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Constant-Q Transform
In mathematics and signal processing, the constant-Q transform and variable-Q transform, simply known as CQT and VQT, transforms a data series to the frequency domain. It is related to the Fourier transformJudith C. BrownCalculation of a constant Q spectral transform ''J. Acoust. Soc. Am.'', 89(1):425–434, 1991. and very closely related to the complex Morlet wavelet transform. Its design is suited for musical representation. The transform can be thought of as a series of filters ''f''''k'', logarithmically spaced in frequency, with the ''k''-th filter having a spectral width ''δf''''k'' equal to a multiple of the previous filter's width: :\delta f_k = 2^ \cdot \delta f_ = \left( 2^ \right)^k \cdot \delta f_\text, where ''δf''''k'' is the bandwidth of the ''k''-th filter, ''f''min is the central frequency of the lowest filter, and ''n'' is the number of filters per octave. Calculation The short-time Fourier transform of ''x'' 'n''for a frame shifted to sample ''m'' is cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signal Processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, Data storage, digital storage efficiency, correcting distorted signals, subjective video quality and to also detect or pinpoint components of interest in a measured signal. History According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. They further state that the digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s. In 1948, Claude Shannon wrote the influential paper "A Mathematical Theory of Communication" which was published in the Bell System Technical Journal. The paper laid the groundwork for later development of information c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Sine
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the unit hyperbola. Also, similarly to how the derivatives of and are and respectively, the derivatives of and are and respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. The basic hyperbolic functions are: * hyperbolic sine "" (), * hyperbolic cosine "" (),''Collins Concise Dictionary'', p. 328 from which are derived: * hyperbolic tangent "" (), * hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Transforms
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, the kernel function, integral kernel or nucleus of the transform. Some kernels have an associated ''inverse kernel'' K^( u,t ) which (roughly speaking) yields an inverse transform: :f(t) = \int_^ (Tf)(u) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
STFT
The short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divide a longer time signal into shorter segments of equal length and then compute the Fourier transform separately on each shorter segment. This reveals the Fourier spectrum on each shorter segment. One then usually plots the changing spectra as a function of time, known as a spectrogram or waterfall plot, such as commonly used in software defined radio (SDR) based spectrum displays. Full bandwidth displays covering the whole range of an SDR commonly use fast Fourier transforms (FFTs) with 2^24 points on desktop computers. Forward STFT Continuous-time STFT Simply, in the continuous-time case, the function to be transformed is multiplied by a window function which is nonzero for only a short period of time. The Fourier transform ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sliding DFT
In applied mathematics, the sliding discrete Fourier transform is a recursive algorithm to compute successive STFTs of input data frames that are a single sample apart (hopsize − 1). Definition Assuming that the hopsize between two consecutive DFTs is 1 sample, then \begin F_(n) &= \sum_^ f_e^\\ &= \sum_^N f_e^ \\ &= e^ \left \sum_^ f_e^ - f_t + f_ \right\\ &= e^ \left _t(n) - f_t + f_ \right \end From this definition, the DFT can be computed recursively thereafter. However, implementing the window function on a sliding DFT is difficult due to its recursive nature, therefore it is done exclusively in a frequency domain. There is an alternate way to implement the sliding DFT, called sliding Goertzel algorithm, which does the same except it can calculate magnitudes directly. In this case, the sGA is an IIR filter bank In signal processing, a filter bank (or filterbank) is an array of bandpass filters that separates the input signal into multiple components, e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (statistics)
The term kernel is used in statistical analysis to refer to a window function. The term "kernel" has several distinct meanings in different branches of statistics. Bayesian statistics In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. Note that such factors may well be functions of the parameters of the pdf or pmf. These factors form part of the normalization factor of the probability distribution, and are unnecessary in many situations. For example, in pseudo-random number sampling, most sampling algorithms ignore the normalization factor. In addition, in Bayesian analysis of conjugate prior distributions, the normalization factors are generally ignored during the calculations, and only the kernel considered. At the end, the form of the kernel is examined, and if it m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Fourier Transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis 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. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from O\left(N^2\right), which arises if one simply applies the definition of DFT, to O(N \log N), where N is the data size. The difference in speed can be enormous, especially for long data sets where ''N'' may be in the thousands or millions. In the presence of round-off error, many FFT algorithm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goertzel Algorithm
The Goertzel algorithm is a technique in digital signal processing (DSP) for efficient evaluation of the individual terms of the discrete Fourier transform (DFT). It is useful in certain practical applications, such as recognition of dual-tone multi-frequency signaling (DTMF) tones produced by the push buttons of the keypad of a traditional analog telephone. The algorithm was first described by Gerald Goertzel in 1958. Like the DFT, the Goertzel algorithm analyses one selectable frequency component from a discrete signal. Unlike direct DFT calculations, the Goertzel algorithm applies a single real-valued coefficient at each iteration, using real-valued arithmetic for real-valued input sequences. For covering a full spectrum, the Goertzel algorithm has a higher order of complexity than fast Fourier transform (FFT) algorithms, but for computing a small number of selected frequency components, it is more numerically efficient. The simple structure of the Goertzel algorithm makes it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Fourier Transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle. The DFT is the most important discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalent Rectangular Bandwidth
The equivalent rectangular bandwidth or ERB is a measure used in psychoacoustics, which gives an approximation to the bandwidths of the filters in human hearing, using the unrealistic but convenient simplification of modeling the filters as rectangular band-pass filters, or band-stop filters, like in tailor-made notched music training (TMNMT). Approximations For moderate sound levels and young listeners, the bandwidth of human auditory filters can be approximated by the polynomial equation: where ''f'' is the center frequency of the filter in kHz and ERB(''f'') is the bandwidth of the filter in Hz. The approximation is based on the results of a number of published simultaneous masking experiments and is valid from 0.1 to 6.5 kHz.B.C.J. Moore and B.R. Glasberg, "Suggested formulae for calculating auditory-filter bandwidths and excitation patterns" Journal of the Acoustical Society of America 74: 750-753, 1983. The above approximation was given in 1983 by Moore and Glasber ... [...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] |
Hamming Window
In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent Discrete Fourier transform, Discrete Fourier Transform. Several ''Window function, window functions'' can be defined, based on a constant (rectangular window), B-splines, other polynomials, sinusoids, cosine-sums, adjustable, hybrid, and other types. The windowing operation consists of multiplying the given sampled signal by the window function. Conventions * w_0(x) is a zero-phase function (symmetrical about x=0), continuous for x \in [-N/2, N/2], where N is a positive integer (even or odd). * The sequence \ is ''symmetric'', of length N+1. * \ is ''DFT-symmetric'', of length N. * The parameter B displayed on each spectral plot is the function's noise equivalent bandwidth metric, in units of ''DFT bins''. The sparse sampling of a DTFT (such as the DFTs in Fig 1) only reveals the leakage into the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
