Step 1 :First, we need to find the pairs of signals that are orthogonal by checking the integral condition for each pair of signals. We will calculate the integral for each pair and check if it is equal to 0.
Step 2 :Let's define the signals as follows: \(s_1(t) = \sqrt{2} \frac{\sqrt{E_s}}{\sqrt{T_s}} \cos(\omega t + \frac{\pi}{4})\), \(s_2(t) = -\sqrt{2} \frac{\sqrt{E_s}}{\sqrt{T_s}} \sin(\omega t + \frac{\pi}{4})\), \(s_3(t) = -\sqrt{2} \frac{\sqrt{E_s}}{\sqrt{T_s}} \cos(\omega t + \frac{\pi}{4})\), and \(s_4(t) = \sqrt{2} \frac{\sqrt{E_s}}{\sqrt{T_s}} \sin(\omega t + \frac{\pi}{4})\).
Step 3 :Calculate the integrals for each pair of signals and check if they are equal to 0.
Step 4 :\(\int_{0}^{2 \pi} s_{1}(t) s_{2}(t) dt = 0\), \(\int_{0}^{2 \pi} s_{1}(t) s_{3}(t) dt = 0\), \(\int_{0}^{2 \pi} s_{1}(t) s_{4}(t) dt = 0\), \(\int_{0}^{2 \pi} s_{2}(t) s_{3}(t) dt = 0\), \(\int_{0}^{2 \pi} s_{2}(t) s_{4}(t) dt = 0\), and \(\int_{0}^{2 \pi} s_{3}(t) s_{4}(t) dt = 0\).
Step 5 :\(\boxed{\text{The orthogonal pairs of signals are }(s_1, s_2), (s_1, s_3), (s_1, s_4), (s_2, s_3), (s_2, s_4), \text{ and } (s_3, s_4). \text{ All pairs of signals are orthogonal.}}\)