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by: Sam Robel


Sam Robel
GPA 3.68


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Class Notes
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This 3 page Class Notes was uploaded by Sam Robel on Monday October 19, 2015. The Class Notes belongs to ECE 351 at Oregon State University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/224423/ece-351-oregon-state-university in Engineering Electrical & Compu at Oregon State University.

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Date Created: 10/19/15
Required to define roots of polynomials zzl0 and solutions of linear differential difference equations 62x0 0 as 963 One viewpoint is as pair of real numbers with linked arithmetic operations xxrxxr139xi yyyyrjy where j is by definition j J 1 Note physicists like to use i rather than j but to engineers this represents the symbol for current Conjugation x xrrxi xr Jxi Addition l Subtraction Multiplication nyxryr xiyixryi xiyr Modulus magnitude xxnx1 x xi212 positive and real x x XE xryr xiyi xiyr xryi By 2 y y y y y y 2 2 yr yi Division WARNING Complex Representation ajb can be ambiguous O O O O C M W M 1 I III Run HI IJI I I II VI ARBITRARY COMPLEX SIGNAL xtxrtbcitxrtxitatexpj6t CARTESIANtoPOLAR COORDINATES alttgtxlttgtxlttgtxlttgtl grggzdgggmzwagmkamagnitude 90 arctaHIXI txrtI phase component aka angle P POLARtoCARTESIAN COORDINATES xrt at cos 6t xi t atsin 6a BASIS FOR STATEMENT THAT IIQ SIGNALS ARE 90 OUT OF PHASE Complex Plane 00 sin6t 900 cos6t EULER39S RELATION FOR COMPLEX EXPONENTIALS expj6t 2 e19 C056tsin6t COSQQ jsin6t FULL COMPLEX EXPONENTIAL REPRESENTATION OF ANY COMPLEX SIGNAL xt atexpj6t atej6 em ej6 exp1nat j6t SUMMARY TWO EQUIVALENT COMPLEX SIGNAL REPRESENTATIONS 7Q xltrgtxltrgt at t 0 Complex Sinusoidal Signal Aexpj27zft of frequency f o ReaI SinusoidaI SignaI Complex Sinusoid Conjugate Complex Sinusoid with frequency f and Acos27rft expj27r exp j27t initial phase a1tAZ a1tAZ 51Y2 51Y 2 0 Real Damped SInUSOidal Aexp at cos27rft exp at j27rft expat j27rft Signal note exp argument A A is linear function of t 18ng 2833 o Arbitrary Real Signal xt x I 700 6XP 190 WU 6XP J399t 39 nOte exp argu ment 395 analytic signal conjugate analytic signal arbitrary fu nction of t contains only frequencies contains only frequencies at envelope function This decomposition in not unique Made unique by forcing separation of positive and negative frequencies


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