Correct choice is (d) X (F) = \(\frac{1}{2} [X_l (F-F_c)+X_l^* (-F-F_c)]\)
For explanation: X (F) = \(\int_{-\infty}^∞ x(t)e^{-j2πFt} dt\)
=\(\int_{-\infty}^∞ \{Re[x_l (t) e^{j2πF_c t}]\}e^{-j2πFt} dt\)
Using the identity, Re(ε)=1/2(ε+ε^*)
X (F) = \(\int_{-\infty}^∞ [x_l (t) e^{j2πF_c t}+x_l^* (t)e^{-j2πF_c t}] e^{-j2πFt} dt\)
=\(\frac{1}{2}[X_l (F-F_c)+X_l^* (-F-F_c)]\).