CU Boulder 09-09-2002Due 09-16-2002Problem 1Problem 2Problem 3CU Boulder 09-09-2002ECE Dept Communication Theory (ECEN 4242)Homework #2Due 09-16-2002Problem 1Consider the signalelsewhere. ,0;2/23 ),2(;232/ ),(;2/0 ,)(tt/tttttgWe wish to find an approximate representation for g(t) of the form)()sin()(1tektctgNNkkwhere eN(t) is the error signal for an approximation of order N and the signals {sin(kt)}, k=1, 2, …, N, represent the basis of the approximation.1) Plot the signal g(t).2) Calculate the energy of g(t).3) Show that the signal set {sin(kt)}, k=1, 2, …, N, forms an orthogonal basis.4) Determine the coefficients {ck}, k=1,2, …, N, so that the error signal eN(t) is orthogonal to thebasis signals {sin(kt)}, k=1,2, …, N.5) Modify the MATLAB script "examp21.m" given in the course web directory, to approximate the signal g(t) given above. The new MATLAB script should - read N- compute the coefficients ck's using the expression found in 4)- evaluate g(t) using the same sampling interval as in "examp21.m"- computeNkkktcts1)sin()(, which represents an approximation to g(t)- compute the error signal eN(t)- compute the energies of the signals g(t), s(t), and eN(t). These energies are Eg, Es, and Ee, respectively. - plot g(t), s(t), and eN(t) on the same figure for comparison purpose.6) Find the minimum N for which Es - 0.95Eg.7) Repeat the execution of the new script for N=1, 3, 5, 7, 9, and 11 and plot Ee versus N, where Ee is the energy of eN(t). Conclusions?Problem 22.6-1Problem