solutions_test1_s1_f05

solutions_test1_s1_f05 - ECE 220 Sections 001 002 Test 1...

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ECE 220, Sections 001, 002, Test 1 Solutions 22 September 2005 Problem 1. (15 points) Consider the graph of the signal s ( t ) shown in figure 1. The graph is accurate. Write down a mathematical expression for this signal, using (time-shifted and scaled, if need be) versions of the δ ( t ) signal. -40 -30 -20 -10 0 10 20 30 40 -1 0 1 2 3 4 5 6 7 8 9 10 t --> s(t) The signal s(t) Figure 1: The signal s ( t ) in problem 1. Solution: The signal s ( t ) contains 6 δ ( · ) impulses, equally spaced; the time spacing is 10 units. The impulses (from left to right) can then be expressed as δ ( t + 20) , δ ( t + 10) , δ ( t ) , δ ( t - 10) , δ ( t - 20) , δ ( t - 30) , or, more compactly, as: δ ( t - 10 k ) , k = - 2 , - 1 , 0 , 1 , 2 , 3 . So, finally, s ( t ) = 8 3 X k = - 2 δ ( t - 10 k ) 1
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Problem 2. (10 points) Determine whether T = 3 is a period of the signal f ( t ) = 3 cos(2 π 100 t + π/ 4) · cos(2 πt - π/ 4) Justify your answer. Guesses will not count! Solution: We need to check whether f ( t + 3) = f ( t ) . We have f ( t + 3) = 3 cos(2 π 100( t + 3) + π/ 4) · cos(2 π ( t + 3) - π/ 4) = 3 cos(2 π 100 t + 600 π + π/ 4) · cos(2 πt + 6 π - π/ 4) = 3 cos(2 π 100 t + π/ 4) · cos(2 πt - π/ 4) = f ( t ) and thus T = 3 is indeed a period.
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