quiz1v1_s09_soln

# One page 8 1 2 00 11 00 of hand written notes

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One page ( 8 1 2 00 11 00 ) of HAND-WRITTEN notes permitted. OK to write on both sides. JUSTIFY your reasoning clearly to receive partial credit. Explanations are also REQUIRED to receive FULL credit for any answer. You must write your answer in the space provided on the exam paper itself. Only these answers will be graded. Circle your answers, or write them in the boxes provided. If space is needed for scratch work, use the backs of previous pages. Problem Value Score 1 40 2 30 3 30 No/Wrong Rec 3

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PROBLEM sp-09-Q.1.1: (a) The signal plotted below is the sum of two sinusoids; determine the amplitudes and phases, as well as the frequencies in rad/s. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -5 -4 -3 -2 -1 0 1 2 3 Time t (sec) 1 C 4 cos ..2 =8/t C 3 =8/ (b) Express this signal as a sinusoid in standard form, i.e., A cos .! 0 t C '/ . s a .t/ D 5e j =3 e j 30 t C 5e j =3 e j 30 t Pick off the values from the positive frequency component to get s a .t/ D 10 cos .30 t =3/ (c) Express this signal as a sinusoid in standard form, i.e., A cos .! 0 t C '/ . s b .t/ D < e n . 10 j 30/e j 631t o The complex amplitude in polar form is . 10 j 30/ D 31:62e j1:893 , so s b .t/ D 31:62 cos .631t 1:893/
PROBLEM sp-09-Q.1.2: The following M ATLAB code defines a periodic signal as the sum of sinusoids: tt = -10:0.00001:10; %- in seconds xxt = -5*ones(size(tt)); kk=12; xxt = xxt + (30/kk)*cos( 1.2*pi*kk*(tt-1/6) ); kk=10; xxt = xxt + real( (-5-12/j)*exp(j*1.6*pi*kk*tt) ); In the following questions, the signal x.t/ corresponds to the M ATLAB vector xxt . (a) Determine the fundamental frequency of x.t/ . ! 0 D 1:6 rad/s The signal x.t/ is the sum of three sinusoids with frequencies: 0 , 12.1:2 / D 14:4 , and 10.1:6 / D 16:0 . Using the greatest common divisor (GCD), we obtain the fundamental frequency as ! 0 D 0:1 gcd .0;144;160/ D 1:6 rad/s (b) The signal x.t/ is periodic, so it has a Fourier Series x.t/ D 1 X k D 1 a k e j! 0 kt Determine the values of all the nonzero coefficients f a k g , and list them in the table below. Write the Fourier coefficients in polar form as a k D k e j' k , where k 0 and < ' k . x.t/ D 5 C 2:5 cos .9.1:6 /t 12.1:2/ =6/ C< e f 13e j1:966 e j10.1:6 /t g We have DC ( k D 0/ , and the ninth and tenth harmonics. The phase of the second term is 2:4 , but must be changed to 0:4 to be in the correct range. Index k th Fourier coefficient k a k in polar form 0 5e j 9 1:25e j 0:4 9 1:25e j 0:4 10 6:5e j1:966 10 6:5e j1:966

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PROBLEM sp-09-Q.1.3: (a) Recall Lab #2 where one sinusoid is broadcast from a transmitter, but a vehicle receives two signals, one
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• Fall '08
• JUANG

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