a. Show that the following two basis functions are orthonormal. (2 pts) $1(t) = 2 (cos...

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a. Show that the following two basis functions are orthonormal. (2 pts) $1(t) = √2 (cos (2nt)) if te [0, 1] otherwise 0 { b. Consider the following modulated waveforms. ro(t) I₁(t) I₂(t) = = = = = { = { 0 10 { √2 (cos (2nt) + sin (2nt)) 10 √2 (sin (2xt)) √2 (cos (2nt) + 3 sin (2xt)) 0 √2 (3 cos (2nt) + sin (2xt)) 0 if te [0, 1] otherwise 0 √2 (3 cos (2πt) + 3 sin (2nt)) √2 (cos (2nt) - sin (2nt)) √2 (cos (2nt) - 3 sin (2xt)) 0 √2 (3 cos (2nt) - sin (2nt)) 0 { 0 Fi+s(t) = -2₁(t) i=0,---,7 if te [0, 1] otherwise √2 (3 cos (2nt) - 3 sin (2nt)) if te [0, 1] otherwise if te [0, 1] otherwise ifte [0, 1] otherwise if te [0, 1] otherwise if te [0, 1] otherwise if te [0, 1] otherwise if te [0, 1] otherwise Draw the constellation points for these waveforms using the basis functions of (a). (2 pts) c. Compute E and Ez (Ez = E/N) where N is the number of dimensions (i) for the case where all signals are equally likely. (2 pts) (ii) for the case where (2 pts) p(x₁) = p(x₁) = P(xs) = P(X12) = d. Let and p(z.) =i=1,2,3,5,6,7,9, 10, 11, 13, 14, 15 y(t) = 2(t) + 403(t) 1 if te [0, 1] ds(t) = { 0 otherwise Compute Ey for the case where all signals are equally likely. (2 pts) where a. Show that the following two basis functions are orthonormal. (2 pts) $1(t) = √2 (cos (2nt)) if te [0, 1] otherwise 0 { b. Consider the following modulated waveforms. ro(t) I₁(t) I₂(t) = = = = = { = { 0 10 { √2 (cos (2nt) + sin (2nt)) 10 √2 (sin (2xt)) √2 (cos (2nt) + 3 sin (2xt)) 0 √2 (3 cos (2nt) + sin (2xt)) 0 if te [0, 1] otherwise 0 √2 (3 cos (2πt) + 3 sin (2nt)) √2 (cos (2nt) - sin (2nt)) √2 (cos (2nt) - 3 sin (2xt)) 0 √2 (3 cos (2nt) - sin (2nt)) 0 { 0 Fi+s(t) = -2₁(t) i=0,---,7 if te [0, 1] otherwise √2 (3 cos (2nt) - 3 sin (2nt)) if te [0, 1] otherwise if te [0, 1] otherwise ifte [0, 1] otherwise if te [0, 1] otherwise if te [0, 1] otherwise if te [0, 1] otherwise if te [0, 1] otherwise Draw the constellation points for these waveforms using the basis functions of (a). (2 pts) c. Compute E and Ez (Ez = E/N) where N is the number of dimensions (i) for the case where all signals are equally likely. (2 pts) (ii) for the case where (2 pts) p(x₁) = p(x₁) = P(xs) = P(X12) = d. Let and p(z.) =i=1,2,3,5,6,7,9, 10, 11, 13, 14, 15 y(t) = 2(t) + 403(t) 1 if te [0, 1] ds(t) = { 0 otherwise Compute Ey for the case where all signals are equally likely. (2 pts) where

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