Problem $2(20\%)$

For a discrete$-$time additive Gaussian noise channel what SNR is needed $($in plain scale and in

dB$)$ to have capacity of $4$ bits per channel use?

For any initial capacity value C$,$ what kind of increase in SNR $($in plain scale and in dB$)$ does

it take to double the capacity to $2$C$?$

Problem $3(20\%)$

$($i$)(10\%)$ What probability distribution function has maximum differential entropy, support

$[0,+$$)$ and mean value $4?$

$($ii$)(10\%)$ What is the entropy rate of a Gaussian process with power specral density S$($$)=$ $2$

for every in $[$$,$ $]?($This is known as a white Gaussian process.$)$

Problem $4(20\%)$

$($i$)(15\%)$ Define in every detail the random process $\{$Xn$\}$ that has maximum entropy rate and

satisfies E$\{$X$2$

n$\}=2$ and E$\{$XnXn$+1\}=1.$

$($ii$)(5\%)$ How much is the entropy rate of this process?

Problem $5(20\%)$

$($i$)(10\%)$ Consider a Gaussian i$.$i$.$d$.$ source output sequence with mean $0$ and variance $2.$ Design

a single$-$symbol mean$-$square$-$optimal rate$-$one$-$bit encoder$/$decoder system $($quantizer$).$

$($ii$)(10\%)$ How much is the mean$-$square distortion of your single$-$symbol mean$-$square$-$optimal

rate$-$one$-$bit encoder$/$decoder system?

$1$