Time–frequency analysisIn 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 whose domain is the real line, obtained from the original via some transform), time–frequency analysis studies a two-dimensional signal – a function whose domain is the two-dimensional real plane, obtained from the signal via a time–frequency transform.
WaveletA wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing. For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly one tenth of a second.
Time–frequency representationA time–frequency representation (TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency. Time–frequency analysis means analysis into the time–frequency domain provided by a TFR. This is achieved by using a formulation often called "Time–Frequency Distribution", abbreviated as TFD. TFRs are often complex-valued fields over time and frequency, where the modulus of the field represents either amplitude or "energy density" (the concentration of the root mean square over time and frequency), and the argument of the field represents phase.
Bilinear time–frequency distributionBilinear time–frequency distributions, or quadratic time–frequency distributions, arise in a sub-field of signal analysis and signal processing called time–frequency signal processing, and, in the statistical analysis of time series data. Such methods are used where one needs to deal with a situation where the frequency composition of a signal may be changing over time; this sub-field used to be called time–frequency signal analysis, and is now more often called time–frequency signal processing due to the progress in using these methods to a wide range of signal-processing problems.
Wavelet transformIn mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. A function is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space of square integrable functions.
Continuous wavelet transformIn mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. The continuous wavelet transform of a function at a scale (a>0) and translational value is expressed by the following integral where is a continuous function in both the time domain and the frequency domain called the mother wavelet and the overline represents operation of complex conjugate.
Fourier analysisIn mathematics, Fourier analysis (ˈfʊrieɪ,_-iər) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics.
Continuous waveletIn numerical analysis, continuous wavelets are functions used by the continuous wavelet transform. These functions are defined as analytical expressions, as functions either of time or of frequency. Most of the continuous wavelets are used for both wavelet decomposition and composition transforms. That is they are the continuous counterpart of orthogonal wavelets. The following continuous wavelets have been invented for various applications: Poisson wavelet Morlet wavelet Modified Morlet wavelet Mexican ha
Fourier transformIn physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.
Discrete wavelet transformIn numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information (location in time). Haar wavelet The first DWT was invented by Hungarian mathematician Alfréd Haar. For an input represented by a list of numbers, the Haar wavelet transform may be considered to pair up input values, storing the difference and passing the sum.
Fourier seriesA Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation.
Wigner distribution functionThe Wigner distribution function (WDF) is used in signal processing as a transform in time-frequency analysis. The WDF was first proposed in physics to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, and it is of importance in quantum mechanics in phase space (see, by way of comparison: Wigner quasi-probability distribution, also called the Wigner function or the Wigner–Ville distribution).
Impact eventAn impact event is a collision between astronomical objects causing measurable effects. Impact events have physical consequences and have been found to regularly occur in planetary systems, though the most frequent involve asteroids, comets or meteoroids and have minimal effect. When large objects impact terrestrial planets such as the Earth, there can be significant physical and biospheric consequences, though atmospheres mitigate many surface impacts through atmospheric entry.
Harmonic analysisHarmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on the real line, or by Fourier series for periodic functions. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis.
Gabor waveletGabor wavelets are wavelets invented by Dennis Gabor using complex functions constructed to serve as a basis for Fourier transforms in information theory applications. They are very similar to Morlet wavelets. They are also closely related to Gabor filters. The important property of the wavelet is that it minimizes the product of its standard deviations in the time and frequency domain. Put another way, the uncertainty in information carried by this wavelet is minimized.
Haar waveletIn mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal basis. The Haar sequence is now recognised as the first known wavelet basis and is extensively used as a teaching example. The Haar sequence was proposed in 1909 by Alfréd Haar.
Fractional Fourier transformIn mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power, where n need not be an integer — thus, it can transform a function to any intermediate domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition.
Frequency domainIn mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how the signal is distributed within different frequency bands over a range of frequencies. A frequency-domain representation consists of both the magnitude and the phase of a set of sinusoids (or other basis waveforms) at the frequency components of the signal.
Morlet waveletIn mathematics, the Morlet wavelet (or Gabor wavelet) is a wavelet composed of a complex exponential (carrier) multiplied by a Gaussian window (envelope). This wavelet is closely related to human perception, both hearing and vision. Wavelet#History In 1946, physicist Dennis Gabor, applying ideas from quantum physics, introduced the use of Gaussian-windowed sinusoids for time-frequency decomposition, which he referred to as atoms, and which provide the best trade-off between spatial and frequency resolution.
Gabor atomIn applied mathematics, Gabor atoms, or Gabor functions, are functions used in the analysis proposed by Dennis Gabor in 1946 in which a family of functions is built from translations and modulations of a generating function. In 1946, Dennis Gabor suggested the idea of using a granular system to produce sound. In his work, Gabor discussed the problems with Fourier analysis. Although he found the mathematics to be correct, it did not reflect the behaviour of sound in the world, because sounds, such as the sound of a siren, have variable frequencies over time.
