$[$LD$]11\mathrm{.}3-1$: In a BPSK baseband system, the discrete time channel is specified by H$($z$)=$

$1+0\mathrm{.}5$z$1.$ The received signal samples are

z$[$k$]=$ h$[$k$]?$ s$[$k$]+$ w$[$k$].(1)$

The BPSK signal is s$[$k$]=1$ with equal probability. The discrete channel noise w$[$k$]$ is additive white

Gaussian with zero mean and variance N $/2$ such that Eb$/$N $=20,$ where Eb is the energy in the baseband

symbol.

$($a$)$ Find the probability of error $($analytically$)$ if z$[$k$]$ is directly sent into a BPSK decision device.

$($b$)$ Find the proability of error $($analytically$)$ if z$[$k$]$ is first sent through a ZF equalizer before a BPSK

decision device