Integrals, Unit Step, etc.
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Signals and Systems – Final Exam
Date Due: April 22, 2013, Monday @4:00 PM
Total = 100 points
Each question has three parts and each student will be answering the part of each question assigned to him as indicated below:
Question 1: Express the following complex numbers in polar form: (10 points)
3 + j4 b. -100 + j46 c. -23 + j7
Question 2: Let Z1 = 7 + j5 and Z2 = -3 + j4.
Determine the following in both Cartesian and Polar form: (10 points)
Z1/Z2 b. Z1*Z2 c. (Z1-Z2)/(Z1+Z2)
Question 3: Classify the signals below as periodic or aperiodic. If periodic, then identify the period. (15 points)
x(t) = cos(4t) + 2sin(8t) b. x(t) = 3cos(4t) + sin(pt) c. x(t) = cos(3pt) + 2cos(4pt)
Question 4: Determine if the following systems are time-invariant, linear, causal, and/or memoryless? (15 points)
a) dy/dt + 6 y(t) = 4 x(t) b) dy/dt + 4 y(t) = 2 x(t) c. y(t) = sin(x(t))
Question 5: Solve the following difference equations using recursion first by hand (for n=0 to n=4) and then plot. Check your solution using Maple or MATLAB (for n=0 to n=30). Attach plots to your solution. (20 points)
a) y[n] + 0.5y[n-1] = 2x[n-1]; x[n] = d[n], y[-1] = 0
b. y[n] + 2y[n-1] = 2x[n-1]; x[n] = d[n], y[-1] = 0
c) y[n] + 1.2y[n-1] + 0.32y[n-2] = x[n]-x[n-1]; x[n] = u[n], y[-2] = 1, y[-1]=2
Question 6. Solve the di?erential equations: (15 points)
x’’ + 4x’ + 13x = 0; x(0) = 3, x’(0) = 0
x’’ + 6x’ + 9x = 50 sin(t); x(0) = 1, x’(0) = 4
x’’ + 4x’ – 3x = 4et; x(0) = 1, x’(0) = -2
Question 7: Find the Fourier series of the function: (15 points)