Electrical Engineering 20N - Fall 1999 - Midterm 1

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

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EECS 20. Solution to Midterm No. 1, October 13, 1999. 1. 20 points Fill in the blanks: (a) If A = { 1 , 2 , 3 } , B = { 2 , 3 , , # } , then A B = { 2 , 3 } and A B = { 1 , 2 , 3 , , # } . (b) If the predicates P, Q, R all evaluate to false , then [ ¬ P Q ] [ ¬ Q R ] [ ¬ R P ] evaluates to false . (c) If f : X Y and g : Y Z , then g f : X Z . (d) Euler’s formula is exp = cos( θ ) + i sin( θ ) . (e) If A cos( ωt + θ ) = cos( ωt + π/ 4) + cos( ωt π/ 4), then A = 2 , θ = 0 . 2. 20 points Determine which of the following functions are periodic and what is their period in seconds or samples. (a) n Ints, x ( n ) = cos(2 πn/ 111) . Periodic, with period 111 samples . (b) n Ints, x ( n ) = cos(2 π 2 n ) . Not periodic . (c) t Reals, x ( t ) = cos(2 π 2 t ) . Periodic with period 1 / 2 sec . (d) t Reals, x ( t ) = exp(2 π 60 t + π/ 4) . Periodic with period 1/60 sec . 1
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3. 20 points Consider a discrete-time LTI system H : [ Ints Comps ] [ Ints Comps ] such that for input signal x , the output signal y is: n Ints, y ( n ) = x ( n ) + x ( n 1) .
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