Electrical Engineering 20N - Spring 1999 - Midterm 1

Electrical Engineering 20N - Spring 1999 - Midterm 1 - EECS...

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EECS 20. Solution to Midterm 1. 5 March 1999 1. 24 points Fill in the blanks: (a) If A = { 1 , 2 , 3 } , B = { 3 , 4 , 5 } ,then A B = { 3 } and A B = { 1 , 2 , 3 , 4 , 5 } (b) If the predicates P, Q, R all evaluate to true then the predicate [ ¬ P ∧¬ Q ] ( ¬ R ) evaluates to false . (c) If f : X Y and g : Y X f g : Y Y and g f : X X 2. 24 points Fill in the blanks: (a) Euler’s formula is exp =cos( θ )+ i sin( θ ) . (b) The four distinct roots of the equation z 4 =1 are 1 , - 1 ,i, - i . (c) If t Reals cos( t + π/ 2) = Re [ A exp( jt )] then A = j. 3. 20 points Find the period and the fundamental frequency of the following functions and then give their Fourier series representation as a sum of complex exponentials. All functions are denoted x : Time Comps , and time is in seconds. (a) t x ( t )=10sin(2 πt )+10cos(2 ) =(5 - 5 j ) e j 2 +(5+5 j ) e - j 2 . The fundamental frequency is 1 Hz and the fundamental period is 1 sec. (b) t x ( t )=(1+ i )sin(2 )+sin( ) = - i/ 2 × e iπt + i/ 2 × e - iπt +(1 - i ) / 2 × e i 2 - (1 - i ) / 2 × e - i 2 .
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Electrical Engineering 20N - Spring 1999 - Midterm 1 - EECS...

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