2. In this problem, we will encode bits into linearly modulated signals using 8-PAM at symbol rate 10kHz.Use the abstract alphabet A = {0,1,..., 7}, the signal constellation C =

{-7,–5, –3, –1, 1,3, 5, 7},and the mapping f(i) = 2i–7. Let p1(t) = I_T/2,T/2(t) and let p2(t) =It/2,T/2(t)(1+cos(2nt/T))/2. (a) (20 pts) Let an E A for n E Z and suppose that a has the empirical frequency properties described in problem 1. Let xn = f(an) and let u;(t) be the linearly modulated signal encoding x with pulse P:(t). Find the power spectral densities of u1 and u2. Create a single log-log plot with both PSDS.The signals are real, so the PSD is even and it is only necessary to plot positive frequencies. (b) (20 pts) Consider the finite bit sequence b = (1,0,0,1,1, 1, 1,0,0, 1, 1,0, 1,1, 1), so b e {0,1}15. (I generated this bit sequence randomly.) Translate this into a symbol sequence in (ao, a1, a2, a3, a4) EAš and a sequence of amplitudes (xo, x1, X2, X3, X4) C5 and let vi(t) be this linear modulated signal with pulse p;(t) that encodes this data. Find the energy spectral densities of vi and v2.Create a single log-log plot with both ESDS. (c) (20 pts) Repeat the previous part for b = (1,1,1, 0,0,0,1, 1, 1,0,0,0, 1, 1, 1). Which of the bit sequences produces an ESD plot more similar to the PSD plot from part (a) and why?