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106-465: 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

212-405: A n ) {\displaystyle (a_{n})} and ( b n ) {\displaystyle (b_{n})} become closer and closer if and only if more and more of their terms agree exactly. Formally, the sequence of partial sums of some infinite summation converges if for every fixed power of X {\displaystyle X} the coefficient stabilizes: there

318-870: A , possibly including some points of the boundary line Re( s ) = a . In the region of convergence Re( s ) > Re( s 0 ) , the Laplace transform of f can be expressed by integrating by parts as the integral F ( s ) = ( s − s 0 ) ∫ 0 ∞ e − ( s − s 0 ) t β ( t ) d t , β ( u ) = ∫ 0 u e − s 0 t f ( t ) d t . {\displaystyle F(s)=(s-s_{0})\int _{0}^{\infty }e^{-(s-s_{0})t}\beta (t)\,dt,\quad \beta (u)=\int _{0}^{u}e^{-s_{0}t}f(t)\,dt.} That is, F ( s ) can effectively be expressed, in

424-421: A Mellin transform , to transform the whole of a difference equation , in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power. Laplace also recognised that Joseph Fourier 's method of Fourier series for solving the diffusion equation could only apply to

530-456: A finite number of coefficients of A and B . For example, the X term is given by For this reason, one may multiply formal power series without worrying about the usual questions of absolute , conditional and uniform convergence which arise in dealing with power series in the setting of analysis . Once we have defined multiplication for formal power series, we can define multiplicative inverses as follows. The multiplicative inverse of

636-487: A formal series is an infinite sum that is considered independently from any notion of convergence , and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums , etc.). A formal power series is a special kind of formal series, of the form where the a n , {\displaystyle a_{n},} called coefficients , are numbers or, more generally, elements of some ring , and

742-724: A one-sided or two-sided transform. (Just like we have the one-sided Laplace transform and the two-sided Laplace transform .) The bilateral or two-sided Z-transform of a discrete-time signal x [ n ] {\displaystyle x[n]} is the formal power series X ( z ) {\displaystyle X(z)} defined as: X ( z ) = Z { x [ n ] } = ∑ n = − ∞ ∞ x [ n ] z − n {\displaystyle X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=-\infty }^{\infty }x[n]z^{-n}} where n {\displaystyle n}

848-421: A Borel measure locally of bounded variation), then the Laplace transform F ( s ) of f converges provided that the limit lim R → ∞ ∫ 0 R f ( t ) e − s t d t {\displaystyle \lim _{R\to \infty }\int _{0}^{R}f(t)e^{-st}\,dt} exists. The Laplace transform converges absolutely if

954-658: A Laplace transform into known transforms of functions obtained from a table and construct the inverse by inspection. In pure and applied probability , the Laplace transform is defined as an expected value . If X is a random variable with probability density function f , then the Laplace transform of f is given by the expectation L { f } ( s ) = E ⁡ [ e − s X ] , {\displaystyle {\mathcal {L}}\{f\}(s)=\operatorname {E} \left[e^{-sX}\right],} where E ⁡ [ r ] {\displaystyle \operatorname {E} [r]}

1060-473: A closely related mathematical technique. However, the explicit formulation and application of what we now understand as the Z-transform were significantly advanced in 1947 by Witold Hurewicz and colleagues. Their work was motivated by the challenges presented by sampled-data control systems, which were becoming increasingly relevant in the context of radar technology during that period. The Z-transform provided

1166-439: A finite number of uniformly spaced z {\displaystyle z} values can be computed efficiently via Bluestein's FFT algorithm . The discrete-time Fourier transform (DTFT)—not to be confused with the discrete Fourier transform (DFT)—is a special case of such a Z-transform obtained by restricting z {\displaystyle z} to lie on the unit circle. Following three methods are often used for

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1272-428: A formal power series A is a formal power series C such that AC = 1, provided that such a formal power series exists. It turns out that if A has a multiplicative inverse, it is unique, and we denote it by A . Now we can define division of formal power series by defining B / A to be the product BA , provided that the inverse of A exists. For example, one can use the definition of multiplication above to verify

1378-414: A generalization of the Laplace transform connected to his work on moments . Other contributors in this time period included Mathias Lerch , Oliver Heaviside , and Thomas Bromwich . In 1934, Raymond Paley and Norbert Wiener published the important work Fourier transforms in the complex domain , about what is now called the Laplace transform (see below). Also during the 30s, the Laplace transform

1484-455: A limited region of space, because those solutions were periodic . In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space. In 1821, Cauchy developed an operational calculus for the Laplace transform that could be used to study linear differential equations in much the same way the transform is now used in basic engineering. This method was popularized, and perhaps rediscovered, by Oliver Heaviside around

1590-426: A more comprehensive framework for the analysis of digital control systems. This advanced formulation has played a pivotal role in the design and stability analysis of discrete-time control systems, contributing significantly to the field of digital signal processing. Interestingly, the conceptual underpinnings of the Z-transform intersect with a broader mathematical concept known as the method of generating functions ,

1696-470: A power series may not represent a function of its variable. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, since the operations that can be applied are different. A formal power series with coefficients in a ring R {\displaystyle R} is called a formal power series over R . {\displaystyle R.} The formal power series over

1802-406: A powerful tool in combinatorics and probability theory. This connection was hinted at as early as 1730 by Abraham de Moivre , a pioneering figure in the development of probability theory. De Moivre utilized generating functions to solve problems in probability, laying the groundwork for what would eventually evolve into the Z-transform. From a mathematical perspective, the Z-transform can be viewed as

1908-445: A purely algebraic manner by applying a field of fractions construction to the convolution ring of functions on the positive half-line. The resulting space of abstract operators is exactly equivalent to Laplace space, but in this construction the forward and reverse transforms never need to be explicitly defined (avoiding the related difficulties with proving convergence). If f is a locally integrable function (or more generally

2014-496: A ring R {\displaystyle R} form a ring, commonly denoted by R [ [ x ] ] . {\displaystyle R[[x]].} (It can be seen as the ( x ) -adic completion of the polynomial ring R [ x ] , {\displaystyle R[x],} in the same way as the p -adic integers are the p -adic completion of the ring of the integers.) Formal powers series in several indeterminates are defined similarly by replacing

2120-407: A sequence of elements of Z [ [ X ] ] [ [ Y ] ] {\displaystyle \mathbb {Z} [[X]][[Y]]} converges if the coefficient of each power of Y {\displaystyle Y} converges to a formal power series in X {\displaystyle X} , a weaker condition than stabilizing entirely. For instance, with this topology, in

2226-431: A specific instance of a Laurent series , where the sequence of numbers under investigation is interpreted as the coefficients in the (Laurent) expansion of an analytic function . This perspective not only highlights the deep mathematical roots of the Z-transform but also illustrates its versatility and broad applicability across different branches of mathematics and engineering. The Z-transform can be defined as either

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2332-638: A standard table of Z-transform pairs. This method is widely used for its efficiency and simplicity, especially when the original function can be easily broken down into recognizable components. A) Determine the inverse Z-transform of the following by series expansion method, X ( z ) = 1 1 − 1.5 z − 1 + 0.5 z − 2 {\displaystyle X(z)={\frac {1}{1-1.5z^{-1}+0.5z^{-2}}}} Solution: Case 1: ROC: | Z | > 1 {\displaystyle \left\vert Z\right\vert >1} Since

2438-854: A system. The Laplace transform's key property is that it converts differentiation and integration in the time domain into multiplication and division by s in the Laplace domain. Thus, the Laplace variable s is also known as an operator variable in the Laplace domain: either the derivative operator or (for s ) the integration operator . Given the functions f ( t ) and g ( t ) , and their respective Laplace transforms F ( s ) and G ( s ) , f ( t ) = L − 1 { F ( s ) } , g ( t ) = L − 1 { G ( s ) } , {\displaystyle {\begin{aligned}f(t)&={\mathcal {L}}^{-1}\{F(s)\},\\g(t)&={\mathcal {L}}^{-1}\{G(s)\},\end{aligned}}}

2544-445: A systematic and effective method for solving linear difference equations with constant coefficients, which are ubiquitous in the analysis of discrete-time signals and systems. The method was further refined and gained its official nomenclature, "the Z-transform," in 1952, thanks to the efforts of John R. Ragazzini and Lotfi A. Zadeh , who were part of the sampled-data control group at Columbia University. Their work not only solidified

2650-406: Is stable if every bounded input produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region Re( s ) ≥ 0 . As a result, LTI systems are stable, provided that the poles of the Laplace transform of the impulse response function have negative real part. This ROC is used in knowing about the causality and stability of

2756-493: Is 1 by the Cauchy–Hadamard theorem . However, as a formal power series, we may ignore this completely; all that is relevant is the sequence of coefficients [1, −3, 5, −7, 9, −11, ...]. In other words, a formal power series is an object that just records a sequence of coefficients. It is perfectly acceptable to consider a formal power series with the factorials [1, 1, 2, 6, 24, 120, 720, 5040, ... ] as coefficients, even though

2862-530: Is a complex frequency-domain parameter s = σ + i ω {\displaystyle s=\sigma +i\omega } with real numbers σ and ω . An alternate notation for the Laplace transform is L { f } {\displaystyle {\mathcal {L}}\{f\}} instead of F , often written as F ( s ) = L { f ( t ) } {\displaystyle F(s)={\mathcal {L}}\{f(t)\}} in an abuse of notation . The meaning of

2968-571: Is a complex number . It is related to many other transforms, most notably the Fourier transform and the Mellin transform . Formally , the Laplace transform is converted into a Fourier transform by the substitution s = i ω {\displaystyle s=i\omega } where ω {\displaystyle \omega } is real. However, unlike the Fourier transform, which gives

3074-466: Is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). In the case where the ROC is causal (see Example 2 ), this means the path C {\displaystyle C} must encircle all of the poles of X ( z ) {\displaystyle X(z)} . A special case of this contour integral occurs when C {\displaystyle C}

3180-445: Is a point beyond which all further partial sums have the same coefficient. This is clearly the case for the right hand side of ( 1 ), regardless of the values a n {\displaystyle a_{n}} , since inclusion of the term for i = n {\displaystyle i=n} gives the last (and in fact only) change to the coefficient of X n {\displaystyle X^{n}} . It

3286-408: Is a real number so that the contour path of integration is in the region of convergence of F ( s ) . In most applications, the contour can be closed, allowing the use of the residue theorem . An alternative formula for the inverse Laplace transform is given by Post's inversion formula . The limit here is interpreted in the weak-* topology . In practice, it is typically more convenient to decompose

Z-transform - Misplaced Pages Continue

3392-414: Is also obvious that the limit of the sequence of partial sums is equal to the left hand side. This topological structure, together with the ring operations described above, form a topological ring. This is called the ring of formal power series over R {\displaystyle R} and is denoted by R [ [ X ] ] {\displaystyle R[[X]]} . The topology has

3498-547: Is also the I {\displaystyle I} -adic topology, where I = ( X , Y ) {\displaystyle I=(X,Y)} is the ideal generated by X {\displaystyle X} and Y {\displaystyle Y} , still enjoys the property that a summation converges if and only if its terms tend to 0. The same principle could be used to make other divergent limits converge. For instance in R [ [ X ] ] {\displaystyle \mathbb {R} [[X]]}

3604-427: Is an integer and z {\displaystyle z} is, in general, a complex number . In polar form , z {\displaystyle z} may be written as: where A {\displaystyle A} is the magnitude of z {\displaystyle z} , j {\displaystyle j} is the imaginary unit , and ϕ {\displaystyle \phi }

3710-403: Is complete, and if x {\displaystyle x} is an element of I {\displaystyle I} , then there is a unique Φ : R [ [ X ] ] → S {\displaystyle \Phi :R[[X]]\to S} with the following properties: One can perform algebraic operations on power series to generate new power series. Besides

3816-433: Is either of the form Re( s ) > a or Re( s ) ≥ a , where a is an extended real constant with −∞ ≤ a ≤ ∞ (a consequence of the dominated convergence theorem ). The constant a is known as the abscissa of absolute convergence, and depends on the growth behavior of f ( t ) . Analogously, the two-sided transform converges absolutely in a strip of the form a < Re( s ) < b , and possibly including

3922-544: Is in fact the same one used to define the product of polynomials in one indeterminate, which suggests using a similar notation. One embeds R {\displaystyle R} into R [ [ X ] ] {\displaystyle R[[X]]} by sending any (constant) a ∈ R {\displaystyle a\in R} to the sequence ( a , 0 , 0 , … ) {\displaystyle (a,0,0,\ldots )} and designates

4028-477: Is like a polynomial , but with infinitely many terms. Alternatively, for those familiar with power series (or Taylor series ), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value). For example, consider the series If we studied this as a power series, its properties would include, for example, that its radius of convergence

4134-431: Is named after mathematician and astronomer Pierre-Simon, Marquis de Laplace , who used a similar transform in his work on probability theory . Laplace wrote extensively about the use of generating functions (1814), and the integral form of the Laplace transform evolved naturally as a result. Laplace's use of generating functions was similar to what is now known as the z-transform , and he gave little attention to

4240-583: Is needed here, and would make formal power series seem more complicated than they are. It is possible to describe R [ [ X ] ] {\displaystyle R[[X]]} more explicitly, and define the ring structure and topological structure separately, as follows. As a set, R [ [ X ] ] {\displaystyle R[[X]]} can be constructed as the set R N {\displaystyle R^{\mathbb {N} }} of all infinite sequences of elements of R {\displaystyle R} , indexed by

4346-507: Is not however the standard when considering formal power series, as it would lead to convergence considerations that are as subtle as they are in analysis , while the philosophy of formal power series is on the contrary to make convergence questions as trivial as they can possibly be. With this topology it would not be the case that a summation converges if and only if its terms tend to 0. The ring R [ [ X ] ] {\displaystyle R[[X]]} may be characterized by

Z-transform - Misplaced Pages Continue

4452-471: Is not necessary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform . When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually intended. The Laplace transform can be alternatively defined as the bilateral Laplace transform , or two-sided Laplace transform , by extending the limits of integration to be

4558-405: Is quite natural and convenient to designate a general sequence ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} by the formal expression ∑ i ∈ N a i X i {\displaystyle \textstyle \sum _{i\in \mathbb {N} }a_{i}X^{i}} , even though

4664-487: Is simple to prove via Poisson summation , to the functional equation. Hjalmar Mellin was among the first to study the Laplace transform, rigorously in the Karl Weierstrass school of analysis, and apply it to the study of differential equations and special functions , at the turn of the 20th century. At around the same time, Heaviside was busy with his operational calculus. Thomas Joannes Stieltjes considered

4770-681: Is the complex argument (also referred to as angle or phase ) in radians . Alternatively, in cases where x [ n ] {\displaystyle x[n]} is defined only for n ≥ 0 {\displaystyle n\geq 0} , the single-sided or unilateral Z-transform is defined as: X ( z ) = Z { x [ n ] } = ∑ n = 0 ∞ x [ n ] z − n . {\displaystyle X(z)={\mathcal {Z}}\{x[n]\}=\sum _{n=0}^{\infty }x[n]z^{-n}.} In signal processing , this definition can be used to evaluate

4876-459: Is the expectation of random variable r {\displaystyle r} . By convention , this is referred to as the Laplace transform of the random variable X itself. Here, replacing s by − t gives the moment generating function of X . The Laplace transform has applications throughout probability theory, including first passage times of stochastic processes such as Markov chains , and renewal theory . Of particular use

4982-480: Is the finest topology for which always converges as a summation to the formal power series designated by the same expression, and it often suffices to give a meaning to infinite sums and products, or other kinds of limits that one wishes to use to designate particular formal power series. It can however happen occasionally that one wishes to use a coarser topology, so that certain expressions become convergent that would otherwise diverge. This applies in particular when

5088-754: Is the ability to recover the cumulative distribution function of a continuous random variable X by means of the Laplace transform as follows: F X ( x ) = L − 1 { 1 s E ⁡ [ e − s X ] } ( x ) = L − 1 { 1 s L { f } ( s ) } ( x ) . {\displaystyle F_{X}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}\operatorname {E} \left[e^{-sX}\right]\right\}(x)={\mathcal {L}}^{-1}\left\{{\frac {1}{s}}{\mathcal {L}}\{f\}(s)\right\}(x).} The Laplace transform can be alternatively defined in

5194-420: Is the unit circle. This contour can be used when the ROC includes the unit circle, which is always guaranteed when X ( z ) {\displaystyle X(z)} is stable, that is, when all the poles are inside the unit circle. With this contour, the inverse Z-transform simplifies to the inverse discrete-time Fourier transform , or Fourier series , of the periodic values of the Z-transform around

5300-613: Is useful for converting differentiation and integration in the time domain into much easier multiplication and division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering , mostly as a tool for solving linear differential equations and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic polynomial equations , and by simplifying convolution into multiplication . Once solved,

5406-427: Is usually no easy characterization of the range. Typical function spaces in which this is true include the spaces of bounded continuous functions, the space L (0, ∞) , or more generally tempered distributions on (0, ∞) . The Laplace transform is also defined and injective for suitable spaces of tempered distributions. In these cases, the image of the Laplace transform lives in a space of analytic functions in

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5512-628: Is where μ is a probability measure , for example, the Dirac delta function . In operational calculus , the Laplace transform of a measure is often treated as though the measure came from a probability density function f . In that case, to avoid potential confusion, one often writes L { f } ( s ) = ∫ 0 − ∞ f ( t ) e − s t d t , {\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0^{-}}^{\infty }f(t)e^{-st}\,dt,} where

5618-430: The x n {\displaystyle x^{n}} are formal powers of the symbol x {\displaystyle x} that is called an indeterminate or, commonly, a variable . Hence, power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that

5724-438: The Laplace transform , named after Pierre-Simon Laplace ( / l ə ˈ p l ɑː s / ), is an integral transform that converts a function of a real variable (usually t {\displaystyle t} , in the time domain ) to a function of a complex variable s {\displaystyle s} (in the complex-valued frequency domain , also known as s -domain , or s -plane ). The transform

5830-410: The completion of the polynomial ring R [ X ] {\displaystyle R[X]} equipped with a particular metric . This automatically gives R [ [ X ] ] {\displaystyle R[[X]]} the structure of a topological ring (and even of a complete metric space). But the general construction of a completion of a metric space is more involved than what

5936-491: The continuous variable case which was discussed by Niels Henrik Abel . From 1744, Leonhard Euler investigated integrals of the form z = ∫ X ( x ) e a x d x  and  z = ∫ X ( x ) x A d x {\displaystyle z=\int X(x)e^{ax}\,dx\quad {\text{ and }}\quad z=\int X(x)x^{A}\,dx} as solutions of differential equations, introducing in particular

6042-474: The gamma function . Joseph-Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions , investigated expressions of the form ∫ X ( x ) e − a x a x d x , {\displaystyle \int X(x)e^{-ax}a^{x}\,dx,} which resembles a Laplace transform. These types of integrals seem first to have attracted Laplace's attention in 1782, where he

6148-452: The natural numbers (taken to include 0). Designating a sequence whose term at index n {\displaystyle n} is a n {\displaystyle a_{n}} by ( a n ) {\displaystyle (a_{n})} , one defines addition of two such sequences by and multiplication by This type of product is called the Cauchy product of

6254-865: The region of convergence . The inverse Laplace transform is given by the following complex integral, which is known by various names (the Bromwich integral , the Fourier–Mellin integral , and Mellin's inverse formula ): f ( t ) = L − 1 { F } ( t ) = 1 2 π i lim T → ∞ ∫ γ − i T γ + i T e s t F ( s ) d s , {\displaystyle f(t)={\mathcal {L}}^{-1}\{F\}(t)={\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{\gamma -iT}^{\gamma +iT}e^{st}F(s)\,ds,}    ( Eq. 3 ) where γ

6360-1493: The ROC is the exterior of a circle, x ( n ) {\displaystyle x(n)} is causal (signal existing for n≥0). X ( z ) = 1 1 − 3 2 z − 1 + 1 2 z − 2 = 1 + 3 2 z − 1 + 7 4 z − 2 + 15 8 z − 3 + 31 16 z − 4 + . . . . {\displaystyle X(z)={1 \over 1-{3 \over 2}z^{-1}+{1 \over 2}z^{-2}}=1+{{3 \over 2}z^{-1}}+{{7 \over 4}z^{-2}}+{{15 \over 8}z^{-3}}+{{31 \over 16}z^{-4}}+....} thus, x ( n ) = { 1 , 3 2 , 7 4 , 15 8 , 31 16 … } ↑ {\displaystyle {\begin{aligned}x(n)&=\left\{1,{\frac {3}{2}},{\frac {7}{4}},{\frac {15}{8}},{\frac {31}{16}}\ldots \right\}\\&\qquad \!\uparrow \\\end{aligned}}} (arrow indicates term at x(0)=1) Note that in each step of long division process we eliminate lowest power term of z − 1 {\displaystyle z^{-1}} . Case 2: ROC: | Z | < 0.5 {\displaystyle \left\vert Z\right\vert <0.5} Since

6466-1126: The ROC is the interior of a circle, x ( n ) {\displaystyle x(n)} is anticausal (signal existing for n<0). By performing long division we get, X ( z ) = 1 1 − 3 2 z − 1 + 1 2 z − 2 = 2 z 2 + 6 z 3 + 14 z 4 + 30 z 5 + … {\displaystyle X(z)={\frac {1}{1-{\frac {3}{2}}z^{-1}+{\frac {1}{2}}z^{-2}}}=2z^{2}+6z^{3}+14z^{4}+30z^{5}+\ldots } x ( n ) = { 30 , 14 , 6 , 2 , 0 , 0 }     ↑ {\displaystyle {\begin{aligned}x(n)&=\{30,14,6,2,0,0\}\\&\qquad \qquad \qquad \quad \ \ \,\uparrow \\\end{aligned}}} (arrow indicates term at x(0)=0) Note that in each step of long division process we eliminate lowest power term of z {\displaystyle z} . Note: B) Determine

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6572-436: The Z-transform is expanded into a power series. This approach is useful when the Z-transform function is rational, allowing for the approximation of the inverse by expanding into a series and determining the signal coefficients term by term. This technique decomposes the Z-transform into a sum of simpler fractions, each corresponding to known Z-transform pairs. The inverse Z-transform is then determined by looking up each term in

6678-423: The Z-transform of the unit impulse response of a discrete-time causal system . An important example of the unilateral Z-transform is the probability-generating function , where the component x [ n ] {\displaystyle x[n]} is the probability that a discrete random variable takes the value. The properties of Z-transforms (listed in § Properties ) have useful interpretations in

6784-466: 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 inspection, manipulation, or numerical approximation. Such methods tend not to be accurate except in

6890-457: The base ring R {\displaystyle R} already comes with a topology other than the discrete one, for instance if it is also a ring of formal power series. In the ring of formal power series Z [ [ X ] ] [ [ Y ] ] {\displaystyle \mathbb {Z} [[X]][[Y]]} , the topology of above construction only relates to the indeterminate Y {\displaystyle Y} , since

6996-511: The bilateral Laplace transform is B { f } {\displaystyle {\mathcal {B}}\{f\}} , instead of F . Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero. This means that, on the range of the transform, there is an inverse transform. In fact, besides integrable functions, the Laplace transform is a one-to-one mapping from one function space into another in many other function spaces as well, although there

7102-415: The coefficient of Y {\displaystyle Y} . This asymmetry disappears if the power series ring in Y {\displaystyle Y} is given the product topology where each copy of Z [ [ X ] ] {\displaystyle \mathbb {Z} [[X]]} is given its topology as a ring of formal power series rather than the discrete topology. With this topology,

7208-460: The context of probability theory. The inverse Z-transform is: x [ n ] = Z − 1 { X ( z ) } = 1 2 π j ∮ C X ( z ) z n − 1 d z {\displaystyle x[n]={\mathcal {Z}}^{-1}\{X(z)\}={\frac {1}{2\pi j}}\oint _{C}X(z)z^{n-1}dz} where C {\displaystyle C}

7314-414: The corresponding power series diverges for any nonzero value of X . Algebra on formal power series is carried out by simply pretending that the series are polynomials. For example, if then we add A and B term by term: We can multiply formal power series, again just by treating them as polynomials (see in particular Cauchy product ): Notice that each coefficient in the product AB only depends on

7420-490: The decomposition of a function into its components in each frequency, the Laplace transform of a function with suitable decay is an analytic function , and so has a convergent power series , the coefficients of which give the decomposition of a function into its moments . Also unlike the Fourier transform, when regarded in this way as an analytic function, the techniques of complex analysis , and especially contour integrals , can be used for calculations. The Laplace transform

7526-607: The earlier Heaviside operational calculus . The advantages of the Laplace transform had been emphasized by Gustav Doetsch , to whom the name Laplace transform is apparently due. The Laplace transform of a function f ( t ) , defined for all real numbers t ≥ 0 , is the function F ( s ) , which is a unilateral transform defined by F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t , {\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,}    ( Eq. 1 ) where s

7632-656: The entire real axis. If that is done, the common unilateral transform simply becomes a special case of the bilateral transform, where the definition of the function being transformed is multiplied by the Heaviside step function . The bilateral Laplace transform F ( s ) is defined as follows: F ( s ) = ∫ − ∞ ∞ e − s t f ( t ) d t . {\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.}    ( Eq. 2 ) An alternate notation for

7738-469: The evaluation of the inverse -transform, This method involves applying the Cauchy Residue Theorem to evaluate the inverse Z-transform. By integrating around a closed contour in the complex plane, the residues at the poles of the Z-transform function inside the ROC are summed. This technique is particularly useful when working with functions expressed in terms of complex variables. In this method,

7844-414: The familiar formula An important operation on formal power series is coefficient extraction. In its most basic form, the coefficient extraction operator [ X n ] {\displaystyle [X^{n}]} applied to a formal power series A {\displaystyle A} in one variable extracts the coefficient of the n {\displaystyle n} th power of

7950-398: The following universal property . If S {\displaystyle S} is a commutative associative algebra over R {\displaystyle R} , if I {\displaystyle I} is an ideal of S {\displaystyle S} such that the I {\displaystyle I} -adic topology on S {\displaystyle S}

8056-870: The following table is a list of properties of unilateral Laplace transform: f ( t ) u ( t − a )   {\displaystyle f(t)u(t-a)\ } e − a s L { f ( t + a ) } {\displaystyle e^{-as}{\mathcal {L}}\{f(t+a)\}} f P ( t ) = ∑ n = 0 ∞ ( − 1 ) n f ( t − T n ) {\displaystyle f_{P}(t)=\sum _{n=0}^{\infty }(-1)^{n}f(t-Tn)} F P ( s ) = 1 1 + e − T s F ( s ) {\displaystyle F_{P}(s)={\frac {1}{1+e^{-Ts}}}F(s)} Formal power series In mathematics ,

8162-471: The integral ∫ 0 ∞ | f ( t ) e − s t | d t {\displaystyle \int _{0}^{\infty }\left|f(t)e^{-st}\right|\,dt} exists as a proper Lebesgue integral. The Laplace transform is usually understood as conditionally convergent , meaning that it converges in the former but not in the latter sense. The set of values for which F ( s ) converges absolutely

8268-754: The integral can be understood to be a (proper) Lebesgue integral . However, for many applications it is necessary to regard it as a conditionally convergent improper integral at ∞ . Still more generally, the integral can be understood in a weak sense , and this is dealt with below. One can define the Laplace transform of a finite Borel measure μ by the Lebesgue integral L { μ } ( s ) = ∫ [ 0 , ∞ ) e − s t d μ ( t ) . {\displaystyle {\mathcal {L}}\{\mu \}(s)=\int _{[0,\infty )}e^{-st}\,d\mu (t).} An important special case

8374-410: The integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on [0, ∞) . For locally integrable functions that decay at infinity or are of exponential type ( | f ( t ) | ≤ A e B | t | {\displaystyle |f(t)|\leq Ae^{B|t|}} ),

8480-455: The inverse Laplace transform reverts to the original domain. The Laplace transform is defined (for suitable functions f {\displaystyle f} ) by the integral L { f } ( s ) = ∫ 0 ∞ f ( t ) e − s t d t , {\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,} where s

8586-1889: The inverse Z-transform of the following by series expansion method, Eliminating negative powers if z {\displaystyle z} and dividing by z {\displaystyle z} , X ( z ) z = z 2 z ( z 2 − 1.5 z + 0.5 ) = z z 2 − 1.5 z + 0.5 {\displaystyle {\frac {X(z)}{z}}={\frac {z^{2}}{z(z^{2}-1.5z+0.5)}}={\frac {z}{z^{2}-1.5z+0.5}}} By Partial Fraction Expansion, X ( z ) z = z ( z − 1 ) ( z − 0.5 ) = A 1 z − 0.5 + A 2 z − 1 A 1 = ( z − 0.5 ) X ( z ) z | z = 0.5 = 0.5 ( 0.5 − 1 ) = − 1 A 2 = ( z − 1 ) X ( z ) z | z = 1 = 1 1 − 0.5 = 2 X ( z ) z = 2 z − 1 − 1 z − 0.5 {\displaystyle {\begin{aligned}{\frac {X(z)}{z}}&={\frac {z}{(z-1)(z-0.5)}}={\frac {A_{1}}{z-0.5}}+{\frac {A_{2}}{z-1}}\\[4pt]&A_{1}=\left.{\frac {(z-0.5)X(z)}{z}}\right\vert _{z=0.5}={\frac {0.5}{(0.5-1)}}=-1\\[4pt]&A_{2}=\left.{\frac {(z-1)X(z)}{z}}\right\vert _{z=1}={\frac {1}{1-0.5}}={2}\\[4pt]{\frac {X(z)}{z}}&={\frac {2}{z-1}}-{\frac {1}{z-0.5}}\end{aligned}}} Laplace transform In mathematics ,

8692-441: The latter is not an expression formed by the operations of addition and multiplication defined above (from which only finite sums can be constructed). This notational convention allows reformulation of the above definitions as and which is quite convenient, but one must be aware of the distinction between formal summation (a mere convention) and actual addition. Having stipulated conventionally that one would like to interpret

8798-509: The limit does not exist, so in particular it does not converge to This is because for i ≥ 2 {\displaystyle i\geq 2} the coefficient ( n i ) / n i {\displaystyle {\tbinom {n}{i}}/n^{i}} of X i {\displaystyle X^{i}} does not stabilize as n → ∞ {\displaystyle n\to \infty } . It does however converge in

8904-450: The lines Re( s ) = a or Re( s ) = b . The subset of values of s for which the Laplace transform converges absolutely is called the region of absolute convergence, or the domain of absolute convergence. In the two-sided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence: this is a consequence of Fubini's theorem and Morera's theorem . Similarly,

9010-423: The lower limit of 0 is shorthand notation for lim ε → 0 + ∫ − ε ∞ . {\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\varepsilon }^{\infty }.} This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it

9116-403: The mathematical framework of the Z-transform but also expanded its application scope, particularly in the field of electrical engineering and control systems. A notable extension, known as the modified or advanced Z-transform , was later introduced by Eliahu I. Jury . Jury's work extended the applicability and robustness of the Z-transform, especially in handling initial conditions and providing

9222-546: The powers of a single indeterminate by monomials in several indeterminates. Formal power series are widely used in combinatorics for representing sequences of integers as generating functions . In this context, a recurrence relation between the elements of a sequence may often be interpreted as a differential equation that the generating function satisfies. This allows using methods of complex analysis for combinatorial problems (see analytic combinatorics ). A formal power series can be loosely thought of as an object that

9328-403: The product topology of R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} where the topology of R {\displaystyle \mathbb {R} } is the usual topology rather than the discrete one, then the above limit would converge to exp ⁡ ( X ) {\displaystyle \exp(X)} . This more permissive approach

9434-415: The region of convergence, as the absolutely convergent Laplace transform of some other function. In particular, it is analytic. There are several Paley–Wiener theorems concerning the relationship between the decay properties of f , and the properties of the Laplace transform within the region of convergence. In engineering applications, a function corresponding to a linear time-invariant (LTI) system

9540-409: The right affect any fixed X n {\displaystyle X^{n}} . Infinite products are also defined by the topological structure; it can be seen that an infinite product converges if and only if the sequence of its factors converges to 1 (in which case the product is nonzero) or infinitely many factors have no constant term (in which case the product is zero). The above topology

9646-407: The right hand side as a well-defined infinite summation. To that end, a notion of convergence in R N {\displaystyle R^{\mathbb {N} }} is defined and a topology on R N {\displaystyle R^{\mathbb {N} }} is constructed. There are several equivalent ways to define the desired topology. Informally, two sequences (

9752-446: The same topology as one would get by taking formal power series in all indeterminates at once. In the above example that would mean constructing Z [ [ X , Y ] ] {\displaystyle \mathbb {Z} [[X,Y]]} and here a sequence converges if and only if the coefficient of every monomial X i Y j {\displaystyle X^{i}Y^{j}} stabilizes. This topology, which

9858-482: The second example given above, the coefficient of Y {\displaystyle Y} converges to 1 1 − X {\displaystyle {\tfrac {1}{1-X}}} , so the whole summation converges to Y 1 − X {\displaystyle {\tfrac {Y}{1-X}}} . This way of defining the topology is in fact the standard one for repeated constructions of rings of formal power series, and gives

9964-426: The sequence ( 0 , 1 , 0 , 0 , … ) {\displaystyle (0,1,0,0,\ldots )} by X {\displaystyle X} ; then using the above definitions every sequence with only finitely many nonzero terms can be expressed in terms of these special elements as these are precisely the polynomials in X {\displaystyle X} . Given this, it

10070-442: The set of all formal power series in X with coefficients in a commutative ring R , the elements of this set collectively constitute another ring which is written R [ [ X ] ] , {\displaystyle R[[X]],} and called the ring of formal power series in the variable  X over R . One can characterize R [ [ X ] ] {\displaystyle R[[X]]} abstractly as

10176-405: The set of values for which F ( s ) converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the region of convergence (ROC). If the Laplace transform converges (conditionally) at s = s 0 , then it automatically converges for all s with Re( s ) > Re( s 0 ) . Therefore, the region of convergence is a half-plane of the form Re( s ) >

10282-429: The topology that was put on Z [ [ X ] ] {\displaystyle \mathbb {Z} [[X]]} has been replaced by the discrete topology when defining the topology of the whole ring. So converges (and its sum can be written as X 1 − Y {\displaystyle {\tfrac {X}{1-Y}}} ); however would be considered to be divergent, since every term affects

10388-559: The turn of the century. Bernhard Riemann used the Laplace transform in his 1859 paper On the Number of Primes Less Than a Given Magnitude , in which he also developed the inversion theorem. Riemann used the Laplace transform to develop the functional equation of the Riemann zeta function , and this method is still used to related the modular transformation law of the Jacobi theta function , which

10494-480: The two sequences of coefficients, and is a sort of discrete convolution . With these operations, R N {\displaystyle R^{\mathbb {N} }} becomes a commutative ring with zero element ( 0 , 0 , 0 , … ) {\displaystyle (0,0,0,\ldots )} and multiplicative identity ( 1 , 0 , 0 , … ) {\displaystyle (1,0,0,\ldots )} . The product

10600-464: The unit circle: x [ n ] = 1 2 π ∫ − π + π X ( e j ω ) e j ω n d ω . {\displaystyle x[n]={\frac {1}{2\pi }}\int _{-\pi }^{+\pi }X(e^{j\omega })e^{j\omega n}d\omega .} The Z-transform with a finite range of n {\displaystyle n} and

10706-423: The useful property that an infinite summation converges if and only if the sequence of its terms converges to 0, which just means that any fixed power of X {\displaystyle X} occurs in only finitely many terms. The topological structure allows much more flexible usage of infinite summations. For instance the rule for multiplication can be restated simply as since only finitely many terms on

10812-399: The usual topology of R {\displaystyle \mathbb {R} } , and in fact to the coefficient 1 i ! {\displaystyle {\tfrac {1}{i!}}} of exp ⁡ ( X ) {\displaystyle \exp(X)} . Therefore, if one would give R [ [ X ] ] {\displaystyle \mathbb {R} [[X]]}

10918-600: The variable, so that [ X 2 ] A = 5 {\displaystyle [X^{2}]A=5} and [ X 5 ] A = − 11 {\displaystyle [X^{5}]A=-11} . Other examples include Similarly, many other operations that are carried out on polynomials can be extended to the formal power series setting, as explained below. Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra If one considers

11024-469: The vicinity of the complex unity, i.e. at low frequencies. The foundational concept now recognized as the Z-transform, which is a cornerstone in the analysis and design of digital control systems, was not entirely novel when it emerged in the mid-20th century. Its embryonic principles can be traced back to the work of the French mathematician Pierre-Simon Laplace , who is better known for the Laplace transform ,

11130-494: Was following in the spirit of Euler in using the integrals themselves as solutions of equations. However, in 1785, Laplace took the critical step forward when, rather than simply looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form ∫ x s φ ( x ) d x , {\displaystyle \int x^{s}\varphi (x)\,dx,} akin to

11236-476: Was instrumental in G H Hardy and John Edensor Littlewood 's study of tauberian theorems , and this application was later expounded on by Widder (1941), who developed other aspects of the theory such as a new method for inversion. Edward Charles Titchmarsh wrote the influential Introduction to the theory of the Fourier integral (1937). The current widespread use of the transform (mainly in engineering) came about during and soon after World War II , replacing

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