Consider transmitting independent and equally likely binary data using the signals

s k$$t$1$k $2$P c cos$2$f c t$2$P s sin$2$f c t$,$ k $1,2.$ The received signal due to transmitting s k$$t$$ is

given by r$$t$1$k $2$P c cos$2$f c t $2$P s sin$2$f c t $$ n$$t$,$ where $$ is independent random

channel phase that is constant over a bit interval, T$,$ and n$$t$$ is AWGN with variance $$n

$2$ N$02.$ Assume that

f c $1$T$.$

This channel is peculiar since the random i$.$i$.$d$.$ channel phase is discrete$-$valued with

Pr$0$ Pr$12.$

The receiver has two objectives: $($i$)$ Estimate the binary data and $($ii$)$ estimate the channel phase.

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$($a$.)$ Provide a complete signal space description of the transmitted signals. Make sure to completely specify

the basis waveforms and the coefficients of expansion over the bit inerval.

$1$t$$ s$1,$

$2$t$$ s$2,$

$($b$.)$ Provide the coefficients of expansion for the signals

y k$$t$1$k $2$P c cos$2$f c t $2$P s sin$2$f c t $$ over the same time interval, in the signal space

above.

y$1,$ y$2,$

$($c$.)$The channel phase is estimated using threshold detection in the quadrature $($sin$)$ carrier dimension.

Determine the probability of channel phase estimation error, P$$ Pr $.$

P$$

$($d$.)$ The binary data are detected using threshold detection in the in$-$phase $($cos$)$ carrier dimension, after

attempting to remove the effects of the channel phase using the phase estimate $.$ Determine the probability of

bit error. $($For simplicity, use the variable P$$ in your answer.$)$

P b $$

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