|
Multidimensional Transform
In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions. Multidimensional Fourier transform One of the more popular multidimensional transforms is the Fourier transform, which converts a signal from a time/space domain representation to a frequency domain representation.Smith, W. Handbook of Real-Time Fast Fourier Transforms:Algorithms to Product Testing, Wiley_IEEE Press, edition 1, pages 73–80, 1995 The discrete-domain multidimensional Fourier transform (FT) can be computed as follows: : F(w_1,w_2,\dots,w_m) = \sum_^\infty \sum_^\infty \cdots \sum_^\infty f(n_1,n_2,\dots,n_m) e^ where ''F'' stands for the multidimensional Fourier transform, ''m'' stands for multidimensional dimension. Define ''f'' as a multidimensional discrete-domain signal. The inverse multidimensional Fourier transform is given by : f(n_1,n_2,\dots,n_m) = \left(\frac\right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multidimensional Modulation
Multidimensional modulation (MD modulation) is modifying or multiplying an MD signal (typically sinusoidal and referred to as the carrier signal) with another signal that carries some information or message. In the frequency domain, the signal is moved from one frequency to another. if then Typically the carrier signal is a sinusoidal signal and in various applications. The figures below illustrate a quick example of a 2-D modulation. The original signal from () is modulated with a sinusoidal signal to get (). The equations () and () are the real and the imaginary components of the modulated signal. Background/Motivation The MD modulation is one of the properties of the Multidimensional Fourier Transform. MD Fourier Transform (FT) Fourier Transform (FT) of multi-dimensional (MD) signal or system is the transform of the MD signal or system that decomposes it into its frequency components. Essentially, it is the frequency response of the MD signal or system, so it de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MJPEG
Motion JPEG (M-JPEG or MJPEG) is a video compression format in which each video frame or interlaced field of a digital video sequence is compressed separately as a JPEG image. Originally developed for multimedia PC applications, Motion JPEG enjoys broad client support: most major web browsers and players provide native support, and plug-ins are available for the rest. Software and devices using the M-JPEG standard include web browsers, media players, game consoles, digital cameras, IP cameras, webcams, streaming servers, video cameras, and non-linear video editors. History Motion JPEG was originally developed for multimedia PC applications. Early implementations of MJPEG were generally implemented in Hardware. C-Cube was an early proponent with their CL550 JPEG codec been used in several hardware implementations. It was announced that the NeXTdimension from NeXT would ship with an onboard CL550 to implement MJPEG. This was however later shelved and wasn't included in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrogram
A 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, and others. Spectrograms of audio can be used to identify spoken words phonetically, and to analyse the various calls of animals. A spectrogram can be generated by an optical spectrometer, a bank of band-pass filters, by Fourier transform or by a wavelet transform (in which case it is also known as a scaleogram or scalogram). A spectrogram is usually depicted as a heat map, i.e., as an image with the intensity shown by varying the colour or brightness. Format A common format is a graph with two geometric dimensions: one axis represents time, and the other axis r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Leakage
The Fourier transform of a function of time, s(t), is a complex-valued function of frequency, S(f), often referred to as a frequency spectrum. Any linear time-invariant operation on s(t) produces a new spectrum of the form H(f)•S(f), which changes the relative magnitudes and/or angles (phase) of the non-zero values of S(f). Any other type of operation creates new frequency components that may be referred to as spectral leakage in the broadest sense. Sampling, for instance, produces leakage, which we call ''aliases'' of the original spectral component. For Fourier transform purposes, sampling is modeled as a product between s(t) and a Dirac comb function. The spectrum of a product is the convolution between S(f) and another function, which inevitably creates the new frequency components. But the term 'leakage' usually refers to the effect of ''windowing'', which is the product of s(t) with a different kind of function, the window function. Window functions happen to have fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nyquist Rate
In signal processing, the Nyquist rate, named after Harry Nyquist, is a value (in units of samples per second or hertz, Hz) equal to twice the highest frequency (bandwidth) of a given function or signal. When the function is digitized at a higher sample rate (see ), the resulting discrete-time sequence is said to be free of the distortion known as aliasing. Conversely, for a given sample-rate the corresponding Nyquist frequency in Hz is one-half the sample-rate. Note that the ''Nyquist rate'' is a property of a continuous-time signal, whereas ''Nyquist frequency'' is a property of a discrete-time system. The term ''Nyquist rate'' is also used in a different context with units of symbols per second, which is actually the field in which Harry Nyquist was working. In that context it is an upper bound for the symbol rate across a bandwidth-limited baseband channel such as a telegraph line or passband channel such as a limited radio frequency band or a frequency division mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aliasing
In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when sampled. It also often refers to the distortion or artifact that results when a signal reconstructed from samples is different from the original continuous signal. Aliasing can occur in signals sampled in time, for instance digital audio, or the stroboscopic effect, and is referred to as temporal aliasing. It can also occur in spatially sampled signals (e.g. moiré patterns in digital images); this type of aliasing is called spatial aliasing. Aliasing is generally avoided by applying low-pass filters or anti-aliasing filters (AAF) to the input signal before sampling and when converting a signal from a higher to a lower sampling rate. Suitable reconstruction filtering should then be used when restoring the sampled signal to the continuous domain or converting a signal from a lower to a higher sampling rate. For spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete-time Fourier Transform
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function. Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the discrete Fourier transform (DFT) (see ), which is by far the most common method of modern Fourier analysis. Both transforms are invertible. The inverse DTFT is the origin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous 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] |
Frequency Spectrum
The power spectrum S_(f) of a time series x(t) describes the distribution of 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) 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 spectral energy distribution that would b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropy Encoding
In information theory, an entropy coding (or entropy encoding) is any lossless data compression method that attempts to approach the lower bound declared by Shannon's source coding theorem, which states that any lossless data compression method must have expected code length greater or equal to the entropy of the source. More precisely, the source coding theorem states that for any source distribution, the expected code length satisfies \mathbb E_ (d(x))\geq \mathbb E_ \log_b(P(x))/math>, where l is the number of symbols in a code word, d is the coding function, b is the number of symbols used to make output codes and P is the probability of the source symbol. An entropy coding attempts to approach this lower bound. Two of the most common entropy coding techniques are Huffman coding and arithmetic coding. If the approximate entropy characteristics of a data stream are known in advance (especially for signal compression), a simpler static code may be useful. These static codes in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantization (signal Processing)
Quantization, in mathematics and digital signal processing, is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements. Rounding and truncation are typical examples of quantization processes. Quantization is involved to some degree in nearly all digital signal processing, as the process of representing a signal in digital form ordinarily involves rounding. Quantization also forms the core of essentially all lossy compression algorithms. The difference between an input value and its quantized value (such as round-off error) is referred to as quantization error. A device or algorithmic function that performs quantization is called a quantizer. An analog-to-digital converter is an example of a quantizer. Example For example, rounding a real number x to the nearest integer value forms a very basic type of quantizer – a ''uniform'' one. A typical (''mid-tread'') uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
