Homework Assignment 2
Due Sept. 10, 1997

1. In the following problems, be sure to normalize discrete-time frequencies so they are between -pi and +pi.
   1. A periodic signal x(t) contains frequencies up to 1 kHz. At what rate must x(t) be sampled in order to reconstruct it unambigouusly?
   2. A signal x(t) = cos(2 pi 2000 t) is sampled at 6 kHz. What is the discrete-time frequency of x(n)?
   3. A signal x(t) = cos(2 pi 2000 t) is sampled at 1.5 kHz. What is the discrete-time frequency of x(n)?
   4. A signal x(t) = sin(2 pi 100 t) is sampled at 1 kHz. Find another signal y(t) which, when sampled at 1 kHz, will have y(n) = x(n).
   5. A discrete-time signal x(n) = cos(0.5 pi) + 2 cos(0.25 pi) is sent to an ideal reconstructor at a rate of 10 kHz. What is the output x(t) of the system?
2. Problem 1.13 from Orfanidis. Plot the spectra in dB.
3. Using the equation for |X(f)|² from the above problem, plot x(t) and |X(f)|² for the following values of a and f_o:
   1. a = 1, f_o=1.
   2. a = 2, f_o=1.
   3. a = 1, f_o=2.
   4. a = 2, f_o=2.
   Discuss the effect of changing a and f_o.