ECE-542 Digital Communication Theory, Spring 2004
Homework Assignment 8, Due date: Friday, Apr. 16, 2004
1. Carefully reconstruct the counting argument presented in class to show that if symbols have
equal priors, then the bit-error probability (Pb) and symbol-error probability (PS) are related as
PS=Pb(2k−1)/(2k−1),where kis the number of bits per symbol. Hint: First establish that the
average number of bit errors per symbol error is k2k−1/(2k−1). Then use the argument that the
average number of bit errors per symbol can be calculated either as kPbor as PS×the average
number of bit errors per symbol error.
2. of Show that in the case of gray-coding, and if we ignore non-adjacent symbol errors, then
Pb=PS/k.
3. Show that if a function from IR to IR is continuous, then the image of any closed and bounded set
is also closed (and bounded). Hint: First show that if xn→xthen f(xn)→f(x)iffis continuous.
4. Show that the set {x∈IR n:xi≥0,x
1+...+xn=1}is closed and bounded.
5. Use problems 3 and 4 to argue that the channel capacity for a discrete channel exists.