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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 number, real-valued coefficient at each iteration, using real-valued arithmetic for real-valued input sequences. For covering a full spectrum (except when using for continuous stream of data where coefficients are reused for subsequent calculations, which has computational complexity equivalent of sliding DFT), the Goertzel algorithm has a Computational complexity theory, higher order of ... [...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 Sampling (signal processing), 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 signal processing, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex valued frequency-domain (the z-domain or z-plane) representation. It can be considered a discrete-time equivalent of the Laplace transform (the ''s-domain'' or ''s-plane''). This similarity is explored in the theory of time-scale calculus. While the continuous-time Fourier transform is evaluated on the s-domain's vertical axis (the imaginary axis), the discrete-time Fourier transform is evaluated along the z-domain's unit circle. The s-domain's left half-plane maps to the area inside the z-domain's unit circle, while the s-domain's right half-plane maps to the area outside of the z-domain's unit circle. In signal processing, one of the means of designing digital filters is to take analog designs, subject them to a bilinear transform which maps them from the s-domain to the z-domain, and then produce the digital filter by in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bluestein's FFT Algorithm
The chirp Z-transform (CZT) is a generalization of the discrete Fourier transform (DFT). While the DFT samples the Z plane at uniformly-spaced points along the unit circle, the chirp Z-transform samples along spiral arcs in the Z-plane, corresponding to straight lines in the S plane.A study of the Chirp Z-transform and its applications - Shilling, Steve Alan The DFT, real DFT, and zoom DFT can be calculated as special cases of the CZT. Specifically, the chirp Z transform calculates the Z transform at a finite number of points zk along a contour, defined as: : |
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). A Fourier transform 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 Matrix decomposition, factorizing the DFT matrix into a product of Sparse matrix, sparse (mostly zero) factors. As a result, it manages to reduce the Computational complexity theory, complexity of computing the DFT from O(n^2), which arises if one simply applies the definition of DFT, to O(n \log n), where is the data size. The difference in speed can be enormous, especially for long data sets where may be in the thousands or millions. ... [...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 Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex number, 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 (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex number, complex Sine wave, 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 fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a member of a #Related asymptotic notations, family of notations invented by German mathematicians Paul Gustav Heinrich Bachmann, Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '':wikt:Ordnung#German, Ordnung'', meaning the order of approximation. In computer science, big O notation is used to Computational complexity theory, classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetic function, arithmetical function and a better understood approximation; one well-known exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Object-oriented Programming
Object-oriented programming (OOP) is a programming paradigm based on the concept of '' objects''. Objects can contain data (called fields, attributes or properties) and have actions they can perform (called procedures or methods and implemented in code). In OOP, computer programs are designed by making them out of objects that interact with one another. Many of the most widely used programming languages (such as C++, Java, and Python) support object-oriented programming to a greater or lesser degree, typically as part of multiple paradigms in combination with others such as imperative programming and declarative programming. Significant object-oriented languages include Ada, ActionScript, C++, Common Lisp, C#, Dart, Eiffel, Fortran 2003, Haxe, Java, JavaScript, Kotlin, Logo, MATLAB, Objective-C, Object Pascal, Perl, PHP, Python, R, Raku, Ruby, Scala, SIMSCRIPT, Simula, Smalltalk, Swift, Vala and Visual Basic.NET. History The idea of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Array Data Type
In computer science, array is a data type that represents a collection of ''elements'' ( values or variables), each selected by one or more indices (identifying keys) that can be computed at run time during program execution. Such a collection is usually called an array variable or array value.Robert W. Sebesta (2001) ''Concepts of Programming Languages''. Addison-Wesley. 4th edition (1998), 5th edition (2001), By analogy with the mathematical concepts vector and matrix, array types with one and two indices are often called vector type and matrix type, respectively. More generally, a multidimensional array type can be called a tensor type, by analogy with the mathematical concept, tensor. Language support for array types may include certain built-in array data types, some syntactic constructions (''array type constructors'') that the programmer may use to define such types and declare array variables, and special notation for indexing array elements. For example, in the P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudocode
In computer science, pseudocode is a description of the steps in an algorithm using a mix of conventions of programming languages (like assignment operator, conditional operator, loop) with informal, usually self-explanatory, notation of actions and conditions. Although pseudocode shares features with regular programming languages, it is intended for human reading rather than machine control. Pseudocode typically omits details that are essential for machine implementation of the algorithm, meaning that pseudocode can only be verified by hand. The programming language is augmented with natural language description details, where convenient, or with compact mathematical notation. The reasons for using pseudocode are that it is easier for people to understand than conventional programming language code and that it is an efficient and environment-independent description of the key principles of an algorithm. It is commonly used in textbooks and scientific publications to document ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christian Reinsch
Christian Reinsch (1934 – October 8, 2022) was a German mathematician who worked in the area of numerical analysis. Reinsch began studying physics in 1953 at the Technical University of Munich, graduating in 1958. He received his doctorate in 1961 studying under Heinz Maier-Leibnitz. Among other things, he worked in the area of numerical linear algebra and interpolation with spline functions. Several of his works have over two thousand citations each. Reinsch died on October 8, 2022. Selected works * * * * * See also *Goertzel algorithm *QR algorithm *Singular value decomposition *Smoothing spline Smoothing splines are function estimates, \hat f(x), obtained from a set of noisy observations y_i of the target f(x_i), in order to balance a measure of goodness of fit of \hat f(x_i) to y_i with a derivative based measure of the smoothness of ... References External links Biography at the TU Munich [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context: one important context is numerical linear algebra, and another is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
