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Log-spectral Distance
The log-spectral distance (LSD), also referred to as log-spectral distortion or root mean square log-spectral distance, is a distance measure between two spectra. The log-spectral distance between spectra P\left(\omega\right) and \hat\left(\omega\right) is defined as p-norm: : D_=^, where P\left(\omega\right) and \hat\left(\omega\right) are power spectra. Unlike the Itakura–Saito distance, the log-spectral distance is symmetric. In speech coding, log spectral distortion for a given frame is defined as the root mean square difference between the original LPC log power spectrum and the quantized or interpolated LPC log power spectrum. Usually the average of spectral distortion over a large number of frames is calculated and that is used as the measure of performance of quantization or interpolation. Meaning When measuring the distortion between signals, the scale or temporality/spatiality of the signals can have different levels of significance to the distortion measures. To ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric (mathematics)
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Spectrum
In physics, a continuous spectrum usually means a set of attainable values for some physical quantity (such as energy or wavelength) that is best described as an interval of real numbers, as opposed to a discrete spectrum, a set of attainable values that is discrete in the mathematical sense, where there is a positive gap between each value and the next one. The classical example of a continuous spectrum, from which the name is derived, is the part of the spectrum of the light emitted by excited atoms of hydrogen that is due to free electrons becoming bound to a hydrogen ion and emitting photons, which are smoothly spread over a wide range of wavelengths, in contrast to the discrete lines due to electrons falling from some bound quantum state to a state of lower energy. As in that classical example, the term is most often used when the range of values of a physical quantity may have both a continuous and a discrete part, whether at the same time or in different situations. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm (mathematics)
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "\,\leq\," in the homogeneity axiom. It can also re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Spectrum
The power spectrum S_(f) of a time series x(t) describes the distribution of Power (physics), power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including Noise (electronics), noise) as analyzed in terms of its frequency content, is called its spectrum. When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (or simply power spectrum), which applies to signals existing over ''all'' time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The power spectral density (PSD) then refers to the spec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Itakura–Saito Distance
The Itakura–Saito distance (or Itakura–Saito divergence) is a measure of the difference between an original spectrum P(\omega) and an approximation \hat(\omega) of that spectrum. Although it is not a perceptual measure, it is intended to reflect perceptual (dis) similarity. It was proposed by Fumitada Itakura and Shuzo Saito in the 1960s while they were with NTT. The distance is defined as: : D_(P(\omega),\hat(\omega))=\frac\int_^ \left \frac-\log \frac - 1 \right\, d\omega The Itakura–Saito distance is a Bregman divergence generated by minus the logarithmic function, but is not a true metric since it is not symmetric and it does not fulfil triangle inequality. In Non-negative matrix factorization, the Itakura-Saito divergence can be used as a measure of the quality of the factorization: this implies a meaningful statistical model of the components and can be solved through an iterative method. The Itakura-Saito distance is the Bregman divergence associated with the Gamma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Quantization
Vector quantization (VQ) is a classical quantization technique from signal processing that allows the modeling of probability density functions by the distribution of prototype vectors. It was originally used for data compression. It works by dividing a large set of points (vectors) into groups having approximately the same number of points closest to them. Each group is represented by its centroid point, as in k-means and some other clustering algorithms. The density matching property of vector quantization is powerful, especially for identifying the density of large and high-dimensional data. Since data points are represented by the index of their closest centroid, commonly occurring data have low error, and rare data high error. This is why VQ is suitable for lossy data compression. It can also be used for lossy data correction and density estimation. Vector quantization is based on the competitive learning paradigm, so it is closely related to the self-organizing map model ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error and give better performance in ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. 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 space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cepstrum
In Fourier analysis, the cepstrum (; plural ''cepstra'', adjective ''cepstral'') is the result of computing the inverse Fourier transform (IFT) of the logarithm of the estimated signal spectrum. The method is a tool for investigating periodic structures in frequency spectra. The ''power cepstrum'' has applications in the analysis of human speech. The term ''cepstrum'' was derived by reversing the first four letters of ''spectrum''. Operations on cepstra are labelled ''quefrency analysis'' (or ''quefrency alanysisB. P. Bogert, M. J. R. Healy, and J. W. Tukey, ''The Quefrency of Time Series for Echoes: Cepstrum, Pseudo Autocovariance, Cross-Cepstrum and Saphe Cracking'', ''Proceedings of the Symposium on Time Series Analysis'' (M. Rosenblatt, Ed) Chapter 15, 209-243. New York: Wiley, 1963.''), ''liftering'', or ''cepstral analysis''. It may be pronounced in the two ways given, the second having the advantage of avoiding confusion with ''kepstrum''. Origin The concept of the cep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parseval's Theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh. Although the term "Parseval's theorem" is often used to describe the unitarity of ''any'' Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem. Statement of Parseval's theorem Suppose that A(x) and B(x) are two complex-valued functions on \mathbb of period 2 \pi that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series :A(x)=\sum_^\infty a_ne^ and :B(x)=\sum_^\infty b_ne^ respective ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
