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ECE 630: Statistical Communication Theory

Prof. B.-P. Paris

Homework 1

Due: February 2, 2017

- Reading
- Madhow: Appendix A and Section 3.1.
Note: the material in sections A.1 and A.2 has been covered in ECE 528 and you are expected to be familiar and comfortable with that material.

- Problems
- These problems are review problems for probability and random
variables.
- A noisy discrete communication channel is available. Once each second
one letter from the three-letter alphabet {a,b,c} can be transmitted
and one letter from the three-letter alphabet {1, 2, 3} is received. The
conditional probabilities of the various received letters, given the
various transmitted letters are specified by the diagram in the
accompanying diagram.
The source sends a, b, and c with the following probabilities:

- Compute all (nine) conditional probabilities of the form P(X|Y ) for X ∈{a,b,c} and Y ∈{1, 2, 3}.
- Compute all (nine) joint probabilities of the form P(X,Y ) for X ∈{a,b,c} and Y ∈{1, 2, 3}.
- A receiver makes decisions as follows:
- If 1 is received, decide a was sent.
- If 2 is received, decide b was sent.
- If 3 is received, decide c was sent.

What is the probability that this receiver makes a wrong decision? (I.e.., its decision is different from what was actually sent.)

- What is the best receiver decision rule (assignment from 1, 2, 3 to a, b, c)?
- What is the resulting probability of error?

- Consider a random variable X having a double-exponential (Laplacian)
density,
where a and b are positive constants.

- Determine the relationship between a and b such that p
_{X}(x) is a valid density function. - Determine the corresponding probability distribution function
P
_{X}(x). - Find the probability that the random variable lies between 2 and 3.
- What is the probability that X lies between 2 and 3 given that the magnitude of X is less than 3.

- Determine the relationship between a and b such that p
- Let x
_{1}, x_{2}, …, x_{N}be a set of N identically distributed statistically independent random variables, each with density function p_{x}and distribution function F_{x}. These variables are applied to a system that selects as its output, y_{N}, the largest of the {x_{i}}, i.e., y_{N}= max{x_{1},x_{2},…,x_{N}}. Clearly, y_{N}is a random variable.- Express p
_{yN}in terms of N, p_{x}, and F_{x}. - Assume now that the x
_{i}are exponentially distributed random variables:Calculate the expectation E[y

_{N}] for N = 1, 2.

- Express p
- Let be a zero mean Gaussian random vector with covariance matrix
K.
- Give an expression for the density function f
_{}(x). - If Y = X
_{1}+ 2X_{2}- X_{3}, find f_{Y }(y). - If the vector has components defined by
determine f

_{}(). What are the properties of the new random vector? - Determine f
_{X1|X2}(x_{1}|x_{2}= β)

- Give an expression for the density function f

- A noisy discrete communication channel is available. Once each second
one letter from the three-letter alphabet {a,b,c} can be transmitted
and one letter from the three-letter alphabet {1, 2, 3} is received. The
conditional probabilities of the various received letters, given the
various transmitted letters are specified by the diagram in the
accompanying diagram.