A source produces 4 symbols with probabilities $${1 \over 2},\,{1 \over 4},\,{1 \over 8}\,\,and\,\,{1 \over 8}.$$ For this source, a practical coding scheme has an average codeword lenght of 2 bits/symbols. The efficiency of the code is :

For the input signal, x(t), shown in Fig.

(a) Sketch the impulse response, h(t), of the matched filter.
(b) Sketch the output, y(t), of the matched filter. Fully label your diagram.
(c) If the input to the matched filter consists of the signal, x(t), plus additive white noise of two sided spectral density of $${10^{ - 6}}$$ watts/Hz, find the maximum output SNR of the matched filter.

In binary data transmission DPSK is preferred to PSK because

Nyquist plot consider a feed back system where the OLTF is: $$G(s) = {1 \over {s\left( {2s + 1} \right)\left( {s + 1} \right)}}.$$ Determine the asymptote which the nyquist plot approaches as $$\omega \to 0.$$ Find also the range of 'k' in terms of the cross over frequency $${\omega _{PC}}$$ for stability.