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Class Note for ECE 6341 with Professor Jackson at UH 2

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This 19 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15
ECE 6341 Spring 2009 Prof David R Jackson ECE Dept Notes 28 Asymptotic Evaluation of Integral Goal evaluate the following integral as Q becomes large gx is a real function here Q gtoo Case 1 g39x 7i 0 x Eab Integration by Parts Case 2 g39x0 0 x0 eab StationaryPhase Method Integration by Parts 19 f fxejggxalx Q gt 00 Note that ej gx Qg39xejf2gx So that we may write Ilts2gt ijltxgtj9x it 8mm dx Integration by Parts 1 d1 6mm dx IltQgt fltxgtmgltxgt x u T udvuvb vdu a la a Hence Integration by Parts cont ReimannLebesgue Lemma fFxengxdx gt O as Q gtoo The integrand oscillates faster so the integral tends to zero Hence Re fxejggx endpoint contributions i equot I n x interior parts cancel StationaryPhase Method 9 ffxejggxdx Assume gxgtgltxogt ogtltxxogt gquotltxogtltxxogt2 Re fxef9gx stationaryphase point Stationary Phase Method cont Conjecture x0A N JQgx I LWA fxe dx A gt 0 slow enough 1 Note We WI show later that III 0 V9 To ensure that the region outside of the SPP gives a negligible contribution we examine the following terms f x exogoc 1 M 198136 N I f x jogoc Stationary Phase Method cont We require that 1 1 left E I right 2 Approximating the first derivative of g by a Taylor series we have 1 1 ltlt 965mm 5 Hence This needs to be one of our assumptions Stationary Phase Method cont Hence we assume that x A A gt0 1 N I0 A fxeJQgXdx X0 A gtoo VVN x qu x A Stati onary Phase Method cont Since A 0 Assume z fx0 A Hence I N fx0 fol JQgxdX x0 Stationary Phase Method cont 0 1 8x8xoXxo gquotxoxxoz H 2 JQgx0 onFA 15g x0QX x0 IQfx0e 6 x0 A A i1 quot Q 2 or 1Qgtfxogtemgltxogt x e 1239g x039 x w dx x0 A where 8quotxo ilgquotxo Stationary Phase Method cont f Qgquotx0 2 AS Q gtOO SL gtoo since Adi gt00 1 l 2 Stationary Phase Method cont Note that SL 2 00 2 L f 6 ds gt J 6 ds AxQ 00 SL 00 2 2 Let M f eJS dSZ J eHs dS C 1m S C1 Deform the path 450 I I gt Res C C1 x Stationary Phase Method cont 7Z39 1 s2 tze 2 t2 7139 q ds 2 die 4 Stationary Phase Method cont 7I oo 2 j Similarly ILzzf e is dse 4J2 X3 1 5 7 Then I z fx0engX0 392 eiJZJ Note 124 Stationary Phase Method cont Hence Example 1 7r 1 Re 7239 005 sin 6d6 Eej singdg Example cont Im 9 g6sin6 2 396cos g gquot6 sme gquot 90 sin490 1lt0 Hence J0 N i Ree 21 614 7239 Example cont W m Willem Hence

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