# HW4 (Due 12/3)

1. (10 points) Try to repeat Q1 and Q4 in the midterm using Lea (or any other probabilistic programming tool).

2. (5 points) Consider a binary channel with a cross over probability of 0.15 and an erasure probability of 0.1, what is the capacity of the channel? In other words, when a bit is sent through the channel, there is a probability of 0.15 that the bit is flipped and then there is an additional probability of 0.1 that the bit got erased (the decoder can't even tell if the received bit is 0 or 1).

3. (5 points) Consider a channel composed of two parallel Gaussian channels with noise powers $${N_1}=1$$ and $${N_2}=5$$. Given a total power $$P$$, you may assign power $$P_1$$ to channel 1 and power $$P_2$$ to channel 2 such that $$P_1+P_2=P$$. By trading off the power between the two channels, compute the (maximum) overall capacity: $$\frac{1}{2} \log(1+\frac{P_1}{N_1}) + \frac{1}{2} \log(1+\frac{P_2}{N_2}$$) for

1. $$P = 13$$

2. $$P = 3$$