Consider the following \( M \)-ary digital communication system with \( \mathrm{M}=4 \). Assume that all symbol are equally probable and that each symbol is transmitted within time duration \( \mathrm{T}=1 \mathrm{~s} \) over an AW channel with double-sided power spectral density \( \mathrm{N}_{0} / 2 \). The expression of the transmit signal with T is shown below.
\[
\begin{array}{ll}
s_{1}(t)=3 \cos (2 \pi t), & s_{2}(t)=3 \cos (4 \pi t) \\
s_{3}(t)=3 \cos (8 \pi t), & s_{4}(t)=3 \cos (16 \pi t)
\end{array}
\]
i) How many bit(s) is/are represented by one symbol? (2 marks)
ii) At least how many basis function(s) is/are needed to represent all symbols? Give the expression(s) of it/them. (3 marks)
iii) The pair-wise error probability between signals j and k is denoted by \( \mathrm{P}_{\mathrm{e}}(\mathrm{j}, \mathrm{k}) \). Find \( \mathrm{P}_{\mathrm{e}}(\mathrm{j}, \mathrm{k}) \). ( 5 marks)
iv) Find the Union bound for the average symbol error rate. (5 marks)