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# 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 frequency domain representation. [1]

67 relations: Abraham de Moivre, Advanced Z-transform, Argument (complex analysis), Autoregressive–moving-average model, Bilinear transform, Causal system, Chirp Z-transform, Complex conjugate, Complex number, Convolution, Cross-correlation, Decimation (signal processing), Dirac delta function, Discrete Fourier transform, Discrete-time Fourier transform, Discrete-time signal, Eliahu I. Jury, Final value theorem, Finite impulse response, Formal power series, Fourier series, Fourier transform, Fraction (mathematics), Frequency domain, Frequency response, Fundamental theorem of algebra, Generating function, Geometric series, Heaviside step function, Hertz, Imaginary unit, Impulse response, Initial value theorem, Integral, Integral transform, John R. Ragazzini, Laplace transform, Laurent series, Linearity, Lotfi A. Zadeh, LTI system theory, Mathematics, Methods of contour integration, Multiplication, Parseval's theorem, Partial fraction decomposition, Periodic summation, Pierre-Simon Laplace, Pole (complex analysis), Pole–zero plot, ... Expand index (17 more) »

## Abraham de Moivre

Abraham de Moivre (26 May 1667 in Vitry-le-François, Champagne, France – 27 November 1754 in London, England) was a French mathematician known for de Moivre's formula, one of those that link complex numbers and trigonometry, and for his work on the normal distribution and probability theory.

In mathematics and signal processing, the advanced Z-transform is an extension of the Z-transform, to incorporate ideal delays that are not multiples of the sampling time.

## Argument (complex analysis)

In mathematics, arg is a function operating on complex numbers (visualized in a complex plane).

## Autoregressive–moving-average model

In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the auto-regression and the second for the moving average.

## Bilinear transform

The bilinear transform (also known as Tustin's method) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.

## Causal system

In control theory, a causal system (also known as a physical or nonanticipative system) is a system where the output depends on past and current inputs but not future inputs i.e. the output y(t_) only depends on the input x(t) for values of t \le t_.

## Chirp Z-transform

The Chirp Z-transform (CZT) is a generalization of the discrete Fourier transform.

## Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers and is the imaginary unit, that satisfies the equation.

## Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated.

## Cross-correlation

In signal processing, cross-correlation is a measure of similarity of two series as a function of the lag of one relative to the other.

## Decimation (signal processing)

In digital signal processing, decimation is the process of reducing the sampling rate of a signal.

## Dirac delta function

In mathematics, the Dirac delta function, or function, is a generalized function, or distribution, on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line.

## Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite list of equally spaced samples of a function into the list of coefficients of a finite combination of complex sinusoids, ordered by their frequencies, that has those same sample values.

## Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to the uniformly-spaced samples of a continuous function.

## Discrete-time signal

A discrete signal or discrete-time signal is a time series consisting of a sequence of quantities.

## Eliahu I. Jury

Eliahu Ibraham Jury (born May 23, 1923) is an American engineer, born in Baghdad, Iraq.

## Final value theorem

In mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity.

## Finite impulse response

In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time.

## Formal power series

In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number.

## Fourier series

In mathematics, a Fourier series is a way to represent a (wave-like) function as the sum of simple sine waves.

## Fourier transform

The Fourier transform decomposes a function of time (a signal) into the frequencies that make it up, similarly to how a musical chord can be expressed as the amplitude (or loudness) of its constituent notes.

## Fraction (mathematics)

A fraction (from fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.

## Frequency domain

In electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time.

## Frequency response

Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system.

## Fundamental theorem of algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

## Generating function

In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers.

## Geometric series

In mathematics, a geometric series is a series with a constant ratio between successive terms.

## Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by H (but sometimes u or θ), is a discontinuous function whose value is zero for negative argument and one for positive argument.

## Hertz

The hertz (symbol Hz) is the unit of frequency in the International System of Units (SI) and is defined as one cycle per second.

## Imaginary unit

The term imaginary unit or unit imaginary number refers to a solution to the equation.

## Impulse response

In signal processing, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse.

## Initial value theorem

In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.

## Integral

The integral is an important concept in mathematics.

## Integral transform

In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf.

## John R. Ragazzini

John Ralph Ragazzini (1912 &ndash; November 22, 1988) was an American electrical engineer and a professor of Electrical Engineering.

## Laplace transform

The Laplace transform is a widely used integral transform in mathematics and electrical engineering named after Pierre-Simon Laplace that transforms a function of time into a function of complex frequency.

## Laurent series

In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree.

## Linearity

In common usage, linearity refers to a mathematical relationship or function that can be graphically represented as a straight line, as in two quantities that are directly proportional to each other, such as voltage and current in an RLC circuit, or the mass and weight of an object.

Lotfali Askar Zadeh (Lütfəli Rəhimoğlu Əsgərzadə; born February 4, 1921), better known as Lotfi A. Zadeh, is a mathematician, computer scientist, electrical engineer, artificial intelligence researcher and professor emeritus of computer science at the University of California, Berkeley.

## LTI system theory

Linear time-invariant theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas.

## Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change.

## Methods of contour integration

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

## Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "·" or by the absence of symbol) is one of the four elementary, mathematical operations of arithmetic; with the others being addition, subtraction and division.

## Parseval's theorem

In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.

## Partial fraction decomposition

In algebra, the partial fraction decomposition or partial fraction expansion of a rational function (that is a fraction such that the numerator and the denominator are both polynomials) is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

## Periodic summation

In signal processing, any periodic function, s_P(t) with period P, can be represented by a summation of an infinite number of instances of an aperiodic function, s(t), that are offset by integer multiples of P. This representation is called periodic summation: When s_P(t) is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or "samples") of the continuous Fourier transform, S(f) \ \stackrel \ \mathcal\, at intervals of 1/P.

## Pierre-Simon Laplace

Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was an influential French scholar whose work was important to the development of mathematics, statistics, physics, and astronomy.

## Pole (complex analysis)

In the mathematical field of complex analysis, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity of \frac at z.

## Pole–zero plot

In mathematics, signal processing and control theory, a pole–zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as.

## Probability-generating function

In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable.

The radian is the standard unit of angular measure, used in many areas of mathematics.

In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges.

## Real number

In mathematics, a real number is a value that represents a quantity along a continuous line.

## Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

## Sequence

In mathematics, a sequence is an ordered collection of objects in which repetitions are allowed.

## Signal processing

Signal processing is an enabling technology that encompasses the fundamental theory, applications, algorithms, and implementations of processing or transferring information contained in many different physical, symbolic, or abstract formats broadly designated as signals.

## Starred transform

In applied mathematics, the starred transform, or star transform, is a discrete-time variation of the Laplace transform, so-named because of the asterisk or "star" in the customary notation of the sampled signals.

## Time-scale calculus

In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical systems.

## Transfer function

In engineering, a transfer function (also known as the system function or network function and, when plotted as a graph, transfer curve) is a mathematical representation for fit or to describe inputs and outputs of black box models.

## Unit circle

In mathematics, a unit circle is a circle with a radius of one.

## Upsampling

Upsampling is interpolation, applied in the context of digital signal processing and sample rate conversion.

## Witold Hurewicz

Witold Hurewicz (June 29, 1904 – September 6, 1956) was a Polish mathematician.

## Zak transform

In mathematics, the Zak transform is a certain operation which takes as input a function of one variable and produces as output a function of two variables.

## Zero (complex analysis)

In complex analysis, a zero (sometimes called a root) of a holomorphic function f is a complex number a such that f(a).

## Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation In other words, a "zero" of a function is an input value that produces an output of zero (0).

## Zeta function regularization

In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators.

## References

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