequency. Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions.
Time signalA time signal is a visible, audible, mechanical, or electronic signal used as a reference to determine the time of day. Church bells or voices announcing hours of prayer gave way to automatically operated chimes on public clocks; however, audible signals (even signal guns) have limited range. Busy seaports used a visual signal, the dropping of a ball, to allow mariners to check the chronometers used for navigation.
Waterfall plotWaterfall plots are often used to show how two-dimensional phenomena change over time. A three-dimensional spectral waterfall plot is a plot in which multiple curves of data, typically spectra, are displayed simultaneously. Typically the curves are staggered both across the screen and vertically, with "nearer" curves masking the ones behind. The result is a series of "mountain" shapes that appear to be side by side.
Complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that studies complex-valued functions of one or more complex variables. It is used in many branches of mathematics, including functional analysis, algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory.
Classification of discontinuitiesWhile continuous functions are important in mathematics, not all functions are continuous. If a function is not continuous at a limit point (also called an "accumulation point" or "cluster point") of its domain, it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function.*As an Amazon Associate I earn from qualifying purchases.