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Convolution - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Sysytems The Dirac Delta Function and Convolution 1 The Dirac Delta (Impulse) Function The Dirac delta function is a non-physical, singularity function with the following deFnition δ ( x )= ( 0f o r x 6 =0 undeFned at x (1) but with the requirement that Z −∞ δ ( x ) dx =1 , (2) that is, the function has unit area. 0 TT T T 1 2 3 4 a) Unit pulses of different extents b) The impulse function 1/T 1 1/T 2 1/T 3 1/T 4 t d (t) 0 t d (x) X ±igure 1: Unit pulses and the Dirac delta function. ±igure 1 shows a unit pulse function δ T ( t ), that is a brief rectangular pulse function of duration T , deFned to have a constant amplitude 1 /T over its extent, so that the area T × 1 /T under the pulse is unity: δ T ( t o r t 0 1 /T 0 <t T o r t> 0. (3) The Dirac delta function (also known as the impulse function) can be deFned as the limiting form of the unit pulse δ T ( t ) as the duration T approaches zero. As the duration T of δ T ( t ) decreases, the amplitude of the pulse increases to maintain the requirement of unit area under the function, and δ ( t ) = lim T 0 δ T ( t ) . (4) The impulse is therefore deFned to exist only at time t = 0, and although its value is strictly undeFned at that time, it must tend toward inFnity so as to maintain the property of unit area in the limit. The strength of a scaled impulse ( t ) is deFned by its area K . The limiting form of many other functions may be used to approximate the impulse. Common functions include triangular, gaussian, and sinc (sin( x )/ x ) functions. 1

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The impulse function is used extensively in the study of linear systems, both spatial and tem- poral. Although true impulse functions are not found in nature, they are approximated by short duration, high amplitude phenomena such as a hammer impact on a structure, or a lightning strike on a radio antenna. As we will see below, the response of a causal linear system to an impulse deFnes its response to all inputs. An impulse occurring at t = a is δ ( t a ). 1.1 The “Sifting” Property of the Impulse When an impulse appears in a product within an integrand, it has the property of ”sifting” out the value of the integrand at the point of its occurrence: Z −∞ f ( t ) δ ( t a ) dt = f ( a )( 5 ) This is easily seen by noting that δ ( t a ) is zero except at t = a
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Convolution - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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