Answer Happy

Do parts 1.4 and 1.5.
- { Consider a binary communication system that conveys bits 0 and 1 by transmiting si(t) and 82(0), respectively. Assumed to be equally likely, these two signals {s:(t)}?=1 are explicitly given by: cos (2*8.4) OSIST, 8 (1) (1) otherwise, where for i = 1, 2. Ji = ' with > 0 being some fixed postive integer. If si(t) is transmitted, the corresponding received signal, denoted as r(t), is given by: r(t) = s(t) + w(t). (2) wherein wt) is a zero-mean white Gaussian process with power spectral density (PSD): S(I) = VER. PART I. In this part, the orthonormal basis vectors/functions are denoted as vi(t) and Vz(t), and you are told that: VE where & is the energy of the signal sz(t). 1.1 Find the distance, dz2, between the two signals su(t) and salt). (5 points). 1.2 Find the second basis vector/function 2(0) and the geometric geometric representation, si and 82, of the two signals in the obtained orthonormal basis (4,(6), 42(0)}. (5 points). 1.3 Find the outputs, yi and 12, of the two matched filters V:(T-1) and (T-), respectively. (5 points). 1.4 Show that under maximum likelihood detection, the optimum decision rule at the receiver is given by: (5 points) > y2 receiver declares the bit 0 was transmitted 91 < y2 + receiver declares the bit I was transmitted 1.5 Find the analytical expressions of the conditional error probabilities. P[errors, and P[errorsa), as well as of the average error probability, P. (5points). 81(0) (4) (5)