Dirac combIn mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic function with the formula for some given period . Here t is a real variable and the sum extends over all integers k. The Dirac delta function and the Dirac comb are tempered distributions. The graph of the function resembles a comb (with the s as the comb's teeth), hence its name and the use of the comb-like Cyrillic letter sha (Ш) to denote the function. The symbol , where the period is omitted, represents a Dirac comb of unit period.
Multidimensional transformIn mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions. 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. The discrete-domain multidimensional Fourier transform (FT) can be computed as follows: where F stands for the multidimensional Fourier transform, m stands for multidimensional dimension.
Multidimensional discrete convolutionIn signal processing, multidimensional discrete convolution refers to the mathematical operation between two functions f and g on an n-dimensional lattice that produces a third function, also of n-dimensions. Multidimensional discrete convolution is the discrete analog of the multidimensional convolution of functions on Euclidean space. It is also a special case of convolution on groups when the group is the group of n-tuples of integers. Similar to the one-dimensional case, an asterisk is used to represent the convolution operation.
Complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
Radius of convergenceIn mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or . When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges.
LinearityIn mathematics, the term linear is used in two distinct senses for two different properties: linearity of a function (or mapping ); linearity of a polynomial. An example of a linear function is the function defined by that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables and is Linearity of a mapping is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight.
Residue (complex analysis)In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function that is holomorphic except at the discrete points {ak}k, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.
Kronecker deltaIn mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: or with use of Iverson brackets: For example, because , whereas because . The Kronecker delta appears naturally in many areas of mathematics, physics, engineering and computer science, as a means of compactly expressing its definition above.
Cutoff frequencyIn physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced (attenuated or reflected) rather than passing through. Typically in electronic systems such as filters and communication channels, cutoff frequency applies to an edge in a lowpass, highpass, bandpass, or band-stop characteristic – a frequency characterizing a boundary between a passband and a stopband.
BIBO stabilityIn signal processing, specifically control theory, bounded-input, bounded-output (BIBO) stability is a form of stability for signals and systems that take inputs. If a system is BIBO stable, then the output will be bounded for every input to the system that is bounded. A signal is bounded if there is a finite value such that the signal magnitude never exceeds , that is For discrete-time signals: For continuous-time signals: For a continuous time linear time-invariant (LTI) system, the condition for BIBO stability is that the impulse response, , be absolutely integrable, i.
Minimum phaseIn control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable. The most general causal LTI transfer function can be uniquely factored into a series of an all-pass and a minimum phase system. The system function is then the product of the two parts, and in the time domain the response of the system is the convolution of the two part responses.
Discrete cosine transformA discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT, first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression. It is used in most digital media, including (such as JPEG and HEIF), digital video (such as MPEG and H.26x), digital audio (such as Dolby Digital, MP3 and AAC), digital television (such as SDTV, HDTV and VOD), digital radio (such as AAC+ and DAB+), and speech coding (such as AAC-LD, Siren and Opus).
