What is the resulting transmission bit rate

Question

Solve the following problem:

A binary signaling scheme over an AWGN channel with noise power spectral density of N₀/2 uses the equiprobable messages shown in Figure and is operating at a bit rate of R bits/s.

1. What is ε_b/N₀ for this system (in terms of N₀ and R)?

2. What is the error probability for this system (in terms of N₀ and R)?

3. By how many decibels does this system underperform a binary antipodal signaling system with the same ε_b/N₀ ?

4. Now assume that this system is augmented with two more signals s₃(t) = -s₁(t) and s₄(t) = -s₂(t) to result in a 4-ary equiprobable system. What is the resulting transmission bit rate?

5. Using the union bound, find a bound on the error probability of the 4-ary system introduced in part 4.