ECE 3084 QUIZ SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY MARCH 3, 07 Name:. The quiz is closed book, closed notes, except for one -sided sheet of handwritten notes.. Turn off your phone and put it away. No tablets/laptops/wifi/etc. No calculators. 3. Final answers must be entered into the answer box. 4. Correct answers must be accompanied by concise justifications to receive full credit. 5. Do not attach additional sheets. If necessary, use the back of the previous page. Problem Points Score 30 40 3 30 TOTAL: 00 0
PROBLEM. (30 points) Consider the following system, where a signal x( t ) (consisting of the sum of a sinc and a sincsquared function) is passed as an input into a back-to-back connection of an ideal analog-to-digital converter (ADC) followed by an ideal digital-to-analog (DAC) converter: x( t ) = sint sin t ------------------ t + ------------------ t x( t ) ADC x[ n ] DAC y( t ) t f s (a) In the space below, sketch the Fourier transform of y( t ) when the sampling rate is f s = 4 Hz: Y(j) 8 4 0 4 8. (b) In the space below, sketch the Fourier transform of y( t ) when the sampling rate is f s = Hz: Y(j) 8 4 0 4 8
PROBLEM. (40 points) Suppose x( t ) = ------------------ sin t is AM modulated onto a 0 carrier, producing x ( t ): t sin t x( t ) = ------------------ t x ( t ) x ( t ) LPF CUTOFF GAIN x 3 ( t ) x 4 ( t ) LPF y( t ) CUTOFF GAIN cos(0t) cos( t) cos(t) (a) Each of the dashed boxes above is an AM demodulator: The first one (by itself) would demodulate when the carrier is rad/s, while the second one (by itself) would demodulate when the carrier is 0 rad/s. Together, with a proper choice for, the two can be used to demodulate when the carrier is 0 rad/s. Find a value for so that the overall output recovers the input, y( t ) = x( t ): = rad/s. (b) Using your answer from part (a), sketch in the space below the Fourier transform of each of the signals x ( t ) through x 4 ( t ) and y( t ): X (j) 4 6 8 0 8 6 4 X (j) 4 6 8 X 3 (j) 0 8 6 4 4 6 8 X 4 (j) 0 8 6 4 4 6 8 Y(j) 0 8 6 4 4 6 8 0 8 6 4
PROBLEM 3. (a) The Laplace transform of x( t ) = ( t ) u( t ) is X( s ) =. (b) Subject to the initial condition y(0 ) = 4, the differential equation -- ---- d y( t ) = 3u( t ) y( t ) dt (where u( t ) is the unit step) has the following solution: y( t ) = (A + Be t + Ce t + De 3t )u( t ), where: A = B = C = D = 3
ECE 3084 QUIZ SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY MARCH 3, 07 Name: SOLUTIONS. The quiz is closed book, closed notes, except for one -sided sheet of handwritten notes.. Turn off your phone and put it away. No tablets/laptops/wifi/etc. No calculators. 3. Final answers must be entered into the answer box. 4. Correct answers must be accompanied by concise justifications to receive full credit. 5. Do not attach additional sheets. If necessary, use the back of the previous page. Problem Points Score 30 40 3 30 TOTAL: 00 0
PROBLEM. (30 points) Consider the following system, where a signal x( t ) (consisting of the sum of a sinc and a sincsquared function) is passed as an input into a back-to-back connection of an ideal analog-to-digital converter (ADC) followed by an ideal digital-to-analog (DAC) converter: sint sin t x( t ) = ------------------ t + ------------------ t x( t ) ADC x[ n ] DAC y( t ) t f s (a) In the space below, sketch the Fourier transform of y( t ) when the sampling rate is f s = 3 Hz: Y(j).5 0.5 8 4 0 4 8 (b) In the space below, sketch the Fourier transform of y( t ) when the sampling rate is f s = Hz: Y(j) 8 4 0 4 8
PROBLEM. (40 points) Suppose x( t ) = ------------------ sin t is AM modulated onto a 8 carrier, producing x ( t ): t sin t x( t ) = ------------------ t x ( t ) x ( t ) LPF CUTOFF GAIN x 3 ( t ) x 4 ( t ) LPF y( t ) CUTOFF GAIN cos(8t) cos( t) cos(t) (a) (b) Each of the dashed boxes above is an AM demodulator: The first one (by itself) would demodulate when the carrier is rad/s, while the second one (by itself) would demodulate when the carrier is 0 rad/s. Together, with a proper choice for, the two can be used to demodulate when the carrier is 8 rad/s. Find a value for so that the overall output recovers the input, y( t ) = x( t ): = 8 Using your answer from part (a), sketch in the space below the Fourier transform of each of the signals x ( t ) through x 4 ( t ) and y( t ): rad/s. X (j) 0.5 4 6 8 0 8 6 4 X (j) 0.5 4 6 8 X 3 (j) 0 8 6 4 0.5 4 6 8 X 4 (j) 0 8 6 4 0.5 0.5 0.5 4 6 8 Y(j) 0 8 6 4 4 6 8 0 8 6 4
PROBLEM 3. (a) The Laplace transform of x( t ) = ( t ) u( t 4) is X( s ) = 6 ----. s 3 + ---- s + 9 -- e 4s s x( t ) = ( t t + )u( t 4) = ( (t 4) + 6t 5)u( t 4) = (t 4) u( t 4) + 6(t 4)u( t 4) + 9u( t 4) X( s ) = ---- s 3 + ---- 6 9 s + -- e 4s s (b) Subject to the initial condition y(0 ) = 4, the differential equation -- ---- d y( t ) = 4u( t ) y( t ) dt (where u( t ) is the unit step) has the following solution: where: y( t ) = (A + Be t + Ce t + De 3t )u( t ), A = 4 B = 0 C = 0 D = 0 ---- d dt y( t ) = 8u( t) y( t) 8 sy( s ) 4 = -- Y( s) s 4 Y( s )(s + ) = -- (s + ) s 4 Y( s ) = --. ---------------- s + 4 = -- s s + s 3