Solve the following problem:
A QPSK communication system over an AWGN channel uses one of the four equiprobable signals si(t) = A cos(2π fct + iπ/2), where i = 0, 1, 2, 3, fc is the carrier frequency, and the duration of each signal is T . The power spectral density of the channel noise is N0/2.
1. Express the message error probability of this system in terms of A, T , and N0 (an approximate expression is sufficient).
2. If Gray coding is used, what is the bit error probability in terms of the same parameters used in part 1?
3. What is the minimum (theoretical minimum) required transmission bandwidth for this communication system?
4. If, instead of QPSK, binary FSK is used with s1(t) = B cos 2π fct and x2(t) = B cos(2π fc + ?f )t where the duration of the signals is now T1 and ?f = 1/2T1 , determine the required T1 and B in terms of T and A to achieve the same bit rate and the same bit error probability as the QPSK system described in parts 1-3.