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Kaiser Window
The Kaiser window, also known as the Kaiser–Bessel window, was developed by James Kaiser at Bell Laboratories. It is a one-parameter family of window functions used in finite impulse response filter design and spectral analysis. The Kaiser window approximates the DPSS window which maximizes the energy concentration in the main lobe but which is difficult to compute. Definition The Kaiser window and its Fourier transform are given by: : w_0(x) \triangleq \left\ \quad \stackrel\quad \frac , where: * is the zeroth-order modified Bessel function of the first kind, * is the window duration, and * is a non-negative real number that determines the shape of the window. In the frequency domain, it determines the trade-off between main-lobe width and side lobe level, which is a central decision in window design. * Sometimes the Kaiser window is parametrized by , where . For digital signal processing, the function can be sampled symmetrically as: :w = L\cdot w_0\left( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modified Discrete Cosine Transform
The modified discrete cosine transform (MDCT) is a transform based on the type-IV discrete cosine transform (DCT-IV), with the additional property of being lapped: it is designed to be performed on consecutive blocks of a larger dataset, where subsequent blocks are overlapped so that the last half of one block coincides with the first half of the next block. This overlapping, in addition to the energy-compaction qualities of the DCT, makes the MDCT especially attractive for signal compression applications, since it helps to avoid artifacts stemming from the block boundaries. As a result of these advantages, the MDCT is the most widely used lossy compression technique in audio data compression. It is employed in most modern audio coding standards, including MP3, Dolby Digital (AC-3), Vorbis (Ogg), Windows Media Audio (WMA), ATRAC, Cook, Advanced Audio Coding (AAC), High-Definition Coding (HDC), LDAC, Dolby AC-4, and MPEG-H 3D Audio, as well as speech coding standards suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric " bell curve" shape. The parameter is the height of the curve's peak, is the position of the center of the peak, and (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are often used to represent the probability density function of a normally distributed random variable with expected value and variance . In this case, the Gaussian is of the form g(x) = \frac \exp\left( -\frac \frac \right). Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define Gaussian filters, in image processing where two-dimen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Window Function
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, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions. The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. The duration of the segments is determined in ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Signal Processing
Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor. Digital signal processing and analog signal processing are subfields of signal processing. DSP applications include audio and speech processing, sonar, radar and other sensor array processing, spectral density estimation, statistical signal processing, digital image processing, data compression, video coding, audio coding, image compression, signal processing for telecommunications, control systems, biomedical engineering, and seismology, among others. DSP can involve linear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Kaiser
James Frederick Kaiser (Dec. 10, 1929 – Feb. 13, 2020) was an American electrical engineer noted for his contributions in signal processing. He was an IEEE Fellow and received many honors and awards, including the IEEE Centennial Medal, the IEEE W.R.G. Baker Award, the Bell Laboratories Distinguished Technical Staff Award, and the IEEE Jack S. Kilby Signal Processing Medal. Biography Kaiser was born in Piqua, Ohio, and earned his electrical engineering degree from the University of Cincinnati in 1952. He then moved to the Massachusetts Institute of Technology and received his masters and doctorate degrees in 1954 and 1959, respectively. Following his doctorate, he received a three-year appointment as an assistant professor at MIT but decided to take a leave of absence to work at Bell Labs. Although the arrangement was due to only last for a year, he enjoyed the work so much that he elected to stay. While at Bell Labs, he worked on a variety of projects in signal processing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Concentration Problem
The spectral concentration problem in Fourier analysis refers to finding a time sequence of a given length whose discrete Fourier transform is maximally localized on a given frequency interval, as measured by the spectral concentration. Spectral concentration The discrete-time Fourier transform (DTFT) ''U''(''f'') of a finite series w_t, t = 1,2,3,4,...,T is defined as :U(f) = \sum_^w_t e^. In the following, the sampling interval will be taken as Δ''t'' = 1, and hence the frequency interval as ''f'' ∈ ½,½ ''U''(''f'') is a periodic function with a period 1. For a given frequency ''W'' such that 0\lambda_), then the eigenvector corresponding to \lambda_ is called ''nth''–order Slepian sequence (DPSS) (0≤''n''≤''N''-1). This ''nth''–order taper also offers the best sidelobe suppression and is pairwise orthogonal to the Slepian sequences of previous orders (0,1,2,3....,n-1). These lower order Slepian sequences form the basis for spectral estimation by multitaper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
