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In this analysis, the Laplace transform is often interpreted as a transformation from the time-domain, in which inputs and outputs are functions of time, to the frequency-domain, where the same inputs and outputs are functions of complex angular frequency, in radians per unit time. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications. Formal definition The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), defined by: The parameter s is a complex number: with real numbers σ and ω. The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that ƒ must be locally integrable on [0,∞). For locally integrable functions that decay at infinity or are of exponential type, the integral can be understood as 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. Properties of the unilateral Laplace transform Time domain Linearity Frequenc y differenti 's' domain Comment Can be proved using basic rules of integration. is the first derivative of . ation Frequenc y differenti ation More general form, (n)th derivative of F(s). ƒ is assumed to be a Differenti ation differentiable function, and its derivative is assumed to be of exponential type. This can then be obtained by integration by parts ƒ is assumed twice Second Differenti ation differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to General Differenti ation . ƒ is assumed to be n-times differentiable, with nth derivative of exponential type. Follow by mathematical induction. Frequenc y integratio n u(t) is the Heaviside step function. Note (u Integratio n * f)(t) is the convolution of u(t) and f(t). Scaling where a is positive. Frequenc y shifting u(t) is the Heaviside Time shifting step function ƒ(t) and g(t) are Convoluti on extended by zero for t<0 in the definition of the convolution. f(t) is a periodic function of period T Periodic Function so that . This is the result of the time shifting property and the geometric series. Initial value theorem: Final value theorem: , if all poles of sF(s) are in the left-hand plane. The final value theorem is useful because it gives the long-term behaviour without having to perform partial fraction decompositions or other difficult algebra. If a t function's poles are in the right-hand plane (e.g. e or sin(t)) the behaviour of this formula is undefined. A causal system is a system where the impulse response h(t) is zero for all time t prior to t = 0. In general, the region of convergence for causal systems is not the same as that of anticausal systems. ID Function 1 ideal delay 1a unit impulse delayed nth power 2 with frequency shift 2a nth power ( for integer n ) 2a. qth power 1 ( for complex q ) 2a. 2 unit step 2b delayed unit step 2c ramp Time domain Laplace s-domain 1 Region of convergence 2d nth power with frequency shift 2d. exponential decay 1 3 exponential approach 4 sine 5 cosine 6 hyperbolic sine 7 hyperbolic cosine Exponentially-dec 8 aying sine wave Exponentially-dec 9 aying cosine wave 10 nth root 11 natural logarithm Bessel function 12 of the first kind, of order n Modified Bessel 13 14 function of the first kind, of order n Bessel function of the second kind, of order 0 15 Modified Bessel function of the second kind, of order 0 16 Error function Explanatory notes: represents the Heaviside step function. represents the Dirac delta function. represents the Gamma function. is the Euler–Mascheroni constant. , a real number, typically represents time, although it can represent any independent dimension. is the complex angular frequency, and Re{s} is its real part. , , , and are real numbers. , is an integer. [edit] s-Domain equivalent circuits and impedances The Laplace transform is often used in circuit analysis, and simple conversions to the s-Domain of circuit elements can be made. Circuit elements can be transformed into impedances, very similar to phasor impedances. Here is a summary of equivalents: Note that the resistor is exactly the same in the time domain and the s-Domain. The sources are put in if there are initial conditions on the circuit elements. For example, if a capacitor has an initial voltage across it, or if the inductor has an initial current through it, the sources inserted in the s-Domain account for that. The equivalents for current and voltage sources are simply derived from the transformations in the table above. Nyquist–Shannon sampling theorem From Wikipedia, the free encyclopedia Jump to:navigation, search Fig.1: Hypothetical spectrum of a bandlimited signal as a function of frequency The Nyquist–Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal (for example, a function of continuous time or space) into a numeric sequence (a function of discrete time or space). Shannon's version of the theorem states:[1] If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart. The theorem is commonly called the Nyquist sampling theorem, and is also known as Nyquist–Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov, Whittaker–Nyquist–Kotelnikov–Shannon, WKS, etc., sampling theorem, as well as the Cardinal Theorem of Interpolation Theory. It is often referred to as simply the sampling theorem. In essence, the theorem shows that a bandlimited analog signal that has been sampled can be perfectly reconstructed from an infinite sequence of samples if the sampling rate exceeds 2B samples per second, where B is the highest frequency in the original signal. If a signal contains a component at exactly B hertz, then samples spaced at exactly 1/(2B) seconds do not completely determine the signal, Shannon's statement notwithstanding. This sufficient condition can be weakened, as discussed at Sampling of non-baseband signals below.