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Electrical Engineering 20N - Fall 2000 - Midterm 2

# Electrical Engineering 20N - Fall 2000 - Midterm 2 - 1...

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1 EECS20n, Solution to Midterm 2, 11/17/00 1. 10 points Write the following in Cartesian coordinates (i.e. in the form x + jy ) (a) 2 points j 3 j 2 + j +1= j +1+ j +1=2 (b) 2 points (1 j 1) / (1 + j 1) = j (c) 2 points p cos π/ 4+ j sin 4= ± (cos 8+ j sin 8) Write the following in polar coordinates (i.e. in the form re ) (a) 2 points 1+ j 1= 2 e jπ/ 4 (b) 2 points (1 + j 1) / (1 j 1) = e 2

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2 2. 10 points Which of the following discrete-time or continuous-time signals is periodic. An- swer yes or no. If the signal is periodic, give its fundamental period and state the units. Suppose that for a discrete-time signal, n denotes seconds , and for a continuous-time signal, t denotes minutes . (a) 2 points n Ints ,x ( n )= e 2 n Periodic NO; (b) 2 points t Reals ( t e 2 t Periodic YES; Period = 2 π/ 2 min (c) 2 points n Ints ( n )=cos3 πn +sin(3 + 7) Periodic YES; Period = 4 sec 4 points Find A,θ,ω in the following expression: A cos( ωt + θ )=c o s ( 2 π × 10 , 000 t + π 4 )+sin(2 π × 10 , 000 t + π 4 ) = 2cos(2 π × 10000 t ) So A = 2 = 20000 =0 .
3 |H 1 ( ω)| |H 2 ( ω)| H 1 ( ω) H 2 ( ω) ω ω ω ω 1 2 1/2 1 π/4 π/2 1 2 π 2 π− periodic Figure 1: Plots for Problem 3 3. 10 points On Figure 1 plot the amplitude and phase response of the following frequency responses. On your plots carefully mark the values for ω =0 and for one other non-zero value of ω . (a) 4 points ω Reals ,H 1 ( ω )=1+ (b) 4 points ω Reals 2 ( ω )=1+cos ω The frequency response is plotted only for ω> 0 since | H ( ω ) | is even and 6 H ( ω ) is odd.

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Electrical Engineering 20N - Fall 2000 - Midterm 2 - 1...

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