University Problem Set #10 ECE 313
of Illinois Page 1 of 3 Spring 1999
Assigned: Wednesday, March 31, 1999
Due: Friday, April 9, 1999
Reading: Ross, Chapter 5, and Appendix of my Lecture Notes for ECE 313
Reminders: There is no class on Wednesday April 7 (time given off in lieu of the upcoming
evening Hour Exam: Wednesday April 14, 7:00 pm — 8:00 pm in 269 EL)
Noncredit Exercises: Ross, pp. 173-184: 17-19, 40-43, 49-56 70, 73; pp. 184-188: 5, 25-28;
pp. 232-237: 30, 31, 35-39; pp. 237-241: 14, 28, 29.
Problems:
1. A signal X is modeled as a unit Gaussian random variable. For some applications,
however, only the quantized value Y (where Y = α if X > 0 and Y = –α if X ≤ 0) is used.
Note that Y is a discrete random variable.
(a) What is the pmf of Y?
(b) Suppose that α = 1. If the signal X happens to have value 1.29, what is the error made in
representing X by Y? What is the squared-error? Repeat for the case when X happens to
have value π/4 and when X happens to have value –π/4.
(c) We wish to design the quantizer so as to minimize the squared-error. However, since X
(and Y) are random, we can only minimize the squared-error in the probabilistic (that is,
average) sense. Now, part (b) shows that the squared-error depends on the value of X,
and can be expressed as Z = (X – Y)2 = g(X) =
(X – α)2if X > 0
(X + α)2if X ≤ 0.
So we want to choose α so that E[Z] is as small as possible. Use LOTUS to easily find
E[Z] as a function of α, and then find the value of α that minimizes E[Z].
(d) We now get more ambitious and use a 3-bit A/D converter which first quantizes X to the
nearest integer W in the range –3 to +3. Thus, W = 3 if X ≥ 2.5, W = 2 if 1.5 ≤ X < 2.5,
etc. Note that W is a discrete random variable. Find the pmf of W.
(e) The output of the A/D converter is a 3-bit 2's complement representation of W. Suppose
that the output is (Z2, Z1, Z0). What is the pmf of Z2? of Z1? of Z0?
(f) Noncredit exercise (but a real-life engineering problem!): Suppose that W takes on
values –3α, –2α, –α, 0, +α, +2α, +3α and quantization is as before: X is mapped to the
nearest W value. What value of α minimizes E[(X – W)2]?
2. [Read Example 3d on pp. 203-204 first.] Let the (straight) line segment ACB be a
diameter of a circle of unit radius and center C. Consider an arc AD of the circle where
the length X of the arc (measured clockwise around the circle) is a random variable
uniformly distributed on [0,2π). Now consider the “random chord” AD.
(a) Find the probability that the length L of the random chord is greater than the side of the
equilateral triangle inscribed in the circle.
(b) Express L as a function of the random variable X, and find the probability density
function for L.
3. X is a continuous random variable with pdf fX(u) = 0.5 exp (– |u|), –∞ < u < ∞.
(a) What is the value of P{X ≤ ln 2}?
(b) Find the conditional probability that P{|X| ≤ ln 2} given that {X ≤ ln 2}.
(c) Now suppose that X denotes the voltage applied to a semiconductor diode, and that the
current Y is given by Y = eX – 1. Find the pdf of Y.
4. Raw scores on the SAT (and GRE) are transformed by a nonlinear function so that the
minimum score is 200 and the maximum is 800. The histogram of scores resembles a
Gaussian pdf with mean 500 and variance ß2 = 1002, that is, the score X of a student
