Problem $1(20$ points$)$

Consider a digital communications system that has to transmit an image which has been encoded, using an optimal source encoder, as a sequence of information bits. In order to achieve error protection, the information bits are encoded with a channel encoder of rate $R_c.$ The coded bits are transmitted using a finite set of $8$ possible waveforms, each of duration T seconds: every T seconds, the channel accepts one of the possible waveforms as input and outputs the input waveform corrupted by noise.

Propose a transmitter that allows the transmission of all the information bits.

For the transmitter proposed in $1)$:

a$.$ How many coded bits and information bits are sent through the channel every time the channel is used?

b$.$ What are the values of R and $R_T,$ if they are defined as the information rates transmitted through the channel in terms of information bits per channel use $($number of information bits transmitted through the channel every time the channel is used$)$ and information bits per second, respectively? Express all the results as a function of $R_c$ and $T.$

c$.$ What is the energy we utilize to transmit one information bit, $E_b,$ as a function of the energy utilized every time the channel is used, $E_s?$

d$.$ If the channel capacity is $C=\mathrm{.}5$ information bits per channel use and L is the total number of information bits we need to transmit, provide a lower bound on the minimum number of coded bits we need to produce $($i$.$e$.,$ #coded bits$>$lower bound$),$ and an upper bound on $R_c($i$.$e$.,R_c{R}_c{R}_c$ and $T$ stay the same