1. Consider the following binary transmission system including channel coding and digital…

1. Consider the following binary transmission system including channel coding and digital modulation over an AWGN channel

Given that

• the 4-PSK modulator has the waveform

with A = 10 mV and Ta = 1µs

• w(t) is a WGN with PSD

• the channel coding is a linear block code with generating matrix

a) Find the bit rate Rb of the system, and the bit error probability Pbit for the digital modulation system (from cn to ˜cn).

b) Show that the minimum distance decoding of c˜k = [100100] is bˆ k = [100].

c) Find minimum Hamming distance of the code, and the minimum Euclidean distance between the sequences of constellation symbols corresponding to two distinct codewords.

d) Decode the received sequence

to the corresponding detected information sequence

e) Compare the above result to that obtained with separate ML detection and hard decoding, as illustrated below:

2. Show that in the low density parity check matrix of any Gilbert code with  there are 4 linearly dependent columns, and hence the code minimum distance is 4.