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.
Butterworth filterThe Butterworth filter is a type of signal processing filter designed to have a frequency response that is as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers". Butterworth had a reputation for solving very complex mathematical problems thought to be 'impossible'. At the time, filter design required a considerable amount of designer experience due to limitations of the .
Discrete time and continuous timeIn mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time views values of variables as occurring at distinct, separate "points in time", or equivalently as being unchanged throughout each non-zero region of time ("time period")—that is, time is viewed as a discrete variable. Thus a non-time variable jumps from one value to another as time moves from one time period to the next.
Fourier transformIn physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.
Exponential functionThe exponential function is a mathematical function denoted by or (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers.
ConvolutionIn mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function () that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity).
Discrete mathematicsDiscrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry.
Continuous or discrete variableIn mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by measuring or counting, respectively. If it can take on two particular real values such that it can also take on all real values between them (even values that are arbitrarily close together), the variable is continuous in that interval. If it can take on a value such that there is a non-infinitesimal gap on each side of it containing no values that the variable can take on, then it is discrete around that value.
Matrix exponentialIn mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let X be an n×n real or complex matrix. The exponential of X, denoted by eX or exp(X), is the n×n matrix given by the power series where is defined to be the identity matrix with the same dimensions as .
ResonanceResonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillating force is applied at a resonant frequency of a dynamic system, the system will oscillate at a higher amplitude than when the same force is applied at other, non-resonant frequencies. Frequencies at which the response amplitude is a relative maximum are also known as resonant frequencies or resonance frequencies of the system.
AmplitudeThe amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplitude (see below), which are all functions of the magnitude of the differences between the variable's extreme values. In older texts, the phase of a periodic function is sometimes called the amplitude. For symmetric periodic waves, like sine waves, square waves or triangle waves peak amplitude and semi amplitude are the same.
Transfer functionIn engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that models the system's output for each possible input. They are widely used in electronic engineering tools like circuit simulators and control systems. In some simple cases, this function can be represented as two-dimensional graph of an independent scalar input versus the dependent scalar output, called a transfer curve or characteristic curve.
