A binary data stream needs to be transmitted at $10$Mbi$\frac{t}{s}$ by means of binary signaling. To minimize ISI, a duobinary pulse will be used. The differential encoding duobinary signaling can be generated using the scheme shown in Figure $1.$

rigure $$ scraminier and vescrampier vidgram

$1$

a$)$ Determine the minimum required bandwidth for this transmission.

b$)$ To make the data more random, a scrambler is inserted before the differential duobinary encoder. Figure $2$a shows the scrambler structure. Consider a random binary sequence is $S=10110010000001101111,$ find the output of the scrambler T

c$)$ The output of scrambler is applied into the differential duobinary encoder, specify the differential encoder output sequence $p_k,$ polar line code output $a_k,$ and the sampled duobinary transmission signal $y_k=(k{T}_b).$

d$)$ Assume there is no noise, the received signal is $y(k{T}_b)=b_k.$ And the receiver decision is

Calculate the receiver output sequence $I_k.$ The output sequence is then applied to the descrambler of Figure $2$b$.$ Verify that the output of the descrambler is the correct sequence.