hw+3+2011 - EECS 70B Spring 2011 Due Wednesday Week 4...

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EECS 70B Spring 2011 Due: Wednesday Week 4 Homework 3 1. Phase lead/lag between 2 sinusoids : Problem 9.6 (a) 2*. Differential & Integral of exponential functions with complex frequency: Carry out the following differentiation/integration by expressing cos Re{ } t zt e t e - = α ϖ and sin Im{ } t zt e t e - = α ϖ where z j = - + α ϖ . Then do the integration/differential with respect to the complex variable z first and then take the real or imaginary part of your results in complex form. * * ( ) cos ( ) sin ( ) cos ( ) sin t t t t d a e t dt d b e t dt c e t dt d e t dt - - - - α α α α ϖ ϖ ϖ ϖ 3*. General form of a Sinusoid (a) Show that cos sin cos( ) A t B t C t + = + ϖ ϖ ϖ φ ( A and B are real number) where 2 2 1 and ( / ) C A B Tan B A φ - = + = - ) Hint: The most elegant way to prove this is to (1) convert time-domain signals cos A t ϖ and sin B t ϖ to their phasor domain representatives, (2) add up these two phasors and express the sum as a new phasor and (3) then convert it back to time-domain representation.
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