\( t \) is very important that you do the homework in order to understand the material of the course. Yot vill benefit most from the homework if you attempt to do the problems before consulting your friend ind your TAs. While it is perfectly reasonable to discuss your approach to solving the problems with friend, the final write-up of the solution should be your own and not a copy of your friend's olution.
Cotal marks: 100 marks
roblem 1 (15 marks)
Consider the following \( M \)-ary digital communication system with \( M=4 \). Assume that all symbols re equally probable and that each symbol is transmitted within time duration \( T=1 \) s over an AWGN hannel with double-sided power spectral density \( \mathrm{N}_{0} / 2 \). The expression of the transmit signal within is shown below.
\[
\begin{array}{ll}
s_{1}(t)=3 \cos (2 \pi t), & s_{2}(t)=3 \cos (4 \pi t), \\
s_{3}(t)=3 \cos (8 \pi t), & s_{4}(t)=3 \cos (16 \pi t) .
\end{array}
\]
How many bit(s) is/are represented by one symbol? (2 marks)
At least how many basis function(s) is/are needed to represent all symbols? Give the expression(s) of it/them. (3 marks)
i) 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)
Find the Union bound for the average symbol error rate. (5 marks)
oblem 2 ( 15 marks)
In an MFSK system with \( M=4 \), calculate the minimum frequency separation \( \Delta f \) required between the carrier frequencies for orthogonality if the symbol duration \( T_{s}=2 \mathrm{~ms} \) ? (2 marks)
For an 8 -ary PSK \( (\mathrm{M}=8) \) system, if the energy per symbol \( E_{s}=10 \mathrm{~mJ} \) and the noise powe spectral density \( N_{0}=2 * 10^{-5} \mathrm{~W} / \mathrm{Hz} \), calculate the probability of symbol error \( P_{s} \) using th
(1) \( 21^{\circ} \mathrm{C} \)
