optimum detection for binary communication in AWGN

I need help with this project in Matlab. a) Develop the optimum detection rule. b) Derive the theoretic bit error rate (number of error bits/total number of transmitted bits). c) Develop a program (use any computer language you feel comfortable) to simulate this system. o Transmitter generate random binary symbols, x, with equal probability and represent them with polar non-return-zero signaling. o The channel will add zero-mean σ2-variance Gaussian distributed noise, n, into the transmitted signal, that is, the output of the channel is y=x+n. o The optimum receiver will compare the received signal, y, with the threshold you derived to estimate which binary symbol is transmitted. d) Plot sample transmitted data, x, and received data, y. e) Plot and compare the theoretic and simulated bit error rate with different σ (different noise power). o Plot 10*log10(1/σ2) versus the BER (in base-10 logarithmic scale) o For each σ value, you may need to generate more than 10,000 binary symbols. o You may use the following σ values in your simulation: 1.2589, 1.0000, 0.7943, 0.6310, 0.5012, 0.3981, 0.3162.