A duobinary line code is ternary scheme similar to bipolar, but requires only half the bandwidth of the latter. In this code, for symbol $"0"$ no pulse is transmitted, and for symbol

$"1"$ either pulse p$($t$)($given as figure below$),$ with Fourier Transform P$($f$),$ or pulse $-$p$($t$)$ is

transmitted using the following rule: A $1$ is encoded by the same pulse as that used to encode

the preceding $1$ if the two $1\text{'}$s are separated by an even number of $0\text{'}$s; A $1$ is encoded by the

negative of the pulse used to encode the preceding $1$ if the two $1\text{'}$s are separated by an odd

number of $0\text{'}$s$.$ Random binary digits are transmitted every T$\_($b$)$ seconds. Assuming that each

information symbol is equally likely to be $0$ or $1$ and is independent of other symbols. The

transmitting signal can be expressed as:

y$($t$)=\backslash$sum$\_($n$=-\backslash$infty $)^(\backslash$infty $)$ a$\_($n$)*$p$($t$-$nT$\_($b$))$

We use R$\_($b$)=(1)/($T$\_($b$))$ to denote the symbol rate.

The PSD of the pulse shaped waveform is

S$\_($y$)($f$)=(|$P$($f$)|^(2))/($T$\_($b$))\backslash$sum$\_($n$=-\backslash$infty $)^($n$=\backslash$infty $)$ R$\_($n$)$e$^(-$j$2\backslash$pi nfT$\_($b$))$

$=(|$P$($f$)|^(2))/($T$\_($b$))($R$\_(0)+2\backslash$sum$\_($n$=1)^(\backslash$infty $)$ R$\_($n$)$cos$(2\backslash$pi nfT$\_($b$)))$

$($a$)$ If the first $10$ binary information digits are b$\_($k$)=[[1,1,0,0,1,0,0,0,1,1]],$ what

is the corresponding sequence a$\_($k$),$ plot the corresponding transmitting waveform y$($t$)$ for

this $10$T$\_($b$)$ interval.

$($b$)$ Find R$\_(0)$ and R$\_(1)$ for sequence a$\_($n$).$

$($c$)$ Show that R$\_($n$)=0$ for n$>=2$ for sequence a$\_($n$).$

$($d$)$ Find P$($f$)$p$($t$)$ S$\_($y$)($f$).$