Problem $2$

Probably the simplest block code of all is the repetition code. Here, a single digitA is mapped into the codeword AAA. Thus, $0$ is mapped into $000$ and $1$ is mapped into $111.$ The codewords AAA are then mapped into waveforms, $s_1(t),s_2(t),s_3(t),$ where $s_1,s_2,s_3$ are orthogonal and each have energy $,$ and $=1$ or $-1,$ depending on whether the binary digit is a $1$ or a $0.$ White Gaussian noise of spectral density $\frac{N_0}{2}$ is added. Each of the three received waveforms are matched filtered to obtain $Y_1,Y_2,Y_3.$ Given $,Y_{i}N(\sqrt[\phantom{2}]{},\frac{N_0}{2})$ for $i=1,2,3.$

$($a$)$ One reasonable decoding algorithm for this code is called "hard decision" decoding. First, we make individual decisions on each digit separately. Then, the final decoded bit is based on a majority vote $($as in slide $2),$ i$.$e$.,$ if two or three of the digits is decided to be $1,$ then the final decoded bit is $1,$ otherwise $0$ is decoded. Find the probability of error for hard decoding, as a function of $\frac{E_b}{N_0}.$

$($b$)$ The optimum decision rule is also referred to as "soft decoding." This makes minimum error probability decision from $Y_1,Y_2,Y_3$ together, assuming that the two hypotheses are equally likely. Find the probability of error for soft decoding, as a function of $\frac{E_b}{N_0}.$

$1$

$($c$)$ Consider an alternative in which no coding at all is used. That is$,A$ is mapped into antipodal signals, $=1,$ and $s(t)$ is transmitted where $s(t)$ has energy $3.$ Find the probability of error, again assuming equally likely signals and WGN of spectral density $\frac{N_0}{2}.$ Compare with parts $($a$)$ and $($b$)$ and explain.